HALF–LINEAR DISCRETE OSCILLATION THEORY

by

Pavel Řehák Masaryk University Brno, Czech Republic

Brno 2000

Author’s address: Department of Mathematics

Faculty of Education Masaryk University

Poříčí 31 60300 Brno

Czech Republic

rehak@math.muni.cz

Preface

The present work is basically my Ph.D. dissertation that was submitted in May 2000 and in which oscillatory properties of the second order half–linear difference equation are investigated.

Primarily, I would like to thank my thesis advisor Ondřej Došlý. I very appreciate his willingness and patience in giving me consultations. I am grateful for his suggestions.

I also give my thanks to Alexander “Bacho” Lomtatidze, with whom I have had very fruitful discussions. Further, I would like to thank Zuzana Došlá for her help on many various occasions (not only concerning this work).

I am obliged to Professor Ludwig Reich for preparing a perfect scientific and social background during my two–month research stay at the University of Graz in autumn 1999, where a great part of this work was prepared. I wish to express my gratitude to Austrian Academic Exchange Service (AKTION) for awarding me a scholarship to support this stay.

Finally, special thanks to my family and friends for their support.

Brno, June 2000 Pavel Řehák

Contents

Preface iii

Chapter 1. Introduction 1 1.1. Motivation and literature overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Problems connected with our approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3. Current state of problems with a literature overview . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4. Notation and basic background used in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 2. Basic concepts and some properties of (HLΔE) 7 2.1. Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Discretization of (HLDE) and one discrete variational problem . . . . . . . . . . . . . 9

Chapter 3. Roundabout theorem 11 3.1. Picone identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2. Roundabout Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3. Sturmian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4. A remark on Casoratian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 4. Methods of half–linear discrete oscillation theory 19 4.1. Riccati technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2. Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3. Reciprocity principle and other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 5. Oscillation criteria 23 5.1. Oscillation criteria based on the variational principle . . . . . . . . . . . . . . . . . . . . . . . 23 5.2. Hartman–Wintner type theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3. Oscillation criteria based on the Riccati technique (part I) . . . . . . . . . . . . . . . . . . 32 5.4. Oscillation criteria based on the Riccati technique (part II) – the case r_{k} ≡ 1 40

Chapter 6. Nonoscillation and conjugacy criteria 53 6.1. Nonoscillation criteria based on the variational principle . . . . . . . . . . . . . . . . . . . . 53 6.2. Nonoscillation criteria based on the Riccati technique . . . . . . . . . . . . . . . . . . . . . . 56 6.3. Conjugacy criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 7. Comparison theorems 66 7.1. λp_{k} – type comparison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.2. ∑ p_{j} ≤ ∑ P_{j} – type comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3. Telescoping principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.4. Comparison theorem for generalized Riccati difference equations . . . . . . . . . . . . 71

Chapter 8. Strong oscillation 72 8.1. Strong oscillation and nonoscillation, conditional oscillation . . . . . . . . . . . . . . . . . 72 8.2. Generalized Euler difference equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 9.  Conclusions and future directions, open problems  76

CONTENTS v

9.1. Motivation by the linear continuous theory, more general equations . . . . . . . . . 76 9.2. Time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Bibliography 79

CHAPTER 1 Introduction

1.1. Motivation and literature overview

We start this introductory section with a submission of the following three facts:

▶ Consider the second order linear differential equation

(r(t)y′)′ + p(t)y = 0,  (LDE)

where r(t) and p(t) are real–valued continuous functions defined on [a, ∞) with r(t) > 0. It is one of the most frequently occurring types of differential equations in mathematics and physical sciences. In general this equation cannot be solved explicitly and hence it is necessary to investigate (LDE) from the qualitative point of view, which means an observation of properties of its solutions. The qualitative study of (LDE) originated in the classic paper of Sturm, see [107], in 1836. From that time, there have been published thousands papers concerning this equation. We mention here only several basic and/or survey papers and books, where, in particular, the so–called oscillatory properties of (LDE) are investigated: [22, 54, 95, 96, 97, 108, 117]. Recall that equation (LDE) can be classified by the behaviour of its real solutions as oscillatory or nonoscillatory. In the first case, one, and hereby every, solution vanishes at an infinite number of isolated points in [a, ∞); in the second case each solution has only a finite number of zeros in [a, ∞). A special instance of nonoscillation is the disconjugacy of (LDE), where every solution has at most one zero in [a, ∞). The Riccati differential equation

w′ + p(t) + r^{−1}(t)w^{2} = 0  (RDE)

and the quadratic functional

F_{cont}(ξ; a, b) := ∫_{a}^{b}[r(t)ξ′^{2}(t) − p(t)ξ^{2}(t)] dt

play very important roles in the oscillation theory of (LDE). In particular, in the so–called (Reid) Roundabout Theorem, see [97], the connection (in fact, there are equivalences) among the following concepts is shown:

∗ disconjugacy of (LDE) in an interval [a, b],

∗ the existence of a nonzero solution of (LDE) in [a, b],

∗ the existence of a solution of (RDE) in [a, b],

∗ positivity of the quadratic functional F_{cont}(ξ; a, b) for every nontrivial, piecewise of the

class C^{1}, function ξ, for which ξ(a) = 0 = ξ(b). This theorem is of basic importance in the oscillation theory of (LDE); as a consequence of it one can easily get comparison and separation theorems of Sturmian type. Moreover, the Roundabout theorem provides very useful tools for the investigation of oscillatory properties of (LDE), namely the Riccati technique and the variational principle. Note that the above mentioned result can be extended to more general “linear” objects, e.g. linear Hamiltonian differential systems and, hence, the oscillation theory of the second order linear differential equations is, among others, a good guide to the investigation of such complicated objects.

▶ It is a well–known fact that there is a striking similarity between qualitative theories of differential equations and difference equations. Moreover, it turns out that it makes a sense to study qualitative properties of difference equations (in spite of the fact that they are actually recurrence relations). Therefore, in the last years, a considerable effort has been made to investigate qualitative properties (including very important oscillation theory) of the discrete counterpart of (LDE), namely the second order linear difference equation

Δ(r_{k}Δy_{k}) + p_{k}y_{k+1} = 0,  (LΔE)

where r_{k} and p_{k} are real valued sequences defined on ℕ with r_{k} ≠ 0. It has been shown that in some aspects these theories are quite analogical, and, on the other hand, that to treat some problems concerning (LΔE), one has to use somewhat different methods than those for differential equations. For example, at the first sight, we can see that the unknown sequence y in the second term of equation (LΔE) has an index k + 1 and that the sequence r_{k} is required only to be nonzero in the contrast to the function r(t) in the continuous case. These discrepancies will be explained in Sections 2.1 and 2.2. Note that equation (LΔE) is frequently considered in the equivalent form of the three–term recurrence relation

r_{k+1}y_{k+2} + q_{k}y_{k+1} + r_{k}y_{k} = 0,

where q_{k} = p_{k} − r_{k+1} − r_{k}. In what follows we refer several papers and books concerning particularly oscillation theory of equation (LΔE). One of the first monographs, where difference equation (LΔE) was quite seriously studied from the qualitative point of view, is [51] (in 1948), although it is said that Sturm proved his famous separation theorem firstly for (LΔE) (with r_{k} > 0) and only then for (LDE). One can also find several papers from the beginning of the 20th century, where the above three term recurrence relation appears. The following important book is [4] (1964). In both of these monographs one can find a number of examples of the application of (LΔE) in a practice. In the above mentioned and in the most of the following literature, equation (LΔE) is considered with the assumption r_{k} > 0, although this requirement is not necessary in the general theory. The discrete version of the Roundabout theorem for equation (LΔE) is given in [3] in a full generality. The Riccati difference equation

Δw_{k} + p_{k} + \frac{w_{k}^{2}}{r_{k} + w_{k}} = 0

and the discrete quadratic functional

F_{disc}(ξ; m, n) := ∑_{k=m}^{n}[r_{k}(Δξ_{k})^{2} − p_{k}ξ_{k+1}^{2}]

play essentially the same role as equation (RDE) and functional F_{cont}, respectively, in the continuous case. Instead of the concept of a zero of a solution we use the concept of the generalized zero of a solution. This is due to the discretization that also causes e.g. the presence of some additional requirements concerning the solution of the Riccati difference equation. For the precise definition see Section 2.1. Note that the Roundabout theorem was extended to the linear Hamiltonian difference system in [5], and also to more general “linear” objects. From further, already quite extensive, literature, where oscillatory properties of (LΔE) (and some related topics) are investigated, we select the following: The monographs [1, 39] and [65] contain (in addition to oscillation theory) many further aspects of theory of various difference equations, introduction to the difference calculus and extensive literature overview. In quite frequently cited paper [55] discrete analogues of the concepts, as e.g. disconjugacy, principal solution and Green’s function, are introduced. Mostly, the oscillatory properties of solutions of equation (LΔE) are investigated in [10, 11, 21, 33, 49, 50, 59, 60, 61, 66, 71, 91, 92, 100]. For some further discrete versions of well–known “continuous” results concerning (LDE) and related topics see [6, 12, 13, 14, 17, 18, 25, 37, 52, 67, 85, 113, 116, 119].

▶ Recently (in 1976 and 1979), it was showed (see [87] and [40], respectively), that the basic oscillatory properties of the second order half–linear differential equation

(r(t)|y′|^{α−1} sgn y′)′ + p(t)|y|^{α−1} sgn y = 0,  (HLDE)

where r(t) > 0 and α > 1, are essentially the same as those of linear equation (LDE), which is a special case of (HLDE) for α = 2. The terminology half–linear equation is justified by the following fact. If a function y is a solution of (HLDE), then for any real constant c the function cy is a solution of the same equation. This means that the space of all solutions of (HLDE) is homogeneous, but not generally additive, and thus, it has only half of properties of linear space. Sometimes, half–linear equations are called “homogeneous (of degree α−1)”. In some papers, one can see equation (HLDE) in somewhat different form (but equivalent for the so–called proper solutions), namely

(r(t)y′)′ + p(t)|y|^{α−1} sgn y|r(t)y′|^{2−α} = 0,

or in a slightly more general form

(r(t)y′)′ + p(t)f(y, r(t)y′) = 0,

where suitable restrictions on the function f make this equation to be half–linear. Note that in the above mentioned papers [40, 87] Sturmian type theorems for (HLDE) were presented. Since that time, many papers have appeared, where the most concepts known from the oscillation theory of (LDE), is extended to (HLDE). E.g., in [63] or in [77] the proofs of some (nontrivial) parts of the half–linear version of the Roundabout theorem concerning the variational principle can be found. Quadratic functional F_{cont} is replaced here by the so–called α–degree functional

F_{cont}^{α}(ξ; a, b) := ∫_{a}^{b}[r(t)|ξ′(t)|^{α} − p(t)|ξ(t)|^{α}] dt.

Furthermore, it is known that in the half–linear theory the role of equation (RDE) is played by the generalized Riccati differential equation

w′ + p(t) + (α − 1)r^{1−β}(t)|w|^{β} = 0,  (GRDE)

where β is the conjugate number of α, i.e., 1/α + 1/β = 1. From an extensive literature concerning half–linear differential equations we select the following: Various oscillation and nonoscillation criteria for equation (HLDE) or for a slightly more general equation can be founded in [23, 24, 26, 28, 35, 42, 45, 64, 68, 69, 70, 75, 80, 88]. For an extension of some further known concepts from theory of linear equation (LDE) to the half–linear case see e.g. [8, 9, 29, 30, 31, 32, 36, 41, 43, 44, 46, 47, 73, 76, 78, 81, 82, 83, 84, 86, 89, 93]. Note that the oscillatory results for (HLDE) find the applications e.g. in the theory of partial differential equations with the so–called α–Laplacian.

Now, let us repeat in a brevity the above formulated facts:

• there is a deeply sophisticated oscillation theory of linear differential equation (LDE),

• qualitative theories of linear differential equation (LDE) and linear difference equation (LΔE) manifest considerable similarities,

• there exists a striking similarity between theories of linear differential equation (LDE)

and of half–linear differential equation (HLDE). Thus, it suggests an idea to develop the oscillation theory of the second order half–linear difference equation

Δ(r_{k}|Δy_{k}|^{α−1} sgn Δy_{k}) + p_{k}|y_{k+1}|^{α−1} sgn y_{k+1} = 0,  (HLΔE)

where r_{k} and p_{k} arereal–valued sequences defined on ℕ with r_{k} ≠ 0 and α > 1, using the approach, which can be represented by the following picture:

LDE ⟶^{HL} HLDE
Δ↓  ↓^{Δ}
LΔE ⟶^{HL} HLΔE,
HL ≡ generalization in a half–linear sense,
Δ ≡ discretization.

This means that we motivate ourselves by the “linear continuous theory”, which offers many interesting topics for the discretization and the extension to the half–linear discrete case. Hence we speak about the half–linear discrete oscillation theory. We see that equation (HLΔE) is a generalization of equation (LΔE) and at the same time a discretization of equation (HLDE). The process of this discretization will be precisely described in Section 2.2. In that section we will also show that equation (HLΔE) can be obtained as the Euler– Lagrange equation of a certain discrete variational problem connected with an extremizing of a hereafter mentioned discrete α–degree functional.

1.2. Problems connected with our approach

The approach described in Section 1.1 brings some interesting problems, especially, of the following two types:

1) The first type is related to the discretization. The techniques of proofs that are needed in the discrete case are often different from the continuous case and mostly more complicated. This is due to the absence of the chain rule for computing the difference of the “composite” sequences (like a Δf(x_{k})) and some other specific properties of difference (and summation) calculus.

2) The second type of problems is related to the extension to the half–linear case (both, continuous and discrete). There exist certain limitations in the use of the linear approach to the investigating of half–linear equations. These limitations are first of all the absence of transformation theory similar to that for linear equations or the impossibility of the extension of the so–called Casoratian to the half–linear discrete case, see also Section 3.4. Casoratian is the discrete counterpart of Wronskian from the theory of linear differential equations.

1.3. Current state of problems with a literature overview

In 90th, several papers appeared, where the equation

Δ(r_{k}(Δy_{k})^{σ}) + p_{k}y_{k+1}^{σ} = 0  (HLΔEσ)

with r_{k} > 0 and (mostly) p_{k} ≥ 0, σ being a quotient of positive odd integers, is investigated, see [16, 19, 20, 74, 79]. The authors of these papers present the results concerning the existence of a positive solution of (HLΔEσ) and some comparison theorems. However, there is an absence of the central statement of half–linear discrete oscillation theory, namely a half–linear discrete extension of the Roundabout theorem. Moreover, equation under their consideration is not so general as (HLΔE).

For the completeness’ sake, let us mention several further papers that are connected with this thesis in a certain sense. From a great amount of works devoted to the oscillation theory of nonlinear difference equation we select the following: Oscillatory behaviour of difference equations, which are slightly more general than (HLΔE), is investigated in [15, 33, 112, 114, 118]. A somewhat different type of nonlinearity in the second order difference equation, which is quite close to that in our equation, is e.g. of the form

Δ(r_{k}Δy_{k}) + p_{k}f(y_{k+1}) = 0

with uf(u) > 0. Oscillatory and monotone properties of this (or of very similar) equation are studied e.g. in [7, 56, 62, 94, 109, 110, 111, 113]. Of course, these results cannot be so strong as for (HLΔE), since such equations have not “so nice – almost linear” properties, that are necessary to use our methods.

As we have suggested at the end of Section 1.1, we would like to develop oscillation theory of (HLΔE) motivating ourselves by theory of (LDE). Thus, the first idea, which comes in mind, is to try to prove a half–linear discrete extension of the Roundabout theorem. This idea was successfully realized in [98], and hereby we have obtained a powerful tool to investigate oscillatory properties of (HLΔE). The principal aim of this work is to present this particular statement along with its applications and related results taking the following author’s papers as basis: [38, 98, 99, 101, 102, 103, 104, 105, 106].

Next we give the contents of this work in a more precise way:

In the last section of this chapter we list the most frequently used notation and we also mention, which parts of mathematics are used in this work.

In Chapter 2 we will define the basic concepts of the half–linear discrete oscillation theory. Further, we will describe connections of (HLΔE) with some other mathematical objects and explain some discrepancies between the continuous and the discrete cases.

Chapter 3 contains the most important result of this work, namely the discrete half–linear version of the Roundabout Theorem. The Picone type identity, presented in Section 3.1, plays a crucial role in the proof of this theorem. As we will see in Section 3.3, the Sturm type comparison and separation theorem can be easily obtained from the Roundabout theorem. The last section of this chapter is devoted to the remark on Casoratian.

In Chapter 4 we will explain what we mean by the terms the “Riccati technique” and the “variational principle”, respectively, and how they can be used in the oscillation theory of (HLΔE). We will also discuss the use of some further methods.

Chapter 5 will be started with oscillation criteria for (HLΔE) based on the variational principle and there it will be also shown the use of the reciprocity principle. This will be followed by the discrete half–linear version of well–known Hartman–Wintner theorem with some applications. The last two sections of this chapter are devoted to the oscillation criteria for (HLΔE), where we suppose the convergence of ∑^{∞} p_{j}, and where ∑^{∞} r_{j}^{1−β} = ∞ with r_{k} > 0 (or r_{k} ≡ 1, respectively) is assumed.

In Chapter 6 we will see that the use of the different methods to extend certain type of nonoscillation criterion gives the different results. Chapter is finished by conjugacy criteria.

Various comparison theorems for (HLΔE) of some other types than that classical Sturm’s, including the telescoping principle, are contained in Chapter 7.

In Chapter 8 it will be shown how the strong oscillation of equation (LDE) and related concepts, introduced by Nehari in [90], can be extended to the case of equation (HLΔE). We will also investigate some oscillatory properties of discrete generalized Euler equation.

The work is finished by Chapter 9, where future directions will be indicated which may be pursued in the context of this research.

1.4. Notation and basic background used in this work

To precede a possible misunderstanding and to make reading more synoptic, we list some notation used in the sequel:

• Δ – usual forward difference operator, i.e., Δx_{k} = x_{k+1} − x_{k}, Δ^{2}x_{k} = Δ(Δx_{k}), • [m, n] – a discrete interval, i.e., [m, n] ≡ {m, m + 1, …, n} for m ≤ n, (m, n] ≡

{m + 1, m + 2, …, n} for m < n,

• α and β – mutually conjugate numbers, i.e., 1/α + 1/β = 1, α > 1, β > 1,

• Φ : ℝ → ℝ – function defined by Φ(x) = |x|^{α−1} sgn x = |x|^{α−2}x, Φ^{−1} being the inverse of Φ, i.e., Φ^{−1}(x) = |x|^{β−1} sgn x,

• l^{∞} – the set of real sequences defined on the set of (positive) integers, where any sequence is bounded with respect to the usual supremum norm.

In what follows we mention the parts of mathematics that are used in this work. Of course, in particular, it is the difference (and the summation) calculus. Since the difference calculus is not so usual as the differential, we recall here the most frequently applied rules: Δ(u_{k} + v_{k}) = Δu_{k} + Δv_{k}, Δ(Cu_{k}) = CΔu_{k} for C ∈ ℝ, Δ(u_{k}v_{k}) = Δu_{k}v_{k+1} + u_{k}Δv_{k} = Δu_{k}v_{k}+u_{k+1}Δv_{k}, Δ(u_{k}/v_{k}) = (Δu_{k}v_{k}−u_{k}Δv_{k})/v_{k}v_{k+1}, ∑(u_{k}+v_{k}) = ∑ u_{k}+∑ v_{k}, ∑ Du_{k} = D ∑ u_{k} for D ∈ ℝ, ∑(u_{k}Δv_{k}) = u_{k}v_{k} − ∑ Δu_{k}v_{k+1} and ∑_{k=m}^{n−1} u_{k} = v_{n} − v_{m} for v_{k} such that Δv_{k} = u_{k}. As we have already mentioned, unfortunately, there is no rule in difference calculus similar to the chain rule from differential calculus. It means that we only know Δf(x_{k}) = f(x_{k+1}) − f(x_{k}). Note that to compute it, the Lagrange mean value theorem is often useful. A great number of various inequalities is a further part of mathematical background that is used and the Schauder fixed point theorem finds also an application in this work. The most of our results is motivated by the theory of linear differential equations and hence one can say that this part is used in a certain sense as well.

CHAPTER 2 Basic concepts and some properties of (HLΔE)

2.1. Basic concepts

In this section we will define basic concepts of half–linear discrete oscillation theory and will compare our equation with its continuous counterpart (HLDE). We investigate the second order half–linear difference equation

Δ(r_{k}Φ(Δy_{k})) + p_{k}Φ(y_{k+1}) = 0,  (2.1)

where p_{k} and r_{k} are real-valued sequences defined on ℕ (or on ℤ) with r_{k} ≠ 0 and Φ(x) := |x|^{α−1} sgn x = |x|^{α−2}x, α > 1. Sometimes, we will deal with a slightly more special equation

Δ(Φ(Δy_{k})) + p_{k}Φ(y_{k+1}) = 0.  (2.2)

Observe that in the continuous case the transformation of the independent variable

s = ∫ ^{t} r^{1−β}(τ) dτ, x(s) = y(t)  (2.3)

transforms (HLDE) into the equation

\frac{d}{ds} (Φ (\frac{d}{ds}x)) + r^{β−1}(t(s))p(t(s))Φ(x) = 0,

where t = t(s) is the inverse function of s = s(t) given by (2.3). In the discrete case we have no analogy of this transformation and hence it is purposeful to investigate half–linear difference equation in a more general form (2.1), if it is possible.

Since (2.1) is in fact a recurrence relation, given real initial values y_{m} and y_{m+1} for some m ∈ ℕ, it is clear that we can inductively obtain y_{m+2}, y_{m+3}, . . .. Hereby, the existence and uniqueness of a solution of the initial value problem (2.1), y_{m} = A and y_{m+1} = B is guaranteed.

Instead of the zero of a solution, known from the continuous case, it is necessary to use the following more complicated concept in the corresponding discrete case.

Definition 2.1.

• An interval (m, m + 1] is said to contain the generalized zero of a solution y of (2.1), if y_{m} ≠ 0 and r_{m}y_{m}y_{m+1} ≤ 0.

• Equation (2.1) is said to be disconjugate on an interval [m, n] provided any solution of this equation has at most one generalized zero on (m, n + 1] and the solution ỹ satisfying ỹ_{m} = 0 has no generalized zeros on (m, n + 1]. Otherwise, (2.1) is said to be conjugate on [m, n].

Next we introduce the so–called discrete α–degree functional.

Definition 2.2.

• Define a class U = U(m, n) of the so–called admissible sequences by

U(m, n) = {ξ | ξ : [m, n + 2] ⟶ ℝ such that ξ_{m} = ξ_{n+1} = 0}.

• Define an “α-degree” functional F on U(m, n) by

F(ξ; m, n) = ∑_{k=m}^{n}[r_{k}|Δξ_{k}|^{α} − p_{k}|ξ_{k+1}|^{α}].

• We say F is positive definite on U provided F(ξ) ≥ 0 for all ξ ∈ U(m, n) and F(ξ) = 0 if and only if ξ = 0.

The concept of (non)oscillation of (2.1) is defined in the following way.

Definition 2.3. Equation (2.1) is said to be nonoscillatory if there exists m ∈ ℕ such that this equation is disconjugate on [m, n] for every n > m. In the opposite case (2.1) is said to be oscillatory. Oscillation of (2.1) may be equivalently defined as follows. A nontrivial solution of (2.1) is called oscillatory if it has infinitely many generalized zeros. In view of the fact that the Sturm type Separation Theorem extends to (2.1), see Theorem 3.3, we have the following: Nontrivial solutions of (2.1) are either all oscillatory or all nonoscillatory. Hence we can speak about oscillation or nonoscillation of equation (2.1).

Along with equation (2.1) consider the so–called generalized Riccati difference equation

R[w_{k}] := Δw_{k} + p_{k} + S(w_{k}, r_{k}) = 0,  (2.4)

or, equivalently,

w_{k+1} = −p_{k} + S̄(w_{k}, r_{k}),  (2.5)

where

S(w_{k}, r_{k}) = w_{k} (1 − \frac{r_{k}}{Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))})  (2.6)

and S̄(w_{k}, r_{k}) = w_{k} − S(w_{k}, r_{k}). Equation (2.4) is related to (2.1) by the Riccati type substitution

w_{k} = \frac{r_{k}Φ(Δy_{k})}{Φ(y_{k})}.

The connection of (2.4) with (2.1) will be given in a more precise way by the Roundabout theorem (Theorem 3.1). The properties of the functions S and S̄ that are needed in the further consideration, will be described by Lemma 4.3 in Section 4.1.

In Section 1.1 we have noted two differences between equation (LDE) and equation (LΔE):

1) Unknown sequence y in the second term of (LΔE) has an index k + 1 while in (LDE) we have y(t). The same holds for half–linear equations. It is due to the discretization, which is described in the next section.

2) The sequence r_{k} in (LΔE) is required only to be nonzero in the contrast to the continuous case where we need r(t) > 0. There exist several viewpoints, which provide an explanation of this discrepancy, e.g. the variational viewpoint, see [3], or the discrete analogy of the rewriting of second order linear differential equation into the self–adjoint form (LDE) – this includes the multiplying of equation by the so–called integrating factor (always positive in the continuous case) that may change the sign in the discrete case. This integrating factor is then just the sequence r_{k}. Note that the authors who studied equation (LΔE) with similar definition of generalized zero as in Definition 2.1 (but with the assumption y_{m}y_{m+1} ≤ 0 instead of r_{m}y_{m}y_{m+1} ≤ 0 – this lead to the Hartman’s definition of the so–called node), had problems with Sturmian theory since there exist simple examples such as the Fibonacci recurrence relation

y_{k+2} − y_{k+1} − y_{k} = 0

with the linearly independent solutions

u_{k} = (\frac{1 − \sqrt{5}}{2})^{k} and v_{k} = (\frac{1 + \sqrt{5}}{2})^{k} ,

where one solution u_{k} seems to be oscillatory while another v_{k} is nonoscillatory (but we know that generalized zeros of one solution would separate generalized zeros of another – independent solution). This fact does not occur in the continuous case and our definition of generalized zero brings these exceptional cases into the general theory. On the other hand, it is obvious that the presence of the sequence r_{k} in Definition 2.1 is the result of the assumption r_{k} ≠ 0 for equation (LΔE). Note that we cannot rewrite the Fibonacci equation into the self–adjoint form (LΔE) with a positive r_{k}. Its self–adjoint form is

Δ((−1)^{k}Δy_{k}) + (−1)^{k}y_{k+1} = 0

and according to our definition it is oscillatory. Since linear equation (LΔE) is the special case of equation (2.1), it seems to be natural to require the assumption r_{k} ≠ 0 also for equation (2.1).

2.2. Discretization of (HLDE) and one discrete variational problem

In Chapter 1 we have explained where equation (2.1) comes from. Recall that it can be viewed either as a half–linear extension of linear difference equation (LΔE), or a discretization of half–linear differential equation (HLDE). In addition to this, (2.1) can be understood as equation that is connected to a certain discrete variational problem, as we will see in this section.

But, first, let us describe the process of discretization of (HLDE). Consider the second order half–linear differential equation

(r̃(t)Φ(z′(t)))′ + p̃(t)Φ(z(t)) = 0,

where r̃(t) and p̃(t) are continuous on [a, b]. For small h = (b − a)/n, n ∈ ℕ, we have

z′(t) ≈ \frac{z(t) − z(t − h)}{h}

and

(r̃(t)Φ(z′(t)))′ ≈ \frac{1}{h} \{\frac{r̃(t + h)Φ[z(t + h) − z(t)]}{h} − \frac{r̃(t)Φ[z(t) − z(t − h)]}{h}\} .

Let t = a + kh, where k is a discrete variable taking on the integer values 0 ≤ k ≤ n. If z(t) is a solution of the above (HLDE) on [a, b], then we have

r̃(a + (k + 1)h)Φ[z(a + (k + 1)h) − z(a + kh)]−

− r̃(a + kh)Φ[z(a + kh) − z(a + (k − 1)h)] + h^{2}p̃(a + kh)Φ(z(a + kh)) ≈ 0.

Now we set y_{k+1} = z(a + kh), r_{k} = r̃(a + kh) and p_{k} = h^{2}p̃(a + kh). Hence we get

r_{k+1}Φ(y_{k+2} − y_{k+1}) − r_{k}Φ(y_{k+1} − y_{k}) + p_{k}Φ(y_{k+1}) ≈ 0

and thus

Δ(r_{k}Φ(Δy_{k})) + p_{k}Φ(y_{k+1}) ≈ 0

for 0 ≤ k ≤ n − 2. Note that y_{k} is defined for 0 ≤ k ≤ n.

In the second part of this section we give a discrete variational problem. Consider the functional F(ξ; m, n) on a set of admissible sequences

Ū = {ξ | ξ : [m, n + 2] ⟶ ℝ such that ξ_{m} = A, and ξ_{n+1} = B}.

We are interested in to extremize F subject to ξ ∈ Ū. For this purpose denote the summand of F by

f(k, ξ_{k+1}, Δξ_{k}) = r_{k}|Δξ_{k}|^{α} − p_{k}|ξ_{k+1}|^{α}.

Next we give a necessary condition for F to have a local extremum at ξ̃ on Ū.

