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

Abstract

The paper investigates a dynamic equation Δ𝑦(𝑡_𝑛)=𝛽(𝑡_𝑛)[𝑦(𝑡_𝑛−𝑗)−𝑦(𝑡_𝑛−𝑘)] for 𝑛→∞, where 𝑘 and 𝑗 are integers such that 𝑘>𝑗≥0, on an arbitrary discrete time scale 𝕋∶={𝑡_𝑛} with 𝑡_𝑛∈ℝ, 𝑛∈ℤ^∞𝑛_0−𝑘={𝑛₀−𝑘,𝑛₀−𝑘+1,…}, 𝑛₀∈ℕ, 𝑡_𝑛<𝑡_𝑛+1, Δ𝑦(𝑡_𝑛)=𝑦(𝑡_𝑛+1)−𝑦(𝑡_𝑛), and lim_𝑛→∞𝑡_𝑛=∞. We assume 𝛽∶𝕋→(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 ℤ^∞𝑠∶={𝑠,𝑠+1,…}, and if an integer 𝑞≥𝑠, we define ℤ^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)∶={𝑡_𝑛}, where 𝑡_𝑛∈ℝ, 𝑛∈ℤ^∞𝑛_0−𝑘, 𝑛₀∈ℕ, 𝑘>0 is an integer, 𝑡_𝑛<𝑡_𝑛+1, and lim_𝑛→∞𝑡_𝑛=∞. For a fixed 𝑣∈ℤ^∞𝑛_0−𝑘, we define a time scale 𝕋_𝑣=𝑇_𝑣(𝑡)∶={𝑡_𝑛}, where 𝑛∈ℤ^∞𝑣. Obviously, 𝑇_𝑛0−𝑘(𝑡)=𝑇(𝑡). In addition, for integers 𝑠, 𝑞, 𝑞≥𝑠≥𝑛₀−𝑘, we define the set 𝕋^𝑞𝑠=𝑇^𝑞𝑠(𝑡)∶={𝑡_𝑠,𝑡_𝑠+1,…,𝑡_𝑞}.

In the paper we study a dynamic equation Δ𝑦𝑡_𝑛=𝛽𝑡_𝑛𝑦𝑡_𝑛−𝑗−𝑦𝑡_𝑛−𝑘(1.1) as 𝑛→∞. The difference is defined as usual: Δ𝑦(𝑡_𝑛)∶=𝑦(𝑡_𝑛+1)−𝑦(𝑡_𝑛), integers 𝑘 and 𝑗 in (1.1) satisfy the inequality 𝑘>𝑗≥0, and 𝛽∶𝕋→ℝ⁺∶=(0,∞). Without loss of generality, we assume that 𝑡_𝑛0−𝑘>0 (this is a technical detail, necessary for some expressions to be well defined). Throughout the paper, we adopt the notation ∑^{𝑘𝑖=𝑘+1}ℬ(𝑡_𝑖)=0 where 𝑘 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 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(𝕋^𝑛0_𝑛0−𝑘,ℝ) be the space of discrete functions mapping the discrete interval 𝕋^𝑛0_𝑛0−𝑘 into ℝ. Let 𝑣∈ℤ^∞𝑛₀ be given. The function 𝑦∶𝕋_𝑣−𝑘→ℝ is said to be a solution of (1.1) on 𝕋_𝑣−𝑘 if it satisfies (1.1) for every 𝑛∈ℤ^∞𝑣. A solution 𝑦 of (1.1) on 𝕋_𝑣−𝑘 is asymptotically convergent if the limit lim_𝑛→∞𝑦(𝑡_𝑛) exists and is finite. For a given 𝑣∈ℤ^∞𝑛₀ and 𝜑∈𝒞, we say that 𝑦=𝑦_{(𝑡𝑣,𝜑)} is a solution of (1.1) defined by the initial conditions (𝑡_𝑣,𝜑) if 𝑦_{(𝑡𝑣,𝜑)} is a solution of (1.1) on 𝕋_𝑣−𝑘 and 𝑦_{(𝑡𝑣,𝜑)}(𝑡_𝑣+𝑚)=𝜑(𝑡_𝑚) for 𝑚∈ℤ^0−𝑘.

2.1. Auxiliary Inequality

The inequalityΔ𝜔𝑡_𝑛≥𝛽𝑡_𝑛𝜔𝑡_𝑛−𝑗−𝜔𝑡_𝑛−𝑘(2.1) is a helpful tool in the analysis of solutions of (1.1). Let 𝑣∈ℤ^∞𝑛₀. The function 𝜔∶𝕋_𝑣−𝑘→ℝ is said to be a solution of (2.1) on 𝕋_𝑣−𝑘 if 𝜔 satisfies (2.1) for 𝑛∈ℤ^∞𝑣. A solution 𝜔 of (2.1) on 𝕋_𝑣−𝑘 is asymptotically convergent if the limit lim_𝑛→∞𝜔(𝑡_𝑛) 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 𝜑∈𝒞 be increasing (nondecreasing, decreasing, nonincreasing) on 𝕋^𝑛0_𝑛0−𝑘. Then the solution 𝑦_(𝑛0,𝜑)(𝑡_𝑛) of (1.1), where 𝑛∈ℤ^∞𝑛₀ is increasing (nondecreasing, decreasing, nonincreasing) on 𝕋_𝑛0, too.

