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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 580750, 20 pages
http://dx.doi.org/10.1155/2012/580750
Research Article

Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic
3Department of Mathematics, University of Ε½ilina, 01026 Ε½ilina, Slovakia

Received 10 September 2011; Accepted 12 November 2011

Academic Editor: TokaΒ Diagana

Copyright Β© 2012 J. DiblΓ­k et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper investigates a dynamic equation Δ𝑦(𝑑𝑛)=𝛽(𝑑𝑛)[𝑦(π‘‘π‘›βˆ’π‘—)βˆ’π‘¦(π‘‘π‘›βˆ’π‘˜)] for π‘›β†’βˆž, where π‘˜ and 𝑗 are integers such that π‘˜>𝑗β‰₯0, on an arbitrary discrete time scale π•‹βˆΆ={𝑑𝑛} with π‘‘π‘›βˆˆβ„, π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜={𝑛0βˆ’π‘˜,𝑛0βˆ’π‘˜+1,…}, 𝑛0βˆˆβ„•, 𝑑𝑛<𝑑𝑛+1, Δ𝑦(𝑑𝑛)=𝑦(𝑑𝑛+1)βˆ’π‘¦(𝑑𝑛), and limπ‘›β†’βˆžπ‘‘π‘›=∞. We assume π›½βˆΆπ•‹β†’(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 π‘›β†’βˆž. 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 β„€βˆžπ‘ βˆΆ={𝑠,𝑠+1,…}, and if an integer π‘žβ‰₯𝑠, we define β„€π‘žπ‘ βˆΆ={𝑠,𝑠+1,…,π‘ž}.

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(𝑑) as an arbitrary unbounded increasing sequence of real numbers, that is, 𝑇(𝑑)∢={𝑑𝑛}, where π‘‘π‘›βˆˆβ„, π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, 𝑛0βˆˆβ„•, π‘˜>0 is an integer, 𝑑𝑛<𝑑𝑛+1, and limπ‘›β†’βˆžπ‘‘π‘›=∞. For a fixed π‘£βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, we define a time scale 𝕋𝑣=𝑇𝑣(𝑑)∢={𝑑𝑛}, where π‘›βˆˆβ„€βˆžπ‘£. Obviously, 𝑇𝑛0βˆ’π‘˜(𝑑)=𝑇(𝑑). In addition, for integers 𝑠, π‘ž, π‘žβ‰₯𝑠β‰₯𝑛0βˆ’π‘˜, we define the set π•‹π‘žπ‘ =π‘‡π‘žπ‘ (𝑑)∢={𝑑𝑠,𝑑𝑠+1,…,π‘‘π‘ž}.

In the paper we study a dynamic equation 𝑑Δ𝑦𝑛𝑑=π›½π‘›π‘¦ξ€·π‘‘ξ€Έξ€Ίπ‘›βˆ’π‘—ξ€Έξ€·π‘‘βˆ’π‘¦π‘›βˆ’π‘˜ξ€Έξ€»(1.1) as π‘›β†’βˆž. The difference is defined as usual: Δ𝑦(𝑑𝑛)∢=𝑦(𝑑𝑛+1)βˆ’π‘¦(𝑑𝑛), integers π‘˜ and 𝑗 in (1.1) satisfy the inequality π‘˜>𝑗β‰₯0, and π›½βˆΆπ•‹β†’β„+∢=(0,∞). Without loss of generality, we assume that 𝑑𝑛0βˆ’π‘˜>0 (this is a technical detail, necessary for some expressions to be well defined). Throughout the paper, we adopt the notation βˆ‘π‘˜π‘–=π‘˜+1ℬ(𝑑𝑖)=0 where π‘˜ is an integer and ℬ 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 [4–11]).

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 [12–19].

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 π‘›β†’βˆž.

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 π’žβˆΆ=π’ž(𝕋𝑛0𝑛0βˆ’π‘˜,ℝ) be the space of discrete functions mapping the discrete interval 𝕋𝑛0𝑛0βˆ’π‘˜ into ℝ. Let π‘£βˆˆβ„€βˆžπ‘›0 be given. The function π‘¦βˆΆπ•‹π‘£βˆ’π‘˜β†’β„ is said to be a solution of (1.1) on π•‹π‘£βˆ’π‘˜ if it satisfies (1.1) for every π‘›βˆˆβ„€βˆžπ‘£. A solution 𝑦 of (1.1) on π•‹π‘£βˆ’π‘˜ is asymptotically convergent if the limit limπ‘›β†’βˆžπ‘¦(𝑑𝑛) exists and is finite. For a given π‘£βˆˆβ„€βˆžπ‘›0 and πœ‘βˆˆπ’ž, we say that 𝑦=𝑦(𝑑𝑣,πœ‘) is a solution of (1.1) defined by the initial conditions (𝑑𝑣,πœ‘) if 𝑦(𝑑𝑣,πœ‘) is a solution of (1.1) on π•‹π‘£βˆ’π‘˜ and 𝑦(𝑑𝑣,πœ‘)(𝑑𝑣+π‘š)=πœ‘(π‘‘π‘š) for π‘šβˆˆβ„€0βˆ’π‘˜.