Proposition 2.1. If F has a local extremum at ξ̃ on Ū, then ξ̃_{k} satisfies equation (2.1) on [m, n].

Proof. By [3, Theorem 4.3], the assumptions of proposition imply that ξ̃ satisfies the Euler–Lagrange equation

\frac{∂f}{∂s} (k, ξ_{k+1}, Δξ_{k}) − Δ\frac{∂f}{∂t} (k, ξ_{k+1}, Δξ_{k}) = 0

(recall that f is the function of three variables, i.e., f = f(r, s, t)) and the simple calculation shows that ξ̃_{k} is a solution of (2.1) for k ∈ [m, n].

Thus we see that (2.1) is in fact Euler–Lagrange equation associated to the above variational problem concerning extremizing of α–degree functional.

CHAPTER 3 Roundabout theorem

3.1. Picone identity

The results of this chapter were established in [98]. We start with a statement playing an important role in the proof of Theorem 3.1. To this purpose, consider the second order difference operators of the form

l[y_{k}] ≡ Δ(r_{k}Φ(Δy_{k})) + p_{k}Φ(y_{k+1})

and

L[z_{k}] ≡ Δ(R_{k}Φ(Δz_{k})) + P_{k}Φ(z_{k+1}),

where k ∈ [m, n], m, n ∈ ℤ, m ≤ n, and p_{k}, P_{k} are real–valued sequences defined on [m, n]. Sequences r_{k}, R_{k} are real–valued and defined on [m, n + 1] with r_{k} ≠ 0, R_{k} ≠ 0 on this interval.

Now we can formulate the discrete half–linear extension of the so–called Picone identity. The idea is to rewrite the functional F associated with the disconjugate equation (2.1), see Definition 2.2, into a form which in the linear case corresponds to the “completion to the square”, which then shows equivalence between the disconjugacy of (2.1) and the positive definiteness of F. Note that our version of the Picone identity is quite general and we will actually use only its special case.

Lemma 3.1 (Picone Type Identity). Let y_{k}, z_{k} be defined on [m, n + 2] and let z_{k} ≠ 0 for k ∈ [m, n + 1]. Then the equality

Δ \{\frac{y_{k}}{Φ(z_{k})}[Φ(z_{k})r_{k}Φ(Δy_{k}) − Φ(y_{k})R_{k}Φ(Δz_{k})]\} =

= (P_{k} − p_{k})|y_{k+1}|^{α} + (r_{k} − R_{k})|Δy_{k}|^{α} + \frac{y_{k+1}}{Φ(z_{k+1})}\{l[y_{k}]Φ(z_{k+1}) − L[z_{k}]Φ(y_{k+1})\} +

+ \{R_{k}|Δy_{k}|^{α} − \frac{R_{k}Φ(Δz_{k})}{Φ(z_{k+1})}|y_{k+1}|^{α} + \frac{R_{k}Φ(Δz_{k})}{Φ(z_{k})}|y_{k}|^{α}\} (3.1)

holds for k ∈ [m, n].

Proof. For k ∈ [m, n] we have

Δ[y_{k}r_{k}Φ(Δy_{k})] − y_{k+1}l[y_{k}]
= y_{k+1}Δ(r_{k}Φ(Δy_{k})) + Δy_{k}r_{k}Φ(Δy_{k}) − y_{k+1}Δ(r_{k}Φ(Δy_{k})) − y_{k+1}p_{k}Φ(y_{k+1})
= −p_{k}|y_{k+1}|^{α} + r_{k}|Δy_{k}|^{α}.

Further, again for k ∈ [m, n], we have

L[z_{k}]Φ(y_{k+1})\frac{y_{k+1}}{Φ(z_{k+1})} − Δ [\frac{y_{k}}{Φ(z_{k})}Φ(y_{k})R_{k}Φ(Δz_{k})] −

−R_{k}|Δy_{k}|^{α} + \frac{R_{k}Φ(Δz_{k})}{Φ(z_{k+1})}y_{k+1}Φ(y_{k+1}) − \frac{R_{k}Φ(Δz_{k})}{Φ(z_{k})}y_{k}Φ(y_{k}) =

= \frac{Δ(R_{k}Φ(Δz_{k}))}{Φ(z_{k+1})}y_{k+1}Φ(y_{k+1}) + P_{k}y_{k+1}Φ(y_{k+1}) − \frac{Δy_{k}Φ(z_{k}) − y_{k}ΔΦ(z_{k})}{Φ(z_{k})Φ(z_{k+1})} ×

×Φ(y_{k})R_{k}Φ(Δz_{k}) − \frac{y_{k+1}}{Φ(z_{k+1})}Φ(y_{k+1})Δ(R_{k}Φ(Δz_{k})) −

−\frac{y_{k+1}}{Φ(z_{k+1})}ΔΦ(y_{k})R_{k}Φ(Δz_{k}) −

−R_{k}|Δy_{k}|^{α} + \frac{R_{k}Φ(Δz_{k})}{Φ(z_{k+1})}y_{k+1}Φ(y_{k+1}) − \frac{R_{k}Φ(Δz_{k})}{Φ(z_{k})}y_{k}Φ(y_{k})

= P_{k}|y_{k+1}|^{α} − R_{k}|Δy_{k}|^{α} +

+\frac{1}{Φ(z_{k})Φ(z_{k+1})}[−R_{k}Δy_{k}Φ(y_{k})Φ(z_{k})Φ(Δz_{k}) +

+R_{k}y_{k}Φ(y_{k})Φ(Δz_{k})ΔΦ(z_{k}) − R_{k}y_{k+1}ΔΦ(y_{k})Φ(z_{k})Φ(Δz_{k}) + +R_{k}y_{k+1}Φ(y_{k+1})Φ(z_{k})Φ(Δz_{k}) − R_{k}y_{k}Φ(y_{k})Φ(z_{k+1})Φ(Δz_{k})]

= P_{k}|y_{k+1}|^{α} − R_{k}|Δy_{k}|^{α} +

+\frac{1}{Φ(z_{k})Φ(z_{k+1})}[R_{k}y_{k}Φ(y_{k})Φ(z_{k})Φ(Δz_{k}) − R_{k}y_{k}Φ(y_{k})Φ(z_{k})Φ(Δz_{k})]

= P_{k}|y_{k+1}|^{α} − R_{k}|Δy_{k}|^{α}.

Combining these two equalities we get the desired result.

The last summand of (3.1) can be rewritten as G(y, z)R_{k}z_{k}/z_{k+1}, where

G(y, z) := \frac{z_{k+1}}{z_{k}} |Δy_{k}|^{α} − \frac{z_{k+1}Φ(Δz_{k})}{z_{k}Φ(z_{k+1})}|y_{k+1}|^{α} + \frac{z_{k+1}Φ(Δz_{k})}{z_{k}Φ(z_{k})}|y_{k}|^{α}.

Using this fact we have the following lemma.

Lemma 3.2. Let y_{k}, z_{k} be defined on [m, n + 1] and let z_{k} ≠ 0 on this interval. Then

G(y, z) ≥ 0

for k ∈ [m, n], where the equality holds if and only if Δy_{k} = y_{k}(Δz_{k}/z_{k}).

Proof. It is sufficient to verify the inequality

\frac{z_{k+1}}{z_{k}} |Δy_{k}|^{α} + \frac{z_{k+1}|Δz_{k}|^{α−2}Δz_{k}}{|z_{k}|^{α−2}z_{k}^{2}} |y_{k}|^{α} ≥ \frac{z_{k+1}|Δz_{k}|^{α−2}Δz_{k}}{z_{k}|z_{k+1}|^{α−2}z_{k+1}} |y_{k+1}|^{α}  (3.2)

for k ∈ [m, n].

Denote u_{k} := z_{k}/z_{k+1}. Then inequality (3.2) assumes the form

\frac{|Δy_{k}|^{α}}{u_{k}} + \frac{|1 − u_{k}|^{α−2}(1 − u_{k})}{|u_{k}|^{α}}|y_{k}|^{α} ≥ \frac{1}{u_{k}} |1 − u_{k}|^{α−2}(1 − u_{k})|y_{k+1}|^{α}.  (3.3)

First we prove the statement of Lemma 3.2 for k ∈ [m, n] such that u_{k} = 1. In this case clearly G(y, z) = |Δy_{k}|^{α} ≥ 0, where equality holds if and only if Δy_{k} = 0, which holds if and only if Δy_{k} = y_{k}(Δz_{k}/z_{k}).

In the remainder of this proof, when we write u we mean u_{k} (the same holds for the sequences a, b, v and v_{0}).

Now, denote

J_{0} := {k ∈ [m, n]; z_{k} ≠ z_{k+1}},
J_{1} := {k ∈ J_{0}; y_{k} = 0},
J_{2} := {k ∈ J_{0}; Δy_{k} = 0}.

Hence we have four cases: Case I: k ∈ J_{0} \ (J_{1} ∪ J_{2}).

Putting a = Δy_{k}, b = y_{k} we get y_{k+1} = a + b. Inequality (3.3) now leads to

\frac{|a|^{α}}{u} + \frac{|1 − u|^{α−2}(1 − u)}{|u|^{α}}|b|^{α} ≥ \frac{1}{u}|1 − u|^{α−2}(1 − u)|a + b|^{α}

and by dividing it by |b|^{α} the inequality (3.3) assumes the form

|\frac{a}{b}|^{α} \frac{1}{u} + \frac{1}{|u|^{α}} |1 − u|^{α−2}(1 − u) ≥ \frac{1}{u}|1 − u|^{α−2}(1 − u) |1 + \frac{a}{b}|^{α} .

Now, denote

H(v; u) := \frac{|v|^{α}}{u} − \frac{1}{u}|1 − u|^{α−2}(1 − u)|1 + v|^{α} + \frac{1}{|u|^{α}} |1 − u|^{α−2}(1 − u),

where v := a/b. For v = v_{0} := (1 − u)/u the following equality holds:

H(v_{0}; u) = \frac{|1 − u|^{α}}{u|u|^{α}} − \frac{1}{u|u|^{α}} |1 − u|^{α−2}(1 − u) + \frac{1}{|u|^{α}} |1 − u|^{α−2}(1 − u)
= \frac{|1 − u|^{α−2}}{u|u|^{α}} ((1 − u)^{2} − (1 − u) + u − u^{2}) = 0.

Differentiating H with respect to v we obtain

H_{v}(v; u) = α\frac{|v|^{α−1} sgn v}{u} − α\frac{1}{u}|1 − u|^{α−2}(1 − u)|1 + v|^{α−1} sgn(1 + v)

and hence

H_{v}(v_{0}; u) = α\frac{|1 − u|^{α−1} sgn(1 − u)}{|u|^{α−1}u sgn u} − α\frac{|1 − u|^{α−2}(1 − u)}{|u|^{α−1}u sgn u} = 0.

Further, we get

H_{vv}(v; u) = α(α − 1)\frac{1}{u} (|v|^{α−2} − |1 − u|^{α−2}(1 − u)|1 + v|^{α−2}) .

Consequently,

H_{vv}(v_{0}; u) = α(α − 1)\frac{1}{u} (\frac{|1 − u|^{α−2}}{|u|^{α−2}} − \frac{|1 − u|^{α−2}(1 − u)}{|u|^{α−2}})
= α(α − 1)\frac{|1 − u|^{α−2}}{u|u|^{α−2}} (1 − 1 + u)
= α(α − 1)\frac{|1 − u|^{α−2}}{|u|^{α−2}} > 0.

Since

H_{v}(v; u) = 0 ⟺ |v|^{α−1} sgn v = |1 − u|^{α−2}(1 − u)|1 + v|^{α−1} sgn(1 + v)
⟺ |v|^{α−1} sgn v = |(1 − u)(1 + v)|^{α−1} sgn[(1 − u)(1 + v)]
⟺ v = 1 + v − u − uv
⟺ v = \frac{1 − u}{u}
⟺ v = v_{0}

holds, hence H_{v} has just a single zero v_{0}. Note that this case occurs if and only if Δy_{k} = y_{k}(Δz_{k}/z_{k}). In the opposite case, H(v; u) > 0.

Case II: k ∈ J_{1} \ J_{2}.

Suppose, by contradiction (see the inequality (3.3)), that

\frac{|Δy_{k}|^{α}}{u} ≤ \frac{1}{u}|1 − u|^{α−2}(1 − u)|Δy_{k}|^{α}.

Therefore

\frac{1}{u} + |1 − u|^{α−2} ≤ \frac{|1 − u|^{α−2}}{u} .

Further, we distinguish the following particular cases:

• if u > 1, then the following inequality holds:

1 + u|1 − u|^{α−2} ≤ |1 − u|^{α−2},
\frac{1}{|1 − u|^{α−1}} ≤ sgn(1 − u),

a contradiction, since the left hand side is positive,

• for 0 < u < 1 the same computation as above holds and hence we get a contradiction, since 1/|1 − u|^{α−1} ≰ 1, where 0 < |1 − u|^{α−1} < 1,

• for u < 0 we have

\frac{1}{|1 − u|^{α−1}} ≥ sgn(1 − u),

again a contradiction, since 1/|1 − u|^{α−1} ≱ 1, where |1 − u|^{α−1} > 1. Case III: k ∈ J_{2} \ J_{1}. Assume, by way of contradiction, that

\frac{|1 − u|^{α−2}(1 − u)}{|u|^{α}}|y_{k}|^{α} ≤ \frac{1}{u}|1 − u|^{α−2}(1 − u)|y_{k}|^{α}.

Consequently,

\frac{1 − u}{|u|^{α}} ≤ \frac{1 − u}{u}.

Similarly as in Case II we have • u > 1 : 1/|u|^{α} ≥ 1/u ⟹ u ≥ |u|^{α},

• 0 < u < 1 : 1/|u|^{α} ≤ 1/u ⟹ u ≤ |u|^{α},

• u < 0 : 1/|u|^{α} ≤ 1/u ⟹ u ≥ |u|^{α}. Obviously in every particular case we again come to a contradiction.

Case IV: k ∈ J_{1} ∩ J_{2}.

Here we clearly see that G(y, z) = 0, since y_{k} = 0 and Δy_{k} = 0. Note that this case occurs if and only if Δy_{k} = y_{k}(Δz_{k}/z_{k}). The proof is complete.

Remark 3.1. (i) (Linear case) If we put α = 2, we get

G(y, z) = (Δy_{k} − \frac{Δz_{k}}{z_{k}} y_{k})^{2} .

(ii) Note that the above Picone identity was used in [33] to investigate oscillatory properties of the forced second order half–linear difference equation

Δ(r_{k}Φ(Δy_{k})) + p_{k}Φ(y_{k+1}) = h_{k}.

(iii) For the half–linear continuous version of the Picone identity and some of its applications see [63].

3.2. Roundabout Theorem

Now we are in a position to formulate the central statement of the oscillation theory of equation (2.1) – the discrete half–linear version of the Roundabout Theorem. Consider equation (2.1) on the interval [m, n] with r_{k} ≠ 0 on [m, n + 1].

Theorem 3.1 (Roundabout Theorem). The following statements are equivalent:

(i) Equation (2.1) is disconjugate on [m, n].

(ii) Equation (2.1) has a solution y without generalized zeros on [m, n + 1]. (iii) The generalized Riccati difference equation (2.4) has a solution w_{k} on [m, n] with r_{k} +

w_{k} > 0 on [m, n]. (iv) The functional F, defined by Definition 2.2, is positive definite on U(m, n).

Proof. (i)⟹(ii): The proof of this implication is essentially the same as in the linear case. Indeed, let z_{k} be a solution of (2.1) given by the initial conditions

z_{m} = 0, z_{m+1} = 1.

It follows that r_{k}z_{k}z_{k+1} > 0 for k ∈ [m + 1, n]. Consider a solution z_{k}^{[ε]} satisfying the initial conditions

z_{m}^{[ε]} = εr_{m}, z_{m+1}^{[ε]} = 1.

Then

z_{k}^{[ε]} → z_{k} as ε → 0.

If we choose ε > 0 sufficiently small, then y_{k} ≡ z_{k}^{[ε]} satisfies

r_{k}y_{k}y_{k+1} > 0 for k ∈ [m, n],

i.e., y has no generalized zero on [m, n + 1].

(ii)⟹(iii): Assume that z_{k} is a solution of (2.1) with r_{k}z_{k}z_{k+1} > 0 on [m, n]. Using the Riccati type substitution w_{k} = r_{k}Φ(Δz_{k})/Φ(z_{k}) we have

\frac{r_{k}z_{k+1}}{z_{k}} = r_{k} (\frac{z_{k} + Δz_{k}}{z_{k}} ) = r_{k} (1 + \frac{Φ^{−1}(w_{k})}{Φ^{−1}(r_{k})})
= \frac{r_{k}}{Φ^{−1}(r_{k})}(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k})).  (3.4)

Since

Φ (\frac{z_{k}}{z_{k+1}} ) = Φ(r_{k})Φ (\frac{z_{k}}{r_{k}z_{k+1}} ) = \frac{r_{k}}{Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))},

we obtain

Δw_{k} = p_{k} − w_{k} (1 − \frac{Φ(z_{k})}{Φ(z_{k+1})}) = p_{k} − w_{k} (1 − \frac{r_{k}}{Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))}) .

Now, (3.4) clearly implies r_{k} + w_{k} > 0 and, hence, (iii) holds.

(iii)⟹(iv): Assume that w_{k} is a solution of (2.4) with r_{k} + w_{k} > 0. Note that then z_{k} given by w_{k} = r_{k}Φ(Δz_{k})/Φ(z_{k}), i.e. Δz_{k} = Φ^{−1}(w_{k}/r_{k})z_{k}, is a solution of (2.1). From the Picone identity (3.1) applied to the case p_{k} ≡ P_{k}, r_{k} ≡ R_{k}, y_{k} = ξ_{k} and w_{k} = r_{k}Φ(Δz_{k})/Φ(z_{k}) we obtain

Δ[ξ_{k}r_{k}Φ(Δξ_{k})] − Δ[|ξ_{k}|^{α}w_{k}] = ξ_{k+1}Δ(r_{k}Φ(Δξ_{k})) + p_{k}|ξ_{k+1}|^{α} + G̃,

where

G̃ = G̃(ξ, w) = r_{k}|Δξ_{k}|^{α} − \frac{w_{k}r_{k}}{Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))}|ξ_{k+1}|^{α} + w_{k}|ξ_{k}|^{α}.

Hence

r_{k}|Δξ_{k}|^{α} − p_{k}|ξ_{k+1}|^{α} = Δ[w_{k}|ξ_{k}|^{α}] + G̃.

The summation of the above given equality from m to n yields

F(ξ) = [w_{k}|ξ_{k}|^{α}]_{k=m}^{n+1} + ∑_{k=m}^{n} G̃(ξ, w).

Then F(ξ) ≥ 0, since r_{k}z_{k+1}/z_{k} > 0 by (3.4) and Lemma 3.2 holds. In addition, if F(ξ) = 0, then, again by Lemma 3.2, Δξ_{k} = ξ_{k}Δz_{k}/z_{k}. Further, we have ξ_{m} = 0 and therefore ξ ≡ 0. Consequently, F(ξ) > 0 for all nontrivial admissible sequences.

(iv)⟹(i): Suppose, by contradiction, that (2.1) is not disconjugate on [m, n]. Then there exists a nontrivial solution y of (2.1) such that

r_{M}y_{M}y_{M+1} ≤ 0, y_{M+1} ≠ 0
r_{N}y_{N}y_{N+1} ≤ 0, y_{N} ≠ 0

where m + 1 ≤ M + 1 < N ≤ n. Define

ξ_{k} = \{ \begin{matrix}0 & for k = m, …, M, \\ y_{k} & for k = M + 1, …, N, \\ 0 & for k = N + 1, …, n + 1. \end{matrix}

Then ξ_{k} is a nontrivial admissible sequence and hence F(ξ) > 0. Using summation by parts we obtain

F(ξ) = ∑_{k=m}^{n}[r_{k}|Δξ_{k}|^{α} − p_{k}|ξ_{k+1}|^{α}]

= [ξ_{k}r_{k}Φ(Δξ_{k})]_{k=m}^{n+1} − ∑_{k=m}^{n} ξ_{k+1}l[ξ_{k}] = − ∑_{k=M}^{N−1} ξ_{k+1}l[ξ_{k}]
= y_{M+1}[−p_{M}Φ(y_{M+1}) − Δ(r_{M}Φ(Δξ_{M}))] +
+y_{N}[−p_{N−1}Φ(y_{N}) − Δ(r_{N−1}Φ(Δξ_{N−1}))]
= y_{M+1}[Δ(r_{M}Φ(Δy_{M})) − r_{M+1}Φ(Δξ_{M+1}) + r_{M}Φ(Δξ_{M})] +
+y_{N}[Δ(r_{N−1}Φ(Δy_{N−1})) − r_{N}Φ(Δξ_{N}) + r_{N−1}Φ(Δξ_{N−1})]
= y_{M+1}[r_{M+1}Φ(Δy_{M+1}) − r_{M}Φ(Δy_{M}) − r_{M+1}Φ(Δy_{M+1}) + r_{M}Φ(y_{M+1})] +
+y_{N}[r_{N}Φ(Δy_{N}) − r_{N−1}Φ(Δy_{N−1}) + r_{N}Φ(y_{N}) + r_{N−1}Φ(Δy_{N−1})]
= G_{1} + G_{2},

where

G_{1} = G_{1}(y_{M}, y_{M+1}; r_{M}) = y_{M+1}r_{M}Φ(y_{M+1}) − y_{M+1}r_{M}Φ(Δy_{M})

and

G_{2} = G_{2}(y_{N}, y_{N+1}; r_{N}) = y_{N}r_{N}Φ(Δy_{N}) + y_{N}r_{N}Φ(y_{N}).

To show that F(ξ) ≤ 0 it remains to verify that G_{1} ≤ 0 and G_{2} ≤ 0. Let us examine for example the function G_{2}. It means that we shall check the inequality

y_{N}r_{N}Φ(Δy_{N}) ≤ −y_{N}r_{N}Φ(y_{N}).

It holds if and only if

r_{N}Φ (\frac{Δy_{N}}{y_{N}} ) ≤ −r_{N}.

Now, if Δy_{N} = 0, then we get G_{2} = r_{N}|y_{N}|^{α}. Hence r_{N} must be negative, since we assume r_{N}y_{N}^{2} ≤ 0. Consequently, G_{2} < 0. Further, let Δy_{N} ≠ 0. Putting x = y_{N+1}/y_{N} we obtain

G̃_{2}(x; r_{N}) := r_{N}|x − 1|^{α−1} sgn(x − 1) + r_{N}.

Note that G_{2} < 0 (G_{2} = 0) if and only if G̃_{2} < 0 (G̃_{2} = 0). If y_{N+1} = 0 then x = 0 and hence G̃_{2}(0; r_{N}) = 0. Differentiating G̃_{2} with respect to x we obtain

\frac{∂G̃_{2}}{∂x} = (α − 1)r_{N}|x − 1|^{α−2}.

Now, we distinguish the following particular cases:

• x > 0 ⟺ y_{N}y_{N+1} > 0 ⟺ r_{N} < 0 ⟺ ∂G̃_{2}/∂x < 0,

• x < 0 ⟺ y_{N}y_{N+1} < 0 ⟺ r_{N} > 0 ⟺ ∂G̃_{2}/∂x > 0. Therefore we get G_{2} ≤ 0.

Similarly one can verify that G_{1} ≤ 0 holds. Summarizing the above computations we have F(ξ) = G_{1} + G_{2} ≤ 0, a contradiction. Hence (i) holds.

Remark 3.2. The Picone identity could be also used to show that absence of generalized zeros of the solution of equation (2.1) implies positive definiteness of the functional F (the implication (ii) ⇒ (iv)). But we include the generalized Riccati equation into the Roundabout theorem since such an equivalence is important for applications of our theory.

3.3. Sturmian theory

This section is devoted to Sturmian theory. Consider two equations l[y_{k}] = 0 and L[z_{k}] = 0 (the operators l, L are defined at the beginning of Section 3.1). Denote

F_{R,P} (ξ) := ∑_{k=m}^{n}[R_{k}|Δξ_{k}|^{α} − P_{k}|ξ_{k+1}|^{α}].

Then we have the following versions of Sturmian theorems for half-linear difference equations.

Theorem 3.2 (Sturm type Comparison Theorem). Suppose that we have R_{k} ≥ r_{k} and p_{k} ≥ P_{k} for k ∈ [m, n]. If l[y_{k}] = 0 is disconjugate on [m, n], then L[z_{k}] = 0 is also disconjugate on [m, n].

Proof. Suppose that l[y_{k}] = 0 is disconjugate on [m, n]. Then Theorem 3.1 yields F(ξ) > 0 for all admissible sequences ξ. For such a ξ we also have

F_{R,P} (ξ) ≥ F(ξ) > 0.

Hence F_{R,P} (ξ) > 0 and thus L[z_{k}] = 0 is disconjugate on [m, n] by Theorem 3.1.

For further comparison theorems see Chapter 7. As far as the separation result is concerned, note that the implication (ii) ⇒ (i) from Theorem 3.1 is in fact a Sturm type separation theorem. Hence we have the following statement. For the related result see also the next section.

Theorem 3.3 (Sturm type Separation Theorem). Two nontrivial solutions y^{[1]} and y^{[2]} of l[y_{k}] = 0, which are not proportional, cannot have a common zero. If y^{[1]} satisfying y_{m}^{[1]} = 0 has a generalized zero in (n, n + 1], then y^{[2]} has a generalized zero in (m, n + 1]. If y^{[1]} has generalized zeros in (m, m + 1] and (n, n + 1], then y^{[2]} has a generalized zero in (m, n + 1].

Proof. It is sufficient to prove the part concerning the common zero of nonproportional solutions since the remaining part follows from Theorem 3.1. Suppose, by contradiction, that y_{l}^{[1]} = 0 = y_{l}^{[2]} for some l ∈ [m, n]. Let ỹ be a solution of l[y_{k}] = 0 such that ỹ_{l} = 0, ỹ_{l+1} = 1. Then y^{[1]} := Aỹ and y^{[2]} := Bỹ, where A, B are suitable nonzero constants, are also nontrivial solutions of l[y_{k}] = 0 satisfying

y_{l}^{[1]} = 0, y_{l+1}^{[1]} = A and y_{l}^{[2]} = 0, y_{l+1}^{[2]} = B,

respectively. Hence y^{[1]} = Cy^{[2]}, where C = A/B, a contradiction.

3.4. A remark on Casoratian

The so–called Casoratian, defined by

ω(u, v)(k) = | \begin{matrix}u_{k} & v_{k} \\ Δu_{k} & Δv_{k} \end{matrix} | = | \begin{matrix}u_{k} & v_{k} \\ u_{k+1} & v_{k+1} \end{matrix} | ,

plays very important role in the oscillation theory of linear difference equation (LΔE); it is a discrete counterpart to Wronskian from the “continuous” theory. Its importance is given by the fact that ω(u, v)(k) is an identically zero/nonzero constant, if u_{k}, v_{k} are linearly dependent/independent solutions of (LΔE). In general, this does not hold for half–linear equation (2.1). In [41] it is proved for equation (HLDE) that the only possibility is α = 2, if we require the “characteristic” properties of Wronskian.

Thus, in what follows, we will see that the Casoratian can be useful in the half–linear discrete theory only as a certain characterization of the (in)dependence of two solutions. It is very easy to prove the next statement using the Riccati type transformation.

Theorem 3.4. Let u_{k}, v_{k} be two solutions of (2.1). Then the following statements are equivalent:

(i) ω(u, v)(T) = 0 for some T, (ii) u_{k} is a constant multiple of v_{k}, (iii) ω(u, v)(k) = 0 for all k.

From the above theorem one can see that Casoratian of two solutions of (2.1) is either identically zero or it is never zero on an interval where equation (2.1) is considered; in the second case such solutions are not proportional.

CHAPTER 4 Methods of half–linear discrete oscillation theory

4.1. Riccati technique

The method of oscillation theory of (2.1) that uses the following idea, is usually referred as the Riccati technique. Suppose that equation (2.1) is nonoscillatory, i.e., there exists a solution y_{k} without generalized zeros (for k sufficiently large) and therefore there exists a solution of equation (2.4) with r_{k} + w_{k} > 0 according to Theorem 3.1 and, vice versa, the existence of a solution of (2.4) with r_{k} + w_{k} > 0 in a neighbourhood of infinity guarantees nonoscillation of (2.1). Actually, according to hereafter given Lemma 4.2, for nonoscillation of (2.1) it is sufficient to find a solution of the generalized Riccati difference inequality

R[w_{k}] ≤ 0,  (4.1)

where the operator R is defined by (2.4), satisfying r_{k} + w_{k} > 0 in a neighbourhood of infinity. Sometimes we will use more special results where the generalized Riccati equation appears in a certain “summation” form, see Sections 5.2, 5.3, 5.4 and 7.2.

First we give auxiliary statement based on the Sturm type separation theorem.

Lemma 4.1. If there exists a sequence u_{k} such that

r_{k}u_{k}u_{k+1} > 0 and u_{k+1}l[u_{k}] ≤ 0  (4.2)

for k ∈ [m, ∞), m ∈ ℕ, where the operator l is defined at the beginning of Section 3.1, then (2.1) is nonoscillatory.