Lemma 2.2. Let 𝜑∈𝒞 be increasing (nondecreasing) and 𝜔∶𝕋→ℝ be a solution of inequality (2.1) with 𝜔(𝑡_𝑚)=𝜑(𝑡_𝑚), 𝑚∈ℤ^𝑛0_𝑛0−𝑘. Then, 𝜔(𝑡_𝑛), where 𝑛∈ℤ^∞𝑛₀ is increasing (nondecreasing).

The proofs of both lemmas above follow directly from the form of (1.1), (2.1), and from the properties 𝛽(𝑡_𝑛)>0, 𝑛∈ℤ^∞𝑛_0−𝑘, 𝑘>𝑗≥0.

Theorem 2.3. Let 𝜔∶𝕋→ℝ be a solution of (2.1) on 𝕋. Then there exists a solution 𝑦∶𝕋→ℝ of (1.1) on 𝕋 such that 𝑦𝑡_𝑛≤𝜔𝑡_𝑛(2.2) holds for every 𝑛∈ℤ^∞𝑛_0−𝑘. In particular, a solution 𝑦_(𝑛0,𝜙) of (1.1) with 𝜙∈𝒞, defined by 𝜙𝑡_𝑛∶=𝜔𝑡_𝑛,𝑛∈ℤ^𝑛0_𝑛0−𝑘,(2.3) is such a solution.

Proof. Let 𝜔(𝑡_𝑛) be a solution of (2.1) defined on 𝕋. We will show that the solution 𝑦(𝑡_𝑛)∶=𝑦_(𝑛0,𝜙)(𝑡_𝑛) of (1.1) with 𝜙 defined by (2.3) satisfies (2.2), that is, 𝑦_(𝑛0,𝜙)𝑡_𝑛≤𝜔𝑡_𝑛(2.4) for every 𝑛∈ℤ^∞𝑛_0−𝑘. Let 𝑊∶𝕋→ℝ be defined by 𝑊𝑡_𝑛∶=𝜔𝑡_𝑛−𝑦𝑡_𝑛.(2.5) Then 𝑊(𝑡_𝑛)=0 if 𝑛∈ℤ^𝑛0_𝑛0−𝑘 and, in addition, 𝑊 is a solution of (2.1) on 𝕋. Lemma 2.2 implies that 𝑊 is nondecreasing. Consequently, 𝑊𝑡_𝑛=𝜔𝑡_𝑛−𝑦𝑡_𝑛≥𝑊𝑡_𝑛0=𝜔𝑡_𝑛0−𝑦𝑡_𝑛0=0,(2.6) and 𝑦(𝑡_𝑛)≤𝜔(𝑡_𝑛) for all 𝑛≥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 𝜀∶𝕋→ℝ⁺ such that 𝜀𝑡_𝑛+1≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖(2.7) for every 𝑛∈ℤ^∞𝑛₀. Then there exists a solution 𝜔=𝜔_𝜀 of (2.1) defined on 𝕋 and having the form 𝜔_𝜀𝑡_𝑛∶=^𝑛_𝑖=𝑛_0−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖.(2.8)

Proof. Assuming that 𝜔_𝜀 defined by (2.8) is a solution of (2.1) for 𝑛∈ℤ^∞𝑛₀, we will deduce the inequality for 𝜀. We get Δ𝜔𝜔_𝜀𝜀𝑡_𝑛=𝜔_𝜀𝑡_𝑛+1−𝜔_𝜀𝑡_𝑛=∑^𝑛+1_{𝑖=𝑛0−𝑘+1}𝛽𝑡_𝑖−1𝜀𝑡_𝑖−∑^𝑛_{𝑖=𝑛0−𝑘+1}𝛽𝑡_𝑖−1𝜀𝑡_𝑖=𝛽𝑡_𝑛𝜀𝑡_𝑛+1,𝑡_𝑛−𝑗−𝜔_𝜀𝑡_𝑛−𝑘=^𝑛−𝑗_{𝑖=𝑛0−𝑘+1}𝛽𝑡_𝑖−1𝜀𝑡_𝑖−^𝑛−𝑘_{𝑖=𝑛0−𝑘+1}𝛽𝑡_𝑖−1𝜀𝑡_𝑖=^𝑛−𝑗_{𝑖=𝑛−𝑘+1}𝛽𝑡_𝑖−1𝜀𝑡_𝑖.(2.9) We substitute 𝜔_𝜀 for 𝜔 in (2.1). Then, using (2.9), (2.1) turns into 𝛽𝑡_𝑛𝜀𝑡_𝑛+1≥𝛽𝑡_𝑛^𝑛−𝑗_𝑛−_𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖.(2.10) Reducing the last inequality by 𝛽(𝑡_𝑛), 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 𝜑∈𝒞 can be decomposed into the difference of two increasing functions 𝜑_𝑗∈𝒞, 𝑗=1,2, that is, 𝜑𝑡_𝑛=𝜑₁𝑡_𝑛−𝜑₂𝑡_𝑛,𝑛∈ℤ^𝑛0_𝑛0−𝑘.(2.11)