2.1. Auxiliary Inequality

The inequalityξ€·π‘‘Ξ”πœ”π‘›ξ€Έξ€·π‘‘β‰₯π›½π‘›πœ”ξ€·π‘‘ξ€Έξ€Ίπ‘›βˆ’π‘—ξ€Έξ€·π‘‘βˆ’πœ”π‘›βˆ’π‘˜ξ€Έξ€»(2.1) is a helpful tool in the analysis of solutions of (1.1). Let π‘£βˆˆβ„€βˆžπ‘›0. The function πœ”βˆΆπ•‹π‘£βˆ’π‘˜β†’β„ is said to be a solution of (2.1) on π•‹π‘£βˆ’π‘˜ if πœ” satisfies (2.1) for π‘›βˆˆβ„€βˆžπ‘£. A solution πœ” of (2.1) on π•‹π‘£βˆ’π‘˜ is asymptotically convergent if the limit limπ‘›β†’βˆžπœ”(𝑑𝑛) 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 πœ‘βˆˆπ’ž be increasing (nondecreasing, decreasing, nonincreasing) on 𝕋𝑛0𝑛0βˆ’π‘˜. Then the solution 𝑦(𝑛0,πœ‘)(𝑑𝑛) of (1.1), where π‘›βˆˆβ„€βˆžπ‘›0 is increasing (nondecreasing, decreasing, nonincreasing) on 𝕋𝑛0, too.

Lemma 2.2. Let πœ‘βˆˆπ’ž be increasing (nondecreasing) and πœ”βˆΆπ•‹β†’β„ be a solution of inequality (2.1) with πœ”(π‘‘π‘š)=πœ‘(π‘‘π‘š), π‘šβˆˆβ„€π‘›0𝑛0βˆ’π‘˜. Then, πœ”(𝑑𝑛), where π‘›βˆˆβ„€βˆžπ‘›0 is increasing (nondecreasing).

The proofs of both lemmas above follow directly from the form of (1.1), (2.1), and from the properties 𝛽(𝑑𝑛)>0, π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, π‘˜>𝑗β‰₯0.

Theorem 2.3. Let πœ”βˆΆπ•‹β†’β„ be a solution of (2.1) on 𝕋. Then there exists a solution π‘¦βˆΆπ•‹β†’β„ of (1.1) on 𝕋 such that π‘¦ξ€·π‘‘π‘›ξ€Έξ€·π‘‘β‰€πœ”π‘›ξ€Έ(2.2) holds for every π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜. In particular, a solution 𝑦(𝑛0,πœ™) of (1.1) with πœ™βˆˆπ’ž, defined by πœ™ξ€·π‘‘π‘›ξ€Έξ€·π‘‘βˆΆ=πœ”π‘›ξ€Έ,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜,(2.3) is such a solution.

Proof. Let πœ”(𝑑𝑛) be a solution of (2.1) defined on 𝕋. We will show that the solution 𝑦(𝑑𝑛)∢=𝑦(𝑛0,πœ™)(𝑑𝑛) of (1.1) with πœ™ defined by (2.3) satisfies (2.2), that is, 𝑦(𝑛0,πœ™)ξ€·π‘‘π‘›ξ€Έξ€·π‘‘β‰€πœ”π‘›ξ€Έ(2.4) for every π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜. Let π‘ŠβˆΆπ•‹β†’β„ be defined by π‘Šξ€·π‘‘π‘›ξ€Έξ€·π‘‘βˆΆ=πœ”π‘›ξ€Έξ€·π‘‘βˆ’π‘¦π‘›ξ€Έ.(2.5) Then π‘Š(𝑑𝑛)=0 if π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜ and, in addition, π‘Š is a solution of (2.1) on 𝕋. Lemma 2.2 implies that π‘Š is nondecreasing. Consequently, π‘Šξ€·π‘‘π‘›ξ€Έξ€·π‘‘=πœ”π‘›ξ€Έξ€·π‘‘βˆ’π‘¦π‘›ξ€Έξ€·π‘‘β‰₯π‘Šπ‘›0𝑑=πœ”π‘›0ξ€Έξ€·π‘‘βˆ’π‘¦π‘›0ξ€Έ=0,(2.6) and 𝑦(𝑑𝑛)β‰€πœ”(𝑑𝑛) for all 𝑛β‰₯𝑛0.

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 πœ€βˆΆπ•‹β†’β„+ such that πœ€ξ€·π‘‘π‘›+1ξ€Έβ‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ(2.7) for every π‘›βˆˆβ„€βˆžπ‘›0. Then there exists a solution πœ”=πœ”πœ€ of (2.1) defined on 𝕋 and having the form πœ”πœ€ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑛𝑖=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ.(2.8)

Proof. Assuming that πœ”πœ€ defined by (2.8) is a solution of (2.1) for π‘›βˆˆβ„€βˆžπ‘›0, we will deduce the inequality for πœ€. We get Ξ”πœ”πœ€ξ€·π‘‘π‘›ξ€Έ=πœ”πœ€ξ€·π‘‘π‘›+1ξ€Έβˆ’πœ”πœ€ξ€·π‘‘π‘›ξ€Έ=𝑛+1βˆ‘π‘–=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έβˆ’π‘›βˆ‘π‘–=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έξ€·π‘‘=π›½π‘›ξ€Έπœ€ξ€·π‘‘π‘›+1ξ€Έ,πœ”πœ€ξ€·π‘‘π‘›βˆ’π‘—ξ€Έβˆ’πœ”πœ€ξ€·π‘‘π‘›βˆ’π‘˜ξ€Έ=π‘›βˆ’π‘—ξ“π‘–=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έβˆ’π‘›βˆ’π‘˜ξ“π‘–=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ=π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ.(2.9) We substitute πœ”πœ€ for πœ” in (2.1). Then, using (2.9), (2.1) turns into π›½ξ€·π‘‘π‘›ξ€Έπœ€ξ€·π‘‘π‘›+1𝑑β‰₯π›½π‘›ξ€Έπ‘›βˆ’π‘—ξ“π‘›βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ.(2.10) Reducing the last inequality by 𝛽(𝑑𝑛), 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 πœ‘βˆˆπ’ž can be decomposed into the difference of two increasing functions πœ‘π‘—βˆˆπ’ž, 𝑗=1,2, that is, πœ‘ξ€·π‘‘π‘›ξ€Έ=πœ‘1ξ€·π‘‘π‘›ξ€Έβˆ’πœ‘2𝑑𝑛,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜.(2.11)

