Abstract

The paper investigates a dynamic equation Δ𝑦(𝑡𝑛)=𝛽(𝑡𝑛)[𝑦(𝑡𝑛𝑗)𝑦(𝑡𝑛𝑘)] for 𝑛, where 𝑘 and 𝑗 are integers such that 𝑘>𝑗0, on an arbitrary discrete time scale 𝕋={𝑡𝑛} with 𝑡𝑛, 𝑛𝑛0𝑘={𝑛0𝑘,𝑛0𝑘+1,}, 𝑛0, 𝑡𝑛<𝑡𝑛+1, Δ𝑦(𝑡𝑛)=𝑦(𝑡𝑛+1)𝑦(𝑡𝑛), and lim𝑛𝑡𝑛=. We assume 𝛽𝕋(0,). It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for 𝑛. The results are presented as inequalities for the function 𝛽. Examples demonstrate that the criteria obtained are sharp in a sense.

1. Introduction

We use the following notation: for an integer 𝑠, we define that 𝑠={𝑠,𝑠+1,}, and if an integer 𝑞𝑠, we define 𝑞𝑠={𝑠,𝑠+1,,𝑞}.

Hilger initiated in [1, 2] the calculus of time scales in order to create a theory that unifies discrete and continuous analyses. He defined a time scale 𝕋 as an arbitrary nonempty closed subset of real numbers. The theoretical background for time scales can be found in [3].

In this paper, we use discrete time scales. To be exact, we define a discrete time scale 𝕋=𝑇(𝑡) as an arbitrary unbounded increasing sequence of real numbers, that is, 𝑇(𝑡)={𝑡𝑛}, where 𝑡𝑛, 𝑛𝑛0𝑘, 𝑛0, 𝑘>0 is an integer, 𝑡𝑛<𝑡𝑛+1, and lim𝑛𝑡𝑛=. For a fixed 𝑣𝑛0𝑘, we define a time scale 𝕋𝑣=𝑇𝑣(𝑡)={𝑡𝑛}, where 𝑛𝑣. Obviously, 𝑇𝑛0𝑘(𝑡)=𝑇(𝑡). In addition, for integers 𝑠, 𝑞, 𝑞𝑠𝑛0𝑘, we define the set 𝕋𝑞𝑠=𝑇𝑞𝑠(𝑡)={𝑡𝑠,𝑡𝑠+1,,𝑡𝑞}.

In the paper we study a dynamic equation 𝑡Δ𝑦𝑛𝑡=𝛽𝑛𝑦𝑡𝑛𝑗𝑡𝑦𝑛𝑘(1.1) as 𝑛. The difference is defined as usual: Δ𝑦(𝑡𝑛)=𝑦(𝑡𝑛+1)𝑦(𝑡𝑛), integers 𝑘 and 𝑗 in (1.1) satisfy the inequality 𝑘>𝑗0, and 𝛽𝕋+=(0,). Without loss of generality, we assume that 𝑡𝑛0𝑘>0 (this is a technical detail, necessary for some expressions to be well defined). Throughout the paper, we adopt the notation 𝑘𝑖=𝑘+1(𝑡𝑖)=0 where 𝑘 is an integer and denotes the function under consideration.

The results concern the asymptotic convergence of all solutions of (1.1). First we prove that, in the general case, the asymptotic convergence of all solutions is determined only by the existence of an increasing and bounded solution. Therefore, our effort is focused on developing criteria guaranteeing the existence of such solutions. The proofs of the results are based on comparing the solutions of (1.1) with those of an auxiliary inequality with the same left-hand and right-hand sides as in (1.1). We also illustrate general results using examples with particular time scales.

The problem concerning the asymptotic convergence of solutions in the continuous case, that is, in the case of delayed differential equations or other classes of equations, is a classical one and has attracted much attention recently (we refer, e.g., to the papers [411]).

The problem of the asymptotic convergence of solutions of discrete and difference equations with delay has not yet received much attention. Some recent results can be found, for example, in [1219].

Comparing the known investigations with the results presented, we can see that our results give sharp sufficient conditions of the asymptotic convergence of solutions. This is illustrated by examples. Nevertheless, we are not concerned with computing the limits of the solutions as 𝑛.

The paper is organized as follows. In Section 2, auxiliary definitions and results are collected. An auxiliary inequality is studied, and the relationship of its solutions with the solutions of (1.1) is derived. Section 3 contains results concerning the convergence of all solutions of (1.1). The criteria of existence of an increasing and convergent solution of (1.1) are established in Section 4. Examples illustrating the sharpness of the results derived are discussed as well.

2. Auxiliary Definitions and Results

Let 𝒞=𝒞(𝕋𝑛0𝑛0𝑘,) be the space of discrete functions mapping the discrete interval 𝕋𝑛0𝑛0𝑘 into . Let 𝑣𝑛0 be given. The function 𝑦𝕋𝑣𝑘 is said to be a solution of (1.1) on 𝕋𝑣𝑘 if it satisfies (1.1) for every 𝑛𝑣. A solution 𝑦 of (1.1) on 𝕋𝑣𝑘 is asymptotically convergent if the limit lim𝑛𝑦(𝑡𝑛) exists and is finite. For a given 𝑣𝑛0 and 𝜑𝒞, we say that 𝑦=𝑦(𝑡𝑣,𝜑) is a solution of (1.1) defined by the initial conditions (𝑡𝑣,𝜑) if 𝑦(𝑡𝑣,𝜑) is a solution of (1.1) on 𝕋𝑣𝑘 and 𝑦(𝑡𝑣,𝜑)(𝑡𝑣+𝑚)=𝜑(𝑡𝑚) for 𝑚0𝑘.