Proof. Suppose that a sequence u_{k} satisfies (4.2) on the interval [m, ∞). Then ϕ_{k} := −u_{k+1}l[u_{k}] is a nonnegative sequence on this discrete interval. Set r̃_{k} = r_{k} and p̃_{k} = p_{k} − Φ_{k}/|u_{k+1}|^{α}, then p̃_{k} ≥ p_{k} and

Δ(r̃_{k}Φ(Δu_{k})) + p̃_{k}Φ(u_{k+1}) = Δ(r_{k}Φ(Δu_{k})) + (p_{k} − \frac{ϕ_{k}}{|u_{k+1}|^{α}} ) Φ(u_{k+1}) = 0.

Thus the equation Δ(r̃_{k}Φ(Δu_{k})) + p̃_{k}Φ(u_{k+1}) = 0 is disconjugate on [m, ∞). Therefore (2.1) is disconjugate on [m, ∞) as well by the Sturm comparison theorem (Theorem 3.2) and hence nonoscillatory.

Lemma 4.2. Equation (2.1) is nonoscillatory if and only if there exists a sequence w_{k} with r_{k} + w_{k} > 0 for large k such that (4.1) holds.

Proof. The part “only if” follows immediately from Theorem 3.1. For the part “if”, let w_{k} satisfies R[w_{k}] ≤ 0 with r_{k} + w_{k} > 0 on [m, ∞) and let

u_{k} = ∏_{j=m}^{k−1} (1 + Φ^{−1}(w_{j}/r_{j})) , k > m,

is a solution of the first order difference equation Δu_{k} = Φ^{−1}(w_{k}/r_{k})u_{k}, u_{m} = 1. Then u_{k} ≠ 0 since

1 + Φ^{−1}(w_{k}/r_{k}) = \frac{1}{Φ^{−1}(r_{k})} (Φ^{−1}(r_{k}) + Φ^{−1}(w_{k})) ≠ 0

and r_{k}u_{k}u_{k+1} > 0 since

\frac{Φ(u_{k})}{Φ(u_{k+1})} = \frac{1}{Φ (1 + Δu_{k}/u_{k})} = \frac{r_{k}}{Φ (Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))}.

Further,

u_{k+1}l[u_{k}] = u_{k+1}(Δ(r_{k}Φ(Δu_{k})) + p_{k}Φ(u_{k+1})) − \frac{|u_{k+1}|^{α}r_{k}Φ(Δu_{k})ΔΦ(u_{k})}{Φ(u_{k})Φ(u_{k+1})}

+\frac{|u_{k+1}|^{α}r_{k}Φ(Δu_{k})ΔΦ(u_{k})}{Φ(u_{k})Φ(u_{k+1})}

= u_{k+1}Φ(u_{k+1})\frac{Δ(r_{k}Φ(Δu_{k}))Φ(u_{k}) − r_{k}Φ(Δu_{k})ΔΦ(u_{k})}{Φ(u_{k})Φ(u_{k+1})}

+|u_{k+1}|^{α}p_{k} + |u_{k+1}|^{α} \frac{r_{k}Φ(Δu_{k})}{Φ(u_{k})} (1 − \frac{Φ(u_{k})}{Φ(u_{k+1})})

= |u_{k+1}|^{α}R[w_{k}] ≤ 0

for k ∈ [m, ∞). The statement now follows from Lemma 4.1.

At the end of this section we describe some properties of the function

S(x, y) = S(x, y, α) = x (1 − \frac{y}{Φ(Φ^{−1}(x) + Φ^{−1}(y))}) ,

appearing in equation (2.4), which we will need in the sequel. Note that the function S is the “half–linear extension” of the function x^{2}/(x + y), appearing in the Riccati difference equation associated to linear difference equation (LΔE), and hence one can expect a similar behaviour of these functions in a certain sense.

Lemma 4.3. The function S(x, y, α) has the following properties:

(i) S(x, y, α) is continuously differentiable on

D := {(x, y, α) ∈ ℝ × ℝ × [1, ∞), x ≠ −y}.

(ii) Let y > 0. Then xS_{x}(x, y, α) ≥ 0 for x + y > 0, where S_{x}(x, y, α) = 0 if and only if

x = 0. (iii) Let x + y > 0. Then S_{y}(x, y, α) ≥ 0, where the equality holds if and only if x = 0. (iv) S(x, y, α) ≥ 0 for x + y > 0, where the equality holds if and only if x = 0.

(v) Suppose that the sequence (x_{k}, y_{k}), k = 1, 2, …, is such that x_{k} + y_{k} > 0 and there exists a constant M > 0 such that y_{k} ≤ M for k = 1, 2, . . .. Then S(x_{k}, y_{k}, α) → 0 implies x_{k} → 0. Moreover, lim inf_{k→0} y_{k} ≥ 0.

(vi) Let S̄(x, y, α) = x − S(x, y, α). Then S̄(x, y, α) = S̄(y, x, α) on D and S̄_{x}(x, y, α) ≥ 0 for x+y > 0, where the equality holds if and only if y = 0. If y ≤ 1, then S̄(x, y, α) < 1 for all x + y > 0.

(vii) Let x, y > 0. Then S_{α}(x, y, α) ≥ 0. (viii) Suppose that sgn y = sgn(x + y) and y ≠ 0. Then

S(x, y, α) = \frac{(α − 1)|x|^{β}|ξ|^{α−2}}{Φ(Φ^{−1}(x) + Φ^{−1}(y))},

where ξ is between Φ^{−1}(y) and Φ^{−1}(x) + Φ^{−1}(y).

Proof. (i) Obvious.

(ii) One can easily compute that

S_{x}(x, y, α) = \frac{(Φ^{−1}(x) + Φ^{−1}(y))^{α} − (|y|^{\frac{1}{α−1}} )^{α}}{(Φ^{−1}(x) + Φ^{−1}(y))^{α}}

on D. Suppose that y > 0 and x ≥ 0. Then clearly S_{x}(x, y, α) ≥ 0. If we suppose y > 0, x ≥ 0 and x + y > 0, then we have

(−|x|^{\frac{1}{α−1}} + |y|^{\frac{1}{α−1}} )^{α} ≤ α (|y|^{\frac{1}{α−1}} − |x|^{\frac{1}{α−1}} )^{α−1} (−|x|^{\frac{1}{α−1}} ) ≤ 0

and hence S_{x}(x, y, α) ≤ 0 in this case.

(iii) Clearly,

S_{y}(x, y, α) = \frac{−|x|^{\frac{α}{α−1}}}{(Φ^{−1}(x) + Φ^{−1}(y))^{α}} ≤ 0

for x + y > 0 and hence (iii) holds.

(iv) For the case y > 0 the statement follows from (ii) and from the fact that S(0, y, α) = 0. One can observe that the function S(x, y, α) with arbitrary fixed y, y < 0 and x+y > 0, is increasing with respect to the first variable for x > 2^{α−1}|y|, decreasing for |y| < x < 2^{α−1}|y| and S(2^{α−1}|y|, y, α) = 2^{α}|y| > 0. The statement now follows from the continuity of S.

(v) Denote I_{xy}^{++} = {k ∈ ℕ, x_{k} > 0, y_{k} > 0}, I_{xy}^{−+} = {k ∈ ℕ, x_{k} < 0, y_{k} > 0} and I_{y}^{−} = {k ∈ ℕ, y_{k} < 0}. Assume, for the contrary, that x_{k} ↛ 0. Then there exists x^{+} > 0 [or x^{−} < 0] such that x_{k_{l}} ≥ x^{+} for {k_{l}} ⊂ I_{xy}^{++} [or x_{k_{l}} ≤ x^{−} for {k_{l}} ⊂ I_{xy}^{−+}]. But, in view of the monotonicity of S and boundedness above of y, we have

S(x_{k_{l}}, y_{k_{l}}, α) ≥ S(x_{k_{l}}, M) ≥ S(x^{+}, M),
[S(x_{k_{l}}, y_{k_{l}}, α) ≥ S(x_{k_{l}}, M) ≥ S(x^{−}, M)].

From this we see that x^{+} = 0 [x^{−} = 0] since S(x_{k_{l}}, y_{k_{l}}, α) → ∞ and S(x^{+}, M, α) = 0 if and only if x^{+} = 0 [S(x^{−}, M, α) = 0 if and only if x^{−} = 0]. Hence, such positive x^{+} and negative x^{−} do not exist. Concerning the behaviour of x_{k_{l}} in the case {k_{l}} ⊂ I_{y}^{−} note only that the condition x_{k} + y_{k} > 0 implies x_{k} > 0 and this is essentially the same situation as above with x^{+} > 0. The fact that lim inf_{k→∞} y_{k} ≥ 0 follows from the inequality x_{k} > −y_{k} > 0.

(vi) For

S̄(x, y, α) = \frac{xy}{Φ(Φ^{−1}(x) + Φ^{−1}(y))}

clearly

S̄_{x}(x, y, α) = \frac{|y|^{\frac{α}{α−1}}}{(Φ^{−1}(x) + Φ^{−1}(y))^{α}} > 0

for x + y > 0 and y ≠ 0. To prove the inequality S̄(x, y) < 1 note that S̄(x, y) ≤ S̄(x, 1) < 1 for x > 0, y > 0 and for other x, y such that x + y > 0, the desired inequality clearly holds.

(vii) If x, y > 0, then the function S can be rewritten as

S(x, y, α) = x [1 − (1 + (x/y)^{\frac{1}{α−1}} )^{1−α}] .

Now, it is easy to see that

S_{α}(x, y, α) = \frac{x [(1 + (x/y)^{\frac{1}{α−1}} ) ln (1 + (x/y)^{\frac{1}{α−1}} ) − (x/y)^{\frac{1}{α−1}} ln(x/y)^{\frac{1}{α−1}} ]}{(1 + (x/y)^{\frac{1}{α−1}} )^{α}} ≥ 0.

From here we have the statement.

(viii) We apply the Lagrange mean value theorem.

4.2. Variational principle

The further method, known from the oscillation theory of (LDE) and now extended also to the discrete half–linear case, is the so–called variational principle. It is based on the equivalence (i)⇔(iv) from the Roundabout theorem. More precisely, we will use the following lemma.

Lemma 4.4. Equation (2.1) is nonoscillatory if and only if there exists m ∈ ℕ such that

F(ξ; m, ∞) = ∑_{k=m}^{∞} [r_{k}|Δξ_{k}|^{α} − p_{k}|ξ_{k+1}|^{α}] > 0

for every nontrivial ξ ∈ U(m), where

U(m) := {ξ = {ξ_{k}}_{k=1}^{∞} | ξ_{k} = 0, k ≤ m, ∃n > m : ξ_{k} = 0, k ≥ n}.

Remark 4.1. It is clear that to prove an oscillation of (2.1) it suffices to find for any m ∈ ℕ a (nontrivial) admissible sequence ξ ∈ U(m), for which F(ξ) ≤ 0.

4.3. Reciprocity principle and other methods

Further well–known method, which is available in the half–linear discrete oscillation theory is the reciprocity principle. Here we suppose that r_{k} > 0 and p_{k} > 0. If we denote u_{k} = r_{k}Φ(Δy_{k}), where y is a solution of (2.1), then (as one can easily verify) u_{k} satisfies the reciprocal equation

Δ(p_{k}^{1−β}Φ_{β}(Δu_{k})) + r_{k+1}^{1−β}Φ_{β}(u_{k+1}) = 0,  (4.3)

where Φ_{β}(x) = |x|^{β−2}x and β is the conjugate number of α, i.e., 1/α + 1/β = 1. Conversely, if y_{k} = p_{k−1}^{1−β}Φ_{β}(Δu_{k−1}), where u_{k} is a solution of (4.3), then y_{k} solves the original equation (2.1). Since the discrete version of the Rolle mean value theorem holds, see e.g. [1], we have the following equivalence: (2.1) is oscillatory [nonoscillatory] if and only if (4.3) is oscillatory [nonoscillatory]. Indeed, if y_{k} is an oscillatory solution of (2.1) then its difference and hence also u_{k} = r_{k}Φ(Δy_{k}) oscillates. Conversely, if u_{k} oscillates then y_{k} = p_{k−1}^{1−β}Φ_{β}(Δu_{k−1}) oscillates as well.

It is suitable to mention also the following important concepts in this chapter, even though they are not in fact the methods, but one can use them in the applications of the Riccati and variational technique to reach stronger results. We will brief, since the following observations only wait for the use in the half–linear discrete oscillation theory and hence they belong to Chapter 9, devoted to open problems, where they are described in a more precise way along with a literature:

• the use of a half–linear extension of the so–called recessive solution. Recessive solution, known from the oscillation theory of (LΔE), is a discrete counterpart of principal solution from the “continuous theory”,

• in our treatment we mostly consider (2.1) as a perturbation of the one–term equation

Δ(r_{k}Φ(Δy_{k})) = 0.

A more general approach is to compare (2.1) with the equation

Δ(r_{k}Φ(Δy_{k})) + p̃_{k}Φ(y_{k+1}) = 0.

CHAPTER 5 Oscillation criteria

5.1. Oscillation criteria based on the variational principle

In this section, equation (2.1) is considered on [m, ∞) with r_{k} > 0 on this interval. We start with the half–linear discrete versions of two well-known oscillation criteria, which were established in [98]. To prove them we use the variational principle. First we give auxiliary statement that is proved e.g. in [27].

Lemma 5.1 (The second mean value theorem of summation calculus, [27]). Let a sequence a_{k} be monotonic for k ∈ [K, L + 1]. Then for any sequence b_{k} there exist N_{1}, N_{2} ∈ [K, L] such that

∑_{j=K}^{L} a_{j+1}b_{j} ≤ a_{K} ∑_{i=K}^{N_{1}−1} b_{i} + a_{L+1} ∑_{i=N_{1}}^{L} b_{i},

∑_{j=K}^{L} a_{j+1}b_{j} ≥ a_{K} ∑_{i=K}^{N_{2}−1} b_{i} + a_{L+1} ∑_{i=N_{2}}^{L} b_{i}.

Theorem 5.1 (Leighton–Wintner type oscillation criterion). Suppose that

r_{k} > 0, ∑_{j=m}^{∞} r_{j}^{1−β} = ∞  (5.1)

and

∑_{j=m}^{∞} p_{j} = ∞.  (5.2)

Then (2.1) is oscillatory.

Proof. According to Remark 4.1, it is sufficient to find for any K ≥ m a sequence y satisfying y_{k} = 0 for k ≤ K and k ≥ N + 1, where N > K (then y is admissible), such that

F(y; K, N) = ∑_{k=K}^{N}[r_{k}|Δy_{k}|^{α} − p_{k}|y_{k+1}|^{α}] ≤ 0.

Let K < L < M < N. Define the sequence y_{k} by

y_{k} = \{ \begin{matrix}0 & for k = m, …, K, \\ (∑_{j=K}^{k−1} r_{j}^{1−β}) (∑_{j=K}^{L} r_{j}^{1−β})^{−1} & for k = K + 1, …, L + 1, \\ 1 & for k = L + 1, …, M, \\ (∑_{j=k}^{N} r_{j}^{1−β}) (∑_{j=M}^{N} r_{j}^{1−β})^{−1} & for k = M, …, N, \\ 0 & k ≥ N + 1. \end{matrix}

5.1. Oscillation criteria – variational principle 24

Using summation by parts we have

F(y; K, N) = ∑_{k=K}^{N}[r_{k}|Δy_{k}|^{α} − p_{k}|y_{k+1}|^{α}]

= ∑_{k=K}^{L−1} r_{k}|Δy_{k}|^{α} + r_{L}|Δy_{L}|^{α} + ∑_{k=L+1}^{M−1} r_{k}|Δy_{k}|^{α} + ∑_{k=M}^{N} r_{k}|Δy_{k}|^{α} − ∑_{k=K}^{N} p_{k}|y_{k+1}|^{α}

= [y_{k}r_{k}Φ(Δy_{k})]_{k=K}^{L} − ∑_{j=K}^{L−1} y_{k+1}Δ(r_{k}Φ(Δy_{k})) + r_{L} (r_{L}^{1−β})^{α} (∑_{j=K}^{L} r_{j}^{1−β})^{−α}+

+[y_{k}r_{k}Φ(Δy_{k})]_{k=M}^{N+1} − ∑_{k=M}^{N} y_{k+1}Δ(r_{k}Φ(Δy_{k})) − ∑_{k=K}^{N} p_{k}|y_{k+1}|^{α}

= y_{L}r_{L}Φ(Δy_{L}) + r_{L}^{1−β} (∑_{j=K}^{L} r_{j}^{1−β})^{−α}+ y_{M}r_{M}Φ(Δy_{M}) − ∑_{k=K}^{N} p_{k}|y_{k+1}|^{α}

= (∑_{k=K}^{L} r_{k}^{1−β})^{1−α}+ (∑_{k=M}^{N} r_{k}^{1−β})^{1−α}− ∑_{k=K}^{L} p_{k}|y_{k+1}|^{α} − ∑_{k=L+1}^{M−1} p_{k} − ∑_{k=M}^{N} p_{k}|y_{k+1}|^{α}.

Further, the sequence y is strictly monotonic on [K, L + 1] and [M, N + 1] since

Δy_{k} = r_{k}^{1−β} (∑_{j=K}^{L} r_{j}^{1−β})^{−1}> 0 for k ∈ [K, L]

and

Δy_{k} = −r_{k}^{1−β} (∑_{j=M}^{N} r_{j}^{1−β})^{−1}< 0 for k ∈ [M, N].

Hence, by Lemma 5.1, there exists N_{1} ∈ [K, L] such that

∑_{k=K}^{L} p_{k}|y_{k+1}|^{α} ≥ |y_{K}|^{α} ∑_{k=K}^{N_{1}−1} p_{k} + |y_{L+1}|^{α} ∑_{k=N_{1}}^{L} p_{k} = ∑_{k=N_{1}}^{L} p_{k},

and similarly there exists N_{2} ∈ [M, N] for which

∑_{k=M}^{N} p_{k}|y_{k+1}|^{α} ≥ |y_{M}|^{α} ∑_{k=M}^{N_{2}−1} p_{k} + |y_{N+1}|^{α} ∑_{k=N_{2}}^{N} p_{k} = ∑_{k=M}^{N_{2}−1} p_{k}.

Using these estimates we have

F(y; K, N) ≤ (∑_{k=K}^{L} r_{k}^{1−β})^{1−α}+ (∑_{k=M}^{N} r_{k}^{1−β})^{1−α}− ∑_{k=N_{1}}^{N_{2}−1} p_{k}.

Now, denote A = (∑_{j=K}^{L} r_{j}^{1−β})^{1−α} and let ε > 0 be arbitrary. According to (5.2), the integer M can be chosen in such a way that

∑_{j=N_{1}}^{k} p_{j} ≥ A + ε

whenever k > M. Since (5.1) holds,

(∑_{k=M}^{N} r_{k}^{1−β})^{1−α} ≤ ε

if N is sufficiently large.

Summarizing the above estimates, if M, N are sufficiently large, then we have

F(y; K, N) ≤ A + ε − (A + ε) = 0,

which completes the proof.

In the case when ∑_{j=m}^{∞} p_{j} is convergent, we can use the following criterion.

Theorem 5.2 (Hinton–Lewis type oscillation criterion). Suppose that (5.1) holds and

lim_{k→∞} (∑_{j=m}^{k} r_{j}^{1−β})^{α−1}(∑_{j=k}^{∞} p_{j}) > 1.  (5.3)

Then (2.1) is oscillatory.

A closer examination of the proof shows that lim in (5.3) can be replaced by lim inf.

Proof. Let the sequence y be the same as in the proof of the previous theorem. Hence we have

F(y; K, N) ≤ (∑_{j=K}^{L} r_{j}^{1−β})^{1−α}+ (∑_{j=M}^{N} r_{j}^{1−β})^{1−α}− ∑_{j=N_{1}}^{N_{2}−1} p_{k}

= (∑_{j=m}^{L} r_{j}^{1−β})^{1−α} [(∑_{j=m}^{L} r_{j}^{1−β})^{α−1}(∑_{j=K}^{L} r_{j}^{1−β})^{1−α}−

− (∑_{j=m}^{L} r_{j}^{1−β})^{α−1}( ∑_{j=N_{1}}^{N_{2}−1} p_{j}) + (∑_{j=m}^{L} r_{j}^{1−β})^{α−1}(∑_{j=M}^{N} r_{j}^{1−β})^{1−α}] ,

where m ≤ K < L < M < N, N_{1} ∈ [K, L] and N_{2} ∈ [M, N].

Now, let ε > 0 be such that lim in (5.3) is greater than or equal to 1 + 4ε. According to (5.3), K may be chosen in such a way that

(∑_{j=m}^{k} r_{j}^{1−β})^{α−1}(∑_{j=k}^{∞} p_{j}) ≥ 1 + 3ε  (5.4)

for k > K. Obviously, there exists L > K such that

(∑_{j=m}^{L} r_{j}^{1−β})^{α−1}(∑_{j=K}^{L} r_{j}^{1−β})^{1−α} ≤ 1 + ε.

In view of the fact that (5.4) holds, there exists M > L such that

(∑_{j=m}^{k} r_{j}^{1−β})^{α−1}(∑_{j=k}^{l} p_{j}) ≥ 1 + 2ε

for l ≥ M. Finally, since ∑_{j=m}^{∞} r_{j} = ∞ holds, we have

(∑_{j=m}^{L} r_{j}^{1−β})^{α−1}(∑_{j=M}^{N} r_{j}^{1−β})^{1−α} ≤ ε

if N > M is sufficiently large.

Using these estimates and the fact that ∑_{j=m}^{k} r_{j}^{1−β} is positive and increasing with respect to k, k ≥ m and ∑_{j=N_{1}}^{N_{2}} p_{j} is positive if N_{1}, N_{2} are sufficiently large, we get

\begin{aligned}F(y) & ≤ (∑_{j=m}^{L} r_{j}^{1−β})^{1−α} [(∑_{j=m}^{L} r_{j}^{1−β})^{α−1}(∑_{j=K}^{L} r_{j}^{1−β})^{1−α}− \\ & − (∑_{j=m}^{N_{1}} r_{j}^{1−β})^{α−1}( ∑_{j=N_{1}}^{N_{2}−1} p_{j}) + (∑_{j=m}^{L} r_{j}^{1−β})^{α−1}(∑_{j=M}^{N} r_{j}^{1−β})^{1−α}] \end{aligned}

≤ (∑_{j=m}^{L} r_{j}^{1−β})^{1−α}[1 + ε − 1 − 2ε + ε] = 0,

which yields the desired result.

We finish this section with a “complementary” case of the previous criterion – in the sense of the convergence of ∑^{∞} r_{j}^{1−β} < ∞. Here we suppose not only r_{k} > 0 but also p_{k} > 0 on [m, ∞). This statement was established in [101].

Theorem 5.3. Suppose p_{k} > 0 on [m, ∞). Further, let

r_{k} > 0, ∑_{j=m}^{∞} r_{j}^{1−β} < ∞  (5.5)

and

lim_{k→∞} ( ∑_{j=k+1}^{∞} r_{j}^{1−β})^{α−1} (∑_{j=m}^{k} p_{j}) > 1.  (5.6)

Then (2.1) is oscillatory.

Proof. We will use the reciprocity principle. From (5.6) one can observe that

∑_{j=m}^{∞} p_{j}^{(1−β)(1−α)} = ∑_{j=m}^{∞} p_{j} = ∞.

Therefore, by Theorem 5.2, equation (4.3) is oscillatory if

lim_{k→∞} (∑_{j=m}^{k} p_{j}^{(1−β)(1−α)})^{β−1}(∑_{j=k}^{∞} r_{j+1}^{1−β}) > 1,

which is equivalent to (5.6). From here (2.1) is also oscillatory.

5.2. Hartman–Wintner type theorem

The results presented in this section were established in [102].

The so–called Hartman–Wintner theorem plays an important role in the oscillation theory of (LDE). It was proved via the Riccati technique in [54, Chapter XI] for equation (LDE) with r(t) ≡ 1 and it describes the connection between certain asymptotic behaviour of the solution of the Riccati differential equation w′ + p(t) + w^{2} = 0 and the existence of lim_{t→∞} \frac{1}{t} ∫^{t} ∫^{s} p(ξ)dξds. There exist various extensions of this result to the half–linear differential equation (HLDE), see [24, 30, 42, 64, 75]. The discrete version of this statement with some applications to the linear difference equation (LΔE) can be found in [11]. In the first part of this section we present the half–linear discrete version of this lemma with some applications. In addition to these results we give a condition, which guarantees that there does not exist a solution of (2.1) without generalized zeros on a given interval. These results find the application to prove some oscillation and conjugacy criteria for equation (2.1), which are given in the second part of this section and in Section 6.3, respectively.

Before we present the main statement of this section, let us mention a remark concerning the comparison of this statement with the continuous case and its possible extension. Some of our statements are new even in the linear case since we require the sequence r_{k} to be only nonzero in contrast to [11]. On the other hand, it is still an open problem to prove the following results for a more general class of sequences r_{k} than those, which are bounded above. Note that even in the linear case we have no “full discrete version” of Hartman–Wintner theorem for equation (LΔE) with the discrete counterpart of the condition ∫^{∞} r^{−1}(s)ds = ∞. At present, the “corresponding” discrete version is known only under the restriction lim sup_{k→∞} k^{−3/2} ∑^{k} r_{j} < ∞. In fact, this can be slightly extended using the “weighted averaging” technique, see [11, 49]. As we have mentioned in Section 1.2, the results in the discrete case are mostly more complicated than those in the corresponding continuous case. However, as we will see, this is not always true. Thanks to the necessary condition for the convergence of an infinite series and to the special form of (2.4) (more precisely, to the special form of the function S), the following two statements – their proofs (and also some of its applications) are simpler (and stronger), in a certain sense, than those for differential equations, cf. [64, 75]. Note that these statements have no continuous analogue. Theorem 5.4. Assume that

there exists M > 0 such that r_{k} ≤ M for k ∈ ℕ.  (5.7)

Further suppose that (2.1) is nonoscillatory. Then the following statements are equivalent: (i)

lim inf_{k→∞} ∑_{j=1}^{k} p_{j} > −∞.  (5.8)

(ii) For any solution y with r_{k}y_{k}y_{k+1} > 0, k ≥ m for some m ∈ ℕ, the sequence w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}), k ≥ m, satisfies

∑_{j=m}^{∞} S(w_{j}, r_{j}) < ∞.  (5.9)

Moreover, this implies lim inf_{k→∞} r_{k} ≥ 0. (iii)

lim_{k→∞} ∑_{j=1}^{k} p_{j} exists (as a finite number).  (5.10)

Proof. (i) ⇒ (ii): In view of the nonnegativity of the function S, see Lemma 4.3 [(iv)], the sequence ∑_{j=m}^{k} S(w_{j}, r_{j}) is nondecreasing for k ≥ m. Therefore there exists the limit of this sequence that is equal either to a finite (positive) number or to infinity. Suppose, for the contrary, that there is a nonoscillatory solution of (2.1) such that

w_{k} = \frac{r_{k}Φ(Δy_{k})}{Φ(y_{k})} > −r_{k}  (5.11)

and

∑_{j=m}^{∞} S(w_{j}, r_{j}) = ∞.  (5.12)

From (2.4) we have

w_{k+1} = w_{m} − ∑_{j=m}^{k} p_{j} − ∑_{j=m}^{k} S(w_{j}, r_{j}), k ≥ m.  (5.13)

Now, from (5.8), (5.12) and (5.13) we obtain lim_{k→∞} w_{k} = −∞. But this contradicts to (5.11) since (5.7) holds. Therefore we must have (5.9). Moreover, a necessary condition for ∑_{j=m}^{∞} S(w_{j}, r_{j}) < ∞ is S(w_{k}, r_{k}) → 0 and this implies w_{k} → 0 by Lemma 4.3 [(v)]. The condition r_{k} + w_{k} > 0 now implies lim inf_{k→∞} r_{k} ≥ 0.

(ii) ⇒ (iii): Let w be a sequence as in (ii). According to the above observation, we know that w_{k} → 0. Now, letting k → ∞ in equation (5.13), we obtain the statement (iii).

(iii) ⇒ (i): Obvious.

The following result is a counterpart to Theorem 5.4.

Theorem 5.5. Assume that (5.7) holds and (2.1) is nonoscillatory. Then the following statements are equivalent:

(i)

lim inf_{k→∞} ∑_{j=1}^{k} p_{j} = −∞.  (5.14)

(ii) There exists a solution y of (2.1) with r_{k}y_{k}y_{k+1} > 0, k ≥ m for some m ∈ ℕ, such

that (5.12) holds, where w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}) > −r_{k} for k ≥ m. (iii)

lim_{k→∞} ∑_{j=1}^{k} p_{j} = −∞.  (5.15)

Proof. (i) ⇒ (ii): This follows from Theorem 5.4.

(ii) ⇒ (iii): From (2.4) using r_{k} + w_{k} > 0, k ≥ m, and (5.7) we get

∑_{j=m}^{k} p_{j} = −w_{k+1} + w_{m} − ∑_{j=m}^{k} S(w_{j}, r_{j}) ≤ M + w_{m} − ∑_{j=m}^{k} S(w_{j}, r_{j}) → −∞.

(iii) ⇒ (i): Trivial.

The following theorem gives a necessary condition for nonoscillation of (2.1) in terms of the existence of a solution of the generalized Riccati difference equation in a “summation” form.