Proof. Let constants 𝑀𝑛>0, π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜ be such that 𝑀𝑛+1>𝑀𝑛𝑑+max0,πœ‘π‘›ξ€Έξ€·π‘‘βˆ’πœ‘π‘›+1ξ€Έξ€Ύ(2.12) is valid for each π‘›βˆˆβ„€π‘›0π‘›βˆ’10βˆ’π‘˜. We set πœ‘1ξ€·π‘‘π‘›ξ€Έξ€·π‘‘βˆΆ=πœ‘π‘›ξ€Έ+𝑀𝑛,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜,πœ‘2ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑀𝑛,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜.(2.13) It is obvious that (2.11) holds. Now we verify that both functions πœ‘π‘—, 𝑗=1,2 are increasing. The first one should satisfy πœ‘1(𝑑𝑛+1)>πœ‘1(𝑑𝑛) for π‘›βˆˆβ„€π‘›0π‘›βˆ’10βˆ’π‘˜, which means that πœ‘ξ€·π‘‘π‘›+1ξ€Έ+𝑀𝑛+1𝑑>πœ‘π‘›ξ€Έ+𝑀𝑛(2.14) or 𝑀𝑛+1>𝑀𝑛𝑑+πœ‘π‘›ξ€Έξ€·π‘‘βˆ’πœ‘π‘›+1ξ€Έ.(2.15) We conclude that the last inequality holds because, due to (2.12), we have 𝑀𝑛+1>𝑀𝑛𝑑+max0,πœ‘π‘›ξ€Έξ€·π‘‘βˆ’πœ‘π‘›+1ξ€Έξ€Ύβ‰₯𝑀𝑛𝑑+πœ‘π‘›ξ€Έξ€·π‘‘βˆ’πœ‘π‘›+1ξ€Έ.(2.16) The inequality πœ‘2(𝑑𝑛+1)>πœ‘2(𝑑𝑛) obviously holds for every π‘›βˆˆβ„€π‘›0π‘›βˆ’10βˆ’π‘˜ due to (2.12) as well.

2.4. Auxiliary Asymptotic Decomposition

The following lemma can be proved easily by induction. The symbol π’ͺ (capital β€œπ‘‚β€) stands for the Landau order symbol.

Lemma 2.6. For fixed π‘Ÿ, πœŽβˆˆβ„β§΅{0}, the asymptotic representation (π‘›βˆ’π‘Ÿ)𝜎=π‘›πœŽξ‚ƒ1βˆ’πœŽπ‘Ÿπ‘›ξ‚€1+π’ͺ𝑛2(2.17) holds for π‘›β†’βˆž.

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 β„€βˆžπ‘›0βˆ’π‘˜, then all the solutions of (1.1) defined on β„€βˆžπ‘›0βˆ’π‘˜ 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 πœ‘βˆˆπ’ž is given. For definiteness, let πœ‘ 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,πœ‘) is monotone, that is, it is either increasing or nondecreasing. We prove that 𝑦(𝑛0,πœ‘) is convergent.
Denote the assumed increasing and asymptotically convergent solution of (1.1) as 𝑦=π‘Œ(𝑑𝑛), π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜. Without loss of generality, we assume that 𝑦(𝑛0,πœ‘)β‰’π‘Œ on β„€βˆžπ‘›0βˆ’π‘˜ since, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume ξ€·π‘‘Ξ”π‘Œπ‘›ξ€Έ>0,π‘›βˆˆβ„€π‘›0π‘›βˆ’10βˆ’π‘˜.(3.1) Hence, there is a constant 𝛾>0 such that ξ€·π‘‘Ξ”π‘Œπ‘›ξ€Έξ€·π‘‘βˆ’π›ΎΞ”π‘¦π‘›ξ€Έ>0,π‘›βˆˆβ„€π‘›0π‘›βˆ’10βˆ’π‘˜(3.2) or Ξ”ξ€·π‘Œξ€·π‘‘π‘›ξ€Έξ€·π‘‘βˆ’π›Ύπ‘¦π‘›ξ€Έξ€Έ>0,π‘›βˆˆβ„€π‘›0π‘›βˆ’10βˆ’π‘˜.(3.3) This implies that the function π‘Œ(𝑑𝑛)βˆ’π›Ύπ‘¦(𝑑𝑛) is increasing on ℀𝑛0π‘›βˆ’10βˆ’π‘˜, and Lemma 2.1 implies that π‘Œ(𝑑𝑛)βˆ’π›Ύπ‘¦(𝑑𝑛) is increasing on β„€βˆžπ‘›0βˆ’π‘˜. Thus, π‘Œξ€·π‘‘π‘›ξ€Έξ€·π‘‘βˆ’π›Ύπ‘¦π‘›ξ€Έξ€·π‘‘>π‘Œπ‘›0ξ€Έξ€·π‘‘βˆ’π›Ύπ‘¦π‘›0ξ€Έ,π‘›βˆˆβ„€βˆžπ‘›0(3.4) or 𝑦𝑑𝑛𝑑<𝑦𝑛0ξ€Έ+1π›Ύξ€·π‘Œξ€·π‘‘π‘›ξ€Έξ€·π‘‘βˆ’π‘Œπ‘›0ξ€Έξ€Έ,π‘›βˆˆβ„€βˆžπ‘›0(3.5) and, consequently, 𝑦(𝑑𝑛) is a bounded function on β„€βˆžπ‘›0βˆ’π‘˜ because of the boundedness of π‘Œ(𝑑𝑛). Obviously, in such a case, 𝑦(𝑑𝑛) 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,πœ‘) generated by a nonmonotone initial function πœ‘βˆˆπ’ž, there are two possibilities: 𝑦(𝑛0,πœ‘) is either eventually monotone and, consequently, convergent, or 𝑦(𝑛0,πœ‘) is eventually nonmonotone.
Now we use the statement of Lemma 2.5 that every discrete function πœ‘βˆˆπ’ž can be decomposed into the difference of two increasing discrete functions πœ‘π‘—βˆˆπ’ž, 𝑗=1,2. In accordance with the previous part of the proof, every function πœ‘π‘—βˆˆπ’ž, 𝑗=1,2 defines an increasing and asymptotically convergent solution 𝑦(𝑛0,πœ‘π‘—). Now it is clear that the solution 𝑦(𝑛0,πœ‘) 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 πœ” 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 β„€βˆžπ‘›0βˆ’π‘˜.(b)All solutions of (1.1) defined on β„€βˆžπ‘›0βˆ’π‘˜ are asymptotically convergent.(c)Inequality (2.1) has a strictly monotone and asymptotically convergent solution on β„€βˆžπ‘›0βˆ’π‘˜.