Proof. Let constants 𝑀_𝑛>0, 𝑛∈ℤ^𝑛0_𝑛0−𝑘 be such that 𝑀_𝑛+1>𝑀_𝑛+max0,𝜑𝑡_𝑛−𝜑𝑡_𝑛+1(2.12) is valid for each 𝑛∈ℤ^{𝑛0𝑛−1}_0−𝑘. We set 𝜑₁𝑡𝜑_𝑛₂∶=𝜑𝑡_𝑛+𝑀_𝑛,𝑛∈ℤ^𝑛0_𝑛0−𝑘,𝑡_𝑛∶=𝑀_𝑛,𝑛∈ℤ^𝑛0_𝑛0−𝑘.(2.13) It is obvious that (2.11) holds. Now we verify that both functions 𝜑_𝑗, 𝑗=1,2 are increasing. The first one should satisfy 𝜑₁(𝑡_𝑛+1)>𝜑₁(𝑡_𝑛) for 𝑛∈ℤ^{𝑛0𝑛−1}_0−𝑘, which means that 𝜑𝑡_𝑛+1+𝑀_𝑛+1>𝜑𝑡_𝑛+𝑀_𝑛(2.14) or 𝑀_𝑛+1>𝑀_𝑛+𝜑𝑡_𝑛−𝜑𝑡_𝑛+1.(2.15) We conclude that the last inequality holds because, due to (2.12), we have 𝑀_𝑛+1>𝑀_𝑛+max0,𝜑𝑡_𝑛−𝜑𝑡_𝑛+1≥𝑀_𝑛+𝜑𝑡_𝑛−𝜑𝑡_𝑛+1.(2.16) The inequality 𝜑₂(𝑡_𝑛+1)>𝜑₂(𝑡_𝑛) obviously holds for every 𝑛∈ℤ^{𝑛0𝑛−1}_0−𝑘 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∈ℝ⧵{0}, the asymptotic representation (𝑛−𝑟)^𝜎=𝑛^𝜎1−𝜎𝑟𝑛+𝒪𝑛1²(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 ℤ^∞𝑛_0−𝑘, then all the solutions of (1.1) defined on ℤ^∞𝑛_0−𝑘 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 𝜑∈𝒞 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(𝑡_𝑛), 𝑛∈ℤ^∞𝑛_0−𝑘. Without loss of generality, we assume that 𝑦_(𝑛0,𝜑)≢𝑌 on ℤ^∞𝑛_0−𝑘 since, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume Δ𝑌𝑡_𝑛>0,𝑛∈ℤ^{𝑛0𝑛−1}_0−𝑘.(3.1) Hence, there is a constant 𝛾>0 such that Δ𝑌𝑡_𝑛−𝛾Δ𝑦𝑡_𝑛>0,𝑛∈ℤ^{𝑛0𝑛−1}_0−𝑘(3.2) or Δ𝑌𝑡_𝑛−𝛾𝑦𝑡_𝑛>0,𝑛∈ℤ^{𝑛0𝑛−1}_0−𝑘.(3.3) This implies that the function 𝑌(𝑡_𝑛)−𝛾𝑦(𝑡_𝑛) is increasing on ℤ^{𝑛0𝑛−1}_0−𝑘, and Lemma 2.1 implies that 𝑌(𝑡_𝑛)−𝛾𝑦(𝑡_𝑛) is increasing on ℤ^∞𝑛_0−𝑘. Thus, 𝑌𝑡_𝑛−𝛾𝑦𝑡_𝑛>𝑌𝑡_𝑛0−𝛾𝑦𝑡_𝑛0,𝑛∈ℤ^∞𝑛₀(3.4) or 𝑦𝑡_𝑛<𝑦𝑡_𝑛0+1𝛾𝑌𝑡^𝑛−𝑌𝑡_𝑛0,𝑛∈ℤ^∞𝑛₀(3.5) and, consequently, 𝑦(𝑡_𝑛) is a bounded function on ℤ^∞𝑛_0−𝑘 because of the boundedness of 𝑌(𝑡_𝑛). Obviously, in such a case, 𝑦(𝑡_𝑛) 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 𝜑∈𝒞, 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 𝜑∈𝒞 can be decomposed into the difference of two increasing discrete functions 𝜑_𝑗∈𝒞, 𝑗=1,2. In accordance with the previous part of the proof, every function 𝜑_𝑗∈𝒞, 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 ℤ^∞𝑛_0−𝑘.(b)All solutions of (1.1) defined on ℤ^∞𝑛_0−𝑘 are asymptotically convergent.(c)Inequality (2.1) has a strictly monotone and asymptotically convergent solution on ℤ^∞𝑛_0−𝑘.