Theorem 5.6. Let (5.7) and (5.8) hold. If (2.1) is nonoscillatory, then there exists a sequence w_{k} such that r_{k} + w_{k} > 0, k ≥ m for some m ∈ ℕ, and

w_{k} = ∑_{j=k}^{∞} p_{j} + ∑_{j=k}^{∞} S(w_{j}, r_{j}).  (5.16)

Proof. In view of the assumptions of theorem, (5.10) holds. From (2.4) we get

w_{k+1} = w_{m} − ∑_{j=m}^{k} p_{j} − ∑_{j=m}^{k} S(w_{j}, r_{j}), k ≥ m,

and letting k → ∞ in this equation, we obtain

0 = w_{m} − ∑_{j=m}^{∞} p_{j} − ∑_{j=m}^{∞} S(w_{j}, r_{j})

by the proof of Theorem 5.4. Replacing m by k we obtain (5.16).

It is clear that the necessary condition in the above theorem is also sufficient for nonoscillation of (2.1). One can easily verify it by applying the difference operator to both sides of (5.16) and taking into account that r_{k} + w_{k} > 0, k ≥ m. Then the statement follows from Theorem 3.1. However, the next theorem shows that such a type of condition guaranteeing nonoscillation can be somewhat weakened.

Theorem 5.7. Let (5.7) and (5.8) hold. Suppose that r_{k} > 0 for large k. If there exists a sequence z_{k} such that r_{k} + z_{k} > 0, k ≥ m for some m ∈ ℕ, and a constant C satisfying

z_{k} ≥ C − ∑_{j=1}^{k−1} p_{j} + ∑_{j=k}^{∞} S(z_{j}, r_{j}) ≥ 0,  (5.17)

or

z_{k} ≤ C − ∑_{j=1}^{k−1} p_{j} + ∑_{j=k}^{∞} S(z_{j}, r_{j}) ≤ 0,  (5.18)

then (2.1) is nonoscillatory.

Proof. Suppose that (5.7) and (5.8) hold and there exists a constant C such that (5.17) or (5.18) holds. Let

w_{k} = C − ∑_{j=1}^{k−1} p_{j} + ∑_{j=k}^{∞} S(w_{j}, r_{j}).

Then Δw_{k} = −p_{k} − S(w_{k}, r_{k}). We have z_{k} ≥ w_{k} ≥ 0 or z_{k} ≤ w_{k} ≤ 0 and hence S(z_{k}, r_{k}) ≥ S(w_{k}, r_{k}), k ≥ m according to Lemma 4.3 [(ii)]. Obviously, r_{k} + w_{k} > 0 and Δw_{k} + p_{k} + S(w_{k}, r_{k}) ≤ 0 for k ≥ m. Now, equation (2.1) is nonoscillatory by Lemma 4.2

The following statement gives a condition (in terms of the sequences r_{k} and p_{k}), which guarantees the nonexistence of the solution of (2.1) without generalized zeros (i.e., the existence of the solution without this property) in a given finite discrete interval.

Theorem 5.8. Let m, n be any integers such that n ≥ m + 1. If

∑_{j=m}^{n−1} p_{j} ≥ r_{m} + r_{n},

then (2.1) possesses no solution without generalized zeros on the interval [m, n].

Proof. Suppose, by the contradiction, that there exists a solution y of (2.1), which has no generalized zeros on the interval [m, n]. Then the sequence w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}) satisfies the equation (2.4) with r_{k} + w_{k} > 0 for k ∈ [m, n − 1]. We will show that

∑_{j=m}^{n−1} p_{j} < r_{m} + r_{n}.  (5.19)

For this purpose, we distinguish two particular cases:

a) Inequality (5.19) holds for n = m + 1: From equation (2.4) we get

p_{m} = w_{m} − w_{m+1} − w_{m} + \frac{w_{m}r_{m}}{(Φ^{−1}(r_{m}) + Φ^{−1}(w_{m}))^{α−1}}
< r_{m+1} + \frac{w_{m}r_{m}}{(Φ^{−1}(r_{m}) + Φ^{−1}(w_{m}))^{α−1}} ,

since r_{m+1} + w_{m+1} > 0. Now, we rewrite the right-hand side of the last inequality by this way:

r_{m+1} + r_{m} + \frac{w_{m}r_{m} − r_{m}(Φ^{−1}(r_{m}) + Φ^{−1}(w_{m}))^{α−1}}{(Φ^{−1}(r_{m}) + Φ^{−1}(w_{m}))^{α−1}} .

The numerator of the last fraction is negative if and only if

r_{m}^{2}Φ (\frac{Δy_{m}}{y_{m}} ) < r_{m}^{2}Φ (\frac{y_{m+1}}{y_{m}} ) ,

but this clearly holds. Hence p_{m} < r_{m+1} + r_{m} and thus (5.19) holds for n = m + 1.

b) Inequality (5.19) holds for n ≥ m + 2: Summation of equation (2.4) from m + 1 to n − 1 yields

∑_{j=m+1}^{n−1} p_{j} = w_{m+1} − w_{n} − ∑_{j=m+1}^{n−1} S(w_{j}, r_{j})
< w_{m+1} − w_{n} < w_{m+1} + r_{n},  (5.20)

since Lemma 4.3 [(iv)] holds. From equation (2.1) we immediately have

r_{m}Φ (\frac{Δy_{m}}{y_{m+1}} ) − p_{m} = r_{m+1} \frac{Φ(Δy_{m+1})}{Φ(y_{m+1})}.

Hence

w_{m+1} = r_{m}Φ (1 − \frac{y_{m}}{y_{m+1}} ) − p_{m} < r_{m} − p_{m},

since we have r_{m} > 0 if and only if y_{m}y_{m+1} > 0. Using this result in (5.20) we get (5.19) and theorem is proved.

From Theorems 5.4 and 5.5 we have that if (5.7) holds and equation (2.1) is nonoscillatory, then the sequence ∑_{j=1}^{k} p_{j} either converges or diverges to −∞. From this we immediately get the following oscillation criteria. Note that criterion (5.21) has no continuous analogue. See also the related remarks at the beginning of Section 5.4.

Corollary 5.1 (of Theorems 5.4 and 5.5). Let (5.7) hold. A sufficient condition for (2.1) to be oscillatory is that either

lim_{k→∞} ∑_{j=1}^{k} p_{j} = ∞,

or

lim inf_{k→∞} ∑_{j=1}^{k} p_{j} < lim sup_{k→∞} ∑_{j=1}^{k} p_{j}.  (5.21)

From the above criteria it is seen that if (5.7) holds then the nonexistence of the limit in (5.10) as a finite number (except in the case (5.15)) implies oscillation. From this point of view, the cases when (5.10) or (5.15) holds seem to be interesting for the examination. More precisely, we look for additional conditions, which guarantee oscillation of (2.1) in these cases. In the case when (5.10) holds, such conditions already exist, see following sections of this chapter. However, concerning the case when (5.15) holds, we still have no conditions of such type. But it really makes a sense to look for them since both oscillation and nonoscillation are possible in this case. For example, if p_{n} ≡ a < 0 and r_{k} ≡ 1, then equation (2.1) is of course nonoscillatory (since Δ(Φ(Δy_{k})) = 0 is nonoscillatory) and (5.15) holds. The next criterion enables to give an example showing oscillatory equation (2.1) with p_{k} satisfying (5.15).

Corollary 5.2 (of Theorem 5.8). If there exist two sequences of integers m_{k} and n_{k}, n_{k} ≥ m_{k} + 1, such that m_{k} → ∞ for k → ∞, and

∑_{j=m_{k}}^{n_{k}−1} p_{j} ≥ r_{m_{k}} + r_{n_{k}},

then equation (2.1) is oscillatory.

Example 5.1. Let m_{k} = 4k, k ∈ ℕ. Define r_{k} ≡ 1 and p_{m_{k}} = 1, p_{m_{k}+1} = 1, p_{m_{k}+2} = 1, p_{m_{k}+3} = −4 for k ∈ ℕ. Then

∑_{j=m_{k}}^{m_{k}+2} p_{j} = 3 > 2 = r_{m_{k}} + r_{m_{k}+3}

for all k ∈ ℕ. Equation (2.1) is oscillatory by Corollary 5.2. It is clear that ∑_{j=1}^{∞} p_{j} = −∞.

We finish this section with a half–linear extension of [65, Theorem 6.21]. In its proof we use essentially the same idea as in the proofs of Theorems 5.4 and 5.5.

Theorem 5.9. Suppose that r_{k} ≤ 1, k ∈ ℕ. If for all m ∈ ℕ there exists n ≥ m such that

lim_{k→∞} ∑_{j=n}^{k} p_{j} ≥ 1,  (5.22)

then (2.1) is oscillatory.

In fact, as we will see in the proof, lim in (5.22) can be replaced by lim sup. However, according to Corollary 5.1, it is superfluous.

Proof. Suppose, for the contrary, that (2.1) is nonoscillatory. Then there exist m ∈ ℕ and a solution of (2.1) with r_{k}y_{k}y_{k+1} > 0 for k ≥ m. Therefore, we can consider (2.4) with r_{k} + w_{k} > 0, where w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}), k ≥ m. Pick n ≥ m such that (5.22) holds. Summing (2.4) from n to k we get

w_{k+1} = w_{n} − ∑_{j=n}^{k} p_{j} − ∑_{j=n}^{k} S(w_{j}, r_{j}), k ≥ n.

Hence

w_{k+1} = S̄(w_{n}, r_{n}) − ∑_{j=n}^{k} p_{j} − ∑_{j=n+1}^{k} S(w_{j}, r_{j}),  (5.23)

where

S̄(w_{n}, r_{n}) = \frac{w_{n}r_{n}}{(Φ^{−1}(r_{n}) + Φ^{−1}(w_{n}))^{α−1}} .

If ∑_{j=n+1}^{∞} S(w_{j}, r_{j}) = ∞, then we get a contradiction from equation (5.23) since w_{k} > −r_{k} ≥ −1, k ≥ m. Thus we can assume ∑_{j=n+1}^{∞} S(w_{j}, r_{j}) < ∞. Then S(w_{k}, r_{k}) → 0 and therefore w_{k} → 0 according to Lemma 4.3 [(v)]. In view of the nonnegativity of S we have

−w_{k+1} ≥ −S̄(w_{n}, r_{n}) + ∑_{j=n}^{k} p_{j}.

Hence, from (5.22) and Lemma 4.3 [(vi)], we have

0 ≥ −S̄(w_{n}, r_{n}) + lim sup_{k→∞} ∑_{j=n}^{k} p_{j} > 0,

a contradiction.

5.3. Oscillation criteria based on the Riccati technique (part I)

The main results of this section are oscillation criteria for equation (2.1) (established in [104]) that are half–linear extensions of those proved in [50] for equation (LΔE). In particular, we consider the case when ∑^{∞} p_{j} is convergent (to a positive number) and ∑^{∞} r_{j}^{1−β} = ∞ (with r_{k} > 0). We will prove that several various additional conditions, which, along with the above assumptions, are sufficient for equation (2.1) to be oscillatory. We also compare some of the following results with those presented in other sections of this chapter.

First we give several auxiliary statements. We start with a technical result concerning certain behaviour of the sequence ∑^{k} p_{j}.

Lemma 5.2 ([50, Lemma 2.1]). Suppose that there exist integers n_{0}, n_{1}, n_{1} ≥ n_{0}, such that

∑_{j=k}^{∞} p_{j} ≥ 0 and ∑_{j=n_{1}}^{∞} p_{j} > 0  (5.24)

for all k ≥ n_{0}. Then there exists m ≥ n_{0} such that

∑_{j=m}^{k} p_{j} ≥ 0  (5.25)

for all k ≥ m.

In the sequel we will frequently assume the condition

(5.24) holds and p_{k} ≢ 0 eventually.  (5.26)

The next lemma claims that, under certain assumptions, a positive solution of equation (2.1) has a positive difference, i.e., one can find a positive solution of (2.4).

Lemma 5.3. Assume that (5.26) holds. Further suppose

r_{k} > 0, k ∈ ℕ, and ∑_{j=1}^{∞} r_{j}^{1−β} = ∞  (5.27)

and let y be a nonoscillatory solution of (2.1) such that y_{k} > 0 for all k ≥ m. Then there exists n, n ≥ m, such that Δy_{k} > 0 for all k ≥ n.

Proof. The proof is by contradiction. We will distinguish two cases:

Case 1: Suppose that Δy_{k} < 0 for large k, say k ≥ n ≥ m (then also ΔΦ(y_{k}) < 0). Without loss of generality, we may suppose that (5.25) holds for k ≥ n and p_{n} ≥ 0. Define φ_{k} = ∑_{j=n}^{k} p_{j} for k ≥ n and φ_{n−1} = 0. Thus we have

∑_{j=n}^{k} p_{j}Φ(y_{j+1}) = ∑_{j=n}^{k} Δφ_{j−1}Φ(y_{j+1}) = φ_{k}Φ(y_{k+2}) − ∑_{j=n}^{k} φ_{j}ΔΦ(y_{j+1}) ≥ 0

for k ≥ n. Therefore

r_{k+1}Φ(Δy_{k+1}) − r_{n}Φ(Δy_{n}) = ∑_{j=n}^{k} Δ(r_{j}Φ(Δy_{j})) ≤ 0

and so Φ(Δy_{k+1}) ≤ r_{n}Φ(Δy_{n})/r_{k+1}. Hence Δy_{k+1} = r_{n}^{β−1}Φ(Δy_{n})r_{k+1}^{1−β} and

y_{k+2} − y_{n+1} = ∑_{j=n}^{k} Δy_{j+1} = r_{n}^{β−1}Φ(Δy_{n}) ∑_{j=n}^{k} r_{j}^{1−β}.

Now we have y_{k} → −∞ as k → ∞, a contradiction.

Case 2: Assume that there exists a subsequence k_{l} → ∞ such that Δy_{k_{l}} ≤ 0 and Δy_{k_{l}+1} ≥ 0. The sequence w_{k} = −r_{k}Φ(Δy_{k})/Φ(y_{k}) is a solution of the generalized Riccati difference equation Δw_{k} = p_{k} + S̃(w_{k}, r_{k}) with r_{k} > w_{k}, where

S̃(w_{k}, r_{k}) = −w_{k} (1 − \frac{r_{k}}{(Φ^{−1}(r_{k}) − Φ^{−1}(w_{k}))^{α−1}} ) ≥ 0.

Hence

0 ≥ w_{k_{(l+1)}+1} − w_{k_{l}} = ∑_{j=k_{l}}^{k_{(l+1)}} p_{j} + ∑_{j=k_{l}}^{k_{(l+1)}} S̃(w_{j}, r_{j}).

Summing the above equation as k_{l+1} → ∞ we obtain ∑_{j=k_{1}}^{∞} p_{j} < 0, since w_{k} ≢ 0 and this contradicts assumption (5.26).

In the next lemma, a necessary condition for nonoscillation of (2.1) (more precisely, for the existence of a positive solution of (2.1)) is given in the case when lim_{k→∞} ∑^{k} p_{j} is convergent and the assumptions of Lemma 5.3 hold. Note that a closer examination of the proof of the below given Theorem 7.4 shows that this condition is also sufficient. Compare these statements also with Theorems 5.6 and 5.7, where similar results are presented under different assumptions.

Lemma 5.4. Let the assumptions of Lemma 5.3 hold and assume further that

∑_{j=1}^{∞} p_{j} = lim_{k→∞} ∑_{j=1}^{k} p_{j} is convergent.  (5.28)

Let y be a nonoscillatory solution of (2.1) such that y_{k} > 0 for all k ≥ m. Then there exists n ≥ m such that

w_{k} ≥ ∑_{j=k}^{∞} p_{j} + ∑_{j=k}^{∞} S(w_{j}, r_{j})  (5.29)

for k ≥ n, where w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}) > 0 and the function S is defined by (2.6).

Proof. From Lemma 5.3 there exists n such that w_{k} > 0 for k ≥ n and w_{k} satisfies the generalized Riccati difference equation (2.4) for k ≥ n. Summing this equation we have

w_{l+1} − w_{k} + ∑_{j=k}^{l} p_{j} + ∑_{j=k}^{l} S(w_{j}, r_{j}) = 0,

where l > k ≥ n. Therefore,

0 < w_{l+1} ≤ w_{n} − ∑_{j=n}^{l} p_{j}

for all l > n and hence

w_{k} ≥ ∑_{j=k}^{l} p_{j} + ∑_{j=k}^{l} S(w_{j}, r_{j})

for l > k ≥ n. Letting l → ∞ we obtain (5.29).

The following statement claims that under slightly stronger assumptions than those of the previous lemma we can estimate from above a positive sequence given by the Riccati type substitution.

Lemma 5.5. Let (5.27) and (5.28) hold. Assume further that p_{k} ≥ 0 (and eventually nontrivial) for all k ≥ m and let y be a nonoscillatory solution of (2.1) such that y_{k} > 0 for all k ≥ m. Then w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}) > 0 for all k ≥ m and satisfies w_{k} → 0 as k → ∞. Further, we have the inequality

w_{k} ≤ (∑_{j=m}^{k−1} r_{j}^{1−β})^{1−α}

for k > m.

Proof. From Lemma 5.3 and (2.4) we have w_{k} > 0 and

Δw_{k} + S(w_{k}, r_{k}) = w_{k+1} − \frac{w_{k}}{(r_{k}^{β−1} + w_{k}^{β−1})^{α−1}} ≤ 0

for k ≥ m. Hence w_{k+1}^{β−1}(r_{k}^{β−1} + w_{k}^{β−1}) − w_{k}^{β−1}r_{k}^{β−1} ≤ 0 and therefore

Δ (∑_{j=1}^{k−1} r_{j}^{1−β} − w_{k}^{1−β}) = \frac{(w_{k}w_{k+1})^{β−1} + (r_{k}w_{k+1})^{β−1} − (r_{k}w_{k})^{β−1}}{(r_{k}w_{k}w_{k+1})^{β−1}} ≤ 0

for k ≥ m. Summation of the above inequality from m to k − 1 gives

∑_{j=1}^{k−1} r_{j}^{1−β} − w_{k}^{1−β} ≤ ∑_{j=1}^{m−1} r_{j}^{1−β} − w_{m}^{1−β} ≤ ∑_{j=1}^{m−1} r_{j}^{1−β}

for k > m and so

w_{k}^{β−1} ≤ (∑_{j=m}^{k−1} r_{j}^{1−β})^{−1}

and w_{k} → 0 as k → ∞ by (5.27).

We finish this part devoted to auxiliary lemmas with a statement that follows from the well–known Young inequality. We will use it in the proof of Theorem 5.13.

Lemma 5.6 ([53]). If α > 1 and β > 1 are mutually conjugate numbers, then

|uv| ≤ \frac{|u|^{α}}{α} + \frac{|v|^{β}}{β}  (5.30)

holds for any u, v ∈ ℝ.

Now we can present main results of this section. We start with a couple of oscillation criteria for equation (2.1) that involve a sequence F^{t}, of which form comes from the form of the generalized Riccati difference equation (2.4). If (5.26) and (5.28) hold and if the series in (5.31) are convergent, for a given m ∈ ℕ we define the sequence F_{k}^{t} as follows

F_{k}^{0} = ∑_{j=k}^{∞} p_{j}, k ≥ m,

F_{k}^{t} = ∑_{j=k}^{∞} F_{j}^{t−1} (1 − \frac{r_{j}}{(Φ^{−1}(F_{j}^{t−1}) + Φ^{−1}(r_{j}))^{α−1}} ) + F_{k}^{0}  (5.31)

= ∑_{j=k}^{∞} S(F_{j}^{t−1}, r_{j}) + F_{k}^{0}, k ≥ m, t ∈ ℕ.

The following theorem shows that (2.1) is oscillatory provided the series in (5.31) is not convergent for some t ∈ ℕ.

Theorem 5.10. Let conditions (5.26), (5.27) and (5.28) hold. Then equation (2.1) is oscillatory if either

(i) there exists a positive integer t such that F_{k}^{0}, …, F_{k}^{t−1} defined by (5.31) satisfy

∑_{j=k}^{∞} S(F_{j}^{t−1}, r_{j}) = ∞,  (5.32)

or (ii) there exists an integer n ≥ m such that

lim sup_{t→∞} F_{n}^{t} = ∞.  (5.33)

Proof. Suppose, by contradiction, that (2.1) is nonoscillatory.

Let (i) hold. If t = 1, let y be a solution of (2.1) such that y_{k} > 0 for k ≥ m. Let w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}), k ≥ m. Then by Lemmas 5.3 and 5.4 we have

w_{k} ≥ ∑_{j=k}^{∞} S(w_{j}, r_{j}) + ∑_{j=k}^{∞} p_{j} ≥ F_{k}^{0}

for k ≥ n ≥ m. Hence,

∑_{j=k}^{∞} S(F_{j}^{0}, r_{j}) ≤ ∑_{j=k}^{∞} S(w_{j}, r_{j}) < ∞, k ≥ n,

since the function S(x, y) is increasing with respect to the first variable for x > 0, y > 0. But the last inequality contradicts assumption (5.32) for t = 1. Similarly, if t > 1 then from (5.29) and (5.31) we have that w_{k} ≥ F_{k}^{i}, i = 0, 1, …, t − 1 and hence

∑_{j=k}^{∞} S(F_{j}^{t−1}, r_{j}) ≤ ∑_{j=k}^{∞} S(w_{j}, r_{j}) < ∞,

which again contradicts (5.32).

Suppose next that (ii) holds. Obviously, in view of w_{k} ≥ F_{k}^{t}, t = 0, 1, 2, …, we have from (5.29)

lim sup_{t→∞} F_{n}^{t} ≤ w_{n} < ∞,

a contradiction.

Remark 5.1. In [104, Corollary 1] it is presented an alternative proof of criterion (5.58) from Corollary 5.6 based on the above statement.

The following criterion completes, in a certain sense, the criterion mentioned in the last remark. However, the assumption r_{k} ≡ 1 is not required here.

Theorem 5.11. Let the assumptions of Lemma 5.5 hold and F_{k}^{t} be defined by (5.31). Assume further that

lim sup_{k→∞} (∑_{j=m}^{k−1} r_{j}^{1−β})^{α−1} F_{k}^{t} > 1  (5.34)

for some t ∈ ℕ ∪ {0}. Then (2.1) is oscillatory.

Proof. If not, then as in the proof of Theorem 5.10 we have

F_{k}^{t} ≤ w_{k}, k ≥ m, t = 0, 1, 2, …,

and hence by Lemma 5.5

F_{k}^{t} ≤ (∑_{j=m}^{k−1} r_{j}^{1−β})^{1−α} , k ≥ m, t = 0, 1, 2, ….

But then

lim sup_{k→∞} (∑_{j=m}^{k−1} r_{j}^{1−β})^{α−1} F_{k}^{t} ≤ 1,

which contradicts (5.34).

Remark 5.2. In particular, under the assumptions of Lemma 5.5, the condition

lim sup_{k→∞} (∑_{j=m}^{k−1} r_{j}^{1−β})^{α−1} ∑_{j=k}^{∞} p_{j} > 1  (5.35)

guarantees oscillation of (2.1). Note that using the variational technique this criterion was proved in Section 5.1 (Theorem 5.2) without the assumption p_{k} ≥ 0, however, with lim inf instead of lim sup in (5.35).

Using the estimate from Lemma 5.5 we have the following criterion (compare with Corollary 5.2).

Theorem 5.12. Let the assumptions of Lemma 5.5 hold and assume further that there exist two sequences of positive integers {m_{l}}, {n_{l}} with m + 1 ≤ m_{l} < n_{l}, m_{l} → ∞ as l → ∞ such that

∑_{j=m_{l}}^{n_{l}} p_{j} ≥ r_{n_{l}+1} + ( ∑_{j=m}^{m_{l}−1} r_{j}^{1−β})^{1−α}  (5.36)

for large k. Then (2.1) is oscillatory.

Proof. Suppose, for the contrary, that (2.1) is nonoscillatory and let y be its solution with y_{k} > 0, k ≥ m. As in the proof of Lemma 5.5 we may suppose that w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}) > 0 for k ≥ n ≥ m and that w_{k} satisfies the generalized Riccati difference equation (2.4). Hence, Δw_{k} + p_{k} < 0, k ≥ m, which yields

w_{n_{l}+1} − w_{m_{l}} + ∑_{j=m_{l}}^{n_{l}} p_{j} < 0, m < m_{l} < n_{l},

so that

∑_{j=m_{l}}^{n_{l}} p_{j} < w_{m_{l}} − w_{n_{l}+1} < r_{n_{l}+1} + w_{m_{l}}.

Thus, by Lemma 5.5, we have

∑_{j=m_{l}}^{n_{l}} p_{j} < r_{n_{l}+1} + ( ∑_{j=m}^{m_{l}−1} r_{j}^{1−β})^{1−α} ,

which contradicts (5.36).

Remark 5.3. Under the assumptions of Lemma 5.5, a necessary condition for the existence of a nonoscillatory solution of equation (2.1) is

p_{k} < (∑_{j=m}^{k−1} r_{j}^{1−β})^{1−α} for all large k.  (5.37)

Indeed, the inequality Δw_{k} + p_{k} < 0, k ≥ m, implies 0 < w_{k+1}/w_{k} < 1 − p_{k}/w_{k}. Thus p_{k} < w_{k} and so Lemma 5.5 implies (5.37).

As a further use of the estimate from Lemma 5.5 we give a criterion, where a “weighted sequence” appears in the assumptions.

Theorem 5.13. In addition to the assumptions of Lemma 5.5 assume that there exists a sequence μ = {μ_{k}}, μ_{k} > 0, k = 1, 2, …, such that

lim_{k→∞} ∑_{j=m}^{k} μ_{j}p_{j} = ∞  (5.38)

and

lim_{k→∞} (∑_{j=m}^{k} μ_{j}p_{j})^{−1} ∑_{j=m}^{k} \frac{|Δμ_{j−1}|^{α}}{μ_{j}^{α−1}} r_{j} [1 + r_{j}^{1−β} (∑_{i=m}^{j−1} r_{i}^{1−β})^{−1}]^{δ} = 0,  (5.39)

where

δ = \{ \begin{matrix}α − 1 & for 1 < α ≤ 2, \\ (α − 1)^{2} & for α ≥ 2. \end{matrix}

Proof. Suppose, for the contrary, that y is eventually positive solution of (2.1). As in Lemma 5.5 we have the generalized Riccati equation (2.4) with w_{k} = r_{k}Φ(Δy_{k})/Φ(y_{k}) > 0 for k ≥ m. Multiplying (2.4) by μ_{k} and summing it from m to k we have

∑_{j=m}^{k} μ_{j}p_{j} = − ∑_{j=m}^{k} μ_{j}Δw_{j} − ∑_{j=m}^{k} μ_{j}S(w_{j}, r_{j}).

Obviously,

− ∑_{j=m}^{k} μ_{j}Δw_{j} = μ_{m}w_{m} − μ_{k+1}w_{k+1} + ∑_{j=m}^{k} w_{j+1}Δμ_{j}

= ∑_{j=m}^{k} w_{j}Δμ_{j−1} − w_{m}μ_{m−1} − w_{k+1}μ_{k}

= ∑_{j=m}^{k} w_{j}Δμ_{j−1} + w_{m}μ_{m−1}.

Observe that from Lemma 5.5,

r_{k}^{β−1} + w_{k}^{β−1} = r_{k}^{β−1} (1 + r_{k}^{1−β}w_{k}^{β−1}) ≤ r_{k} (1 + \frac{r_{k}^{1−β}}{∑_{j=m}^{k−1} r_{j}^{1−β}} ) ,  (5.40)

and by the Lagrange mean value theorem,

S(w_{k}, r_{k}) = \frac{(α − 1)w_{k}^{β}ξ_{k}^{α−2}}{(r_{k}^{β−1} + w_{k}^{β−1})^{α−1}} ,  (5.41)

where r_{k}^{β−1} ≤ ξ_{k} ≤ r_{k}^{β−1} + w_{k}^{β−1}. Using (5.41) and the inequality (5.30), where we set

u = w_{k}Δμ_{k} [βμ_{k}S(w_{k}, r_{k})]^{1−β} and v = [βμ_{k}S(w_{k}, r_{k})]^{β−1} ,

we get

∑_{j=m}^{k} μ_{j}p_{j} = w_{m}μ_{m−1} + ∑_{j=m}^{k} w_{j}Δμ_{j−1} − ∑_{j=m}^{k} μ_{j}S(w_{j}, r_{j})

≤ w_{m}μ_{m−1} + \frac{1}{αβ^{α/β}(α − 1)^{α/β}} ∑_{j=m}^{k} \frac{|Δμ_{j−1}|^{α} (r_{j}^{β−1} + w_{j}^{β−1})^{(α−1)^{2}}}{μ_{j}^{α−1}ξ_{j}^{(α−2)(α−1)}} ,

so by (5.40),

∑_{j=m}^{k} μ_{j}p_{j} ≤ w_{m}μ_{m−1} + \frac{1}{α^{α}} ∑_{j=m}^{k} \frac{|Δμ_{j−1}|^{α}}{μ_{j}^{α−1}} r_{j} (1 + \frac{r_{j}^{1−β}}{∑_{i=m}^{j−1} r_{i}^{1−β}} )^{δ} .  (5.42)

Combining (5.38), (5.39) and (5.42) we now get the desired contradiction.