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 πœ€βˆΆπ•‹β†’β„+ satisfying βˆžξ“π‘–=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έπœ€ξ€·π‘‘<∞,𝑛+1ξ€Έβ‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ(4.1) for every π‘›βˆˆβ„€βˆžπ‘›0. Then the initial function πœ‘ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑛𝑖=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜(4.2) defines an increasing and asymptotically convergent solution 𝑦(𝑑𝑛0,πœ‘)(𝑑𝑛) of (1.1) on 𝕋 satisfying 𝑦(𝑑𝑛0,πœ‘)𝑑𝑛≀𝑛𝑖=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ(4.3) for every π‘›βˆˆβ„€βˆžπ‘›0.

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 π›½βˆ—βˆΆπ•‹β†’β„+ and πœ€βˆΆπ•‹β†’β„+ such that the inequalities π›½ξ€·π‘‘π‘›ξ€Έβ‰€π›½βˆ—ξ€·π‘‘π‘›ξ€Έπœ€ξ€·π‘‘,(4.4)𝑛+1ξ€Έβ‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ(4.5) hold for all π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, and moreover βˆžξ“π‘–=𝑛0βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ<∞.(4.6) Then there exists an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„ of (1.1) satisfying 𝑦𝑑𝑛≀𝑛𝑖=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ(4.7) for every π‘›βˆˆβ„€βˆžπ‘›0. Such a solution is defined, for example, by the initial function πœ‘ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑛𝑖=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜.(4.8)

Proof. From (4.5) and (4.6), we get πœ€ξ€·π‘‘π‘›+1ξ€Έβ‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έβ‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ,∞>βˆžξ“π‘–=𝑛0βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έβ‰₯βˆžξ“π‘–=𝑛0βˆ’π‘˜+1π›½ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ.(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 π›½βˆ— mentioned in Theorem 4.2 can have the form π›½βˆ—ξ€·π‘‘π‘›ξ€Έξ€·π‘‘=π‘βˆ’π›Ύπ‘›ξ€Έ,(4.10) where 𝑐 is a positive constant and π›ΎβˆΆπ•‹β†’β„+ is a suitable function such that 𝛾(𝑑𝑛)<𝑐 (at least for all sufficiently large 𝑛) and limπ‘›β†’βˆžπ›Ύξ€·π‘‘π‘›ξ€Έ=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 π›½ξ€·π‘‘π‘›ξ€Έπ‘β‰€π‘βˆ’π‘‘π‘›,1(4.12)𝑑𝛼𝑛+1β‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’π‘‘π‘–βˆ’1ξ‚Ή1𝑑𝛼𝑖(4.13) hold for all π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, and moreover βˆžξ“π‘–=𝑛0βˆ’π‘˜+11𝑑𝛼𝑖<∞.(4.14) Then there exists an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) satisfying 𝑦𝑑𝑛≀𝑛𝑖=𝑛0βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’π‘‘π‘–βˆ’1ξ‚Ή1𝑑𝛼𝑖(4.15) for every π‘›βˆˆβ„€βˆžπ‘›0. Such a solution is defined, for example, by the initial function πœ‘ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑛𝑖=𝑛0βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’π‘‘π‘–βˆ’1ξ‚Ή1𝑑𝛼𝑖,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜.(4.16)

Proof. We will apply Theorem 4.2 with π›½βˆ—ξ€·π‘‘π‘›ξ€Έπ‘βˆΆ=π‘βˆ’π‘‘π‘›ξ€·π‘‘,πœ€π‘›ξ€Έ1∢=𝑑𝛼𝑛.(4.17) Inequality (4.5) turns into πœ€ξ€·π‘‘π‘›+1ξ€Έ=1𝑑𝛼𝑛+1β‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ=π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’π‘‘π‘–βˆ’1ξ‚Ή1𝑑𝛼𝑖(4.18) and is true due to (4.13). Inequality (4.6) holds due to assumption (4.14) as well because limπ‘›β†’βˆžπ‘‘π‘›=∞ and βˆžξ“π‘–=𝑛0βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ=βˆžξ“π‘–=𝑛0βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’π‘‘π‘–βˆ’1ξ‚Ή1𝑑𝛼𝑖<∞.(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 π›½ξ€·π‘‘π‘›ξ€Έπ‘β‰€π‘βˆ’ln𝑑𝑛,1(4.20)ξ€·ln𝑑𝑛+1𝛼β‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’lnπ‘‘π‘–βˆ’1ξ‚Ή1ξ€·ln𝑑𝑖𝛼(4.21) hold for all π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, and moreover βˆžξ“π‘–=𝑛0βˆ’π‘˜+11ξ€·ln𝑑𝑖𝛼<∞.(4.22) Then there exists an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) satisfying 𝑦𝑑𝑛≀𝑛𝑖=𝑛0βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’lnπ‘‘π‘–βˆ’1ξ‚Ή1ξ€·ln𝑑𝑖𝛼(4.23) for every π‘›βˆˆβ„€βˆžπ‘›0. Such a solution is defined, for example, by the initial function πœ‘ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑛𝑖=𝑛0βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’lnπ‘‘π‘–βˆ’1ξ‚Ή1ξ€·ln𝑑𝑖𝛼,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜.(4.24)