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 𝜀∶𝕋→ℝ⁺ satisfying ^∞_𝑖=_{𝑛0−𝑘+1}𝜀𝑡𝛽𝑡_𝑖−1𝜀𝑡_𝑖<∞,_𝑛+1≥^𝑛−𝑗_{𝑖=𝑛−𝑘+1}𝛽𝑡_𝑖−1𝜀𝑡_𝑖(4.1) for every 𝑛∈ℤ^∞𝑛₀. Then the initial function 𝜑𝑡_𝑛∶=^𝑛_𝑖=𝑛_0−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖,𝑛∈ℤ^𝑛0_𝑛0−𝑘(4.2) defines an increasing and asymptotically convergent solution 𝑦_{(𝑡𝑛0,𝜑)}(𝑡_𝑛) of (1.1) on 𝕋 satisfying 𝑦_{(𝑡𝑛0,𝜑)}𝑡_𝑛≤^𝑛_𝑖=𝑛_0−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖(4.3) for every 𝑛∈ℤ^∞𝑛₀.

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 𝛽^∗∶𝕋→ℝ⁺ and 𝜀∶𝕋→ℝ⁺ such that the inequalities 𝜀𝑡𝛽𝑡_𝑛≤𝛽^∗𝑡_𝑛,(4.4)_𝑛+1≥^𝑛−𝑗_{𝑖=𝑛−𝑘+1}𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖(4.5) hold for all 𝑛∈ℤ^∞𝑛_0−𝑘, and moreover ^∞_𝑖=_{𝑛0−𝑘+1}𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖<∞.(4.6) Then there exists an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ of (1.1) satisfying 𝑦𝑡_𝑛≤^𝑛_𝑖=𝑛_0−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖(4.7) for every 𝑛∈ℤ^∞𝑛₀. Such a solution is defined, for example, by the initial function 𝜑𝑡_𝑛∶=^𝑛_𝑖=𝑛_0−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖,𝑛∈ℤ^𝑛0_𝑛0−𝑘.(4.8)

Proof. From (4.5) and (4.6), we get 𝜀𝑡_𝑛+1≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1∞>_∞𝛽𝑡_𝑖−1𝜀𝑡_𝑖,_𝑖=𝑛_0−𝑘+1𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖≥^∞_𝑖=𝑛_0−𝑘+1𝛽𝑡_𝑖−1𝜀𝑡_𝑖.(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 𝛽^∗𝑡_𝑛=𝑐−𝛾𝑡_𝑛,(4.10) where 𝑐 is a positive constant and 𝛾∶𝕋→ℝ⁺ is a suitable function such that 𝛾(𝑡_𝑛)<𝑐 (at least for all sufficiently large 𝑛) and lim_𝑛→∞𝛾𝑡_𝑛=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 𝛽𝑡_𝑛≤𝑐−𝑝𝑡_𝑛1,(4.12)𝑡^𝛼𝑛+1≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑐−𝑡𝑝_𝑖−1𝑡1^𝛼𝑖(4.13) hold for all 𝑛∈ℤ^∞𝑛_0−𝑘, and moreover ^∞_𝑖=_{𝑛0−𝑘+1}1𝑡^𝛼𝑖<∞.(4.14) Then there exists an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) satisfying 𝑦𝑡_𝑛≤^𝑛_𝑖=𝑛_0−𝑘+1𝑐−𝑡𝑝_𝑖−1𝑡1^𝛼𝑖(4.15) for every 𝑛∈ℤ^∞𝑛₀. Such a solution is defined, for example, by the initial function 𝜑𝑡_𝑛∶=^𝑛_𝑖=𝑛_0−𝑘+1𝑐−𝑡𝑝_𝑖−1𝑡1^𝛼𝑖,𝑛∈ℤ^𝑛0_𝑛0−𝑘.(4.16)