2.1. Auxiliary Inequality

The inequality𝑡Δ𝜔𝑛𝑡𝛽𝑛𝜔𝑡𝑛𝑗𝑡𝜔𝑛𝑘(2.1) is a helpful tool in the analysis of solutions of (1.1). Let 𝑣𝑛0. The function 𝜔𝕋𝑣𝑘 is said to be a solution of (2.1) on 𝕋𝑣𝑘 if 𝜔 satisfies (2.1) for 𝑛𝑣. A solution 𝜔 of (2.1) on 𝕋𝑣𝑘 is asymptotically convergent if the limit lim𝑛𝜔(𝑡𝑛) exists and is finite.

We give some properties of solutions of inequalities of type (2.1) to be used later on. We will also compare the solutions of (1.1) with those of (2.1).

Lemma 2.1. Let 𝜑𝒞 be increasing (nondecreasing, decreasing, nonincreasing) on 𝕋𝑛0𝑛0𝑘. Then the solution 𝑦(𝑛0,𝜑)(𝑡𝑛) of (1.1), where 𝑛𝑛0 is increasing (nondecreasing, decreasing, nonincreasing) on 𝕋𝑛0, too.

Lemma 2.2. Let 𝜑𝒞 be increasing (nondecreasing) and 𝜔𝕋 be a solution of inequality (2.1) with 𝜔(𝑡𝑚)=𝜑(𝑡𝑚), 𝑚𝑛0𝑛0𝑘. Then, 𝜔(𝑡𝑛), where 𝑛𝑛0 is increasing (nondecreasing).

The proofs of both lemmas above follow directly from the form of (1.1), (2.1), and from the properties 𝛽(𝑡𝑛)>0, 𝑛𝑛0𝑘, 𝑘>𝑗0.

Theorem 2.3. Let 𝜔𝕋 be a solution of (2.1) on 𝕋. Then there exists a solution 𝑦𝕋 of (1.1) on 𝕋 such that 𝑦𝑡𝑛𝑡𝜔𝑛(2.2) holds for every 𝑛𝑛0𝑘. In particular, a solution 𝑦(𝑛0,𝜙) of (1.1) with 𝜙𝒞, defined by 𝜙𝑡𝑛𝑡=𝜔𝑛,𝑛𝑛0𝑛0𝑘,(2.3) is such a solution.

Proof. Let 𝜔(𝑡𝑛) be a solution of (2.1) defined on 𝕋. We will show that the solution 𝑦(𝑡𝑛)=𝑦(𝑛0,𝜙)(𝑡𝑛) of (1.1) with 𝜙 defined by (2.3) satisfies (2.2), that is, 𝑦(𝑛0,𝜙)𝑡𝑛𝑡𝜔𝑛(2.4) for every 𝑛𝑛0𝑘. Let 𝑊𝕋 be defined by 𝑊𝑡𝑛𝑡=𝜔𝑛𝑡𝑦𝑛.(2.5) Then 𝑊(𝑡𝑛)=0 if 𝑛𝑛0𝑛0𝑘 and, in addition, 𝑊 is a solution of (2.1) on 𝕋. Lemma 2.2 implies that 𝑊 is nondecreasing. Consequently, 𝑊𝑡𝑛𝑡=𝜔𝑛𝑡𝑦𝑛𝑡𝑊𝑛0𝑡=𝜔𝑛0𝑡𝑦𝑛0=0,(2.6) and 𝑦(𝑡𝑛)𝜔(𝑡𝑛) for all 𝑛𝑛0.

2.2. A Solution of Inequality (2.1)

Now we will construct a solution of (2.1). The result obtained will help us obtain sufficient conditions for the existence of an increasing and asymptotically convergent solution of (1.1) (see Theorem 4.1 below).

Lemma 2.4. Let there exists a function 𝜀𝕋+ such that 𝜀𝑡𝑛+1𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖(2.7) for every 𝑛𝑛0. Then there exists a solution 𝜔=𝜔𝜀 of (2.1) defined on 𝕋 and having the form 𝜔𝜀𝑡𝑛=𝑛𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖.(2.8)

Proof. Assuming that 𝜔𝜀 defined by (2.8) is a solution of (2.1) for 𝑛𝑛0, we will deduce the inequality for 𝜀. We get Δ𝜔𝜀𝑡𝑛=𝜔𝜀𝑡𝑛+1𝜔𝜀𝑡𝑛=𝑛+1𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖𝑛𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖𝑡=𝛽𝑛𝜀𝑡𝑛+1,𝜔𝜀𝑡𝑛𝑗𝜔𝜀𝑡𝑛𝑘=𝑛𝑗𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖𝑛𝑘𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖=𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖.(2.9) We substitute 𝜔𝜀 for 𝜔 in (2.1). Then, using (2.9), (2.1) turns into 𝛽𝑡𝑛𝜀𝑡𝑛+1𝑡𝛽𝑛𝑛𝑗𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖.(2.10) Reducing the last inequality by 𝛽(𝑡𝑛), we obtain the desired inequality.

2.3. Decomposition of a Function into the Difference of Two Increasing Functions

It is well-known that every absolutely continuous function is representable as the difference of two increasing absolutely continuous functions [20, page 318]. We will need a simple analogue of this result on discrete time scales under consideration.