Proof. We will apply Theorem 4.2 with π›½βˆ—ξ€·π‘‘π‘›ξ€Έπ‘βˆΆ=π‘βˆ’ln𝑑𝑛𝑑,πœ€π‘›ξ€Έ1∢=ξ€·ln𝑑n𝛼.(4.25) Inequality (4.5) turns into πœ€ξ€·π‘‘π‘›+1ξ€Έ=1ξ€·ln𝑑𝑛+1𝛼β‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ=π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’lnπ‘‘π‘–βˆ’1ξ‚Ή1ξ€·ln𝑑𝑖𝛼(4.26) and is true due to (4.21). Inequality (4.6) holds due to assumption (4.22) as well because limπ‘›β†’βˆžπ‘‘π‘›=∞ and βˆžξ“π‘–=𝑛0βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ=βˆžξ“π‘–=𝑛0βˆ’π‘˜+1ξ‚Έπ‘π‘βˆ’lnπ‘‘π‘–βˆ’1ξ‚Ή1ξ€·ln𝑑𝑖𝛼<∞.(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(𝑑)∢={𝑛(1+𝛿(𝑛))}

Now, using Theorem 4.3, we derive sufficient conditions for the existence of an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) in the case when the time scale is defined as 𝕋=𝑇(𝑑)={𝑑𝑛},π‘‘π‘›βˆΆ=𝑛(1+𝛿(𝑛)), where π›ΏβˆΆπ•‹β†’β„, |𝛿(𝑛)|β‰€π›Ώβˆ—, π›Ώβˆ—βˆˆ(0,1), π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜, and ξ‚€1𝛿(𝑛)=π’ͺ𝑛2.(4.28)

Theorem 4.5. Let (4.12) be true for 1π‘βˆΆ=π‘π‘˜βˆ’π‘—,π‘βˆΆ=βˆ—(π‘˜+𝑗+1),2(π‘˜βˆ’π‘—)(4.29) where π‘βˆ—>1, that is, 𝛽𝑑𝑛≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑑𝑛(4.30) holds for all π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜. Let, moreover, π›Όβˆˆ(1,π‘βˆ—). Then there exists an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) satisfying (4.15) for π‘›βˆˆβ„€βˆžπ‘›0. 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 β„›(𝑑𝑛) of (4.13), we have ℛ𝑑𝑛=π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έ1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)π‘‘π‘–βˆ’1ξ‚Ή1𝑑𝛼𝑖=1π‘˜βˆ’π‘—π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11π‘‘π›Όπ‘–βˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11π‘‘π‘–βˆ’1𝑑𝛼𝑖=1π‘˜βˆ’π‘—π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11𝑖𝛼(1+𝛿(𝑖))π›Όβˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11(π‘–βˆ’1)(1+𝛿(π‘–βˆ’1))𝑖𝛼(1+𝛿(𝑖))𝛼.(4.31) Since π‘–βˆˆ{π‘›βˆ’π‘˜+1,π‘›βˆ’π‘˜+2,…,π‘›βˆ’π‘—} and π‘›β†’βˆž, we can asymptotically decompose β„›(𝑑𝑛) as π‘›β†’βˆž using decomposition formula (2.17) in Lemma 2.6. Applying this formula to the term π‘–βˆ’π›Ό in the first sum with 𝜎=βˆ’π›Ό and with π‘Ÿ=π‘›βˆ’π‘–, we get 1𝑖𝛼=1(π‘›βˆ’(π‘›βˆ’π‘–))𝛼=1𝑛𝛼1+𝛼(π‘›βˆ’π‘–)𝑛1+π’ͺ𝑛2.(4.32) In addition to this, we have 1(1+𝛿(𝑖))𝛼1=1+π’ͺ𝑖2ξ‚Άξ‚€1=1+π’ͺ𝑛2.(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=1π‘–βˆ’1=1π‘›βˆ’(π‘›βˆ’π‘–+1)𝑛⋅1=11+(π‘›βˆ’π‘–+1)/𝑛𝑛⋅11+π’ͺ𝑛1,(4.34)ξ‚΅11+𝛿(π‘–βˆ’1)=1+π’ͺ(π‘–βˆ’1)2ξ‚Άξ‚€1=1+π’ͺ𝑛2.(4.35) Then we get ℛ𝑑𝑛=1(π‘˜βˆ’π‘—)𝑛𝛼11+π’ͺ𝑛2ξ‚ξ‚„π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έ1+𝛼(π‘›βˆ’π‘–)𝑛1+π’ͺ𝑛2ξ‚ξ‚Ήβˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛𝛼+111+π’ͺπ‘›ξ‚ξ‚„π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+111+π’ͺ𝑛=1(π‘˜βˆ’π‘—)𝑛𝛼1+𝛼(π‘˜βˆ’1)𝑛+1+𝛼(π‘˜βˆ’2)𝑛+β‹―+1+𝛼𝑗𝑛1+π’ͺ𝑛2ξ‚ξ‚Ήβˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛𝛼+111+1+β‹―+1+π’ͺ𝑛=1(π‘˜βˆ’π‘—)𝑛𝛼+11(π‘˜βˆ’π‘—)𝑛+𝛼(π‘˜βˆ’1)+𝛼(π‘˜βˆ’2)+β‹―+𝛼𝑗+π’ͺπ‘›βˆ’π‘ξ‚ξ‚„βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛𝛼+11(π‘˜βˆ’π‘—)+π’ͺ𝑛=1𝑛𝛼+𝛼(π‘˜βˆ’π‘—)𝑛𝛼+1(π‘˜+π‘—βˆ’1)(π‘˜βˆ’π‘—)2βˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛𝛼+1ξ‚€1(π‘˜βˆ’π‘—)+π’ͺ𝑛𝛼+2,(4.36) and, finally, ℛ𝑑𝑛=1𝑛𝛼+𝛼2𝑛𝛼+1𝑝(π‘˜+π‘—βˆ’1)βˆ’βˆ—(π‘˜+𝑗+1)2𝑛𝛼+1ξ‚€1+π’ͺ𝑛𝛼+2.(4.37) A similar decomposition of the left-hand side β„’(𝑑𝑛) in (4.13) leads to (we use the decomposition formula (2.17) in Lemma 2.6 with 𝜎=βˆ’π›Ό and π‘Ÿ=βˆ’1) ℒ𝑑𝑛=1𝑑𝛼𝑛+1=1(𝑛+1)𝛼(1+𝛿(𝑛+1))𝛼=1𝑛𝛼𝛼1βˆ’π‘›ξ‚€1+𝑂𝑛2ξ‚€11+𝑂𝑛2=1ξ‚ξ‚„π‘›π›Όβˆ’π›Όπ‘›π›Ό+1ξ‚€1+π’ͺ𝑛𝛼+2.(4.38) Comparing β„’(𝑑𝑛) and β„›(𝑑𝑛), we see that, for β„’(𝑑𝑛)β‰₯β„›(𝑑𝑛), it is necessary to match the coefficients of the terms π‘›βˆ’π›Όβˆ’1 because the coefficients of the terms π‘›βˆ’π›Ό are equal. It means that we need 1βˆ’π›Ό>21𝛼(π‘˜+π‘—βˆ’1)βˆ’2π‘βˆ—(π‘˜+𝑗+1).(4.39) Simplifying this inequality, we get 12π‘βˆ—1(π‘˜+𝑗+1)>𝛼+2𝛼(π‘˜+π‘—βˆ’1),(4.40) and, finally, π‘βˆ—>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 βˆžξ“π‘–=𝑛0βˆ’π‘˜+11𝑑𝛼𝑖=βˆžξ“π‘–=𝑛0βˆ’π‘˜+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(𝑑)∢={𝑛}