Proof. We will apply Theorem 4.2 with 𝛽^∗𝑡_𝑛∶=𝑐−𝑝𝑡_𝑛,𝜀𝑡_𝑛∶=1𝑡^𝛼𝑛.(4.17) Inequality (4.5) turns into 𝜀𝑡_𝑛+1=1𝑡^𝛼𝑛+1≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖=^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑐−𝑡𝑝_𝑖−1𝑡1^𝛼𝑖(4.18) and is true due to (4.13). Inequality (4.6) holds due to assumption (4.14) as well because lim_𝑛→∞𝑡_𝑛=∞ and ^∞_𝑖=_{𝑛0−𝑘+1}𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖=^∞_𝑖=𝑛_0−𝑘+1𝑐−𝑡𝑝_𝑖−1𝑡1^𝛼𝑖<∞.(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 𝛽𝑡_𝑛≤𝑐−𝑝ln𝑡_𝑛1,(4.20)ln𝑡_𝑛+1^𝛼≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑐−ln𝑝𝑡_𝑖−11ln𝑡_𝑖^𝛼(4.21) hold for all 𝑛∈ℤ^∞𝑛_0−𝑘, and moreover ^∞_𝑖=_{𝑛0−𝑘+1}1ln𝑡_𝑖^𝛼<∞.(4.22) Then there exists an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) satisfying 𝑦𝑡_𝑛≤^𝑛_𝑖=𝑛_0−𝑘+1𝑐−ln𝑝𝑡_𝑖−11ln𝑡_𝑖^𝛼(4.23) for every 𝑛∈ℤ^∞𝑛₀. Such a solution is defined, for example, by the initial function 𝜑𝑡_𝑛∶=^𝑛_𝑖=𝑛_0−𝑘+1𝑐−ln𝑝𝑡_𝑖−11ln𝑡_𝑖^𝛼,𝑛∈ℤ^𝑛0_𝑛0−𝑘.(4.24)

Proof. We will apply Theorem 4.2 with 𝛽^∗𝑡_𝑛∶=𝑐−𝑝ln𝑡_𝑛,𝜀𝑡_𝑛∶=1ln𝑡_n^𝛼.(4.25) Inequality (4.5) turns into 𝜀𝑡_𝑛+1=1ln𝑡_𝑛+1^𝛼≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖=^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑐−ln𝑝𝑡_𝑖−11ln𝑡_𝑖^𝛼(4.26) and is true due to (4.21). Inequality (4.6) holds due to assumption (4.22) as well because lim_𝑛→∞𝑡_𝑛=∞ and ^∞_𝑖=_{𝑛0−𝑘+1}𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖=^∞_𝑖=𝑛_0−𝑘+1𝑐−ln𝑝𝑡_𝑖−11ln𝑡_𝑖^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(𝑡)∶={𝑛(1+𝛿(𝑛))}

Now, using Theorem 4.3, we derive sufficient conditions for the existence of an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) in the case when the time scale is defined as 𝕋=𝑇(𝑡)={𝑡_𝑛},𝑡_𝑛∶=𝑛(1+𝛿(𝑛)), where 𝛿∶𝕋→ℝ, |𝛿(𝑛)|≤𝛿^∗, 𝛿^∗∈(0,1), 𝑛∈ℤ^∞𝑛_0−𝑘, and 𝛿(𝑛)=𝒪1𝑛².(4.28)