Lemma 2.5. Every function 𝜑𝒞 can be decomposed into the difference of two increasing functions 𝜑𝑗𝒞, 𝑗=1,2, that is, 𝜑𝑡𝑛=𝜑1𝑡𝑛𝜑2𝑡𝑛,𝑛𝑛0𝑛0𝑘.(2.11)

Proof. Let constants 𝑀𝑛>0, 𝑛𝑛0𝑛0𝑘 be such that 𝑀𝑛+1>𝑀𝑛𝑡+max0,𝜑𝑛𝑡𝜑𝑛+1(2.12) is valid for each 𝑛𝑛0𝑛10𝑘. We set 𝜑1𝑡𝑛𝑡=𝜑𝑛+𝑀𝑛,𝑛𝑛0𝑛0𝑘,𝜑2𝑡𝑛=𝑀𝑛,𝑛𝑛0𝑛0𝑘.(2.13) It is obvious that (2.11) holds. Now we verify that both functions 𝜑𝑗, 𝑗=1,2 are increasing. The first one should satisfy 𝜑1(𝑡𝑛+1)>𝜑1(𝑡𝑛) for 𝑛𝑛0𝑛10𝑘, which means that 𝜑𝑡𝑛+1+𝑀𝑛+1𝑡>𝜑𝑛+𝑀𝑛(2.14) or 𝑀𝑛+1>𝑀𝑛𝑡+𝜑𝑛𝑡𝜑𝑛+1.(2.15) We conclude that the last inequality holds because, due to (2.12), we have 𝑀𝑛+1>𝑀𝑛𝑡+max0,𝜑𝑛𝑡𝜑𝑛+1𝑀𝑛𝑡+𝜑𝑛𝑡𝜑𝑛+1.(2.16) The inequality 𝜑2(𝑡𝑛+1)>𝜑2(𝑡𝑛) obviously holds for every 𝑛𝑛0𝑛10𝑘 due to (2.12) as well.

2.4. Auxiliary Asymptotic Decomposition

The following lemma can be proved easily by induction. The symbol 𝒪 (capital “𝑂”) stands for the Landau order symbol.

Lemma 2.6. For fixed 𝑟, 𝜎{0}, the asymptotic representation (𝑛𝑟)𝜎=𝑛𝜎1𝜎𝑟𝑛1+𝒪𝑛2(2.17) holds for 𝑛.

3. Convergence of All Solutions

The main result of this part is the statement that the existence of an increasing and asymptotically convergent solution of (1.1) implies the asymptotical convergence of all solutions.

Theorem 3.1. If (1.1) has an increasing and asymptotically convergent solution on 𝑛0𝑘, then all the solutions of (1.1) defined on 𝑛0𝑘 are asymptotically convergent.

Proof. First we prove that every solution defined by a monotone initial function is convergent. We will assume that a monotone initial function 𝜑𝒞 is given. For definiteness, let 𝜑 be increasing or nondecreasing (the case when it is decreasing or nonincreasing can be considered in much the same way). By Lemma 2.1, the solution 𝑦(𝑛0,𝜑) is monotone, that is, it is either increasing or nondecreasing. We prove that 𝑦(𝑛0,𝜑) is convergent.
Denote the assumed increasing and asymptotically convergent solution of (1.1) as 𝑦=𝑌(𝑡𝑛), 𝑛𝑛0𝑘. Without loss of generality, we assume that 𝑦(𝑛0,𝜑)𝑌 on 𝑛0𝑘 since, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume 𝑡Δ𝑌𝑛>0,𝑛𝑛0𝑛10𝑘.(3.1) Hence, there is a constant 𝛾>0 such that 𝑡Δ𝑌𝑛𝑡𝛾Δ𝑦𝑛>0,𝑛𝑛0𝑛10𝑘(3.2) or Δ𝑌𝑡𝑛𝑡𝛾𝑦𝑛>0,𝑛𝑛0𝑛10𝑘.(3.3) This implies that the function 𝑌(𝑡𝑛)𝛾𝑦(𝑡𝑛) is increasing on 𝑛0𝑛10𝑘, and Lemma 2.1 implies that 𝑌(𝑡𝑛)𝛾𝑦(𝑡𝑛) is increasing on 𝑛0𝑘. Thus, 𝑌𝑡𝑛𝑡𝛾𝑦𝑛𝑡>𝑌𝑛0𝑡𝛾𝑦𝑛0,𝑛𝑛0(3.4) or 𝑦𝑡𝑛𝑡<𝑦𝑛0+1𝛾𝑌𝑡𝑛𝑡𝑌𝑛0,𝑛𝑛0(3.5) and, consequently, 𝑦(𝑡𝑛) is a bounded function on 𝑛0𝑘 because of the boundedness of 𝑌(𝑡𝑛). Obviously, in such a case, 𝑦(𝑡𝑛) is asymptotically convergent and has a finite limit.
Summarizing the previous section, we state that every monotone solution is convergent. It remains to consider a class of all nonmonotone initial functions. For the behavior of a solution 𝑦(𝑛0,𝜑) generated by a nonmonotone initial function 𝜑𝒞, there are two possibilities: 𝑦(𝑛0,𝜑) is either eventually monotone and, consequently, convergent, or 𝑦(𝑛0,𝜑) is eventually nonmonotone.
Now we use the statement of Lemma 2.5 that every discrete function 𝜑𝒞 can be decomposed into the difference of two increasing discrete functions 𝜑𝑗𝒞, 𝑗=1,2. In accordance with the previous part of the proof, every function 𝜑𝑗𝒞, 𝑗=1,2 defines an increasing and asymptotically convergent solution 𝑦(𝑛0,𝜑𝑗). Now it is clear that the solution 𝑦(𝑛0,𝜑) is asymptotically convergent.

From Theorem 3.1, it follows that a crucial property assuring the asymptotical convergence of all solutions of (1.1) is the existence of a strictly monotone and asymptotically convergent solution. In the next part, we will focus our attention on the relevant criteria. Now, in order to finish this section, we need an obvious statement concerning the asymptotic convergence. From Lemma 2.1 and Theorem 2.3, we immediately derive the following result.

Theorem 3.2. Let 𝜔 be an increasing and bounded solution of (2.1) on 𝕋. Then there exists an increasing and asymptotically convergent solution 𝑦 of (1.1) on 𝕋.

Combining the statements of Theorems 2.3, 3.1, and 3.2, we get a series of equivalent statements.