The time scale 𝕋=𝑇(𝑑)={𝑑𝑛},π‘‘π‘›βˆΆ=𝑛, where π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜ is a particular case of the previous time scale defined in Section 4.2 if 𝛿(𝑛)=0 for every π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜. Then (1.1) turns into[]Δ𝑦(𝑛)=𝛽(𝑛)𝑦(π‘›βˆ’π‘—)βˆ’π‘¦(π‘›βˆ’π‘˜)(4.42) and (4.30), which is crucial for the existence of an increasing and asymptotically convergent solution, takes the form𝛽1(𝑛)β‰€βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛,π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜(4.43) with a π‘βˆ—>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)∢=(βˆ‘π‘›+1)π‘›βˆ’π‘—π‘–=π‘›βˆ’π‘˜+1.1/𝑖(4.44) It is easy to verify that (4.42) has a solution βˆ‘π‘¦(𝑛)=𝑛𝑖=11/𝑖, which is the 𝑛th partial sum of harmonic series and, therefore, is divergent as π‘›β†’βˆž. Now we asymptotically compare the function 𝛽 with the right-hand side of (4.43). First we develop an asymptotic decomposition of 𝛽 when π‘›β†’βˆž. We get 1𝛽(𝑛)=(βˆ‘π‘›+1)π‘›βˆ’π‘—π‘–=π‘›βˆ’π‘˜+1=11/𝑖⋅11+1/π‘›βˆ‘π‘˜βˆ’π‘—π‘–=1=1(1/(1+((π‘–βˆ’π‘˜)/𝑛)))β‹…11+1/π‘›βˆ‘π‘˜βˆ’π‘—π‘–=1ξ€Ίξ€·1βˆ’(π‘–βˆ’π‘˜)/𝑛+π’ͺ1/𝑛2=1ξ€Έξ€»1βˆ’π‘›ξ‚€1+π’ͺ𝑛2β‹…1⋅1π‘˜βˆ’π‘—βˆ‘1βˆ’π‘˜βˆ’π‘—π‘–=1ξ€Ίξ€·(π‘–βˆ’π‘˜)/(π‘˜βˆ’π‘—)𝑛+π’ͺ1/𝑛2=1⋅1π‘˜βˆ’π‘—1βˆ’π‘›ξ‚€1+π’ͺ𝑛2⋅1+π‘˜βˆ’π‘—ξ“π‘–=1π‘–βˆ’π‘˜ξ‚€1(π‘˜βˆ’π‘—)𝑛+π’ͺ𝑛2=1⋅1π‘˜βˆ’π‘—1βˆ’π‘›ξ‚€1+π’ͺ𝑛2β‹…ξ‚Έπ‘˜ξ‚ξ‚„1βˆ’π‘›+π‘˜βˆ’π‘—+1ξ‚€12𝑛+π’ͺ𝑛2=1β‹…ξ‚Έπ‘˜π‘˜βˆ’π‘—1βˆ’π‘›+π‘˜βˆ’π‘—+1βˆ’12𝑛𝑛1+π’ͺ𝑛2=1βˆ’π‘˜βˆ’π‘—π‘˜+𝑗+1ξ‚€12(π‘˜βˆ’π‘—)𝑛+π’ͺ𝑛2.(4.45) Now, (4.43) requires that 𝛽1(𝑛)=βˆ’π‘˜βˆ’π‘—π‘˜+𝑗+1ξ‚€12(π‘˜βˆ’π‘—)𝑛+π’ͺ𝑛2≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1).2(π‘˜βˆ’π‘—)𝑛(4.46) The last will hold if βˆ’π‘˜+𝑗+1𝑝2(π‘˜βˆ’π‘—)𝑛<βˆ’βˆ—(π‘˜+𝑗+1),2(π‘˜βˆ’π‘—)𝑛(4.47) that is, if π‘βˆ—<1. This inequality is the opposite to π‘βˆ—>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 π‘˜=1 was considered.