Theorem 4.5. Let (4.12) be true for 𝑐∶=1𝑘−𝑗,𝑝∶=𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗),(4.29) where 𝑝^∗>1, that is, 𝛽𝑡_𝑛≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑡_𝑛(4.30) holds for all 𝑛∈ℤ^∞𝑛_0−𝑘. Let, moreover, 𝛼∈(1,𝑝^∗). Then there exists an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) satisfying (4.15) for 𝑛∈ℤ^∞𝑛₀. 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 ℛ(𝑡_𝑛) of (4.13), we have ℛ𝑡_𝑛=^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑡_𝑖−1𝑡1^𝛼𝑖=1𝑘−𝑗^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1𝑡^𝛼𝑖−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1𝑡_𝑖−1𝑡^𝛼𝑖=1𝑘−𝑗^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1𝑖^𝛼(1+𝛿(𝑖))^𝛼−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1(𝑖−1)(1+𝛿(𝑖−1))𝑖^𝛼(1+𝛿(𝑖))^𝛼.(4.31) Since 𝑖∈{𝑛−𝑘+1,𝑛−𝑘+2,…,𝑛−𝑗} and 𝑛→∞, we can asymptotically decompose ℛ(𝑡_𝑛) as 𝑛→∞ using decomposition formula (2.17) in Lemma 2.6. Applying this formula to the term 𝑖^−𝛼 in the first sum with 𝜎=−𝛼 and with 𝑟=𝑛−𝑖, we get 1𝑖^𝛼=1(𝑛−(𝑛−𝑖))^𝛼=1𝑛^𝛼1+𝛼(𝑛𝑛−𝑖)+𝒪𝑛1².(4.32) In addition to this, we have 1(1+𝛿(𝑖))^𝛼=1+𝒪1𝑖²=1+𝒪𝑛1².(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 1𝑖−1=𝑛−(𝑛1−1𝑖+1)=1𝑛⋅1+(𝑛−1𝑖+1)/𝑛=1𝑛⋅1+𝒪1𝑛,(4.34)1+𝛿(𝑖−1)=1+𝒪(𝑖−11)²=1+𝒪𝑛1².(4.35) Then we get ℛ𝑡_𝑛=1−(𝑘𝑝−_∗𝑗)𝑛^𝛼1+𝒪𝑛1₂^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1+𝛼(𝑛𝑛−𝑖)+𝒪𝑛1²(𝑘+𝑗+1)=2(1𝑘−𝑗)𝑛^𝛼+11+𝒪1𝑛^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1+𝒪1𝑛−(𝑘𝑝−_∗𝑗)𝑛^𝛼1+𝛼(𝑘𝑛−1)+1+𝛼(𝑘𝑛−2)+⋯+1+𝛼𝑛𝑗+𝒪𝑛1²(𝑘+𝑗+1)=2(𝑘1−𝑗)𝑛^𝛼+11+1+⋯+1+𝒪1𝑛−(𝑘𝑝−𝑗)𝑛^𝛼+1(𝑘−𝑗)𝑛+𝛼(𝑘−1)+𝛼(𝑘−2)+⋯+𝛼𝑗+𝒪1𝑛_∗(𝑘+𝑗+1)=12(𝑘−𝑗)𝑛^𝛼+1(𝑘−𝑗)+𝒪1𝑛𝑛^𝛼+𝛼(𝑘−𝑗)𝑛^𝛼+1(𝑘+𝑗−1)(𝑘−𝑗)2−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛^𝛼+1(𝑘−𝑗)+𝒪1𝑛^𝛼+2,(4.36) and, finally, ℛ𝑡_𝑛=1𝑛^𝛼+𝛼2𝑛^𝛼+1(𝑘+𝑗−1)−𝑝^∗(𝑘+𝑗+1)2𝑛^𝛼+1+𝒪1𝑛^𝛼+2.(4.37) A similar decomposition of the left-hand side ℒ(𝑡_𝑛) in (4.13) leads to (we use the decomposition formula (2.17) in Lemma 2.6 with 𝜎=−𝛼 and 𝑟=−1) ℒ𝑡_𝑛=1𝑡^𝛼𝑛+1=1(𝑛+1)^𝛼(1+𝛿(𝑛+1))^𝛼=1𝑛^𝛼1−𝛼𝑛+𝑂𝑛1²1+𝑂𝑛1₂=𝑛1_𝛼−𝛼𝑛^𝛼+1+𝒪1𝑛^𝛼+2.(4.38) Comparing ℒ(𝑡_𝑛) and ℛ(𝑡_𝑛), we see that, for ℒ(𝑡_𝑛)≥ℛ(𝑡_𝑛), it is necessary to match the coefficients of the terms 𝑛^−𝛼−1 because the coefficients of the terms 𝑛^−𝛼 are equal. It means that we need −𝛼>12𝛼(𝑘+𝑗−1)−12𝑝^∗(𝑘+𝑗+1).(4.39) Simplifying this inequality, we get 12𝑝^∗(𝑘+𝑗+1)>𝛼+12𝛼(𝑘+𝑗−1),(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=_{𝑛0−𝑘+1}1𝑡^𝛼𝑖=^∞_𝑖=𝑛_0−𝑘+11𝑖^𝛼(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)={𝑡_𝑛},𝑡_𝑛∶=𝑛, where 𝑛∈ℤ^∞𝑛_0−𝑘 is a particular case of the previous time scale defined in Section 4.2 if 𝛿(𝑛)=0 for every 𝑛∈ℤ^∞𝑛_0−𝑘. Then (1.1) turns intoΔ𝑦(𝑛)=𝛽(𝑛)[𝑦(𝑛−𝑗)−𝑦(𝑛−𝑘)](4.42) and (4.30), which is crucial for the existence of an increasing and asymptotically convergent solution, takes the form𝛽(𝑛)≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛,𝑛∈ℤ^{∞𝑛0−𝑘}(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 𝛽(𝑛)∶=1(𝑛+1)∑^{𝑛−𝑗𝑖=𝑛−𝑘+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 𝛽(𝑛)=1=(𝑛+11)∑^{𝑛−𝑗𝑖=𝑛−𝑘+1}1/𝑖=1+11/𝑛⋅∑^{𝑘−𝑗𝑖=1}(1/(1+1((𝑖−𝑘)/𝑛)))=11+−11/𝑛⋅∑^{𝑘−𝑗𝑖=1}1−(𝑖−𝑘1)/𝑛+𝒪1/𝑛²=1𝑛+𝒪𝑛1²⋅𝑘−1𝑗⋅1−∑^{𝑘−𝑗𝑖=1}(𝑖−𝑘)/(1𝑘−𝑗)𝑛+𝒪1/𝑛²𝑘−𝑗⋅1−1𝑛+𝒪𝑛1²⋅1+^𝑘−𝑗_𝑖=1𝑖−𝑘=1(𝑘−𝑗)𝑛+𝒪𝑛1²=𝑘−1𝑗⋅1−1𝑛+𝒪𝑛1²⋅1−𝑘𝑛+𝑘−2𝑗𝑛+1+𝒪𝑛1²=𝑘−1𝑗⋅1−𝑘𝑛+𝑘−2𝑗𝑛+1−1𝑛+𝒪𝑛1²𝑘−𝑗−2𝑘(+𝑘−𝑗+𝑗)1𝑛+𝒪𝑛1².(4.45) Now, (4.43) requires that 𝛽(𝑛)=1𝑘−𝑗−2𝑘(+𝑘−𝑗+𝑗)1𝑛+𝒪𝑛1²≤𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛.(4.46) The last will hold if −𝑘+𝑗+12(𝑘−𝑗)𝑛<−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)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)∶={𝑞^𝑛}, 𝑞>1