Theorem 3.3. The following three statements are equivalent. (a)Equation (1.1) has a strictly monotone and asymptotically convergent solution on 𝑛0𝑘.(b)All solutions of (1.1) defined on 𝑛0𝑘 are asymptotically convergent.(c)Inequality (2.1) has a strictly monotone and asymptotically convergent solution on 𝑛0𝑘.

4. Increasing Convergent Solutions of (1.1)

This part deals with the problem of detecting the existence of asymptotically convergent increasing solutions. We provide sufficient conditions for the existence of such solutions of (1.1).

The important theorem below is a consequence of Lemma 2.1, Theorem 2.3, and Lemma 2.4.

Theorem 4.1. Let there exists a function 𝜀𝕋+ satisfying 𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖𝜀𝑡<,𝑛+1𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖(4.1) for every 𝑛𝑛0. Then the initial function 𝜑𝑡𝑛=𝑛𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖,𝑛𝑛0𝑛0𝑘(4.2) defines an increasing and asymptotically convergent solution 𝑦(𝑡𝑛0,𝜑)(𝑡𝑛) of (1.1) on 𝕋 satisfying 𝑦(𝑡𝑛0,𝜑)𝑡𝑛𝑛𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖(4.3) for every 𝑛𝑛0.

Although Theorem 4.1 itself can serve as a source of various concrete criteria, later we will apply its following modification which can be used easily. Namely, assuming that 𝛽 in (1.1) can be estimated by a suitable function, we can deduce that (1.1) has an increasing asymptotically convergent solution. We consider such a case.

Theorem 4.2. Let there exist functions 𝛽𝕋+ and 𝜀𝕋+ such that the inequalities 𝛽𝑡𝑛𝛽𝑡𝑛𝜀𝑡,(4.4)𝑛+1𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖(4.5) hold for all 𝑛𝑛0𝑘, and moreover 𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖<.(4.6) Then there exists an increasing and asymptotically convergent solution 𝑦𝕋 of (1.1) satisfying 𝑦𝑡𝑛𝑛𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖(4.7) for every 𝑛𝑛0. Such a solution is defined, for example, by the initial function 𝜑𝑡𝑛=𝑛𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖,𝑛𝑛0𝑛0𝑘.(4.8)

Proof. From (4.5) and (4.6), we get 𝜀𝑡𝑛+1𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖,>𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖.(4.9) Then all assumptions of Theorem 4.1 are true. From its conclusion now follows the statement of Theorem 4.2.

4.1. Some Special Criteria

It will be demonstrated by examples that, in many applications, the function 𝛽 mentioned in Theorem 4.2 can have the form 𝛽𝑡𝑛𝑡=𝑐𝛾𝑛,(4.10) where 𝑐 is a positive constant and 𝛾𝕋+ is a suitable function such that 𝛾(𝑡𝑛)<𝑐 (at least for all sufficiently large 𝑛) and lim𝑛𝛾𝑡𝑛=0.(4.11) Below we carry on in this way and give sufficient conditions for the existence of increasing and asymptotically convergent solutions of (1.1) for general discrete time scale under consideration. For several special time scales, we derive such criteria in subsequent sections.

Theorem 4.3. Let there exist constants 𝑐>0, 𝑝>0 and 𝛼>0 such that 𝛽𝑡𝑛𝑝𝑐𝑡𝑛,1(4.12)𝑡𝛼𝑛+1𝑛𝑗𝑖=𝑛𝑘+1𝑝𝑐𝑡𝑖11𝑡𝛼𝑖(4.13) hold for all 𝑛𝑛0𝑘, and moreover 𝑖=𝑛0𝑘+11𝑡𝛼𝑖<.(4.14) Then there exists an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) satisfying 𝑦𝑡𝑛𝑛𝑖=𝑛0𝑘+1𝑝𝑐𝑡𝑖11𝑡𝛼𝑖(4.15) for every 𝑛𝑛0. Such a solution is defined, for example, by the initial function 𝜑𝑡𝑛=𝑛𝑖=𝑛0𝑘+1𝑝𝑐𝑡𝑖11𝑡𝛼𝑖,𝑛𝑛0𝑛0𝑘.(4.16)

Proof. We will apply Theorem 4.2 with 𝛽𝑡𝑛𝑝=𝑐𝑡𝑛𝑡,𝜀𝑛1=𝑡𝛼𝑛.(4.17) Inequality (4.5) turns into 𝜀𝑡𝑛+1=1𝑡𝛼𝑛+1𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖=𝑛𝑗𝑖=𝑛𝑘+1𝑝𝑐𝑡𝑖11𝑡𝛼𝑖(4.18) and is true due to (4.13). Inequality (4.6) holds due to assumption (4.14) as well because lim𝑛𝑡𝑛= and 𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖=𝑖=𝑛0𝑘+1𝑝𝑐𝑡𝑖11𝑡𝛼𝑖<.(4.19) Now, all assumptions of Theorem 4.2 are true, and its statement gives the statement of Theorem 4.3.

Theorem 4.4. Let there exist constants 𝑐>0, 𝑝>0 and 𝛼>0 such that the inequalities 𝛽𝑡𝑛𝑝𝑐ln𝑡𝑛,1(4.20)ln𝑡𝑛+1𝛼𝑛𝑗𝑖=𝑛𝑘+1𝑝𝑐ln𝑡𝑖11ln𝑡𝑖𝛼(4.21) hold for all 𝑛𝑛0𝑘, and moreover 𝑖=𝑛0𝑘+11ln𝑡𝑖𝛼<.(4.22) Then there exists an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) satisfying 𝑦𝑡𝑛𝑛𝑖=𝑛0𝑘+1𝑝𝑐ln𝑡𝑖11ln𝑡𝑖𝛼(4.23) for every 𝑛𝑛0. Such a solution is defined, for example, by the initial function 𝜑𝑡𝑛=𝑛𝑖=𝑛0𝑘+1𝑝𝑐ln𝑡𝑖11ln𝑡𝑖𝛼,𝑛𝑛0𝑛0𝑘.(4.24)