Remark 5.4. If we assume that

lim inf_{k→∞} r_{k}^{β−1} ∑_{j=m}^{k−1} r_{j}^{1−β} > 0,  (5.43)

then (5.39) reduces to

lim_{k→∞} (∑_{j=m}^{k} μ_{j}p_{j})^{−1} ∑_{j=m}^{k} \frac{|Δμ_{j−1}|^{α}}{μ_{j}^{α−1}} r_{j} = 0.  (5.44)

Indeed, if (5.43) holds, then

lim sup_{k→∞} \frac{r_{k}^{1−β}}{∑_{j=m}^{k−1} r_{j}^{1−β}} < ∞

and in (5.42) we have

(1 + \frac{r_{k}^{1−β}}{∑_{j=m}^{k−1} r_{j}^{1−β}} )^{δ} ≤ (1 + M)^{δ}

for each k > m and a suitable constant M.

One can observe that (5.43) is not a serious restriction. For example, r_{k} = k^{α−1}, r_{k} = k^{−(α−1)}, r_{k} = e^{−k(α−1)} and r_{k} = 1 all satisfy (5.43).

Choosing a suitable weighted sequence in Theorem 5.13 we get the following oscillation criterion.

Corollary 5.3. Let the assumption of Lemma 5.5 hold along with condition (5.43). Assume further that there exists ν > 0 such that α − ν > 1 and

lim_{k→∞} ∑_{j=m}^{k} (∑_{i=1}^{j} r_{i}^{1−β})^{ν} p_{j} = ∞.

Then (2.1) is oscillatory.

Proof. Define μ_{k} from Theorem 5.13 as

μ_{k} = (∑_{j=1}^{k} r_{j}^{1−β})^{ν} .

We show that (5.44) holds. First suppose that 0 < ν < 1. Then from the Lagrange mean value theorem

0 < Δμ_{k−1} ≤ νr_{k}^{1−β} (∑_{j=1}^{k} r_{j}^{1−β})^{ν−1} .

From here we have

\frac{(Δμ_{k−1})^{α}}{μ_{k}^{α−1}} r_{k} ≤ \frac{ν^{α}r_{k}^{α(1−β)+1} (∑_{j=1}^{k−1} r_{j}^{1−β})^{α(ν−1)}}{(∑_{j=1}^{k} r_{j}^{1−β})^{(α−1)ν}} ≤ ν^{α}r_{k}^{1−β} (∑_{j=1}^{k−1} r_{j}^{1−β})^{ν−α} .

Similarly, one can show that for ν ≥ 1 we get

\frac{(Δμ_{k−1})^{α}}{μ_{k}^{α−1}} r_{k} ≤ ν^{α}r_{k}^{1−β} (∑_{j=1}^{k} r_{j}^{1−β})^{ν−α} .

But, from (5.43) there exists a positive constant M such that

∑_{j=1}^{k} r_{j}^{1−β} ≤ M ∑_{j=1}^{k−1} r_{j}^{1−β}, k = 2, 3, …

and hence (for ν satisfying the conditions from corollary, regardless whether or not ν < 1 holds) we have

∑_{j=m}^{k} \frac{(Δμ_{j−1})^{α}}{μ_{j}^{α−1}} r_{j} ≤ ν^{α}M ∑_{j=m}^{k} (∑_{i=1}^{j} r_{i}^{1−β})^{ν−α} Δ (∑_{i=1}^{j−1} r_{i}^{1−β}) .

Setting

ω_{k} = ∑_{j=1}^{k} r_{j}^{1−β}

we get

∑_{j=m}^{k} \frac{(Δμ_{j−1})^{α}}{μ_{j}^{α−1}} r_{j} ≤ ν^{α}M ∑_{j=m−1}^{k−1} \frac{Δω_{j}}{ω_{j+1}^{α−ν}} .

Now we show that the sum on the right-hand side of the last inequality is convergent. Let

f(x) = ω_{k} + Δω_{k}(x − k) for k ≤ x ≤ k + 1.

Then f′(x) = Δω_{k}, f(x) ≤ ω_{k+1} for k < ω < k + 1 and hence we get

\frac{Δω_{k}}{ω_{k+1}^{α−ν}} = ∫_{k}^{k+1} \frac{Δω_{k}}{ω_{k+1}^{α−ν}} dx ≤ ∫_{k}^{k+1} \frac{f′(x)}{(f(x))^{α−ν}} dx.

It follows that

∑_{j=m−1}^{k−1} \frac{Δω_{k}}{ω_{k+1}^{α−ν}} ≤ ∫_{m−1}^{k} \frac{f′(x)}{(f(x))^{α−ν}} dx

= \frac{1}{α − ν − 1} [(∑_{j=1}^{m−1} r_{j}^{1−β})^{1+ν−α} − (∑_{j=1}^{k} r_{j}^{1−β})^{1+ν−α}] .

Therefore the condition (5.44) from Remark 5.4 holds and so the statement follows from Theorem 5.13.

5.4. Oscillation criteria based on the Riccati technique (part II) – the case r_{k} ≡ 1

The results presented in this section were established in [103].

In [54, Chap. XI] it was proved that if

lim_{t→∞} c(t) = ∞ or − ∞ < lim inf_{t→∞} c(t) < lim sup_{t→∞} c(t) ≤ ∞,

where

c(t) = \frac{1}{t} ∫_{1}^{t} ∫_{1}^{s} p(τ)dτds,

then (LDE) (with r(t) ≡ 1) is oscillatory. Using the results of [11], in [100] we have observed that in the parallel statement to the above mentioned, for equation (LΔE) (with r_{k} ≡ 1) it is superfluous, in a certain sense, to consider a “direct” discrete counterpart of the function c(t), i.e., the sequence

γ_{k} = \frac{1}{k} ∑_{j=1}^{k} ∑_{i=1}^{j} p_{j},

since the assumption

lim inf_{k→∞} ∑_{j=1}^{k} p_{j} < lim sup_{k→∞} ∑_{j=1}^{k} p_{j}  (5.45)

guarantees oscillation of (LΔE) (with r_{k} ≡ 1) (this criterion has no continuous analogue) and the existence of a finite limit

lim_{k→∞} ∑_{j=1}^{k} p_{j} = C  (5.46)

implies the existence of lim_{k→∞} γ_{k} (hence, lim inf_{k→∞} γ_{k} < lim sup_{k→∞} γ_{k} implies (5.45)). Moreover, we can have lim inf_{k→∞} ∑_{j=1}^{k} p_{j} = −∞ in criterion (5.45). These discrepancies between the discrete and the continuous cases are due to some specific properties of difference calculus. In the half–linear case we have a similar situation. E.g., in [64], the authors investigate the oscillation properties of (HLDE) (with r(t) ≡ 1) in the cases when

lim_{t→∞} c_{α}(t), where c_{α}(t) = \frac{α − 1}{t^{α−1}} ∫_{1}^{t} s^{α−2} ∫_{1}^{s} p(τ)dτds,  (5.47)

exists or not. In particular, lim_{t→∞} c_{α}(t) = ∞ or

−∞ < lim inf_{t→∞} c_{α}(t) < lim sup_{t→∞} c_{α}(t) ≤ ∞

are sufficient conditions for oscillation of (HLDE) (with r(t) ≡ 1). For a similar result, where a more general weight function appears in (5.47), see [75]. From Corollary 5.1 clearly follows that in the case of equation (2.2), similarly as of the linear equation, it suffices to deal with the limit of the sequence ∑_{j=1}^{k} p_{j} in this connection since (5.46) implies the existence of the discrete counterpart of (5.47), namely

lim_{k→∞} \frac{α − 1}{k^{α−1}} ∑_{j=1}^{k} j^{α−2} ∑_{i=1}^{j} p_{i}.

Hence, in this section we will always assume that (5.46) holds.

The following notation will be used.

Q_{k} = (k + 1)^{α−1} ∑_{j=k+1}^{∞} p_{j}, H_{k} = \frac{1}{k + 1} ∑_{j=1}^{k} j^{α}p_{j},

Q_{∗} = lim inf_{k→∞} Q_{k}, Q^{∗} = lim sup_{k→∞} Q_{k},

H_{∗} = lim inf_{k→∞} H_{k}, H^{∗} = lim sup_{k→∞} H_{k}

and we will abbreviate S_{k} instead of S(w_{k}, 1) in this section.

Now we give oscillation criteria for (2.2), considered on ℕ, that are half–linear extensions of those for equation (LΔE) (with r_{k} ≡ 1), see [100], and, at the same time, a discretization of the results for equation (HLDE) (with r(t) ≡ 1) from [64].

Theorem 5.14. Suppose that (5.46) holds and

lim sup_{k→∞} \frac{∑_{j=1}^{k}(j + 1)^{α−2} ∑_{i=j+1}^{∞} p_{i}}{∑_{j=1}^{k} 1/(j + 1)} > \frac{1}{α} (\frac{α − 1}{α})^{α−1} .  (5.48)

Proof. Suppose, by contradiction, that (2.2) is nonoscillatory. Then, according to Theorem 5.6, there exists a sequence w_{k} with 1 + w_{k} > 0, k ≥ m for some m ∈ ℕ, satisfying

w_{k+1} = ∑_{j=k+1}^{∞} p_{j} + ∑_{j=k+1}^{∞} S_{j} = C − ∑_{j=1}^{k} p_{j} + ∑_{j=k+1}^{∞} S_{j}  (5.49)

for k ≥ m, where C is defined by (5.46). Let ε̄ > 0 be given and let ε > 0 be such that

(\frac{1 + ε}{1 − ε})^{α−1} ≤ 1 + ε̄.

Obviously, since w_{k} → 0 as k → ∞, there exists m̄ = m̄(ε̄) ∈ ℕ such that |w_{k}| < 1,

(1 − |w_{k}|^{1/(α−1)})^{α−2} ≥ 1 − ε with α ≥ 2,
(1 + |w_{k}|^{1/(α−1)})^{α−2} ≥ 1 − ε with 1 < α ≤ 2,
(1 + Φ^{−1}(w_{k}))^{α−1} ≤ 1 + ε

for k ≥ m̄. We can suppose that m ≥ m̄.

First assume that α ≥ 2. Multiplying both sides of (5.49) by (k + 1)^{α−2} and applying of the summation from m to k we get

∑_{j=m}^{k}(j + 1)^{α−2}w_{j+1} = C ∑_{j=m}^{k}(j + 1)^{α−2} − ∑_{j=m}^{k}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i} + ∑_{j=m}^{k}(j + 1)^{α−2} ∑_{i=j+1}^{∞} S_{i} (5.50)

for k ≥ m. Set

c_{k} = \frac{1}{k^{α−1}} ∑_{j=1}^{k}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i},

(one can see that lim_{k→∞} c_{k} = C/(α − 1)). Clearly,

k^{α−1}c_{k} = ∑_{j=1}^{m−1}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i} + ∑_{j=m}^{k}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i}

and, using the Lagrange mean value theorem,

∑_{j=m}^{k}(j + 1)^{α−2} ∑_{i=j+1}^{∞} S_{i} ≥ ∑_{j=m}^{k} η_{j}^{α−2} ∑_{i=j+1}^{∞} S_{i}

= \frac{(k + 1)^{α−1}}{α − 1} ∑_{j=k+2}^{∞} S_{j} − \frac{m^{α−1}}{α − 1} ∑_{j=m+1}^{∞} S_{j} +

+ ∑_{j=m}^{k} \frac{(j + 1)^{α−1}}{α − 1} S_{j+1}

≥ \frac{(k + 1)^{α−1}}{α − 1} ∑_{j=k+2}^{∞} S_{j} − \frac{m^{α−1}}{α − 1} ∑_{j=m+1}^{∞} S_{j},

where k ≤ η_{k} ≤ k + 1. Now, from (5.50), we have

∑_{j=m}^{k} [(j + 1)^{α−2}w_{j+1}− \frac{(j + 1)^{α−1}S_{j+1}}{α − 1} ]≥C ∑_{j=1}^{k}(j + 1)^{α−2}− k^{α−1}c_{k}− K_{1},  (5.51)

k ≥ m, where

K_{1} = − ∑_{j=1}^{m−1}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i} + \frac{m^{α−1}}{α − 1} ∑_{j=m+1}^{∞} S_{j} + C ∑_{j=1}^{m−1}(j + 1)^{α−2}.

By Lemma 4.3 [(viii)], the summand on the left–hand side of the last inequality can be rewritten as

\frac{1}{j + 1} (x − \frac{|x|^{β}ξ_{j+1}^{α−2}}{(1 + Φ^{−1}(w_{j+1}))^{α−1}} ) ≤ \frac{1}{j + 1} (x − \frac{1 − ε}{1 + ε}|x|^{β}) = \frac{1}{j + 1}f(x),

where

x = (j + 1)^{α−1}w_{j+1}, f(x) = x − \frac{1 − ε}{1 + ε}|x|^{β}.

One can easily compute that

f(x) ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1} (\frac{1 + ε}{1 − ε})^{α−1} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1} (1 + ε̄)

for any x. Using this fact in (5.51), we obtain

C ∑_{j=1}^{k}(j + 1)^{α−2} − k^{α−1}c_{k} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1} (1 + ε̄) ∑_{j=1}^{k} \frac{1}{j + 1} + K_{1},  (5.52)

k ≥ m.

Assume that 1 < α < 2 and denote

c̄_{k} = \frac{1}{k^{α−1}} ∑_{j=1}^{k} j^{α−2} ∑_{i=1}^{j} p_{i}.

Multiplication of both sides of (5.49) by k^{α−1}, the summation from m to k and similar arrangements as above yields

C ∑_{j=1}^{k} j^{α−2} − k^{α−1}c̄_{k} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1}(1 + ε̄) ∑_{j=1}^{k} \frac{1}{j + 1} + K_{2},  (5.53)

k ≥ m, where

K_{2} = − ∑_{j=1}^{m−1} j^{α−2} ∑_{i=1}^{j} p_{i} + \frac{m^{α−1}}{α − 1} ∑_{j=m+1}^{∞} S_{j} + C ∑_{j=1}^{m−1} j^{α−2} + m(1 + ε̄)\frac{1}{α} (\frac{α − 1}{α})^{α−1} .

Now we would like to rewrite the left–hand side of (5.53) into the same form as of (5.52) because of the next consideration with the common value α > 1. For this purpose, denote the left–hand side of (5.52) as A_{k} and that of (5.53) as B_{k}. Obviously, there exist constants

5.4. Oscillation criteria – Riccati technique, II 44

M, N > 0 (we still suppose 1 < α < 2) such that

A_{k} = B_{k} + C ∑_{j=1}^{k} [(j + 1)^{α−2} − j^{α−2}] + ∑_{j=1}^{k} [j^{α−2} − (j + 1)^{α−2}] ∑_{i=1}^{j} p_{i}
≤ B_{k} + M ∑_{j=1}^{k} [j^{α−2} − (j + 1)^{α−2}]
≤ B_{k} + M(2 − α) ∑_{j=1}^{k} \frac{1}{j^{3−α}}
≤ B_{k} + N.

Now, from (5.52), (5.53) and the last inequality, one can observe that there exists a constant K > 0 such that

(∑_{j=1}^{k} \frac{1}{j + 1})^{−1} A_{k} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1}(1 + ε̄) + K (∑_{j=1}^{k} \frac{1}{j + 1})^{−1}  (5.54)

for α > 1 and sufficiently large k. But we have

A_{k} = (∑_{j=1}^{∞} p_{j}) ∑_{j=1}^{k}(j + 1)^{α−2} − ∑_{j=1}^{k}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i} = ∑_{j=1}^{k}(j + 1)^{α−2} ∑_{i=j+1}^{∞} p_{i}.

Hence (5.54) contradicts (5.48). Corollary 5.4. Suppose that

lim inf_{k→∞} (∑_{j=1}^{k}(j + 1)^{α−2}) ∑_{j=k+1}^{∞} p_{j} > −∞  (5.55)

and

lim sup_{k→∞} \frac{∑_{j=1}^{k} p_{j} ∑_{i=1}^{j−1}(i + 1)^{α−2}}{∑_{j=1}^{k} 1/(j + 1)} > \frac{1}{α} (\frac{α − 1}{α})^{α−1} .  (5.56)

Then (2.2) is oscillatory.

Proof. Define

φ_{k} = (∑_{j=1}^{k}(j + 1)^{α−2}) ∑_{j=1}^{k} p_{j}

for k ≥ 1 and φ_{0} = 0. The left–hand side of (5.54) can be rewritten in the following way:

(∑_{j=1}^{k} \frac{1}{j + 1})^{−1} [C ∑_{j=1}^{k}(j + 1)^{α−2} − ∑_{j=1}^{k}(j + 1)^{α−2} ∑_{i=1}^{j} p_{i}]

= (∑_{j=1}^{k} \frac{1}{j + 1})^{−1} [C ∑_{j=1}^{k}(j + 1)^{α−2} − φ_{k} + φ_{0} + ∑_{j=1}^{k} (p_{j} ∑_{i=1}^{j−1}(i + 1)^{α−2})]

= (∑_{j=1}^{k} \frac{1}{j + 1})^{−1} [(∑_{j=1}^{k}(j + 1)^{α−2}) ∑_{j=k+1}^{∞} p_{j} + ∑_{j=1}^{k} (p_{j} ∑_{i=1}^{j−1}(i + 1)^{α−2})] .

Now, using (5.55) and (5.56), we see that the assumption of Theorem 5.14 is fulfilled.

The next statement claims that under the condition (5.26) the left–hand side of (5.48) can be replaced by a somewhat simpler expression.

Corollary 5.5. Suppose that (5.26) holds. If

lim sup_{k→∞} \frac{∑_{j=1}^{k} j^{α−1}p_{j}}{∑_{j=1}^{k} 1/(j + 1)} > (\frac{α − 1}{α})^{α} ,  (5.57)

then (2.2) is oscillatory.

Proof. Let m ∈ ℕ be such that ∑_{j=k}^{∞} p_{j} ≥ 0 for k ≥ m. Denote

Z_{m} = ∑_{j=1}^{m−1}(j + 1)^{α−2} ∑_{i=j+1}^{∞} p_{i}.

Using the Lagrange mean value theorem, the sequence A_{k} on the left–hand side of (5.54) can be rewritten in the following way for α ≥ 2:

A_{k} = Z_{m} + ∑_{j=m}^{k}(j + 1)^{α−2} ∑_{i=j+1}^{∞} p_{i}

≥ Z_{m} + ∑_{j=m}^{k} ξ_{j}^{α−2} ∑_{i=j+1}^{∞} p_{i}

= Z_{m} + ∑_{j=m}^{k} \frac{Δj^{α−1}}{α − 1} ∑_{i=j+1}^{∞} p_{i}

≥ Z_{m} + ∑_{j=m}^{k} \frac{j^{α−1}}{α − 1}p_{j} − \frac{m^{α−1}}{α − 1} ∑_{j=m}^{∞} p_{j}

for k ≥ m, where k ≤ ξ_{k} ≤ k + 1. Similarly as above, the left–hand side of (5.53) can be rewritten for 1 < α < 2 in the following way:

B_{k} = ∑_{j=1}^{k} j^{α−2} ∑_{i=j+1}^{∞} p_{i}

≥ ∑_{j=1}^{m−1} j^{α−2} ∑_{i=j+1}^{∞} p_{j} + ∑_{j=m}^{k} \frac{j^{α−1}}{α − 1}p_{j} − \frac{m^{α−1}}{α − 1} ∑_{j=m}^{∞} p_{j}

for k ≥ m. But both these inequalities contradict (5.57) and hence (2.2) is oscillatory.

Remark 5.5. One can show that ∑_{j=1}^{k} 1/(j + 1) in (5.48), (5.56) and (5.57) can be replaced by ln k.

Corollary 5.6. The sufficient condition for equation (2.2) to be oscillatory is that either

Q_{∗} > \frac{1}{α} (\frac{α − 1}{α})^{α−1} ,  (5.58)

or that (5.26) holds and

H_{∗} > (\frac{α − 1}{α})^{α} .  (5.59)

Proof. Suppose that (5.58) holds. Using the summation by parts we rewrite the sequence A_{k} on the left–hand side of (5.54) by the following way:

A_{k} = ∑_{j=1}^{k}(j + 1)^{α−2} ∑_{i=j+1}^{∞} p_{i} = ∑_{j=1}^{k} \frac{Q_{j}}{j + 1}

= \frac{1}{k + 1} ∑_{j=1}^{k} Q_{j} + ∑_{j=1}^{k} \frac{1}{j(j + 1)} ∑_{i=1}^{j−1} Q_{i}.  (5.60)

From here we immediately get the statement by Theorem 5.14. Note that the other possibility to prove this part of the statement is to apply the discrete L’Hospital rule to the expression in (5.48).

Suppose that (5.59) holds. Applying the difference operator to H_{k} we have

ΔH_{k} = Δ (\frac{1}{k + 1}) ∑_{j=1}^{k+1} j^{α}p_{j} + \frac{1}{k + 1}(k + 1)^{α}p_{k+1}.

The summation from 1 to k yields

∑_{j=1}^{k}(j + 1)^{α−1}p_{j+1} = ∑_{j=1}^{k} \frac{H_{j+1}}{j + 1} + H_{k+1} − H_{1}

and the statement now follows from Corollary 5.5 since (5.26) holds.

Corollary 5.7. Suppose that (5.26) holds. If

lim inf_{k→∞}(Q_{k} + H_{k}) > (\frac{α − 1}{α})^{α−1} ,

then (2.2) is oscillatory.

Proof. Let m ∈ ℕ be such that ∑_{j=k}^{∞} p_{j} ≥ 0 for k ≥ m. From (5.60) we have

A_{k} = \frac{1}{k + 1} ∑_{j=1}^{k} Q_{j} + ∑_{j=1}^{k} \frac{1}{j(j + 1)} ∑_{i=1}^{j−1} Q_{i},

where A_{k} is the same as in (5.54). Further, using the Lagrange mean value theorem we have

\frac{α}{k + 1} ∑_{j=1}^{k} Q_{j} = \frac{α}{k + 1} ∑_{j=1}^{m−1} Q_{j} + \frac{α}{k + 1} ∑_{j=m}^{k}(j + 1)^{α−1} ∑_{i=j+1}^{∞} p_{i}

≥ \frac{α}{k + 1} ∑_{j=1}^{m−1} Q_{j} + \frac{α}{k + 1} ∑_{j=m}^{k} ξ_{j}^{α−1} ∑_{i=j+1}^{∞} p_{i}

= Q_{k} + H_{k} + \frac{α}{k + 1} ∑_{j=1}^{m−1} Q_{j} − \frac{m^{α}}{k + 1} ∑_{j=m}^{∞} p_{j}.

Now, it is easy to see that the assumption of Theorem 5.14 is fulfilled

The following criterion completes in a certain sense the previous statement.

Theorem 5.15. Suppose that (5.26) holds. If

lim sup_{k→∞}(Q_{k} + H_{k}) > 1,  (5.61)

Proof. Suppose, by contradiction, that y is a nonoscillatory solution of (2.2) such that y_{k} > 0 for all k ≥ m_{1} for some m_{1} ∈ ℕ. By Lemma 5.3 there exists m_{2} ≥ m_{1} such that Δy_{k} > 0 for k ≥ m_{2}. We can make the Riccati type substitution w_{k} = Φ(Δy_{k}/y_{k}) > 0, k ≥ m_{2}, and then w satisfies

Δw_{k} + p_{k} + S_{k} = 0.  (5.62)

Let ε > 0 be given and ε̄ > 0 be such that α(1 − ε̄)^{−1/(β−1)} − (α − 1)(1 − ε̄)^{1/(β−1)} ≤ 1 + ε. Let m_{3} = m_{3}(ε) ∈ ℕ be such that

\frac{1}{(1 + |w_{k}|^{1/(α−1)})^{α−1}} ≥ 1 − ε̄, k ≥ m_{3}.

Set m = max{m_{2}, m_{3}}. Multiplying both sides of (5.62) by k^{α}, the summation from m to k yields

∑_{j=m}^{k} j^{α}Δw_{j} = − ∑_{j=m}^{k} j^{α}p_{j} − ∑_{j=m}^{k} j^{α}S_{j}.

Using the summation by parts, the left–hand side has the form

k^{α}w_{k+1} − (m − 1)^{α}w_{m} − ∑_{j=m}^{k} Δ(j − 1)^{α}w_{j}.

Now, by the Lagrange mean value theorem, we obtain

k^{α}w_{k+1} − (m − 1)^{α}w_{m} + ∑_{j=m}^{k} j^{α}p_{j} = ∑_{j=m}^{k} [Δ(j − 1)^{α}w_{j} − j^{α}S_{j}]

= ∑_{j=m}^{k} [αη_{j}^{α−1}w_{j} − j^{α}S_{j}] ≤ ∑_{j=m}^{k} [αj^{α−1}w_{j} − j^{α} \frac{(α − 1)w_{j}^{β}ξ_{j}^{α−2}}{(1 + Φ^{−1}(w_{j}))^{α−1}} ]

≤ ∑_{j=m}^{k} [αj^{α−1}w_{j} − (α − 1) (j^{α−1}w_{j})^{β} (1 − ε̄)]  (5.63)

for k ≥ m. The summand in the last row of (5.63) can be rewritten as f(x) = αx − (α − 1)x^{β}(1 − ε̄), where x = j^{α−1}w_{j} > 0. Now, it is easy to see that

f(x) ≤ α(1 − ε̄)^{−1/(β−1)} − (α − 1)(1 − ε̄)^{1/(β−1)} ≤ 1 + ε.

Hence, from (5.63) we get

k^{α}w_{k+1} +∑_{j=1}^{k} j^{α}p_{j} ≤ (1 + ε)(k + 1 − m) + (m − 1)^{α}w_{m} +∑_{j=1}^{m−1} j^{α}p_{j}.  (5.64)

Dividing (5.64) by k + 1 and using (5.49) we obtain

\frac{k^{α}}{k + 1} ∑_{j=k+1}^{∞} p_{j} + H_{k} ≤ 1 + ε + \frac{1}{k + 1} [−(1 + ε)m + (m − 1)^{α}w_{m} + ∑_{j=1}^{m−1} j^{α}p_{j}] .

From here we have (1 − ε)Q_{k} + H_{k} ≤ 1 + 2ε for large k and this contradicts (5.61), since ε > 0 is arbitrary.

Let

0 < λ < \frac{1}{α} (\frac{α − 1}{α})^{α−1}

and denote by

ϱ_{min}(λ) and ϱ_{max}(λ)  (5.65)

the least root and the greatest root, respectively, of the equation

|x|^{β} − x + λ = 0.

To prove the next oscillation criteria we need the following two lemmas.

Lemma 5.7. Suppose that equation (2.2) is nonoscillatory,

0 ≤ Q_{∗} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1}  (5.66)

and y is a solution of (2.2) such that y_{k}y_{k+1} > 0, k ≥ m for some m ∈ ℕ. Then

lim inf_{k→∞} \frac{k^{α−1}Φ(Δy_{k})}{Φ(y_{k})} ≥ ϱ_{min}(Q_{∗}),  (5.67)

where ϱ_{min} is defined by (5.65).

Proof. Let (2.2) be nonoscillatory. Then there exists m ∈ ℕ such that y_{k}y_{k+1} > 0, k ≥ m. Set A = lim inf_{k→∞} k^{α−1}w_{k}, where w_{k} = Φ(Δy_{k}/y_{k}). If A = ∞, then obviously there is nothing to prove. So, let A < ∞. If Q_{∗} = 0, then (5.67) is trivial in view of (5.49). Thus, let us suppose Q_{∗} > 0. For arbitrary ε ∈ (0, Q_{∗}) there exists k^{[ε]} > m such that

Q_{k} > Q_{∗} − ε, k ≥ k^{[ε]}.  (5.68)

From (5.49) and (5.68) we have k^{α−1}w_{k} ≥ Q_{∗} − ε, k ≥ k^{[ε]}. Hence A ≥ Q_{∗}. Now, we choose k̄^{[ε]} > k^{[ε]} such that k^{α−1}w_{k} ≥ A − ε and

\frac{ξ_{k}^{α−2}}{(1 + w_{k}^{1/(α−1)})^{α−1}} ≥ 1 − ε  (5.69)

for k ≥ k̄^{[ε]}, where 1 ≤ ξ_{k} ≤ 1 + w_{k}^{1/(α−1)}. Taking into account inequalities (5.68) and (5.69), from (5.49) we get

k^{α−1} ≥ Q_{∗} − ε + k^{α−1} ∑_{j=k}^{∞} S_{j}

= Q_{∗} − ε + ∑_{j=k}^{∞} \frac{(α − 1)w_{j}^{β}ξ_{j}^{α−2}}{(1 + Φ^{−1}(w_{j}))^{α−1}}

≥ Q_{∗} − ε + (1 − ε)k^{α−1}(α − 1) ∑_{j=k}^{∞} w_{j}^{β}

≥ Q_{∗} − ε + (1 − ε)k^{α−1}(α − 1) ∑_{j=k}^{∞} \frac{(A − ε)^{β}}{j^{α}} .  (5.70)

for k ≥ k̄^{[ε]}. It is easy to see that −Δ (1/k^{α−1}) ≤ (α − 1)/k^{α} for α > 1. Hence we get k^{α−1}w_{k} ≥ Q_{∗} − ε + (1 + ε)(A − ε)^{β}, k ≥ k̄^{[ε]}. Therefore A ≥ Q_{∗} − ε + (1 − ε)(A − ε)^{β}. Thus we have A^{β} − A + Q_{∗} ≤ 0, since ε > 0 is arbitrary and consequently A ≥ ϱ_{min}(Q_{∗}).

