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

Authors: J. Diblík; M. Růžičková; Z. Šmarda; Z. Šutá
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The paper investigates a dynamic equationΔy(tn)=β(tn)[y(tn−j)−y(tn−k)] for n→∞, where k and j are integers such that k>j≥0, on an arbitrary discrete time scale 𝕋:={tn} with tn∈ℝ, n∈ℤn0−k∞={n0−k,n0−k+1,…}, n0∈ℕ, tn<tn+1, Δy(tn)=y(tn+1)−y(tn), and limn→∞tn=∞. 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.


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## Body

## 1. Introduction

We use the following notation: for an integers, we define that ℤs∞:={s,s+1,…}, and if an integer q≥s, we define ℤsq:={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 scale𝕋=T(t) as an arbitrary unbounded increasing sequence of real numbers, that is, T(t):={tn}, where tn∈ℝ, n∈ℤn0-k∞, n0∈ℕ, k>0 is an integer, tn<tn+1, and limn→∞tn=∞. For a fixed v∈ℤn0-k∞, we define a time scale 𝕋v=Tv(t):={tn}, where n∈ℤv∞. Obviously, Tn0-k(t)=T(t). In addition, for integers s, q, q≥s≥n0-k, we define the set 𝕋sq=Tsq(t):={ts,ts+1,…,tq}.In the paper we study a dynamic equation(1.1)Δy(tn)=β(tn)[y(tn-j)-y(tn-k)]
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>j≥0, and β:𝕋→ℝ+:=(0,∞). Without loss of generality, we assume that tn0-k>0 (this is a technical detail, necessary for some expressions to be well defined). Throughout the paper, we adopt the notation ∑i=k+1kℬ(ti)=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 asn→∞.The paper is organized as follows. In Section2, 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

Let𝒞:=𝒞(𝕋n0-kn0,ℝ) be the space of discrete functions mapping the discrete interval 𝕋n0-kn0 into ℝ. Let v∈ℤn0∞ be given. The function y:𝕋v-k→ℝ is said to be a solution of (1.1) on𝕋v-k if it satisfies (1.1) for every n∈ℤv∞. A solution y of (1.1) on 𝕋v-k is asymptotically convergent if the limit limn→∞y(tn) exists and is finite. For a given v∈ℤn0∞ and φ∈𝒞, 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 𝕋v-k and y(tv,φ)(tv+m)=φ(tm) for m∈ℤ-k0.


### 2.1. Auxiliary Inequality

The inequality(2.1)Δω(tn)≥β(tn)[ω(tn-j)-ω(tn-k)]
is a helpful tool in the analysis of solutions of (1.1). Let v∈ℤn0∞. The function ω:𝕋v-k→ℝ is said to be a solution of (2.1) on 𝕋v-k if ω satisfies (2.1) for n∈ℤv∞. A solution ω of (2.1) on 𝕋v-k is asymptotically convergent if the limit limn→∞ω(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.
Letφ∈𝒞 be increasing (nondecreasing, decreasing, nonincreasing) on 𝕋n0-kn0. Then the solution y(n0,φ)(tn) of (1.1), where n∈ℤn0∞ is increasing (nondecreasing, decreasing, nonincreasing) on 𝕋n0, too.Lemma 2.2.
Letφ∈𝒞 be increasing (nondecreasing) and ω:𝕋→ℝ be a solution of inequality (2.1) with ω(tm)=φ(tm), m∈ℤn0-kn0. Then, ω(tn), where n∈ℤn0∞ 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, n∈ℤn0-k∞, k>j≥0.Theorem 2.3.
Letω: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
(2.2)y(tn)≤ω(tn)
holds for every n∈ℤn0-k∞. In particular, a solution y(n0,ϕ) of (1.1) with ϕ∈𝒞, defined by
(2.3)ϕ(tn):=ω(tn),n∈Zn0-kn0,
is such a solution.Proof.
Letω(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,
(2.4)y(n0,ϕ)(tn)≤ω(tn)
for every n∈ℤn0-k∞. Let W:𝕋→ℝ be defined by
(2.5)W(tn):=ω(tn)-y(tn).
Then W(tn)=0 if n∈ℤn0-kn0 and, in addition, W is a solution of (2.1) on 𝕋. Lemma 2.2 implies that W is nondecreasing. Consequently,
(2.6)W(tn)=ω(tn)-y(tn)≥W(tn0)=ω(tn0)-y(tn0)=0,
and y(tn)≤ω(tn) 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 functionɛ:𝕋→ℝ+ such that
(2.7)ɛ(tn+1)≥∑i=n-k+1n-jβ(ti-1)ɛ(ti)
for every n∈ℤn0∞. Then there exists a solution ω=ωɛ of (2.1) defined on 𝕋 and having the form
(2.8)ωɛ(tn):=∑i=n0-k+1nβ(ti-1)ɛ(ti).Proof.
Assuming thatωɛ defined by (2.8) is a solution of (2.1) for n∈ℤn0∞, we will deduce the inequality for ɛ. We get
(2.9)Δωɛ(tn)=ωɛ(tn+1)-ωɛ(tn)=∑i=n0-k+1n+1β(ti-1)ɛ(ti)-∑i=n0-k+1nβ(ti-1)ɛ(ti)=β(tn)ɛ(tn+1),ωɛ(tn-j)-ωɛ(tn-k)=∑i=n0-k+1n-jβ(ti-1)ɛ(ti)-∑i=n0-k+1n-kβ(ti-1)ɛ(ti)=∑i=n-k+1n-jβ(ti-1)ɛ(ti).
We substitute ωɛ for ω in (2.1). Then, using (2.9), (2.1) turns into
(2.10)β(tn)ɛ(tn+1)≥β(tn)∑n-k+1n-jβ(ti-1)ɛ(ti).
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 functionφ∈𝒞 can be decomposed into the difference of two increasing functions φj∈𝒞, j=1,2, that is,
(2.11)φ(tn)=φ1(tn)-φ2(tn),n∈Zn0-kn0.Proof.
Let constantsMn>0, n∈ℤn0-kn0 be such that
(2.12)Mn+1>Mn+max⁡{0,φ(tn)-φ(tn+1)}
is valid for each n∈ℤn0-kn0-1. We set
(2.13)φ1(tn):=φ(tn)+Mn,n∈Zn0-kn0,φ2(tn):=Mn,n∈Zn0-kn0.
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 n∈ℤn0-kn0-1, which means that
(2.14)φ(tn+1)+Mn+1>φ(tn)+Mn
or
(2.15)Mn+1>Mn+φ(tn)-φ(tn+1).
We conclude that the last inequality holds because, due to (2.12), we have
(2.16)Mn+1>Mn+max⁡{0,φ(tn)-φ(tn+1)}≥Mn+φ(tn)-φ(tn+1).
The inequality φ2(tn+1)>φ2(tn) obviously holds for every n∈ℤn0-kn0-1 due to (2.12) as well.


### 2.4. Auxiliary Asymptotic Decomposition

The following lemma can be proved easily by induction. The symbol𝒪 (capital “O”) stands for the Landau order symbol.Lemma 2.6.
For fixedr, σ∈ℝ∖{0}, the asymptotic representation
(2.17)(n-r)σ=nσ[1-σrn+O(1n2)]
holds for n→∞.


## 2.1. Auxiliary Inequality

The inequality(2.1)Δω(tn)≥β(tn)[ω(tn-j)-ω(tn-k)]
is a helpful tool in the analysis of solutions of (1.1). Let v∈ℤn0∞. The function ω:