Proof. We will apply Theorem 4.2 with 𝛽𝑡𝑛𝑝=𝑐ln𝑡𝑛𝑡,𝜀𝑛1=ln𝑡n𝛼.(4.25) Inequality (4.5) turns into 𝜀𝑡𝑛+1=1ln𝑡𝑛+1𝛼𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖=𝑛𝑗𝑖=𝑛𝑘+1𝑝𝑐ln𝑡𝑖11ln𝑡𝑖𝛼(4.26) and is true due to (4.21). Inequality (4.6) holds due to assumption (4.22) as well because lim𝑛𝑡𝑛= and 𝑖=𝑛0𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖=𝑖=𝑛0𝑘+1𝑝𝑐ln𝑡𝑖11ln𝑡𝑖𝛼<.(4.27) Now, all assumptions of Theorem 4.2 are true, and its statement gives the statement of Theorem 4.4.

4.2. Time Scale 𝑇(𝑡)={𝑛(1+𝛿(𝑛))}

Now, using Theorem 4.3, we derive sufficient conditions for the existence of an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) in the case when the time scale is defined as 𝕋=𝑇(𝑡)={𝑡𝑛},𝑡𝑛=𝑛(1+𝛿(𝑛)), where 𝛿𝕋, |𝛿(𝑛)|𝛿, 𝛿(0,1), 𝑛𝑛0𝑘, and 1𝛿(𝑛)=𝒪𝑛2.(4.28)

Theorem 4.5. Let (4.12) be true for 1𝑐=𝑝𝑘𝑗,𝑝=(𝑘+𝑗+1),2(𝑘𝑗)(4.29) where 𝑝>1, that is, 𝛽𝑡𝑛1𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑡𝑛(4.30) holds for all 𝑛𝑛0𝑘. Let, moreover, 𝛼(1,𝑝). Then there exists an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) satisfying (4.15) for 𝑛𝑛0. Such a solution is defined, for example, by the initial function (4.16).

Proof. We use Theorem 4.3 and assume (without loss of generality) that 𝑛0 is sufficiently large for the asymptotic computations performed below to be correct. Let us verify that (4.13) holds. For the right-hand side (𝑡𝑛) of (4.13), we have 𝑡𝑛=𝑛𝑗𝑖=𝑛𝑘+11𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑡𝑖11𝑡𝛼𝑖=1𝑘𝑗𝑛𝑗𝑖=𝑛𝑘+11𝑡𝛼𝑖𝑝(𝑘+𝑗+1)2(𝑘𝑗)𝑛𝑗𝑖=𝑛𝑘+11𝑡𝑖1𝑡𝛼𝑖=1𝑘𝑗𝑛𝑗𝑖=𝑛𝑘+11𝑖𝛼(1+𝛿(𝑖))𝛼𝑝(𝑘+𝑗+1)2(𝑘𝑗)𝑛𝑗𝑖=𝑛𝑘+11(𝑖1)(1+𝛿(𝑖1))𝑖𝛼(1+𝛿(𝑖))𝛼.(4.31) Since 𝑖{𝑛𝑘+1,𝑛𝑘+2,,𝑛𝑗} and 𝑛, we can asymptotically decompose (𝑡𝑛) as 𝑛 using decomposition formula (2.17) in Lemma 2.6. Applying this formula to the term 𝑖𝛼 in the first sum with 𝜎=𝛼 and with 𝑟=𝑛𝑖, we get 1𝑖𝛼=1(𝑛(𝑛𝑖))𝛼=1𝑛𝛼1+𝛼(𝑛𝑖)𝑛1+𝒪𝑛2.(4.32) In addition to this, we have 1(1+𝛿(𝑖))𝛼1=1+𝒪𝑖21=1+𝒪𝑛2.(4.33) To estimate the second sum, we need only the first terms of the asymptotic decomposition and the order of accuracy, which can be computed easily without using Lemma 2.6. We also take into account that 1=1𝑖1=1𝑛(𝑛𝑖+1)𝑛1=11+(𝑛𝑖+1)/𝑛𝑛11+𝒪𝑛1,(4.34)11+𝛿(𝑖1)=1+𝒪(𝑖1)21=1+𝒪𝑛2.(4.35) Then we get 𝑡𝑛=1(𝑘𝑗)𝑛𝛼11+𝒪𝑛2𝑛𝑗𝑖=𝑛𝑘+11+𝛼(𝑛𝑖)𝑛1+𝒪𝑛2𝑝(𝑘+𝑗+1)2(𝑘𝑗)𝑛𝛼+111+𝒪𝑛𝑛𝑗𝑖=𝑛𝑘+111+𝒪𝑛=1(𝑘𝑗)𝑛𝛼1+𝛼(𝑘1)𝑛+1+𝛼(𝑘2)𝑛++1+𝛼𝑗𝑛1+𝒪𝑛2𝑝(𝑘+𝑗+1)2(𝑘𝑗)𝑛𝛼+111+1++1+𝒪𝑛=1(𝑘𝑗)𝑛𝛼+11(𝑘𝑗)𝑛+𝛼(𝑘1)+𝛼(𝑘2)++𝛼𝑗+𝒪𝑛𝑝(𝑘+𝑗+1)2(𝑘𝑗)𝑛𝛼+11(𝑘𝑗)+𝒪𝑛=1𝑛𝛼+𝛼(𝑘𝑗)𝑛𝛼+1(𝑘+𝑗1)(𝑘𝑗)2𝑝(𝑘+𝑗+1)2(𝑘𝑗)𝑛𝛼+11(𝑘𝑗)+𝒪𝑛𝛼+2,(4.36) and, finally, 𝑡𝑛=1𝑛𝛼+𝛼2𝑛𝛼+1𝑝(𝑘+𝑗1)(𝑘+𝑗+1)2𝑛𝛼+11+𝒪𝑛𝛼+2.(4.37) A similar decomposition of the left-hand side (𝑡𝑛) in (4.13) leads to (we use the decomposition formula (2.17) in Lemma 2.6 with 𝜎=𝛼 and 𝑟=1) 𝑡𝑛=1𝑡𝛼𝑛+1=1(𝑛+1)𝛼(1+𝛿(𝑛+1))𝛼=1𝑛𝛼𝛼1𝑛1+𝑂𝑛211+𝑂𝑛2=1𝑛𝛼𝛼𝑛𝛼+11+𝒪𝑛𝛼+2.(4.38) Comparing (𝑡𝑛) and (𝑡𝑛), we see that, for (𝑡𝑛)(𝑡𝑛), it is necessary to match the coefficients of the terms 𝑛𝛼1 because the coefficients of the terms 𝑛𝛼 are equal. It means that we need 1𝛼>21𝛼(𝑘+𝑗1)2𝑝(𝑘+𝑗+1).(4.39) Simplifying this inequality, we get 12𝑝1(𝑘+𝑗+1)>𝛼+2𝛼(𝑘+𝑗1),(4.40) and, finally, 𝑝>𝛼. This inequality is assumed, and therefore (4.13) that holds 𝑛0 is sufficiently large.
It remains to prove that (4.14) holds for 𝛼>1. But it is a well-known fact that the series 𝑖=𝑛0𝑘+11𝑡𝛼𝑖=𝑖=𝑛0𝑘+11𝑖𝛼(1+𝛿(𝑖))𝛼(4.41) is convergent for 𝛼>1.
Thus, all assumptions of Theorem 4.3 are fulfilled and, from the conclusions, we deduce that all conclusions of Theorem 4.5 hold.