Lemma 5.8. Assume that (5.26) holds. Suppose that equation (2.2) is nonoscillatory,

0 ≤ H_{∗} ≤ (\frac{α − 1}{α})^{α}  (5.71)

and y is a solution of (2.2) such that y_{k}y_{k+1} > 0, k ≥ m for some m ∈ ℕ. Then

lim sup_{k→∞} \frac{k^{α−1}Φ(Δy_{k})}{Φ(y_{k})} ≤ ϱ_{max} (H_{∗}/(α − 1)) ,  (5.72)

where ϱ_{max} is defined by (5.65).

Proof. Let (2.2) be nonoscillatory. Then there exists m ∈ ℕ such that y_{k}y_{k+1} > 0, k ≥ m. Set B = lim sup_{k→∞} k^{α−1}w_{k}, where w_{k} = Φ(Δy_{k}/y_{k}). If B = −∞, then obviously there is nothing to prove. The inequality (5.72) is valid also for B ≤ 0, since ϱ_{max} (H_{∗}/(α − 1)) ≥ 0, and hence we can assume that B > 0. From (5.64) we have

\frac{k^{α}}{k + 1}w_{k+1} ≤ 1 + ε − H_{k} + K_{k^{[ε]},k}, k ≥ k^{[ε]},

where

K_{k^{[ε]},k} = \frac{1}{k + 1} [(k^{[ε]} − 1)^{α}w_{k^{[ε]}} − (1 + ε)k^{[ε]} + ∑_{j=1}^{k^{[ε]}−1} j^{α}p_{j}]  (5.73)

for a given ε > 0 and sufficiently large k^{[ε]}. Hence B ≤ 1 − H_{∗}. Thus the estimate (5.72) holds for H_{∗} = 0 and hence we proceed with H_{∗} > 0. The inequality (5.63) can be rewritten in the following way:

\frac{k^{α}}{k + 1}w_{k+1} ≤ −H_{k} + K_{k^{[ε]},k} + \frac{(1 + ε)k^{[ε]}}{k + 1} +

+\frac{1}{k + 1} ∑_{j=k^{[ε]}}^{k} [αj^{α−1}w_{j} − (α − 1) (j^{α−1}w_{j})^{β} (1 − ε)]

for k ≥ k^{[ε]}. At the same time we can suppose that k^{[ε]} is so large that k^{α−1}w_{k} ≤ B + ε and H_{k} > H_{∗} − ε for k ≥ k^{[ε]}. Hence we get

\frac{k}{k + 1}w_{k+1} ≤ −H_{k} + K_{k^{[ε]},k} + \frac{(1 + ε)k^{[ε]}}{k + 1} +

+\frac{k − k^{[ε]} + 1}{k + 1} [α(B + ε) − (α − 1)(B + ε)^{β}(1 − ε)] ,

since the behaviour of the function αx − (α − 1)|x|^{β}(1 − ε) is similar to that of the function αx − (α − 1)|x|^{β} for small ε > 0. In particular, it is increasing for x ∈ (−∞, 1) and we have B ∈ (0, 1). Further,

B ≤ −H_{∗} + ε + α(B + ε) − (α − 1)(B + ε)^{β}(1 − ε).

Therefore (α − 1)B^{β} − (α − 1)B + H_{∗} ≤ 0, since ε > 0 is arbitrary. Consequently, B ≤ ϱ_{max} (H_{∗}/(α − 1)).

In view of the assumptions of Corollary 5.6 it is purposeful to look for additional conditions, which guarantee oscillation of equation (2.2) in the complementary cases (5.66) or/and (5.71).

Theorem 5.16. A sufficient condition for (2.2) to be oscillatory is that either

0 ≤ Q_{∗} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1} and H^{∗} > 1 − ϱ_{min}(Q_{∗}),  (5.74)

or that (5.26) holds,

0 ≤ H_{∗} ≤ (\frac{α − 1}{α})^{α} and Q^{∗} > ϱ_{max} (H_{∗}/(α − 1)) ,  (5.75)

where ϱ_{min} and ϱ_{max} are defined by (5.65).

Proof. Suppose, by contradiction, that equation (2.2) is nonoscillatory, i.e., there exists m ∈ ℕ such that y_{k}y_{k+1} > 0 for k ≥ m, where y is a solution of equation (2.2). By Lemma 5.7 (in the case that (5.74) holds) and by Lemma 5.8 (in the case that (5.75) holds), for arbitrary ε > 0 there exists k^{[ε]} ∈ ℕ such that

\frac{k^{α}}{k + 1}w_{k} > ϱ_{min}(Q_{∗}) − ε and k^{α−1}w_{k} < ϱ_{max} (H_{∗}/(α − 1)) + ε

for k ≥ k^{[ε]}, where w_{k} = Φ(Δy_{k}/y_{k}). Suppose that k^{[ε]} is at the same time so large that (5.64) with k^{[ε]} instead of m holds and K_{k^{[ε]},k} < ε for k^{[ε]} ≥ k, where K_{k^{[ε]},k} is defined by (5.73).

Now, if (5.74) holds, then from (5.64) we get

H_{k} ≤ −\frac{k^{α}}{k + 1}w_{k+1} + 1 + ε + K_{k^{[ε]},k}
≤ −ϱ_{min}(Q_{∗}) + 1 + 2ε + K_{k^{[ε]},k}
≤ −ϱ_{min}(Q_{∗}) + 1 + 3ε

for k ≥ k^{[ε]}, which contradicts (5.74). If (5.75) holds, then from (5.49) we have

k^{α−1}w_{k} = Q_{k−1} + k^{α−1} ∑_{j=k}^{∞} S_{j}.

Thus Q_{k} ≤ ϱ_{max} (H_{∗}/(α − 1)) + ε, which contradicts (5.75) and proves the theorem.

The following statement claims that if the assumptions

0 ≤ Q_{∗} ≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1} and 0 ≤ H_{∗} ≤ (\frac{α − 1}{α})^{α}  (5.76)

hold at the same time, then the right–hand sides of the second inequalities in (5.74) and (5.75) can be diminished.

Theorem 5.17. Let (5.76) hold. Then the sufficient condition for (2.2) to be oscillatory is that either

H^{∗} > H_{∗} + ϱ_{max} (H_{∗}/(α − 1)) − ϱ_{min}(Q_{∗}),  (5.77)

or that

Q^{∗} > Q_{∗} + ϱ_{max} (H_{∗}/(α − 1)) − ϱ_{min}(Q_{∗}),  (5.78)

where ϱ_{min} and ϱ_{max} are defined by (5.65).

Proof. Suppose, by contradiction, that equation (2.2) is nonoscillatory, i.e., there exists m ∈ ℕ such that y_{k}y_{k+1} > 0 for k ≥ m, where y is a solution of equation (2.2). Note that the condition Q_{∗} ≥ 0 implies that (5.25) holds. Denote

M = min {1 − ϱ_{max} (H_{∗}/(α − 1)) , ϱ_{min}(Q_{∗})} .

By Lemmas 5.7 and 5.8, for arbitrary ε ∈ (1, M) there exists k^{[ε]} > m such that

\frac{k^{α}}{k + 1}w_{k} > ϱ_{min}(Q_{∗}) − ε and k^{α−1}w_{k} < ϱ_{max} (H_{∗}/(α − 1)) + ε,  (5.79)

respectively, for k ≥ k^{[ε]}. Let k^{[ε]} be at the same time so large that (5.63) (with ε instead of ε̄ and k^{[ε]} instead of m) and (5.69) hold for k ≥ k^{[ε]}.

Assume that (5.77) is fulfilled. We shall proceed with H_{∗} > 0, since for H_{∗} = 0 the condition (5.77) is equivalent to the condition (5.74). Using the first inequality in (5.79) and the fact that ϱ_{max} (H_{∗}/(α − 1)) + ε < 1, from (5.63) we obtain

H_{k} ≤ −ϱ_{min}(Q_{∗}) + ε + K_{k^{[ε]},k} + \frac{(1 + ε)k^{[ε]}}{k + 1} +
+α [ϱ_{max} (H_{∗}/(α − 1)) + ε] − (α − 1) [ϱ_{max} (H_{∗}/(α − 1)) + ε]^{β} (1 − ε),

where K_{k^{[ε]},k} is defined by (5.73). Hence

H^{∗} ≤ −ϱ_{min}(Q_{∗}) + αϱ_{max} (H_{∗}/(α − 1)) − (α − 1) [ϱ_{max} (H_{∗}/(α − 1))]^{β}
= −ϱ_{min}(Q_{∗}) + ϱ_{max} (H_{∗}/(α − 1)) +
+(α − 1) {ϱ_{max} (H_{∗}/(α − 1)) − [ϱ_{max} (H_{∗}/(α − 1))]^{β}}
= −ϱ_{min}(Q_{∗}) + H_{∗} + ϱ_{max} (H_{∗}/(α − 1)) ,

which contradicts (5.77).

Suppose that the condition (5.78) holds. Using inequalities (5.79), the properties of the function S and the same estimates as those in (5.70), we have from (5.49)

Q_{k−1} ≤ ϱ_{max} (H_{∗}/(α − 1)) + ε − k^{α−1} ∑_{j=k}^{∞} S(w_{j}, 1)

≤ ϱ_{max} (H_{∗}/(α − 1)) + ε − k^{α−1} ∑_{j=k}^{∞} S (\frac{ϱ_{min}(Q_{∗})}{j^{α−1}}, 1)
≤ ϱ_{max} (H_{∗}/(α − 1)) + ε −

−(1 − ε)k^{α−1}(α − 1) [ϱ_{min}(Q_{∗}) − ε]^{β} ∑_{j=k}^{∞} Δ (−\frac{1}{j^{α−1}(α − 1)})

= ϱ_{max} (H_{∗}/(α − 1)) + ε − (1 − ε) [ϱ_{min}(Q_{∗}) − ε]^{β}

for k ≥ k^{[ε]}. Since ε > 0 is arbitrary, we have

Q^{∗} ≤ ϱ_{max} (H_{∗}/(α − 1)) − [ϱ_{min}(Q_{∗})]^{β} = ϱ_{max} (H_{∗}/(α − 1)) − ϱ_{min}(Q_{∗}) + Q_{∗},

which contradicts (5.78).

The following examples show that for any pair of numbers (X, Y), where X ≤ Y, there exists a sequence p_{k} such that (5.46) holds and Q_{∗} = X, Q^{∗} = Y (H_{∗} = X, H^{∗} = Y). Therefore Theorems 5.16 and 5.17 are meaningful.

5.4. Oscillation criteria – Riccati technique, II 52

Example 5.2. Let X, Y ∈ ℝ be arbitrary. Set

M = \frac{X + Y}{2} , N = \frac{Y − X}{2}

and p_{k} = Δh_{k}, where

h_{k} = −\frac{M}{k^{α−1}} + \frac{N(−1)^{k}}{k^{α−1}} .

Then

∑_{j=1}^{k−1} p_{j} = h_{k} and lim_{k→∞} ∑_{j=1}^{k} p_{j} = 0.

Hence Q_{k} = (k+1)^{α−1}(−h_{k+1}) = M −N(−1)^{k+1} and thus Q_{∗} = M −N = X, Q^{∗} = M +N = Y. Example 5.3. Let X, Y ∈ ℝ be arbitrary. Set

M = \frac{X + Y}{2} , N = \frac{Y − X}{2}

and p_{k} = Δh_{k}, where

h_{k} = −\frac{M}{(α − 1)k^{α−1}} + \frac{N(−1)^{k}}{k^{α−1}} + \frac{M/(α − 1) + N}{k^{α}} .

Then

∑_{j=1}^{k−1} p_{j} = h_{k} and lim_{k→∞} ∑_{j=1}^{k} p_{j} = 0.

Hence

H_{k} = \frac{1}{k + 1} [j^{α}h_{j}]_{1}^{k+1} − \frac{1}{k + 1} ∑_{j=1}^{k} Δj^{α}h_{j+1}
= (k + 1)^{α−1}h_{k+1} − \frac{α}{k + 1} ∑_{j=1}^{k} ξ_{j}^{α−1}h_{j+1}
= −\frac{M}{α − 1} + N(−1)^{k} + \frac{M/(α − 1) + N}{k + 1} + \frac{αM}{(α − 1)k + 1} ∑_{j=1}^{k} (\frac{ξ_{j}}{j + 1})^{α−1}−

−\frac{αN}{k + 1} ∑_{j=1}^{k}(−1)^{j} (\frac{ξ_{j}}{j + 1})^{α−1} − \frac{α [M/(α − 1) + N]}{k + 1} ∑_{j=1}^{k} \frac{ξ_{j}^{α−1}}{(j + 1)^{α}} ,

where j ≤ ξ_{j} ≤ j + 1. Now, it is easy to see that H_{∗} = M − N = X and H^{∗} = M + N = Y.

CHAPTER 6 Nonoscillation and conjugacy criteria

In the first two sections of this chapter, equation (2.1) is considered on [m, ∞) with r_{k} > 0 on this interval. We prove nonoscillation criteria, which are discrete versions of nonoscillation criteria for (HLDE) established in the recent papers [28, 64, 68, 69]. We will compare our discrete nonoscillation criteria with those for differential equation (HLDE) and we will also see that the use of different methods gives sometimes different result. The last section of this chapter is devoted to conjugacy criteria for equation (2.1).

6.1. Nonoscillation criteria based on the variational principle

First we present nonoscillation criteria based on the variational method that were established in [38]. The crucial role is played by the half-linear discrete version of the Wirtinger type inequality. In the proof of this inequality we need the following technical result.

Lemma 6.1. Let

μ := \{ \begin{matrix}sup_{t>s>0} \frac{1}{t−s} [Φ^{−1} (\frac{t^{α}−s^{p}}{α(t−s)}) − s] , & α ≥ 2, \\ sup_{t>s>0} \frac{1}{t−s} [t − Φ^{−1} (\frac{t^{α}−s^{α}}{α(t−s)})] , & α ≤ 2. \end{matrix}  (6.1)

Then for given B > A > 0 and for ξ = λA + (1 − λ)B given by the Lagrange mean value theorem B^{α} − A^{α} = αΦ(ξ)(B − A) we have max{λ, (1 − λ)} ≤ μ.

Proof. If α ≥ 2, then λ ≥ 1/2, i.e. max{λ, 1 − λ} = λ and for α ≤ 2 we have λ ≤ 1/2. The conclusion now can be easily verified by a direct computation via the Lagrange mean value theorem applied to the function t → t^{α}, t ≥ 0.

Lemma 6.2 (Wirtinger type inequality). Let M_{k} be a positive sequence such that ΔM_{k} is of one sign for k ≥ N ∈ ℕ. Then for every y ∈ U(N) we have

∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α} ≤ α^{α}[μ(1 + ψ_{N})]^{α−1} ∑_{k=N}^{∞} \frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α},  (6.2)

where

ψ_{N} = sup_{k≥N} \frac{|ΔM_{k}|}{|ΔM_{k−1}|},  (6.3)

μ is given in the previous lemma and U(N) is defined in Lemma 4.4.

Proof. Suppose that ΔM_{k} > 0 for k ≥ N, in case ΔM_{k} < 0 we would proceed in the same way. Using summation by part, the Hölder inequality, the Lagrange mean value theorem

6.1. Nonoscillation criteria – variational principle 54

and the Jensen inequality for the convex function ξ → |ξ|^{α}, we have

∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α} = ∑_{k=N}^{∞} M_{k}Δ(|y_{k}|^{α})

= α ∑_{k=N}^{∞} M_{k}|Φ(ξ_{k})||Δy_{k}|
≤ α (∑_{k=N}^{∞} \frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α})^{\frac{1}{α}} (∑_{k=N}^{∞} |ΔM_{k}||Φ(ξ_{k})|^{β})^{\frac{1}{β}}
= α (∑_{k=N}^{∞} \frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α})^{\frac{1}{α}} (∑_{k=N}^{∞} |ΔM_{k}| [λ_{k}|y_{k}|^{α} + (1 − λ_{k})|y_{k+1}|^{α}])^{\frac{1}{β}}
≤ α (∑_{k=N}^{∞} \frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α})^{\frac{1}{α}} (max\{λ_{k}, (1 − λ_{k})\})^{\frac{1}{β}} ×
× (∑_{k=N}^{∞} \frac{|ΔM_{k}|}{|ΔM_{k−1}|}Δ|M_{k−1}||y_{k}|^{α} + ∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α})^{\frac{1}{β}} ,

where ξ_{k} = λ_{k}y_{k} + (1 − λ_{k})y_{k+1} is a number between y_{k}, y_{k+1}, i.e. λ_{k} ∈ [0, 1] Now, by Lemma 6.1, max{λ_{k}, 1 − λ_{k}} ≤ μ and since y_{k} = 0 for k ≤ N, we have

∑_{k=N}^{∞} \frac{|ΔM_{k}|}{|ΔM_{k−1}|}|ΔM_{k−1}||y_{k}|^{α} ≤ (sup_{k≥N} \frac{|ΔM_{k}|}{|ΔM_{k−1}|}) ∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α}.

Consequently, we have

∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α} ≤ α [μ (1 + sup_{k≥N} \frac{|ΔM_{k}|}{|ΔM_{k−1}|}) ∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α}]^{\frac{1}{β}} (∑_{k=N}^{∞} \frac{M_{k}^{α}|Δy_{k}|^{α}}{|ΔM_{k}|^{α−1}} )^{\frac{1}{α}} ,

and hence

∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α} ≤ α^{α} [μ(1 + ψ_{N})]^{α−1} ∑_{k=N}^{∞} \frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α}

what we needed to prove. Theorem 6.1. Suppose that (5.1) holds, ∑^{∞} p_{k}^{+} < ∞, p^{+} := max{0, p},

φ_{N} := [sup_{k≥N} \frac{∑^{k} r_{j}^{1−β}}{∑^{k−1} r_{j}^{1−β}} ]^{α(α−1)} < ∞, ψ_{N} := [sup_{k≥N} \frac{r_{k}}{r_{k−1}} ]^{1−β} < ∞.  (6.4)

Further suppose that

0 < lim sup_{N→∞}(1 + ψ_{N})^{α−1}φ_{N} =: Ψ < ∞.  (6.5)

If

lim sup_{k→∞} (∑^{k−1} r_{j}^{1−β})^{α−1} ∑_{j=k}^{∞} p_{j}^{+} < \frac{1}{αμ^{α−1}} (\frac{α − 1}{α})^{α−1} \frac{1}{Ψ}  (6.6)

then (2.1) is nonoscillatory.

Proof. According to Lemma 4.4, it suffices to find N ∈ ℕ such that

∑_{k=N}^{∞} [r_{k}|Δy_{k}|^{α} − p_{k}|y_{k+1}|^{α}] > 0

for any nontrivial y ∈ U(N). To this end, let M_{k} = 1/ (∑^{k−1} r_{j}^{1−β})^{α−1} . Then, using the Lagrange mean value theorem,

|ΔM_{k}| = \frac{α − 1}{ξ_{k}^{α}}Δ (∑^{k−1} r_{j}^{1−β}) = \frac{α − 1}{ξ_{k}^{α}}r_{k}^{1−β},

where ξ_{k} ∈ (∑^{k−1} r_{j}^{1−β}, ∑^{k} r_{j}^{1−β}). Hence

\frac{α − 1}{(∑^{k} r^{1−β})^{α}} r_{k}^{1−β} ≤ |ΔM_{k}| ≤ \frac{α − 1}{(∑^{k−1} r^{1−β})^{α}} r_{k}^{1−β},  (6.7)

thus

\frac{|ΔM_{k}|}{|ΔM_{k−1}|} ≤ (\frac{r_{k}}{r_{k−1}} )^{1−β}

and

\frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} ≤ r_{k} (\frac{1}{α − 1})^{α−1} [\frac{∑^{k} r^{1−β}}{∑^{k−1} r^{1−β}} ]^{α(α−1)} .

Now, according to (6.6), there exists ε > 0 such that

lim sup_{k→∞} (∑^{k−1} r_{j}^{1−β})^{α−1} ∑_{j=k}^{∞} p_{j}^{+} < \frac{1}{αμ^{α−1}} (\frac{α − 1}{α})^{α−1} \frac{1}{Ψ + ε}.

and (6.5) implies the existence of N_{0} ∈ ℕ such that (1+ψ_{N})^{α−1}φ_{N} < Ψ+ε for N ≥ N_{0}. Now, using the summation by parts and applying the same idea as in the proof of Lemma 6.2, we have for any nontrivial y ∈ U(N)

∑_{k=N}^{∞} p_{k}|y_{k+1}|^{α} ≤ ∑_{k=N}^{∞} p_{k}^{+}|y_{k+1}|^{α} = ∑_{k=N}^{∞} (∑_{j=N}^{∞} p_{j}^{+}) Δ (|y_{k}|^{α}) = ∑_{k=N}^{∞} \frac{1}{M_{k}} (∑_{j=N}^{∞} p_{j}^{+}) M_{k}Δ(|y_{k}|^{α})

< (\frac{α − 1}{α})^{α−1} \frac{1}{μ^{α−1}(Ψ + ε)} ∑_{k=N}^{∞} M_{k}|Φ(ξ_{k})||Δy_{k}|

≤ (\frac{α − 1}{α})^{α−1} \frac{1}{μ^{α−1}(Ψ + ε)} (∑_{k=N}^{∞} \frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α})^{\frac{1}{α}} ×

× [μ(1 + ψ_{N}) ∑_{k=N}^{∞} |ΔM_{k}||y_{k+1}|^{α}]^{\frac{1}{β}}

≤ (\frac{α − 1}{α})^{α−1} \frac{1}{μ^{α−1}(Ψ + ε)} [μ(1 + ψ_{N})]^{\frac{1}{β}} α^{\frac{α}{β}} [μ(1 + ψ_{N})]^{\frac{α−1}{β}} ×

× (∑_{k=N}^{∞} \frac{M_{k}}{|ΔM_{k}|^{α−1}} |Δy_{k}|^{α})^{\frac{1}{α} + \frac{1}{β}}

≤ (1 + ψ_{N})^{α−1}φ_{N} \frac{1}{Ψ + ε} ∑_{k=N}^{∞} r_{k}|Δy_{k}|^{α} ≤ ∑_{k=N}^{∞} r_{k}|Δy_{k}|^{α}.

Hence

∑_{k=N}^{∞} [r_{k}|Δy_{k}|^{α} − p_{k}|y_{k+1}|^{α}] > 0

for every nontrivial y ∈ U(N) what we needed to prove.

The next theorem treats the case when ∑^{∞} r_{k}^{1−β} < ∞.

Theorem 6.2. Suppose that (5.5) holds,

φ̃_{N} = sup_{k≥N} (\frac{∑_{k+1}^{∞} r_{j}^{1−β}}{∑_{k−1}^{∞} r_{j}^{1−β}} )^{α} (\frac{r_{k}}{r_{k−1}} )^{1−β} < ∞  (6.8)

and

0 < lim sup_{N→∞}(1 + φ̃_{N})^{α−1} =: Ψ̃ < ∞.  (6.9)

If

lim sup_{k→∞} (∑_{k}^{∞} r_{j}^{1−β})^{α−1} (∑^{k−1} p_{j}^{+}) < \frac{1}{αμ^{α−1}} (\frac{α − 1}{α})^{α−1} \frac{1}{Ψ̃}  (6.10)

then equation (2.1) is nonoscillatory.

Proof. Let M_{k} = 1/ (∑_{k}^{∞} r_{j}^{1−β})^{α} Then, similarly as in the previous proof,

\frac{|ΔM_{k}|}{|ΔM_{k−1}|} ≤ φ̃_{N} := (\frac{∑_{k−1}^{∞} r_{j}^{1−β}}{∑_{k+1}^{∞} r_{j}^{1−β}} )^{α} (\frac{r_{k}}{r_{k−1}} )^{1−β} ,

\frac{M_{k}^{α}}{|ΔM_{k}|^{α−1}} ≤ (\frac{1}{α − 1})^{α−1} r_{k}

and the remaining part of the proof is the same as for Theorem 6.1.

6.2. Nonoscillation criteria based on the Riccati technique

Now we turn our attention to nonoscillation criteria, which are proved using the Riccati technique. In contrast to theorems in previous section, these criteria contain the sequence p and not only its nonnegative part p^{+}. These criteria impose restrictions on the sequence r which differ from those contained in (6.4), (6.5) and (6.8), (6.9). The first two statements were established in [38]

Theorem 6.3. Suppose that (5.1) holds, ∑^{∞} p_{k} = lim_{k→∞} ∑^{k} p_{j} is convergent and

lim_{k→∞} \frac{r_{k}^{1−β}}{∑^{k−1} r_{j}^{1−β}} = 0.  (6.11)

If

lim sup_{k→∞} (∑^{k−1} r_{j}^{1−β})^{α−1} (∑_{j=k}^{∞} p_{j}) < \frac{1}{α} (\frac{α − 1}{α})^{α−1}  (6.12)

and

lim inf_{k→∞} (∑^{k−1} r_{j}^{1−β})^{α−1} (∑_{j=k}^{∞} p_{j}) > −\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} ,  (6.13)

then (2.1) is nonoscillatory.

Proof. By Lemma 4.2 it is sufficient to show that the generalized Riccati inequality (4.1) has a solution w with r_{k} + w_{k} > 0 in a neighbourhood of infinity. Set

w_{k} = C (∑^{k−1} r_{j}^{1−β})^{1−α}+ ∑_{j=k}^{∞} p_{j},  (6.14)

where C is a suitable constant which will be specified later. By the Lagrange mean value theorem we have

Δ (∑^{k−1} r_{j}^{1−β})^{1−α} = (1 − α)r_{k}^{1−β}η_{k}^{−α},

where ∑^{k−1} r_{j}^{1−β} ≤ η_{k} ≤ ∑^{k} r_{j}^{1−β}. Similarly,

w_{k} [Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k})) − r_{k}] = w_{k} [Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k})) − Φ(Φ^{−1}(r_{k}))]
= (α − 1)|ξ_{k}|^{α−2}Φ^{−1}(w_{k})w_{k} = (α − 1)|ξ_{k}|^{α−2}|w_{k}|^{β},

where ξ_{k} is between Φ^{−1}(r_{k}) and Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}), hence r_{k}^{β−1} − |w_{k}|^{β−1} ≤ ξ_{k} ≤ r_{k}^{β−1} + |w_{k}|^{β−1}. Denote

A_{k} = (∑^{k−1} r_{j}^{1−β})^{α−1} ∑_{j=k}^{∞} p_{j}  (6.15)

and let C = ((α − 1)/α)^{α}. Then

\frac{|w_{k}|}{r_{k}} = \frac{|C (∑^{k−1} r_{j}^{1−β})^{1−α}+ ∑_{j=k}^{∞} p_{j}|}{r_{k}} = (\frac{r_{k}^{1−β}}{∑^{k−1} r_{j}^{1−β}} )^{α−1} |A_{k} + C| → 0

as k → ∞ according to (6.11) and hence r_{k} + w_{k} = r_{k}(1 + w_{k}/r_{k}) > 0 for large k.

Further, the assumptions of the theorem (inequalities (6.12), (6.13)) imply the existence of ε̄ > 0 such that

−\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} + 2ε̄ < A_{k} < \frac{1}{α} (\frac{α − 1}{α})^{α−1} − 2ε̄

for k sufficiently large, say k ≥ K_{1}. Therefore, |C + A_{k}| + ε̄ < C^{\frac{1}{β}} − ε̄. This clearly implies the existence of ε > 0 such that |C + A_{k}|(1 + ε)^{\frac{1}{β}} < C^{\frac{1}{β}} (1 − ε)^{\frac{1}{β}} and hence

|C + A_{k}|^{β}(1 + ε) < C(1 − ε)  (6.16)

for k ≥ K_{1}.

Now, for a given ε > 0, there exists K_{2} ∈ ℕ such that r_{k} > |w_{k}| for k ≥ K_{2},

(\frac{∑^{k−1} r_{j}^{1−β}}{η_{k}} )^{α} ≥ (\frac{∑^{k−1} r_{j}^{1−β}}{∑^{k} r_{j}^{1−β}} )^{α} > 1 − ε,

since (6.11) implies

lim_{k→∞} \frac{∑^{k} r_{j}^{1−β}}{∑^{k−1} r_{j}^{1−β}} = lim_{k→∞} \frac{r_{k}^{1−β} + ∑^{k−1} r_{j}^{1−β}}{∑^{k−1} r_{j}^{1−β}} = 1.