4.4. Time Scale 𝑇(𝑑)∢={π‘žπ‘›}, π‘ž>1

We will focus our attention on the sufficient conditions for the existence of an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) if the time scale is defined as 𝕋=𝑇(𝑑)={𝑑𝑛},π‘‘π‘›βˆΆ=π‘žπ‘›, where π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜ and π‘ž>1. We will apply Theorem 4.4.

Theorem 4.7. Let (4.20) hold for 1π‘βˆΆ=π‘π‘˜βˆ’π‘—,π‘βˆΆ=βˆ—(π‘˜+𝑗+1)lnπ‘ž,2(π‘˜βˆ’π‘—)(4.48) where π‘βˆ—>1, that is, the inequality 𝛽𝑑𝑛≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)lnπ‘ž2(π‘˜βˆ’π‘—)ln𝑑𝑛=1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛(4.49) holds for all π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜. Let, moreover, π›Όβˆˆ(1,π‘βˆ—). Then there exists an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) satisfying (4.23) for π‘›βˆˆβ„€βˆžπ‘›0. 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 β„›(𝑑𝑛) of (4.21), we have ℛ𝑑𝑛=π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έ1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)lnπ‘ž2(π‘˜βˆ’π‘—)lnπ‘‘π‘–βˆ’1ξ‚Ή1ξ€·ln𝑑𝑖𝛼=1π‘˜βˆ’π‘—π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11ξ€·lnπ‘‘π‘–ξ€Έπ›Όβˆ’π‘βˆ—(π‘˜+𝑗+1)lnπ‘ž2(π‘˜βˆ’π‘—)π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11ξ€·lnπ‘‘π‘–βˆ’1ξ€Έξ€·ln𝑑𝑖𝛼=1π‘˜βˆ’π‘—π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11𝑖𝛼(lnπ‘ž)π›Όβˆ’π‘βˆ—(π‘˜+𝑗+1)lnπ‘ž2(π‘˜βˆ’π‘—)π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11(π‘–βˆ’1)𝑖𝛼(lnπ‘ž)𝛼+1=1(π‘˜βˆ’π‘—)(lnπ‘ž)π›Όπ‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11π‘–π›Όβˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)(lnπ‘ž)π›Όπ‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+11(π‘–βˆ’1)𝑖𝛼=ξ€Ίξ€»=1weapplydecompositions(4.32)and(4.34)(π‘˜βˆ’π‘—)(lnπ‘ž)π›Όπ‘›π›Όπ‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έ1+𝛼(π‘›βˆ’π‘–)𝑛1+π’ͺ𝑛2ξ‚ξ‚Ήβˆ’π‘βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)(lnπ‘ž)𝛼𝑛𝛼+1π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+111+π’ͺ𝑛.(4.50) Finally, applying some of the computations from the proof of Theorem 4.5, we get ℛ𝑑𝑛=1(lnπ‘ž)𝛼𝑛𝛼+𝛼2(lnπ‘ž)𝛼𝑛𝛼+1𝑝(π‘˜+π‘—βˆ’1)βˆ’βˆ—(π‘˜+𝑗+1)2(lnπ‘ž)𝛼𝑛𝛼+1ξ‚€1+π’ͺ𝑛𝛼+2.(4.51) and, for the left-hand side β„’(𝑑𝑛) of (4.20), ℒ𝑑𝑛=1(lnπ‘ž)𝛼(𝑛+1)𝛼=1(lnπ‘ž)π›Όπ‘›π›Όβˆ’π›Ό(lnπ‘ž)𝛼𝑛𝛼+1ξ‚€1+π’ͺ𝑛𝛼+2.(4.52) Comparing β„’(𝑑𝑛) and β„›(𝑑𝑛), we see that, for β„’(𝑑𝑛)β‰₯β„›(𝑑𝑛), βˆ’π›Ό(lnπ‘ž)𝛼>𝛼(π‘˜+π‘—βˆ’1)2(lnπ‘ž)π›Όβˆ’π‘βˆ—(π‘˜+𝑗+1)2(lnπ‘ž)𝛼(4.53) is sufficient. Simplifying it, we get π‘βˆ—(π‘˜+𝑗+1)>𝛼(π‘˜+𝑗+1),(4.54) and, finally, π‘βˆ—>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 βˆžξ“π‘–=𝑛0βˆ’π‘˜+11ξ€·ln𝑑𝑖𝛼=βˆžξ“π‘–=𝑛0βˆ’π‘˜+11𝑖𝛼(lnπ‘ž)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 𝛽𝑑𝑛1∢=(βˆ‘π‘›+1)π‘›βˆ’π‘—π‘–=π‘›βˆ’π‘˜+1(ξ€·lnπ‘ž)/ln𝑑𝑖=1(βˆ‘π‘›+1)π‘›βˆ’π‘—π‘–=π‘›βˆ’π‘˜+1.1/𝑖(4.56) Then it is easy to verify that (1.1) has a solution 𝑦𝑑𝑛=𝑛𝑖=1lnπ‘žln𝑑𝑖=𝑛𝑖=11𝑖,(4.57) which is the 𝑛th partial sum of harmonic series and, as such, is divergent as π‘›β†’βˆž. Now we asymptotically compare the function 𝛽 with the right-hand side of (4.49). Proceeding as in Example 4.6, we get 𝛽𝑑𝑛=1βˆ’π‘˜βˆ’π‘—π‘˜+𝑗+1ξ‚€12(π‘˜βˆ’π‘—)𝑛+π’ͺ𝑛2.(4.58) Inequality (4.49) is valid if 𝛽𝑑𝑛=1βˆ’π‘˜βˆ’π‘—π‘˜+𝑗+1ξ‚€12(π‘˜βˆ’π‘—)𝑛+π’ͺ𝑛2≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1),2(π‘˜βˆ’π‘—)𝑛(4.59) that is if π‘βˆ—<1. This inequality is the opposite to π‘βˆ—>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 π‘¦βˆΆπ•‹β†’β„+ of (1.1) expressed by (4.12) and (4.20), that is, by inequalities π›½ξ€·π‘‘π‘›ξ€Έπ‘β‰€π‘βˆ’π‘‘π‘›,π›½ξ€·π‘‘π‘›ξ€Έπ‘β‰€π‘βˆ’ln𝑑𝑛(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(𝑑)={𝑑𝑛}, where 𝑑𝑛=𝑛. In this case, as stated in Theorem 4.5, (4.30), that is, 𝛽𝑑𝑛≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑑𝑛=1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛(4.61) is assumed with a π‘βˆ—>1.