4.3. Time Scale 𝑇(𝑡)={𝑛}

The time scale 𝕋=𝑇(𝑡)={𝑡𝑛},𝑡𝑛=𝑛, where 𝑛𝑛0𝑘 is a particular case of the previous time scale defined in Section 4.2 if 𝛿(𝑛)=0 for every 𝑛𝑛0𝑘. Then (1.1) turns into[]Δ𝑦(𝑛)=𝛽(𝑛)𝑦(𝑛𝑗)𝑦(𝑛𝑘)(4.42) and (4.30), which is crucial for the existence of an increasing and asymptotically convergent solution, takes the form𝛽1(𝑛)𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑛,𝑛𝑛0𝑘(4.43) with a 𝑝>1. Equation (4.42) has recently been considered in [12] and (4.43) coincides with (3.4) in [12, Theorem 3.3]. Thus, Theorem 4.5 can be viewed as a generalization of Theorem 3.3 in [12]. Moreover, using the following example, we will demonstrate that (4.43) is, in a sense, the best one.

Example 4.6. Consider (4.42), where 1𝛽(𝑛)=(𝑛+1)𝑛𝑗𝑖=𝑛𝑘+1.1/𝑖(4.44) It is easy to verify that (4.42) has a solution 𝑦(𝑛)=𝑛𝑖=11/𝑖, which is the 𝑛th partial sum of harmonic series and, therefore, is divergent as 𝑛. Now we asymptotically compare the function 𝛽 with the right-hand side of (4.43). First we develop an asymptotic decomposition of 𝛽 when 𝑛. We get 1𝛽(𝑛)=(𝑛+1)𝑛𝑗𝑖=𝑛𝑘+1=11/𝑖11+1/𝑛𝑘𝑗𝑖=1=1(1/(1+((𝑖𝑘)/𝑛)))11+1/𝑛𝑘𝑗𝑖=11(𝑖𝑘)/𝑛+𝒪1/𝑛2=11𝑛1+𝒪𝑛211𝑘𝑗1𝑘𝑗𝑖=1(𝑖𝑘)/(𝑘𝑗)𝑛+𝒪1/𝑛2=11𝑘𝑗1𝑛1+𝒪𝑛21+𝑘𝑗𝑖=1𝑖𝑘1(𝑘𝑗)𝑛+𝒪𝑛2=11𝑘𝑗1𝑛1+𝒪𝑛2𝑘1𝑛+𝑘𝑗+112𝑛+𝒪𝑛2=1𝑘𝑘𝑗1𝑛+𝑘𝑗+112𝑛𝑛1+𝒪𝑛2=1𝑘𝑗𝑘+𝑗+112(𝑘𝑗)𝑛+𝒪𝑛2.(4.45) Now, (4.43) requires that 𝛽1(𝑛)=𝑘𝑗𝑘+𝑗+112(𝑘𝑗)𝑛+𝒪𝑛21𝑝𝑘𝑗(𝑘+𝑗+1).2(𝑘𝑗)𝑛(4.46) The last will hold if 𝑘+𝑗+1𝑝2(𝑘𝑗)𝑛<(𝑘+𝑗+1),2(𝑘𝑗)𝑛(4.47) that is, if 𝑝<1. This inequality is the opposite to 𝑝>1 guaranteeing the existence of an increasing and asymptotically convergent solution. The example also shows that the criterion (4.43) is sharp in a sense. We end this part with a remark that Example 4.6 corrects the Example 4.4 in [12], where the case 𝑗=0 and 𝑘=1 was considered.

4.4. Time Scale 𝑇(𝑡)={𝑞𝑛}, 𝑞>1

We will focus our attention on the sufficient conditions for the existence of an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) if the time scale is defined as 𝕋=𝑇(𝑡)={𝑡𝑛},𝑡𝑛=𝑞𝑛, where 𝑛𝑛0𝑘 and 𝑞>1. We will apply Theorem 4.4.