Further,

\frac{|ξ_{k}|^{α−2}r_{k}^{β−1}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}} ≤ \frac{r_{k}^{β−1} (Φ^{−1}(r_{k}) + Φ^{−1}(|w_{k}|))^{α−2}}{r_{k} (1 + Φ^{−1}(w_{k}/r_{k}))^{α−1}}

= \begin{matrix}(1 + Φ^{−1}(|w_{k}|/r_{k})) & α−2 \\ (1 + Φ^{−1}(w_{k}/r_{k})) & α−1 \end{matrix} < 1 + ε,

considering the fact that w_{k}/r_{k} → 0 as k → ∞. Multiplying (6.16) by

(α − 1)r_{k}^{1−β} (∑^{k−1} r_{j}^{1−β})^{−α}

and using the above estimates we obtain

0 > −(α − 1)Cr_{k}^{1−β} (∑^{k−1} r_{j}^{1−β})^{−α} (1 − ε) +

+(α − 1)|C + A_{k}|^{β}r_{k}^{1−β} (∑^{k−1} r_{j}^{1−β})^{−α} (1 + ε)

> (1 − α)Cr_{k}^{1−β}η_{k}^{−α} − p_{k} + p_{k} + \frac{(α − 1)|C + A_{k}|^{β} (∑^{k−1} r_{j}^{1−β})^{−α} |ξ_{k}|^{α−2}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}}

= Δw_{k} + p_{k} + \frac{(α − 1)|ξ_{k}|^{α−2}|w_{k}|^{β}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}}
= Δw_{k} + p_{k} + w_{k} (1 − \frac{r_{k}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}} )

for k ≥ max{K_{1}, K_{2}}, what completes the proof.

Remark 6.1. If we consider equation (2.1) with r_{k} ≡ 1, then (6.11) clearly holds. Note that the continuous counterpart (HLDE) of equation (2.1) is often studied under such assumption. On the other hand, by the discrete L’Hospital’s rule, under the assumption ∑^{∞} r_{k}^{1−β} = ∞, condition (6.11) can be replaced by a simpler condition lim_{k→∞} r_{k+1}/r_{k} = 1, if this limit exists. Indeed, we have

lim_{k→∞} \frac{r_{k}^{1−β}}{∑^{k−1} r_{j}^{1−β}} = lim_{k→∞} \frac{r_{k+1}^{1−β} − r_{k}^{1−β}}{r_{k}^{1−β}} = 0.

However, there exist simple examples of the sequence r_{k}, e.g.

r_{k} = {1, 1, 2^{1−α}, 2^{2(1−α)}, 3^{1−α}, 3^{2(1−α)}, …, k^{1−α}, k^{2(1−α)}, … } ,

which shows that this limit does not exist, while (6.11) does.

The following theorem completes the previous statement and treats the “complementary” case ∑^{∞} r_{j}^{1−β} < ∞.

Theorem 6.4. Suppose that (5.5) holds and

lim_{k→∞} \frac{r_{k}^{1−β}}{∑_{j=k}^{∞} r_{j}^{1−β}} = 0.  (6.17)

If

lim sup_{k→∞} (∑_{j=k}^{∞} r_{j}^{1−β})^{α−1} (∑^{k−1} p_{j}) < \frac{1}{α} (\frac{α − 1}{α})^{α−1}

and

lim inf_{k→∞} (∑_{j=k}^{∞} r_{j}^{1−β})^{α−1} (∑^{k−1} p_{j}) > −\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1}  (6.18)

then (2.1) is nonoscillatory.

Proof. One can show by a similar way as in the proof of the previous theorem that the sequence

w_{k} = C (∑_{j=k}^{∞} r_{j}^{1−β})^{1−α}+ ∑_{j=k}^{∞} p_{j},

where C = ((α − 1)/α)^{α}, satisfies the inequality

|C + (∑_{j=k}^{∞} r_{j}^{1−β})^{α−1} (∑^{k−1} p_{j})|^{β} < C.

This fact implies that the generalized Riccati difference inequality (4.1) has a solution such that r_{k} + w_{k} > 0 holds in a neighbourhood of infinity.

Remark 6.2.

(i) Similarly as in the previous theorem one can observe that (6.17) can be replaced by the stronger condition lim_{k→∞} r_{k+1}/r_{k} = 1, if this limit exists.

(ii) For the sake of comparison, recall the continuous counterparts of Theorems 6.1 and 6.3:

Proposition 6.1. Suppose that ∫^{∞} r^{1−β}(t) dt = ∞ and ∫^{∞} p(t) dt is convergent. If

lim sup_{t→∞} (∫ ^{t} r(s)^{1−β} ds)^{α−1} (∫_{t}^{∞} p(s) ds) < \frac{1}{α} (\frac{α − 1}{α})  (6.19)

and

lim inf_{t→∞} (∫ ^{t} r^{1−β}(s) ds)^{α−1} (∫_{t}^{∞} p(s) ds) > −\frac{2α − 1}{α} (\frac{α − 1}{α})  (6.20)

then (HLDE) is nonoscillatory.

This criterion is proved in [28] via the Riccati technique using essentially the same idea as in the proof of Theorem 6.3. In the case ∫^{∞} r^{1−β}(t) dt < ∞, conditions (6.19) and (6.20) are to be replaced by the conditions

lim sup_{t→∞} (∫_{t}^{∞} r^{1−β}(s) ds)^{α−1} (∫ ^{t} p(s) ds) < \frac{1}{α} (\frac{α − 1}{α})

and

lim inf_{t→∞} (∫_{t}^{∞} r^{1−β}(s) ds)^{α−1} (∫ ^{t} p(s) ds) > −\frac{2α − 1}{α} (\frac{α − 1}{α}) ,

respectively, and the statement of the above proposition remains valid.

Note also that (6.19) is proved to be sufficient for nonoscillation of (HLDE) under the assumption p(t) ≥ 0 in [69] (again using the Riccati technique) and in [26] it is proved using the variational method that (6.19) with p(s) replaced by p^{+}(s) := max{0, p(s)} is also sufficient for nonoscillation of (HLDE).

The above Proposition shows that assumptions (6.4), (6.5), (6.8), (6.9), (6.11), (6.17) have no continuous analogue. However, we needed these conditions in our proofs and it is an open problem whether they are really necessary or only needed because of the methods we used. A closer examination of the proofs of Theorems 6.1, 6.2, 6.3 and 6.4 reveals that the main reason why we needed these assumptions is the absence of the chain rule for computing the difference of the “composed” sequences like M_{k} in the proofs of Theorems 6.1, 6.2 and of w_{k} in Theorems 6.3, 6.4.

(iii) One can see that various methods (variational and Riccati) give different results. For example, the first requirement in (6.4) on the sequence r is weaker than (6.11), however, on the other hand, the constant at the right–hand side of (6.6) is less than that in (6.12).

Criteria in the rest of this section were established in [101]. Let λ ≤ 0. Denote by ω_{max}(λ) the greatest root of the equation

|x|^{\frac{1}{β}} + x + λ = 0.

In the case when (6.13) does not hold, we can use the following criterion that completes Theorem 6.3.

Theorem 6.5. Let ∑^{∞} p_{k} = lim_{k→∞} ∑^{k} p_{j} be convergent, (5.1) and (6.11) hold. If

lim sup_{k→∞} A_{k} < [ω_{max} (lim inf_{k→∞} A_{k})]^{\frac{1}{β}} − ω_{max} (lim inf_{k→∞} A_{k})

and

−∞ < lim inf_{k→∞} A_{k} ≤ −\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} ,

where A is defined by (6.15), then (2.1) is nonoscillatory.

Proof. The proof is similar to that of the continuous case, see [64].

Remark 6.3.

(i) It is clear that in the same way as above one can show that there exists a similar complementary case (i.e., when (6.18) fails to hold) also to Theorem 6.4.

(ii) Observe that for r_{k} ≡ 1 the sequence A_{k} reduces to the sequence Q_{k} from Section 5.4.

The following statement is essentially a generalization of the criterion for (LΔE) (with r_{k} ≡ 1) presented in [100] and, at the same time, a discretization of the criterion for (HLDE) (with r(t) ≡ 1) presented in [64]. First we introduce some notation. Set

B_{k} := (∑^{k−1} r_{j}^{1−β})^{−1} ∑^{k−1} [(∑^{j−1} r_{i}^{1−β})^{α} p_{j}] .

For r_{k} ≡ 1 the sequence B_{k} reduces to the sequence H_{k} from Section 5.4. Let

ω̄(α) = γ_{min} + \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} ,

where γ_{min} is the least root of the equation

(α − 1)|x|^{β} + αx + \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} = 0.  (6.21)

Theorem 6.6. Suppose that ∑^{∞} p_{k} = lim_{k→∞} ∑^{k} p_{j} is convergent, (5.1) and (6.11) hold. If

lim sup_{k→∞} B_{k} < (\frac{α − 1}{α})^{α}  (6.22)

and

lim inf_{k→∞} B_{k} > ω̄(α),  (6.23)

then (2.1) is nonoscillatory.

Proof. We show that the generalized Riccati difference inequality (4.1) has a solution w with r_{k} + w_{k} > 0 in a neighbourhood of infinity. Set

w_{k} = (∑^{k−1} r_{j}^{1−β})^{1−α} (C − B_{k}).

By the same way as in the proof of Theorem 6.3 we have

Δ (∑^{k−1} r_{j}^{1−β})^{1−α} = (1 − α)r_{k}^{1−β}η_{k}^{−α}

and

Δ (∑^{k−1} r_{j}^{1−β})^{α} = αr_{k}^{1−β}μ_{k}^{α−1},

where η_{k} and μ_{k}, respectively, are between ∑^{k−1} r_{j}^{1−β} and ∑^{k} r_{j}^{1−β}. Similarly,

w_{k} [Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k})) − r_{k}] = (α − 1)|ξ_{k}|^{α−2}|w_{k}|^{β},

where ξ_{k} is between Φ^{−1}(r_{k}) and Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}), hence

r_{k}^{β−1} − |w_{k}|^{β−1} ≤ ξ_{k} ≤ r_{k}^{β−1} + |w_{k}|^{β−1}.

Let

C = \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} .

Then

\frac{|w_{k}|}{r_{k}} = \frac{|(∑^{k−1} r_{j}^{1−β})^{1−α} (C − B_{k})|}{r_{k}} = (\frac{r_{k}^{1−β}}{∑^{k−1} r_{j}^{1−β}} )^{α−1} |C − B_{k}| → 0

as k → ∞ according to (6.11) and hence r_{k} + w_{k} = r_{k}(1 + w_{k}/r_{k}) > 0 for large k. Further, assumptions (6.22) and (6.23) imply the existence of ε_{1} > 0 such that

ω̄(α) + ε_{1} < B_{k} < (\frac{α − 1}{α})^{α} − ε_{1}

for k sufficiently large, say k ≥ K_{1}. From here,

ω̄(α) − C + ε_{1} = γ_{min} < B_{k} − C < γ_{max} − ε̄_{1},  (6.24)

where γ_{max} = − ((α − 1)/α)^{α−1} is the greatest root of equation (6.21). Therefore, from (6.24) it follows that there exists ε_{2} > 0 such that

(α − 1)|B_{k} − C|^{β} + α(B_{k} − C) + C + ε_{2} < 0

for k ≥ K_{1}, and, finally, this implies the existence of ε > 0 such that

(α − 1)|C − B_{k}|^{β}(1 + ε) − (α − 1)C(1 − ε) + αB_{k}(1 + ε sgn B_{k}) < 0.

Multiplying this inequality by r_{k}^{1−β} (∑^{k−1} r_{j}^{1−β})^{−α} we obtain

(α − 1)r_{k}^{1−β} (∑^{k−1} r_{j}^{1−β})^{−α} |C − B_{k}|^{β}(1 + ε) − (α − 1)r_{k}^{1−β}C (∑^{k−1} r_{j}^{1−β})^{−α} (1 − ε) +

+ αr_{k}^{1−β}B_{k} (∑^{k−1} r_{j}^{1−β})^{−α} (1 + ε sgn B_{k}) < 0 (6.25)

for k ≥ K_{1}. We can suppose that ε is at the same time such that we may add the term p_{k} − p_{k}(1 − ε sgn p_{k}) to the left–hand side of (6.25).

Now, for a given ε > 0 there exists K_{2} ∈ ℕ such that we have (similarly as in the proof of Theorem 6.3) r_{k} > |w_{k}|,

(\frac{∑^{k−1} r_{j}^{1−β}}{η_{k}} )^{α} > 1 − ε, (sgn p_{k})(\frac{∑^{k−1} r_{j}^{1−β}}{∑^{k} r_{j}^{1−β}} )^{α}> (sgn p_{k})(1 − ε sgn p_{k}),

(sgn B_{k})(1 + ε sgn B_{k}) > (sgn B_{k}) (\frac{μ_{k}}{∑^{k} r_{j}^{1−β}} )^{α}= \frac{(sgn B_{k}) (∑^{k−1} r_{j}^{1−β})^{α} μ_{k}^{α−1}}{(∑^{k} r_{j}^{1−β})^{α} (∑^{k−1} r_{j}^{1−β})^{α−1}}

and

\frac{|ξ_{k}|^{α−2}r_{k}^{β−1}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}} < \begin{matrix}(1 + Φ^{−1}(|w_{k}|/r_{k})) & α−2 \\ (1 + Φ^{−1}(w_{k}/r_{k})) & α−1 \end{matrix} < 1 + ε

for k ≥ K_{2}. Using the above estimates, we obtain from (6.25)

0 > (α − 1)r_{k}^{1−β} (∑^{k−1} r_{j}^{1−β})^{−α} |C − B_{k}|^{β}(1 + ε) −

−(α − 1)r_{k}^{1−β}C (∑^{k−1} r_{j}^{1−β})^{−α} (1 − ε) +

+αr_{k}^{1−β}B_{k} (∑^{k−1} r_{j}^{1−β})^{−α} (1 + ε sgn B_{k}) + p_{k} − p_{k}(1 − ε sgn p_{k})

> −(α − 1)r_{k}^{1−β}Cη_{k}^{−α} − p_{k} \begin{matrix}(∑^{k−1} r_{j}^{1−β}) & 2α \\ (∑^{k−1} r_{j}^{1−β}) & α & (∑^{k} r_{j}^{1−β}) & α \end{matrix} +

+\frac{αr_{k}^{1−β}μ_{k}^{α−1}B_{k}}{(∑^{k−1} r_{j}^{1−β})^{α−1} (∑^{k} r_{j}^{1−β})^{α}} + p_{k} +

+\frac{(α − 1) (∑^{k−1} r_{j}^{1−β})^{−α} |C − B_{k}|^{β}|ξ_{k}|^{α−2}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}}

= Δw_{k} + p_{k} + \frac{(α − 1)|ξ_{k}|^{α−2}|w_{k}|^{β}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}}
= Δw_{k} + p_{k} + w_{k} (1 − \frac{r_{k}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}} ) ,

which means that inequality (4.1) has a solution w with r_{k} + w_{k} > 0 in a neighbourhood of infinity and hence equation (2.1) is nonoscillatory by Lemma 4.2.

The following statement completes the previous one in the sense of the “complementary” case ∑^{∞} r_{j}^{1−β} < ∞.

Theorem 6.7. Suppose that (5.5) and (6.17) hold. If

lim sup_{k→∞} (∑_{j=k}^{∞} r_{j}^{1−β})^{−1} ∑_{j=k}^{∞} [(∑_{i=j}^{∞} r_{i}^{1−β})^{α} p_{j}] < (\frac{α − 1}{α})^{α}  (6.26)

and

lim inf_{k→∞} (∑_{j=k}^{∞} r_{j}^{1−β})^{−1} ∑_{j=k}^{∞} [(∑_{i=j}^{∞} r_{i}^{1−β})^{α} p_{j}] > ω̄(α),  (6.27)

then (2.1) is nonoscillatory.

Proof. One can show by the similar way as in the proof of the previous theorem that under the assumptions of theorem the sequence

w_{k} = − (∑_{j=k}^{∞} r_{j}^{1−β})^{1−α} {C − (∑_{j=k}^{∞} r_{j}^{1−β})^{−1} ∑_{j=k}^{∞} [(∑_{i=j}^{∞} r_{i}^{1−β})^{α} p_{j}]} ,

where

C = \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} ,

solves the inequality (4.1) with r_{k} + w_{k} > 0.

Let λ_{1} > −1 and 0 < λ_{2} < ((α − 1)/α))^{α}. Denote by σ_{max}(λ_{1}) the greatest root of the equation

(α − 1)|x|^{β} + αx + λ_{1} = 0

and by ρ̄(λ_{2}) the greatest root of the equation

(α − 1)|x|^{β} − (α − 1)x + λ_{2} = 0.

The following statement is a complement to Theorem 6.6 in the case when (6.23) fails to hold.

Theorem 6.8. Suppose that ∑^{∞} p_{j} = lim_{k→∞} ∑^{k} p_{j} is convergent, (5.1) and (6.11) hold. If

lim sup_{k→∞} B_{k} < lim inf_{k→∞} B_{k} + ρ̄(lim inf_{k→∞} B_{k}) + σ_{max} [lim inf_{k→∞} B_{k} + ρ̄(lim inf_{k→∞} B_{k})]

and

−∞ < lim inf_{k→∞} B_{k} ≤ ω̄(α),

then (2.1) is nonoscillatory.

Proof. The proof is similar to that of the continuous case, see [64].

Remark 6.4.

(i) By the same way as above one can show that there exists a similar complementary statement (i.e., when (6.27) fails to hold) also for Theorem 6.7.

(ii) Similarly as in Theorems 6.3 and 6.4, conditions (6.11) and (6.17) in Theorems 6.5, 6.6, 6.7 and 6.8 can be replaced by the stronger one lim_{k→∞} r_{k+1}/r_{k} = 1 if this limit exists.

6.3. Conjugacy criteria

In this section we give several sufficient conditions for equation (2.1) to possess a nontrivial solution having at least two generalized zeros on a given discrete interval. The results of this section were established in [102].

We start with criterion that uses the idea of Theorem 5.4.

Theorem 6.9. Let (5.7) hold for k ∈ ℤ. Suppose that p_{k} ≢ 0 and

∑_{j=−∞}^{∞} p_{j} = lim_{k→∞} ∑_{j=−k}^{k} p_{j}

exists (as a finite number). If

∑_{j=−∞}^{∞} p_{j} ≥ 0,  (6.28)

then equation (2.1) is conjugate on ℤ.

Proof. Suppose, for the contrary, that equation (2.1) is not conjugate on ℤ. Then there exists a solution y of (2.1) such that r_{k}y_{k}y_{k+1} > 0 for all k ∈ ℤ, see [106]. Hence we can assume that there exists a solution w of the generalized Riccati difference equation (2.4) with r_{k} + w_{k} > 0, k ∈ ℤ, which is related to (2.1) by the substitution w_{k} = r_{k}Φ(Δy_{k}/y_{k}). Let m, n be integers such that 0 ≤ m < n. The summation of (2.4) from m to n − 1 yields

w_{n} − w_{m} = − ∑_{j=m}^{n−1} p_{j} − ∑_{j=m}^{n−1} S(w_{j}, r_{j})

and, similarly, by the summation of (2.4) from −n to −m − 1, we get

w_{−m} − w_{−n} = − ∑_{j=−n}^{m−1} p_{j} − ∑_{j=−n}^{m−1} S(w_{j}, r_{j}).

Letting n → ∞ and putting m = 0 in both above equations we obtain

−w_{0} = − ∑_{j=0}^{∞} p_{j} − ∑_{j=0}^{∞} S(w_{j}, r_{j})

and

w_{0} = − ∑_{j=−∞}^{−1} p_{j} − ∑_{j=−∞}^{−1} S(w_{j}, r_{j}),

since w_{k} and also w_{−k} tend to zero as k → ∞ by Theorem 5.4. Now, the addition of the last two equations yields

∑_{j=−∞}^{∞} p_{j} = − ∑_{j=−∞}^{∞} S(w_{j}, r_{j}),

which contradicts (6.28).

The next statement immediately follows from Theorem 5.8. Note that the following two sufficient conditions do not imply each other.

Corollary 6.1 (of Theorem 5.8). Let m_{1}, m_{2} ∈ ℕ be such that m_{2} ≥ m_{1} + 2. A sufficient condition for (2.1) to be conjugate on the interval [m_{1}, m_{2}] is that either

∑_{j=m_{1}}^{m_{2}} p_{j} ≥ r_{m_{1}} + r_{m_{2}+1},

or

p_{m_{i}} ≥ r_{m_{i}} + r_{m_{i}+1}, i = 1, 2.

The last criterion of this section gives a sufficient condition for conjugacy of (2.1) with r_{k} ≡ 1, i.e., of equation (2.2) in a half–bounded interval [n, ∞). This criterion is a half–linear extension of [37, Theorem 2] that is, in fact, a discretization of [115, Theorem 6].

Theorem 6.10. Suppose that p_{k} ≥ 0 for k ∈ ℕ. A sufficient condition for conjugacy of (2.2) on an interval [n, +∞), n ∈ ℕ, is that there exist integers l, m with n < l < m such that

\frac{1}{(l − n)^{α−1}} < ∑_{k=l}^{m} p_{k}.

Proof. We will show that the solution x of (2.2) given by the initial condition x_{n} = 0, x_{n+1} = 1 has a generalized zero in (n, ∞). For assume it does not. Then without loss of generality we can assume x_{k} > 0 in (n, ∞) and Δx_{k} ≥ 0 in [n, ∞), since if Δx_{k} < 0 at some point in (n, ∞), we would have a generalized zero in (n, ∞) by the condition p_{k} ≥ 0 (which implies Δ^{2}x_{k} ≤ 0). From (2.2) we obtain

(Δx_{m+1})^{α−1} = (Δx_{l})^{α−1} − ∑_{k=l}^{m}(x_{k+1})^{α−1}p_{k}.

Since p_{k} ≥ 0, using the discrete version of the mean value theorem we have

\frac{x_{l}}{l − n} = \frac{x_{l} − x_{n}}{l − n} ≥ Δx_{k} ≥ Δx_{l}

with some k ∈ [n + 1, l − 1]. Thus (x_{l})^{α−1} ≥ (l − n)^{α−1}(Δx_{l})^{α−1}. Hence

(Δx_{l})^{α−1} − ∑_{k=l}^{m}(x_{k+1})^{α−1}p_{k} ≤ (Δx_{l})^{α−1} − ∑_{k=l}^{m}(x_{l})^{α−1}p_{k} ≤

≤ (Δx_{l})^{α−1} − (l − n)^{α−1}(Δx_{l})^{α−1} ∑_{k=l}^{m} p_{k}

= (Δx_{l})^{α−1} [1 − (l − n)^{α−1} ∑_{k=l}^{m} p_{k}] .

By hypothesis, the factor in brackets is negative. If Δx_{l} > 0, then Δx_{m+1} < 0, implying a generalized zero in (m + 1, ∞). If Δx_{l} = 0, then Δx_{m} < 0 since ∑_{k=l}^{m}(x_{k+1})^{α−1}p_{k} > 0 by the assumption. In either case, x has a generalized zero in (m + 1, ∞) and so (2.2) is conjugate on [n, ∞).

CHAPTER 7 Comparison theorems

In this chapter we present comparison theorems for equation (2.1) of other types than that classical of the Sturm type and also comparison theorem for generalized Riccati difference equation, which were established in [105].

7.1. λp_{k} – type comparison theorems

We start with two statements, which claim that (non)oscillation of equation (2.1) is preserved if the sequence p_{k} is multiplied by a suitable real number. Along with equation (2.1) consider the equation

Δ(R_{k}Φ(Δx_{k})) + λp_{k}Φ(x_{k+1}) = 0,  (7.1)

where R_{k} > 0 and λ ∈ ℝ.

Theorem 7.1. Suppose that 0 < r_{k} ≤ R_{k} and λ ∈ [0, 1]. If (2.1) is nonoscillatory, then so is equation (7.1).

Proof. Suppose that (2.1) is nonoscillatory. Let y be its solution such that y_{k} > 0, k ≥ m for some m ∈ ℕ. Set x_{k} = y_{k}^{ν}, where ν = Φ^{−1}(λ). Then Δx_{k} ≤ νy_{k}^{ν−1}Δy_{k} and Δx_{k} ≥ νy_{k+1}^{ν−1}Δy_{k} by the Lagrange mean value theorem, since Δy_{k} ≥ 0 if and only if y_{k+1}^{ν−1} ≤ y_{k}^{ν−1} and Δy_{k} ≤ 0 if and only if y_{k+1}^{ν−1} ≥ y_{k}^{ν−1}. Further,

Δ(r_{k}Φ(Δx_{k})) = r_{k+1}Φ(Δx_{k+1}) − r_{k}Φ(Δx_{k})
≤ r_{k+1}Φ(νy_{k+1}^{ν−1}Δy_{k+1}) − r_{k}Φ(νy_{k+1}^{ν−1}Δy_{k})
= Φ(νy_{k+1}^{ν−1})Δ(r_{k}Φ(Δy_{k}))
= −Φ(νy_{k+1}^{ν−1})p_{k}Φ(y_{k+1})
= −Φ(ν)p_{k}Φ(y_{k+1}^{ν})
= −λp_{k}Φ(x_{k+1}).

This implies that equation (7.1) is nonoscillatory by Lemma 4.1 and Theorem 3.2.

Remark 7.1. The “oscillatory counterpart” to Theorem 7.1 is immediate: Suppose that r_{k} > 0. Let r_{k} ≥ R_{k} and λ ∈ ℝ be such that λ ≥ 1. If (2.1) is oscillatory, then so is equation (7.1).

It is not difficult to prove via the Riccati technique a direct half–linear extension of comparison theorems for linear difference equation (LΔE) given in [50]. Those results are of the similar type as ours (of course, for the case of linear equation (LΔE)). However, there are required some additional conditions and hence they are not so strong. We present their half–linear extension without proof. For this purpose, along with equation (2.1), consider the equation

Δ(R_{k}Φ(Δx_{k})) + q_{k}P_{k}Φ(x_{k+1}) = 0,  (7.2)

where R_{k} > 0.

7.2. ∑ p_{j} ≤ ∑ P_{j} – type comparison theorem 67

Theorem 7.2. Assume that 0 < r_{k} ≤ R_{k}, P_{k} ≤ p_{k}, 0 < q_{k} ≤ 1, Δq_{k} ≤ 0, k ≥ m. Further suppose that ∑_{j=m}^{∞} r_{j}^{1−β} = ∞ and (5.26) holds. Then nonoscillation of (2.1) implies nonoscillation of (7.2).

Theorem 7.3. Assume r_{k} ≥ R_{k}, P_{k} ≥ p_{k}, q_{k} ≥ 1 and Δq_{k} ≥ 0, k ≥ m. Further suppose that ∑_{j=m}^{∞} R_{j}^{1−β} = ∞ and (5.26) holds, where p_{k} is replaced by q_{k}P_{k}. Then oscillation of (2.1) implies oscillation of (7.2).

7.2. ∑ p_{j} ≤ ∑ P_{j} – type comparison theorem

In what follows, we partially use an idea from [79], where equation (2.1) is considered under stronger assumptions. A similar result (however, without the change of the constant α) for the continuous case, i.e., for the case of the equation

(Φ(y′))′ + p(t)Φ(y) = 0,  (7.3)

can be found in [69], where also somewhat stronger additional conditions are required (by comparison with our result).

Along with equation (2.1) consider the equation

Δ(R_{k}Φᾱ(Δx_{k})) + P_{k}Φᾱ(x_{k+1}) = 0,  (7.4)

where Φᾱ(x) = |x|^{ᾱ−1} sgn x, ᾱ > 1.

Theorem 7.4. Assume that the sequences p_{k} and P_{k} satisfies (5.26). Let the series ∑_{j=m}^{∞} p_{j} and ∑_{j=m}^{∞} P_{j} be convergent and

∑_{j=k}^{∞} p_{j} ≤ ∑_{j=k}^{∞} P_{j} for all large k.  (7.5)

Further, suppose that 0 < R_{k} ≤ r_{k}, ∑_{j=1}^{∞} R_{j}^{1−β} = ∞ and 1 < α ≤ ᾱ. If equation (7.4) is nonoscillatory, then so is equation (2.1).

Proof. By Lemma 5.4, nonoscillation of (7.4) implies the existence of m_{1} ∈ ℕ such that

z_{k} ≥ ∑_{j=k}^{∞} P_{j} + ∑_{j=k}^{∞} S(z_{j}, R_{j}, ᾱ) =: Z_{k}  (7.6)

for k ≥ m_{1} (clearly, with z_{k} +R_{k} > 0). Let m_{2} ∈ ℕ be such that (7.5) holds and ∑_{j=k}^{∞} p_{j} ≥ 0 for k ≥ m_{2}. Set m = max{m_{1}, m_{2}} and define the set Ω and the mapping T by

Ω = {w ∈ l^{∞}, 0 ≤ w_{k} ≤ Z_{k} for k ≥ m}

and

(T w)_{k} = ∑_{j=k}^{∞} p_{j} + ∑_{j=k}^{∞} S(w_{j}, R_{j}, α), k ≥ m, w ∈ Ω.