The second criterion (4.20) can successfully be used, for example, for the time scale 𝑇(𝑑)={𝑑𝑛} where 𝑑𝑛=π‘žπ‘› and π‘ž>1. Then, as stated in Theorem 4.7, (4.49), that is, 𝛽𝑑𝑛≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)lnπ‘ž2(π‘˜βˆ’π‘—)ln𝑑𝑛=1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛(4.62) is assumed with a π‘βˆ—>1. Comparing (4.61) and (4.62), we see that, although their left-hand sides are different due to different meaning of 𝑑𝑛 in every case, their right-hand sides are identical.

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

Theorem 4.9. Let 𝛽𝑑𝑛≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛(4.63) holds for all π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜ and for a fixed π‘βˆ—>1. Let, moreover, π›Όβˆˆ(1,π‘βˆ—). Then there exists an increasing and asymptotically convergent solution π‘¦βˆΆπ•‹β†’β„+ of (1.1) satisfying 𝑦𝑑𝑛≀𝑛𝑖=𝑛0βˆ’π‘˜+1ξ‚Έ1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)ξ‚Ή12(π‘˜βˆ’π‘—)(π‘–βˆ’1)𝑖𝛼(4.64) for every π‘›βˆˆβ„€βˆžπ‘›0. Such a solution is defined, for example, by the initial function πœ‘ξ€·π‘‘π‘›ξ€ΈβˆΆ=𝑛𝑖=𝑛0βˆ’π‘˜+1ξ‚Έ1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)ξ‚Ή12(π‘˜βˆ’π‘—)(π‘–βˆ’1)𝑖𝛼,π‘›βˆˆβ„€π‘›0𝑛0βˆ’π‘˜.(4.65)

Proof. We will apply Theorem 4.2 with π›½βˆ—ξ€·π‘‘π‘›ξ€Έ1∢=βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)𝑑2(π‘˜βˆ’π‘—)𝑛,πœ€π‘›ξ€Έ1∢=𝑛𝛼.(4.66) Inequality (4.5) turns into πœ€ξ€·π‘‘π‘›+1ξ€Έ=1(𝑛+1)𝛼β‰₯π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1π›½βˆ—ξ€·π‘‘π‘–βˆ’1ξ€Έπœ€ξ€·π‘‘π‘–ξ€Έ=π‘›βˆ’π‘—ξ“π‘–=π‘›βˆ’π‘˜+1ξ‚Έ1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)ξ‚Ή12(π‘˜βˆ’π‘—)(π‘–βˆ’1)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., 𝑑𝑛=𝑛 for every π‘›βˆˆβ„€βˆžπ‘›0βˆ’π‘˜) 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 π‘βˆ—>𝛼. (4.6) holds as well because the series βˆžξ“π‘–=𝑛0βˆ’π‘˜+1ξ‚Έ1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)ξ‚Ή12(π‘˜βˆ’π‘—)(π‘–βˆ’1)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 𝑑𝑛, the right-hand side of the crucial inequality (4.63) cannot, in a general situation of arbitrary time scale {𝑑𝑛}, be explicitly expressed using only the 𝑑𝑛 terms. This is only possible if, for a given time scale, a function 𝑓 is explicitly known such that 𝑓(𝑑𝑛)=𝑛. Then, (4.63) can be written in the form 𝛽𝑑𝑛≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)𝑑2(π‘˜βˆ’π‘—)𝑓𝑛=1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1).2(π‘˜βˆ’π‘—)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 𝛽𝑑𝑛1∢=(βˆ‘π‘›+1)π‘›βˆ’π‘—π‘–=π‘›βˆ’π‘˜+11/𝑖(4.70) has an increasing asymptotically divergent solution 𝑦(π‘‘π‘›βˆ‘)=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, 𝛽𝑑𝑛=1βˆ‘(𝑛+1)π‘›βˆ’π‘—π‘–=π‘›βˆ’π‘˜+1≀11/π‘–βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1)2(π‘˜βˆ’π‘—)𝑛(4.71) holds, or if 𝛽𝑑𝑛=1βˆ’π‘˜βˆ’π‘—π‘˜+𝑗+1ξ‚€12(π‘˜βˆ’π‘—)𝑛+π’ͺ𝑛2≀1βˆ’π‘π‘˜βˆ’π‘—βˆ—(π‘˜+𝑗+1).2(π‘˜βˆ’π‘—)𝑛(4.72) The last holds for π‘βˆ—<1. This inequality is the opposite to π‘βˆ—>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. RůžičkovÑ and Z. Suta were supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).

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