We will focus our attention on the sufficient conditions for the existence of an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) if the time scale is defined as 𝕋=𝑇(𝑡)={𝑡_𝑛},𝑡_𝑛∶=𝑞^𝑛, where 𝑛∈ℤ^∞𝑛_0−𝑘 and 𝑞>1. We will apply Theorem 4.4.

Theorem 4.7. Let (4.20) hold for 𝑐∶=1𝑘−𝑗,𝑝∶=𝑝^∗(𝑘+𝑗+1)ln𝑞2(𝑘−𝑗),(4.48) where 𝑝^∗>1, that is, the inequality 𝛽𝑡_𝑛≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)ln𝑞2(𝑘−𝑗)ln𝑡_𝑛=1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛(4.49) holds for all 𝑛∈ℤ^∞𝑛_0−𝑘. Let, moreover, 𝛼∈(1,𝑝^∗). Then there exists an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) satisfying (4.23) for 𝑛∈ℤ^∞𝑛₀. 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 ℛ(𝑡_𝑛) of (4.21), we have ℛ𝑡_𝑛=^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)ln𝑞2(𝑘−𝑗)ln𝑡_𝑖−1ln1𝑡_𝑖^𝛼=1𝑘−𝑗^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1ln𝑡_𝑖^𝛼−𝑝^∗(𝑘+𝑗+1)ln𝑞2(𝑘−𝑗)^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1ln𝑡_𝑖−1ln𝑡_𝑖^𝛼=1𝑘−𝑗^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1𝑖^𝛼(ln𝑞)^𝛼−𝑝^∗(𝑘+𝑗+1)ln𝑞2(𝑘−𝑗)^𝑛−𝑗_{𝑖=𝑛−𝑘+1}1(𝑖−1)𝑖^𝛼(ln𝑞)^𝛼+1=1(𝑘−𝑗)(ln𝑞)^{𝛼𝑛−𝑗}_{𝑖=𝑛−𝑘+1}1𝑖^𝛼−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)(ln𝑞)^{𝛼𝑛−𝑗}_{𝑖=𝑛−𝑘+1}1(𝑖−1)𝑖^𝛼==weapp1lydecompositions(4.32)and(4.34)−(𝑘−𝑝𝑗_∗)(ln𝑞)^𝛼𝑛^{𝛼𝑛−𝑗}_{𝑖=𝑛−𝑘+1}1+𝛼(𝑛𝑛−𝑖)+𝒪𝑛1²(𝑘+𝑗+1)2(𝑘−𝑗)(ln𝑞)^𝛼𝑛^{𝛼+1𝑛−𝑗}_{𝑖=𝑛−𝑘+1}1+𝒪1𝑛.(4.50) Finally, applying some of the computations from the proof of Theorem 4.5, we get ℛ𝑡_𝑛=1(ln𝑞)^𝛼𝑛^𝛼+𝛼2(ln𝑞)^𝛼𝑛^𝛼+1(𝑘+𝑗−1)−𝑝^∗(𝑘+𝑗+1)2(ln𝑞)^𝛼𝑛^𝛼+1+𝒪1𝑛^𝛼+2.(4.51) and, for the left-hand side ℒ(𝑡_𝑛) of (4.20), ℒ𝑡_𝑛=1(ln𝑞)^𝛼(𝑛+1)^𝛼=1(ln𝑞)^𝛼𝑛^𝛼−𝛼(ln𝑞)^𝛼𝑛^𝛼+1+𝒪1𝑛^𝛼+2.(4.52) Comparing ℒ(𝑡_𝑛) and ℛ(𝑡_𝑛), we see that, for ℒ(𝑡_𝑛)≥ℛ(𝑡_𝑛), −𝛼(ln𝑞)^𝛼>𝛼(𝑘+𝑗−1)2(ln𝑞)^𝛼−𝑝^∗(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=_{𝑛0−𝑘+1}1ln𝑡_𝑖^𝛼=^∞_𝑖=𝑛_0−𝑘+11𝑖^𝛼(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 𝛽𝑡_𝑛∶=1(𝑛+1)∑^{𝑛−𝑗𝑖=𝑛−𝑘+1}(ln𝑞)/ln𝑡_𝑖=(𝑛+1)∑1^{𝑛−𝑗𝑖=𝑛−𝑘+1}1/𝑖.(4.56) Then it is easy to verify that (1.1) has a solution 𝑦𝑡_𝑛=^𝑛_𝑖=1ln𝑞ln𝑡_𝑖=^𝑛_𝑖=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 𝛽𝑡_𝑛=1𝑘−𝑗−2𝑘(+𝑘−𝑗+𝑗)1𝑛+𝒪𝑛1².(4.58) Inequality (4.49) is valid if 𝛽𝑡_𝑛=1𝑘−𝑗−2𝑘(+𝑘−𝑗+𝑗)1𝑛+𝒪𝑛1²≤𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)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 𝑦∶𝕋→ℝ⁺ of (1.1) expressed by (4.12) and (4.20), that is, by inequalities 𝛽𝑡_𝑛≤𝑐−𝑝𝑡_𝑛𝛽𝑡_𝑛,≤𝑐−ln𝑝𝑡_𝑛(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)={𝑡_𝑛}, where 𝑡_𝑛=𝑛. In this case, as stated in Theorem 4.5, (4.30), that is, 𝛽𝑡_𝑛≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑡_𝑛=1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)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)={𝑡_𝑛} where 𝑡_𝑛=𝑞^𝑛 and 𝑞>1. Then, as stated in Theorem 4.7, (4.49), that is, 𝛽𝑡_𝑛≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)ln𝑞2(𝑘−𝑗)ln𝑡_𝑛=1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)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 𝑡_𝑛 in every case, their right-hand sides are identical.