We show that T has a fixed point in Ω. We must verify that 1) Ω is bounded, closed and convex subset of l^{∞},

2) T maps Ω into itself,

3) T Ω is relatively compact,

4) T is continuous.

ad 1) Clearly, Ω is bounded and convex. Let x^{t} = {x_{k}^{t}}, t = 1, 2, …, be any sequence in Ω such that x^{t} tends to x as t → ∞ (in the sup norm). From our assumptions, for any ε > 0 there exists n ∈ ℕ such that sup_{k≥m} |x_{k}^{t} − x_{k}| < ε for all t ≥ n. Thus, for any fixed k, we have lim_{t→∞} x_{k}^{t} = x_{k}. Since 0 ≤ x_{k}^{t} ≤ z_{k} for all t, then 0 ≤ x_{k} ≤ z_{k}. We have k ≥ m arbitrary and hence x belongs to Ω

ad 2) Suppose that w ∈ Ω and define x_{k} = (T w)_{k}, k ≥ m. Obviously, x_{k} ≥ 0 for k ≥ m. We must show that x_{k} ≤ Z_{k}, k ≥ m. We have

x_{k} = ∑_{j=k}^{∞} p_{j} + ∑_{j=k}^{∞} S(w_{j}, R_{j}, α) ≤ ∑_{j=k}^{∞} P_{j} + ∑_{j=k}^{∞} S(w_{j}, R_{j}, α)

≤ ∑_{j=k}^{∞} P_{j} + ∑_{j=k}^{∞} S(w_{j}, R_{j}, ᾱ) ≤ ∑_{j=k}^{∞} P_{j} + ∑_{j=k}^{∞} S(z_{j}, R_{j}, ᾱ)

by the assumptions of theorem and by Lemma 4.3. Hence, T Ω ⊂ Ω.

ad 3) According to [20, Theorem 3.3] it suffices to show that T Ω is uniformly Cauchy. Let ε > 0 be given. We show that there exists N ∈ ℕ such that for any k, l > N |(T x)_{k}−(T x)_{l}| < ε for any x ∈ Ω. Without loss of generality, suppose k < l. Then we have

|(T x)_{k} − (T x)_{l}| = |∑_{j=k}^{l−1} p_{j} + ∑_{j=k}^{l−1} S(x_{j}, R_{j}, α)|

= ∑_{j=k}^{l−1} p_{j} + ∑_{j=k}^{l−1} S(x_{j}, R_{j}, α)  (7.7)

for large k by Lemma 5.2. Taking into account the properties of S(x_{k}, R_{k}, α) and p_{k}, for any ε > 0 one can find N ∈ ℕ such that

∑_{j=k}^{l−1} p_{j} < \frac{ε}{2} and ∑_{j=k}^{l−1} S(x_{j}, R_{j}, α) < \frac{ε}{2} for l > k > N.

From here and (7.7), |(T x)_{k} − (T x)_{l}| < ε, hence T Ω is relatively compact.

ad 4) Let x^{t} = {x_{k}^{t}}, k ≥ m, be a sequence in Ω converging to x. We must show that T x^{t} converges to T x. Clearly, T x^{t} ∈ T Ω ⊂ Ω for any t and also T x ∈ T Ω ⊂ Ω. For any ε > 0 one can choose M ≥ m such that (T x^{t})_{k} < ε/2 and (T x)_{k} < ε/2 for k > M and for each t ∈ ℕ. Define

(T̃x)_{k,l} = ∑_{j=k}^{l} p_{j} + ∑_{j=k}^{l} S(x_{j}, R_{j}, α) for l ≥ k ≥ m

and (T̃x)_{k,l} = 0 for l < k. Mapping T̃ is obviously continuous. Therefore, for a given ε/2 > 0, there exists N ∈ ℕ such that

|(T̃x^{t})_{k,l} − (T̃x)_{k,l}| < \frac{ε}{2} for t ≥ N and k ≥ m.

Now, having chosen such M, N as above, the following estimates hold for any k ≥ m:

|(T x^{t})_{k} − (T x)_{k}| = |(T̃x^{t})_{k,M} + (T x^{t})_{M+1} − (T̃x)_{k,M} − (T x)_{M+1}|
≤ |(T̃x^{t})_{k,M} − (T̃x)_{k,M} | + |(T x^{t})_{M+1} − (T x)_{M+1}|
< \frac{ε}{2} + \frac{ε}{2} = ε.

Since k ≥ m is arbitrary, we have that T x^{t} converges to T x.

Therefore, it follows from the Schauder fixed point theorem that there exists an element w ∈ Ω such that w = T w. In view of the definition of T , this (positive) sequence w satisfies the equation

w_{k} = ∑_{j=k}^{∞} p_{j} + ∑_{j=k}^{∞} S(w_{j}, R_{j}, α), k ≥ m,

and hence also the equation (2.4). Consequently, the sequence y given by

y_{m} = m_{0} ≠ 0 and y_{k+1} = (1 + (w_{k}/R_{k})^{β−1}) y_{k}, k ≥ m,

is a nonoscillatory solution of

Δ(R_{k}Φ(Δy_{k})) + p_{k}Φ(y_{k+1}) = 0

and hence this equation is nonoscillatory. The statement now follows from Theorem 3.2.

Remark 7.2. A closer examination of the above proof shows that the necessary condition for nonoscillation of equation (2.1) in Lemma 5.4 is also sufficient.

7.3. Telescoping principle

The result presented in this section is a half–linear extension of the so–called telescoping principle, which was introduced in [66] for the second order linear difference equation (LΔE). In fact, it was introduced firstly for second order linear differential equation in [72]. Note that in [66] the authors consider equation (LΔE) only under the assumption r_{k} > 0 and hereby our result with r_{k} ≠ 0 is new even in the linear case (in spite of the fact that an idea of the proof remains quite the same).

Before we present the main result, let us introduce some concepts and assumptions. Denote by S the set of all real sequences y = {y_{k}, k ∈ ℕ}. Assume

I = ⋃_{i=1}^{j} I_{i}, I_{i} = (m_{i}, n_{i}], i = 1, …, j, j ≤ ∞,  (7.8)

where m_{i}, n_{i} ∈ ℕ, i = 1, …, j, are such that m_{i} < n_{i} < m_{i+1} and card(ℕ \ I) = ∞. Based on the set I, we define an interval shrinking transformation τ = τ_{I} : ℕ → ℕ as follows:

K = τ(k) = card([1, k] ∩ I^{C}),

where I^{C} = ℕ \ I. Let M_{i} = τ(m_{i}). Then M_{i} = τ(k) for k ∈ [m_{i}, n_{i}], i = 1, …, j. This transformation τ induces a transformation T = T_{I} : S → S defined as follows: For y ∈ S

T y = Y = {Y_{K}, K ∈ ℕ} with Y_{K} = y_{k} when τ(k) = K.

Theorem 7.5. Let r_{k} ≠ 0, k ∈ ℕ, and assume that (7.8) holds. Let R = T r and P = T p for T = T_{I}. Assume

∑_{k=m_{i}+1}^{n_{i}} p_{k} ≥ 0, i = 1, …, j.  (7.9)

Suppose X = {X_{K}, K ∈ ℕ} is a solution of the equation

Δ(R_{K}Φ(ΔX_{K})) + P_{K}Φ(X_{K+1}) = 0,  (7.10)

such that R_{K}X_{K}X_{K+1} > 0 for K < N and R_{N}X_{N}X_{N+1} ≤ 0. If the sequence y is a solution of equation (2.1) such that y_{1} ≠ 0 and r_{1}Φ(Δy_{1})/Φ(y_{1}) ≤ R_{1}Φ(ΔX_{1})/Φ(X_{1}), then there exists l ≤ n such that r_{l}y_{l}y_{l+1} ≤ 0, where N = τ(n). More precisely, if N ≤ M_{i}, then there exists l ≤ m_{i} such that r_{l}y_{l}y_{l+1} ≤ 0, i = 1, 2, …, j.

Proof. In this proof, by y ⊵ X we mean either y ≥ X or y does not exist. The proof is by induction. Assume that the conclusion is not true. Then w_{k} = −r_{k}Φ(Δy_{k})/Φ(y_{k}) satisfies

Δw_{k} = p_{k} + w_{k} (\frac{r_{k}}{(Φ^{−1}(r_{k}) − Φ^{−1}(w_{k}))^{α−1}} − 1)  (7.11)

or, equivalently,

w_{k+1} = p_{k} + S̃(w_{k}, r_{k}), k = 1, …, n,  (7.12)

and w_{k} < r_{k}, k = 1, …, n, where

S̃(w_{k}, r_{k}) = \frac{w_{k}r_{k}}{(Φ^{−1}(r_{k}) − Φ^{−1}(w_{k}))^{α−1}} .

Observe that the behaviour of the function S̃ is similar to that of the function S̄ from Lemma 4.3. In particular, S̃(w_{k}, r_{k}) is nondecreasing with respect to the first variable for r_{k} > w_{k}. Let V_{K} = −R_{K}Φ(ΔX_{K})/Φ(X_{K}). Then

V_{K+1} = P_{K} + S̃(V_{K}, R_{K}), K = 1, …, N − 1,  (7.13)

V_{K} < R_{K}, K = 1, …, N − 1 and V_{N} ⊵ R_{N}.

If N ≤ M_{1} = m_{1}, then for k = 1, …, N, K = k and hence R_{K} = r_{k}, P_{K} = p_{k} and equation (7.13) is the same as (7.12). By the hypothesis w_{1} ≥ V_{1}, comparing (7.12) and (7.13) step by step (using the above property of S̃), we find that w_{k+1} ≥ V_{k+1}, k = 1, …, N − 1. In particular,

w_{n} = w_{N} ≥ V_{N} ⊵ R_{N} = r_{n}.

This implies that w_{n} ⊵ r_{n}, contradicting the assumption.

If M_{1} < N ≤ M_{2}, then arguing as above we see that w_{m_{1}+1} = w_{M_{1}+1} ≥ V_{M_{1}+1}. Summation of (7.11) from m_{1} + 1 to n_{1} and using (7.9) we obtain

w_{n_{1}+1} − w_{m_{1}+1} = ∑_{k=m_{1}+1}^{n_{1}} p_{k} + ∑_{k=m_{1}+1}^{n_{1}} S̃(w_{k}, r_{k}) ≥ 0,

hence w_{n_{1}+1} ≥ w_{m_{1}+1} ≥ V_{m_{1}+1}. Noting that τ(n_{1} + 1) = N_{1}, we see that w_{k}, V_{K} satisfy the same generalized Riccati equation for n_{1} +1 ≤ k ≤ n and M_{1} +1 ≤ K ≤ N, respectively. As before, we see that w_{n} ≥ V_{N} ⊵ R_{N} = r_{n} and, again, this implies that w_{n} ⊵ r_{n}, contradicting the assumption. The proof of the inductive step from i to i + 1 is similar and hence is omitted.

Theorem 7.6 (Telescoping principle). Under the conditions and with the notation of Theorem 7.5, if equation (7.10) is oscillatory, then equation (2.1) is oscillatory.

Proof. Let X_{k} be a solution of (7.10) with X_{1} ≠ 0. Let y_{k} be a solution of (2.1) satisfying y_{1} ≠ 0, r_{1}Φ(Δy_{1})/Φ(y_{1}) ≤ R_{1}Φ(ΔX_{1})/Φ(X_{1}). By Theorem 7.5, there exists l_{1} > 0 such that r_{l_{1}}y_{l_{1}}y_{l_{1}+1} ≤ 0. Now, working on the solution for k ≥ l_{1} + 1 instead of k ≥ 1 and proceeding as before, we show that there exists l_{2} ≥ l_{1} + 1 such that r_{l_{2}}y_{l_{2}}y_{l_{2}+1} ≤ 0. Continuation of this process leads to the conclusion that y is oscillatory, hence (2.1) is oscillatory.

This principle can be applied to get many new examples of oscillatory equations. We use a process that is the reverse of the construction in Theorem 7.5. Start with any known oscillatory equation (7.10). Choose a sequence of integers M_{i} → ∞. Cut the plane at each vertical line k = M_{i} and pull the two half–planes apart to form a gap of arbitrary finite length. Now fill the gap with an arbitrary nonzero r_{i} and any p_{i} whose sum over the length of the gap is nonnegative. Do this at each M_{i} and denote the new coefficient sequences by r, p. Then equation (2.1) is oscillatory.

It means that any sufficient conditions for oscillation need only be verified on intervals, for example, on ℕ \ I, where I is defined by (7.8). On I, r_{k} and p_{k} can be arbitrary as long as r_{k} ≠ 0 and the p_{k} have a nonnegative sum over each interval (m_{i}, n_{i}] of I, i = 1, …, j.

7.4. Comparison theorem for generalized Riccati difference equations

Along with equation (2.4) consider the generalized Riccati difference equation

Δv_{k} + P_{k} + S(v_{k}, R_{k}) = 0  (7.14)

Theorem 7.7. Suppose that R_{k} ≥ r_{k}, P_{k} ≤ p_{k} for k ∈ [m, n] and let w_{k}, v_{k} be solutions of equations (2.4) and (7.14), respectively, defined on [m, n]. If r_{k} + w_{k} > 0 on [m, n] and v_{m} ≥ w_{m}, then v_{k} ≥ w_{k} and R_{k} + v_{k} > 0 for k ∈ [m, n].

Proof. Let w_{k} and v_{k} be solutions of equations (2.4) and (7.14), respectively, such that v_{m} ≥ w_{m} (then R_{m}+v_{m} > 0) and w_{k} +r_{k} > 0 for k ∈ [m, n]. One can rewrite these equations as

w_{k+1} = −p_{k} + S̄(w_{k}, r_{k})

and

v_{k+1} = −P_{k} + S̄(v_{k}, R_{k}),

respectively, where the function S̄ is defined in Lemma 4.3. According to this lemma, S̄(x, y) ≤ S̄(X, Y) for x ≤ X and y ≤ Y with x + y > 0. Hence,

w_{m+1} = −p_{m} + S̄(w_{m}, r_{m}) ≤ −P_{m} + S̄(v_{m}, R_{m}) = v_{m+1}

and R_{m+1} + v_{m+1} > 0. Continuing this process step by step we find that v_{k} ≥ w_{k} and R_{k} + v_{k} > 0 for k ∈ [m, n]. Theorem is proved.

Remark 7.3. Note that the other possibility (that is more complicated) how to prove the above theorem is to use the generalized Picone identity.

CHAPTER 8 Strong oscillation

8.1. Strong oscillation and nonoscillation, conditional oscillation

The results of this chapter were established in [105]. This section concerns an extension of the so–called strong oscillation to the case of

equation (2.1) and some further related concepts introduced in [90] for the equation

y′′ + p(t)y = 0.

The class of equations (2.1) can be divided according to the following definition.

Definition 8.1.

• Equation (2.1) is said to be strongly oscillatory if the equation

Δ(r_{k}Φ(Δy_{k})) + λp_{k}Φ(y_{k+1}) = 0  (8.1)

is oscillatory for all λ > 0.

• Equation (2.1) is said to be strongly nonoscillatory if equation (8.1) is nonoscillatory for all λ > 0.

• Equation (2.1) is said to be conditionally oscillatory if (8.1) is oscillatory for some λ > 0 and nonoscillatory for some other λ > 0. By Theorem 3.2 it follows that in this case there must exist a positive number γ(p) such that (2.1) is oscillatory for λ > γ(p) and nonoscillatory for λ < γ(p). This number γ(p) is called the oscillation constant of the sequence p_{k} (with respect to the sequence r_{k}).

Now we can present strongly oscillation and nonoscillation criteria. Note that here we consider only the case when ∑^{∞} p_{j} is convergent since if ∑^{∞} p_{j} = ∞ (and ∑^{∞} r_{j}^{1−β} = ∞ with r_{k} > 0), then (2.1) is oscillatory by Theorem 5.1 and obviously also strong oscillatory.

Theorem 8.1. Assume that (5.1) is fulfilled and ∑^{∞} p_{j} is convergent. Then the following statements hold:

(i) Suppose in addition that (6.11) holds and ∑^{∞} p_{j} ≥ 0. If (2.1) is strongly oscillatory, then

lim sup_{k→∞} (∑_{j=m}^{k−1} r_{j}^{1−β})^{α−1} ∑_{j=k}^{∞} p_{j} = ∞.  (8.2)

(ii) Suppose in addition that p_{k} ≥ 0. If (8.2) is fulfilled, then (2.1) is strongly oscillatory. (iii) Suppose in addition that p_{k} ≥ 0. If (2.1) is strongly nonoscillatory, then

lim_{k→∞} (∑_{j=m}^{k−1} r_{j}^{1−β})^{α−1} ∑_{j=k}^{∞} p_{j} = 0.  (8.3)

(iv) Suppose in addition that (6.11) holds. If (8.3) is fulfilled, then (2.1) is strongly nonoscillatory.

The proof of this theorem (and also of the next theorem) is essentially the same as in the half–linear continuous case, see [69], and hence is omitted, in spite of the fact that in some cases we do not require the sequence p_{k} to be nonnegative. We refer also the paper [74], where a result similar to the above theorem is presented for somewhat more special form of equation (2.1). Note that to prove our statements we use criteria Corollary 5.6, Theorem 5.11 and Theorem 6.3.

The following theorem provides an information about the oscillation constant of conditionally oscillatory equation (2.2). We will use the symbols Q_{∗} and Q^{∗} introduced at the beginning of Section 5.3.

Theorem 8.2. Suppose that 0 < Q_{∗} ≤ Q^{∗} < ∞ and p_{k} ≥ 0. Then the oscillation constant γ(p) of the equation (2.2) satisfies

\frac{1}{α} (\frac{α − 1}{α})^{α−1} \frac{1}{Q^{∗}} ≤ γ(p) ≤ min \{\frac{1}{Q^{∗}} , \frac{1}{α} (\frac{α − 1}{α})^{α−1} \frac{1}{Q_{∗}} \} .

In particular, if Q_{∗} = Q^{∗}, then

γ(p) = \frac{1}{α} (\frac{α − 1}{α})^{α−1} \frac{1}{Q^{∗}} .

The proofs of the following two theorems are also omitted since they are similar to the linear discrete case, see [21]. However, note that in the contrast to those results we do not need the sequences under consideration to be necessary nonnegative. To prove our statements we use oscillation criterion Corollary 5.6 and nonoscillation criterion Theorem 6.3.

Theorem 8.3. Let p_{k} and p̃_{k}, k ∈ ℕ, be two sequences such that (5.26) holds and ∑_{j=k}^{∞} p̃_{j} be positive for large k. Further, let γ(p̃), 0 < γ(p̃) < ∞, be the oscillation constant of p̃_{k}. If

Ψ := lim inf_{k→∞} \frac{∑_{j=k}^{∞} p_{j}}{∑_{j=k}^{∞} p̃_{j}} > γ(p̃),  (8.4)

then (2.2) is oscillatory.

Theorem 8.4. Let p_{k} and p̃_{k}, k ∈ ℕ, be two sequences such that (5.26) holds and ∑_{j=k}^{∞} p̃_{j} be positive for large k. Further, let γ(p), 0 < γ(p) < ∞, and γ(p̃), 0 < γ(p̃) < ∞, be the oscillation constants of p_{k} and p̃_{k}, respectively. Then Ψ ≤ γ(p̃)/γ(p), where Ψ is defined by (8.4), and if

lim sup_{k→∞} \frac{∑_{j=k}^{∞} p_{j}}{∑_{j=k}^{∞} p̃_{j}} < γ(p̃),

then (2.2) is nonoscillatory.

8.2. Generalized Euler difference equation

In this section we will investigate some oscillatory properties of the discrete generalized Euler equation

Δ(Φ(Δy_{k})) + \frac{γ}{(k + 1)^{α}} Φ(y_{k+1}) = 0,  (8.5)

where γ ∈ ℝ. Among others, we will need the following auxiliary result, which is a generalization of the well–known Hardy inequality.

Lemma 8.1 ([53]). If α > 1 and η_{k} ≥ 0 for k ∈ ℕ, then

∑_{k=1}^{n} k^{−α} (∑_{j=1}^{k} η_{j})^{α} ≤ (\frac{α}{α − 1})^{α} ∑_{k=1}^{n} η_{k}^{α},

where the equality is possible only with η_{k} = 0 for k = 1, 2, …, n.

To show that (8.5) is (non)oscillatory we will distinguish the following four cases:

1) If γ > ((α − 1)/α)^{α}, then (8.5) is oscillatory by Corollary 5.6 since

k^{α−1} ∑_{j=k}^{∞} \frac{γ}{(j + 1)^{α}} ≥ k^{α−1}γ ∫_{k+1}^{∞} \frac{1}{x^{α}} dx = \frac{γk^{α−1}}{(α − 1)(k + 1)^{α−1}}

≥ \frac{1}{α} (\frac{α − 1}{α})^{α−1} + ε,

for sufficiently small ε > 0.

2) If 0 ≤ γ < ((α − 1)/α)^{α}, then (8.5) is nonoscillatory by Theorem 6.3 since

k^{α−1} ∑_{j=k}^{∞} \frac{γ}{(j + 1)^{α}} ≤ k^{α−1}γ ∫_{k}^{∞} \frac{1}{x^{α}} dx = \frac{γk^{α−1}}{(α − 1)k^{α−1}}

≤ \frac{1}{α} (\frac{α − 1}{α})^{α−1} − ε,

for sufficiently small ε > 0.

3) If γ < 0, then (8.5) is nonoscillatory by the Sturm type comparison theorem (Theorem 3.2) and the result in the case 2).

4) If γ = ((α − 1)/α)^{α}, then (8.5) is nonoscillatory. Indeed, according to Lemma 4.4 it is sufficient to find m ∈ ℕ such that F(ξ; m, ∞) > 0 for any nontrivial ξ ∈ U(m). Put m = 1 and ξ_{k+1} = ∑_{j=1}^{k} η_{j}, k ∈ ℕ, where η is such that ξ is admissible. Clearly, Δξ_{k} = η_{k}. Now, there exists n ∈ ℕ such that

F(ξ; 1, ∞) = ∑_{k=1}^{n} [|η_{k}|^{α} − \frac{γ}{(k + 1)^{α}} |∑_{j=1}^{k} η_{j}|^{α}]

≥ ∑_{k=1}^{n} [|η_{k}|^{α} − \frac{γ}{k^{α}} (∑_{j=1}^{k} |η_{j}|)^{α}]
> 0

by Lemma 8.1, and hence (8.5) is nonoscillatory.

Altogether, equation (8.5) is oscillatory for γ > ((α − 1)/α)^{α} and nonoscillatory otherwise. Thus, if we consider (8.5) as an equation of the form (8.1), more precisely, if λ = γ and p_{k} = (k + 1)^{−α}, then it is easy to see that the oscillation constant of such sequence p_{k} is equal to ((α − 1)/α)^{α}.

On the other hand, the above oscillatory properties of (8.5), can be used to establish criteria (5.58) and (6.12) (with r_{k} ≡ 1) for equation (2.2) by comparing (2.2) with (8.5) using Theorem 7.4 and the above estimates. Similar method was used in [69] to prove these types of criteria for (7.3), however, only under the assumption p(t) ≥ 0. One can also show, either as a consequence of the more general criteria (5.58) and (6.12) or by comparing of (2.2) with (8.5) using Theorem 3.2, that the condition

(\frac{α − 1}{α})^{α} < lim inf_{k→∞}(k + 1)^{α}p_{k} ≤ ∞

8.2. Generalized Euler difference equation 75

implies oscillation of (2.2) and the condition

−∞ ≤ lim sup_{k→∞}(k + 1)^{α}p_{k} < (\frac{α − 1}{α})^{α}

implies nonoscillation of (2.2).

CHAPTER 9 Conclusions and future directions, open problems

9.1. Motivation by the linear continuous theory, more general equations

In this chapter we indicate future directions which may be pursued in the context of our research.

Having considered approach described in Chapter 1, it is clear that there is a great number of open problems in this direction, since the qualitative theory of linear differential equation (LDE) offers many interesting topics for the discretization and the generalization (in the half–linear sense). From those we mention only the following:

1) Very important role in the oscillation theory of (LDE) is played by the so–called principal solution, see e.g. [54]. An extension of this concept to the half–linear differential equation (HLDE) has been already partly done. In [29, 46, 89] the construction of this solution was made and it is based either on the minimality of the solution of generalized Riccati differential equation (GRDE) (since in the linear case the principal solution of (LDE) generates a minimal solution (near ∞) of (RDE) – the so–called distinguished solution), or it is based on the generalized Prüfer transformation (see below). For further application of this concept in the oscillation theory of (HLDE) see [35]. In [32] the integral characterization of principal solution can be found. Recall that the discrete counterpart of principal solution (of (LDE)) is called recessive solution (of (LΔE)). Taking into account the above facts we would like to construct recessive solution for equation (2.1) (in fact, it is not so difficult via the minimal solution of generalized Riccati difference equation, see [106]) and possibly apply it.

2) In our treatment we mostly consider (2.1) as a perturbation of the one term equation

Δ(r_{k}Φ(Δy_{k})) = 0  (9.1)

and this equation is nonoscillatory if and only if r_{k} > 0 eventually (otherwise y_{k} ≡ 1 would be its oscillatory solution). Some oscillation and nonoscillation theorems of Chapters 5 and 6 then state, roughly speaking, that the “perturbed” equation (2.1) becomes oscillatory (remains nonoscillatory) if the sequence p_{k} in the perturbation term p_{k}Φ(y_{k+1}) is “sufficiently positive” (is not “too positive”). This observation suggests the following general approach to the oscillation theory of (2.1). Suppose that p̃_{k} is a sequence such that the equation

Δ(r_{k}Φ(Δy_{k})) + p̃_{k}Φ(p_{k+1}) = 0  (9.2)

is nonoscillatory and consider (2.1) as a perturbation of this equation, i.e., we write this equation in the form

Δ(r_{k}Φ(Δy_{k})) + p̃_{k}Φ(y_{k+1}) + (p_{k} − p̃_{k})Φ(y_{k+1}) = 0.

Now we look for conditions on the difference p − p̃ which guarantee that (2.1) becomes oscillatory (remains nonoscillatory). Note that in the case of half–linear differential equation (HLDE) this approach has been used e.g. in the paper [35] and quite important role is here played by the above mentioned principal solution.

3) The Prüfer transformation is further useful tool in the oscillation theory of equation (LDE). Recall that by this transformation, any nontrivial solution y of (LDE) and its quasiderivative ry′ can be expressed in the form

y(t) = ρ(t) sin φ(t),
r(t)y′(t) = ρ(t) cos φ(t),

where ρ and φ satisfy a certain first order system. In [40] it was showed that this transformation can be extended to equation (HLDE), where the so–called generalized sine and generalized cosine play the role of the usual sine and cosine, respectively. The discrete counterpart of the Prüfer transformation for equation (LΔE) was presented in very recent paper [6]. It could be interesting and useful to extend this concept to the case of equation (2.1).

4) The asymptotic behaviour of solutions of (LDE) is also frequently studied part of the qualitative theory. By this we mean, in particular, looking for conditions guaranteeing certain behaviour of (nonoscillatory) solution, e.g. monotonicity, boundedness or convergence to any number. Of course, such properties are investigated also in the case of difference equations and even of more general form than (2.1) is. On the other hand, there cannot be used “nice – almost linear” properties of half–linear equations. Hence, here is a good opportunity to investigate (2.1) just from this point of view. For example, it is still an open problem (also in the continuous case) to answer questions as what is a half–linear extension of a solution that is summable (integrable) with its square and what is an extension of limit–point and limit–circle classification.

It is clear that another from the possible future directions concerning our work is to investigate more general equations than (2.1) or other types of equations similar to (2.1), for example, equation with a slightly “greater” nonlinearity

Δ(r_{k}Φ(Δy_{k})) + p_{k}Φᾱ(y_{k+1}) = 0,

Φᾱ being the same function as Φ with ᾱ > 1 instead of α (generally, ᾱ ≠ α), or half–linear difference equations with deviating arguments and many others. It could be interesting to observe how results presented in this thesis can be extended or used in the investigation of such equations.

9.2. Time scales

This last remark concerns the fact that we have parallel theories for half–linear differential and difference equations. This fact suggests an idea to develop a unified theory for these equations on arbitrary time scales. A time scale (or a measure chain) 𝕋 is a relatively new mathematical object introduced in [57] (1988). In fact, it is any closed subset of ℝ and calculus on this time scale has been prepared in such a way that it unifies differential and difference calculus. For instance, the symbol of derivative f_{t}^{Δ} on time scale 𝕋 means usual derivative f′(t) for 𝕋 = ℝ and difference Δf_{t} for 𝕋 = ℤ that are two most important examples of time scales. Except of this unification aspect, the theory on time scales enables to explain the discrepancies occurring in parallel statements in the continuous and the discrete cases and therefore helps with better understanding them. Since 1988 many papers have appeared, where, in particular, the so–called dynamic equations on time scales are investigated. A Sturmian theory, (non)oscillation and Riccati technique for the second order dynamic equation

(r(t)y^{Δ}(t))^{Δ} + p(t)y(σ(t)) = 0

is developed in [48]. We do not define here the so–called jump operator σ(t), we only mention that σ(t) ≡ t for 𝕋 = ℝ and σ(t) = t + 1 for 𝕋 = ℤ. Clearly, this equation covers (LDE) and also (LΔE) for 𝕋 = ℝ and 𝕋 = ℤ, respectively. The Roundabout theorem for this equation (even in a matrix form) was established in [2]. Note that in [34, 58] this statement was extended to very general “linear” object, namely to the so–called symplectic dynamic system. Thus, taking into account the above facts, we hope that it is possible to develop a unified (oscillation) theory for half–linear dynamic equation

(r(t)Φ (y^{Δ}(t)))^{Δ} + p(t)Φ(y(σ(t))) = 0.

Note that to know “the form of the discrete results” is sometimes very good direction for such development.

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