Theorem 4.7. Let (4.20) hold for 1𝑐=𝑝𝑘𝑗,𝑝=(𝑘+𝑗+1)ln𝑞,2(𝑘𝑗)(4.48) where 𝑝>1, that is, the inequality 𝛽𝑡𝑛1𝑝𝑘𝑗(𝑘+𝑗+1)ln𝑞2(𝑘𝑗)ln𝑡𝑛=1𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑛(4.49) holds for all 𝑛𝑛0𝑘. Let, moreover, 𝛼(1,𝑝). Then there exists an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) satisfying (4.23) for 𝑛𝑛0. Such a solution is defined, for example, by the initial function (4.24).

Proof. We use Theorem 4.4 and assume (without loss of generality) that 𝑛0 is sufficiently large for the asymptotic computations performed below to be correct. Let us verify (4.21). For the right-hand side (𝑡𝑛) of (4.21), we have 𝑡𝑛=𝑛𝑗𝑖=𝑛𝑘+11𝑝𝑘𝑗(𝑘+𝑗+1)ln𝑞2(𝑘𝑗)ln𝑡𝑖11ln𝑡𝑖𝛼=1𝑘𝑗𝑛𝑗𝑖=𝑛𝑘+11ln𝑡𝑖𝛼𝑝(𝑘+𝑗+1)ln𝑞2(𝑘𝑗)𝑛𝑗𝑖=𝑛𝑘+11ln𝑡𝑖1ln𝑡𝑖𝛼=1𝑘𝑗𝑛𝑗𝑖=𝑛𝑘+11𝑖𝛼(ln𝑞)𝛼𝑝(𝑘+𝑗+1)ln𝑞2(𝑘𝑗)𝑛𝑗𝑖=𝑛𝑘+11(𝑖1)𝑖𝛼(ln𝑞)𝛼+1=1(𝑘𝑗)(ln𝑞)𝛼𝑛𝑗𝑖=𝑛𝑘+11𝑖𝛼𝑝(𝑘+𝑗+1)2(𝑘𝑗)(ln𝑞)𝛼𝑛𝑗𝑖=𝑛𝑘+11(𝑖1)𝑖𝛼==1weapplydecompositions(4.32)and(4.34)(𝑘𝑗)(ln𝑞)𝛼𝑛𝛼𝑛𝑗𝑖=𝑛𝑘+11+𝛼(𝑛𝑖)𝑛1+𝒪𝑛2𝑝(𝑘+𝑗+1)2(𝑘𝑗)(ln𝑞)𝛼𝑛𝛼+1𝑛𝑗𝑖=𝑛𝑘+111+𝒪𝑛.(4.50) Finally, applying some of the computations from the proof of Theorem 4.5, we get 𝑡𝑛=1(ln𝑞)𝛼𝑛𝛼+𝛼2(ln𝑞)𝛼𝑛𝛼+1𝑝(𝑘+𝑗1)(𝑘+𝑗+1)2(ln𝑞)𝛼𝑛𝛼+11+𝒪𝑛𝛼+2.(4.51) and, for the left-hand side (𝑡𝑛) of (4.20), 𝑡𝑛=1(ln𝑞)𝛼(𝑛+1)𝛼=1(ln𝑞)𝛼𝑛𝛼𝛼(ln𝑞)𝛼𝑛𝛼+11+𝒪𝑛𝛼+2.(4.52) Comparing (𝑡𝑛) and (𝑡𝑛), we see that, for (𝑡𝑛)(𝑡𝑛), 𝛼(ln𝑞)𝛼>𝛼(𝑘+𝑗1)2(ln𝑞)𝛼𝑝(𝑘+𝑗+1)2(ln𝑞)𝛼(4.53) is sufficient. Simplifying it, we get 𝑝(𝑘+𝑗+1)>𝛼(𝑘+𝑗+1),(4.54) and, finally, 𝑝>𝛼. This inequality is assumed, and therefore (4.21) is valid if 𝑛0 is sufficiently large.
It remains to prove that (4.22) holds for 𝛼>1. But it is a well-known fact that the series 𝑖=𝑛0𝑘+11ln𝑡𝑖𝛼=𝑖=𝑛0𝑘+11𝑖𝛼(ln𝑞)𝛼(4.55) is convergent for 𝛼>1.
Thus, all assumptions of Theorem 4.4 are true, and, from its conclusions, we deduce that all conclusions of Theorem 4.7 are true.

Example 4.8. Consider (1.1), where 𝛽𝑡𝑛1=(𝑛+1)𝑛𝑗𝑖=𝑛𝑘+1(ln𝑞)/ln𝑡𝑖=1(𝑛+1)𝑛𝑗𝑖=𝑛𝑘+1.1/𝑖(4.56) Then it is easy to verify that (1.1) has a solution 𝑦𝑡𝑛=𝑛𝑖=1ln𝑞ln𝑡𝑖=𝑛𝑖=11𝑖,(4.57) which is the 𝑛th partial sum of harmonic series and, as such, is divergent as 𝑛. Now we asymptotically compare the function 𝛽 with the right-hand side of (4.49). Proceeding as in Example 4.6, we get 𝛽𝑡𝑛=1𝑘𝑗𝑘+𝑗+112(𝑘𝑗)𝑛+𝒪𝑛2.(4.58) Inequality (4.49) is valid if 𝛽𝑡𝑛=1𝑘𝑗𝑘+𝑗+112(𝑘𝑗)𝑛+𝒪𝑛21𝑝𝑘𝑗(𝑘+𝑗+1),2(𝑘𝑗)𝑛(4.59) that is if 𝑝<1. This inequality is the opposite to 𝑝>1 guaranteeing the existence of an increasing and asymptotically convergent solution. Thus, the example also shows that criterion (4.49) is sharp in a sense.