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

Theorem 4.9. Let 𝛽𝑡_𝑛≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛(4.63) holds for all 𝑛∈ℤ^∞𝑛_0−𝑘 and for a fixed 𝑝^∗>1. Let, moreover, 𝛼∈(1,𝑝^∗). Then there exists an increasing and asymptotically convergent solution 𝑦∶𝕋→ℝ⁺ of (1.1) satisfying 𝑦𝑡_𝑛≤^𝑛_𝑖=𝑛_0−𝑘+1𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)(𝑖−1)𝑖1^𝛼(4.64) for every 𝑛∈ℤ^∞𝑛₀. Such a solution is defined, for example, by the initial function 𝜑𝑡_𝑛∶=^𝑛_𝑖=𝑛_0−𝑘+1𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)(𝑖−1)𝑖1^𝛼,𝑛∈ℤ^𝑛0_𝑛0−𝑘.(4.65)

Proof. We will apply Theorem 4.2 with 𝛽^∗𝑡_𝑛∶=1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛,𝜀𝑡^𝑛∶=𝑛1_𝛼.(4.66) Inequality (4.5) turns into 𝜀𝑡_𝑛+1=1(𝑛+1)^𝛼≥^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝛽^∗𝑡_𝑖−1𝜀𝑡_𝑖=^𝑛−𝑗_𝑖=𝑛_−𝑘+1𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)(𝑖−1)𝑖1^𝛼.(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., 𝑡_𝑛=𝑛 for every 𝑛∈ℤ^∞𝑛_0−𝑘) 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=_{𝑛0−𝑘+1}𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)(𝑖−1)𝑖1^𝛼(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 𝑡_𝑛, the right-hand side of the crucial inequality (4.63) cannot, in a general situation of arbitrary time scale {𝑡_𝑛}, be explicitly expressed using only the 𝑡_𝑛 terms. This is only possible if, for a given time scale, a function 𝑓 is explicitly known such that 𝑓(𝑡_𝑛)=𝑛. Then, (4.63) can be written in the form 𝛽𝑡_𝑛≤1𝑘−𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑓𝑡_𝑛=𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)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𝑡_𝑛∶=1(𝑛+1)∑^{𝑛−𝑗𝑖=𝑛−𝑘+1}1/𝑖(4.70) has an increasing asymptotically divergent solution 𝑦(𝑡_𝑛)=∑^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𝑡_𝑛=1(𝑛+1)∑^{𝑛−𝑗𝑖=𝑛−𝑘+1}1/𝑖≤𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)𝑛(4.71) holds, or if 𝛽𝑡_𝑛=1𝑘−𝑗−2𝑘(+𝑘−𝑗+𝑗)1𝑛+𝒪𝑛1²≤𝑘−1𝑗−𝑝^∗(𝑘+𝑗+1)2(𝑘−𝑗)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).