4.5. A General Criterion for the Existence of an Increasing and Asymptotically Convergent Solution

Analysing two criteria for the existence of an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) expressed by (4.12) and (4.20), that is, by inequalities 𝛽𝑡𝑛𝑝𝑐𝑡𝑛,𝛽𝑡𝑛𝑝𝑐ln𝑡𝑛(4.60) with suitable constants 𝑐 and 𝑝, we can state the following. The first criterion (4.12) can successfully be used, for example, for the time scale 𝑇(𝑡)={𝑡𝑛}, where 𝑡𝑛=𝑛. In this case, as stated in Theorem 4.5, (4.30), that is, 𝛽𝑡𝑛1𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑡𝑛=1𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑛(4.61) is assumed with a 𝑝>1.

The second criterion (4.20) can successfully be used, for example, for the time scale 𝑇(𝑡)={𝑡𝑛} where 𝑡𝑛=𝑞𝑛 and 𝑞>1. Then, as stated in Theorem 4.7, (4.49), that is, 𝛽𝑡𝑛1𝑝𝑘𝑗(𝑘+𝑗+1)ln𝑞2(𝑘𝑗)ln𝑡𝑛=1𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑛(4.62) is assumed with a 𝑝>1. Comparing (4.61) and (4.62), we see that, although their left-hand sides are different due to different meaning of 𝑡𝑛 in every case, their right-hand sides are identical.

The following result gives a criterion for every discrete time scale 𝑇(𝑡)={𝑡𝑛} with properties described in introduction.

Theorem 4.9. Let 𝛽𝑡𝑛1𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑛(4.63) holds for all 𝑛𝑛0𝑘 and for a fixed 𝑝>1. Let, moreover, 𝛼(1,𝑝). Then there exists an increasing and asymptotically convergent solution 𝑦𝕋+ of (1.1) satisfying 𝑦𝑡𝑛𝑛𝑖=𝑛0𝑘+11𝑝𝑘𝑗(𝑘+𝑗+1)12(𝑘𝑗)(𝑖1)𝑖𝛼(4.64) for every 𝑛𝑛0. Such a solution is defined, for example, by the initial function 𝜑𝑡𝑛=𝑛𝑖=𝑛0𝑘+11𝑝𝑘𝑗(𝑘+𝑗+1)12(𝑘𝑗)(𝑖1)𝑖𝛼,𝑛𝑛0𝑛0𝑘.(4.65)

Proof. We will apply Theorem 4.2 with 𝛽𝑡𝑛1=𝑝𝑘𝑗(𝑘+𝑗+1)𝑡2(𝑘𝑗)𝑛,𝜀𝑛1=𝑛𝛼.(4.66) Inequality (4.5) turns into 𝜀𝑡𝑛+1=1(𝑛+1)𝛼𝑛𝑗𝑖=𝑛𝑘+1𝛽𝑡𝑖1𝜀𝑡𝑖=𝑛𝑗𝑖=𝑛𝑘+11𝑝𝑘𝑗(𝑘+𝑗+1)12(𝑘𝑗)(𝑖1)𝑖𝛼.(4.67) Asymptotic decompositions of the left-hand and right-hand sides were used in the proof of Theorem 4.5 (if 𝛿(𝑛)=0, i.e., 𝑡𝑛=𝑛 for every 𝑛𝑛0𝑘) and a similar decomposition was used in the proof of Theorem 4.7. Therefore, we will not repeat it. We will only state that the above inequality holds for 𝑝>𝛼. (4.6) holds as well because the series 𝑖=𝑛0𝑘+11𝑝𝑘𝑗(𝑘+𝑗+1)12(𝑘𝑗)(𝑖1)𝑖𝛼(4.68) is obviously convergent.

Remark 4.10. Although Theorem 4.9 is a general result, it has a disadvantage in applications because of its implicit character. Unlike (4.61) and (4.62), where the left-hand and middle parts are explicitly expressed in terms of 𝑡𝑛, the right-hand side of the crucial inequality (4.63) cannot, in a general situation of arbitrary time scale {𝑡𝑛}, be explicitly expressed using only the 𝑡𝑛 terms. This is only possible if, for a given time scale, a function 𝑓 is explicitly known such that 𝑓(𝑡𝑛)=𝑛. Then, (4.63) can be written in the form 𝛽𝑡𝑛1𝑝𝑘𝑗(𝑘+𝑗+1)𝑡2(𝑘𝑗)𝑓𝑛=1𝑝𝑘𝑗(𝑘+𝑗+1).2(𝑘𝑗)𝑛(4.69)

Remark 4.11. On the other hand, in a sense, Theorem 4.9 gives the best possible result. Indeed, (1.1) with 𝛽𝑡𝑛1=(𝑛+1)𝑛𝑗𝑖=𝑛𝑘+11/𝑖(4.70) has an increasing asymptotically divergent solution 𝑦(𝑡𝑛)=𝑛𝑖=11/𝑖. An asymptotic decomposition of the right-hand side of (4.70) was performed in Example 4.6 and an increasing and asymptotically convergent solution exists if (4.63), that is, 𝛽𝑡𝑛=1(𝑛+1)𝑛𝑗𝑖=𝑛𝑘+111/𝑖𝑝𝑘𝑗(𝑘+𝑗+1)2(𝑘𝑗)𝑛(4.71) holds, or if 𝛽𝑡𝑛=1𝑘𝑗𝑘+𝑗+112(𝑘𝑗)𝑛+𝒪𝑛21𝑝𝑘𝑗(𝑘+𝑗+1).2(𝑘𝑗)𝑛(4.72) The last holds for 𝑝<1. This inequality is the opposite to 𝑝>1 guaranteeing the existence of an increasing and asymptotically convergent solution. Thus, the example shows that our general criterion is sharp in a sense.

Acknowledgments

This research was supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague), by the project FEKT-S-11-2(921) and by the Council of Czech Government MSM 00216 30503. M. Růžičková and Z. Suta were supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).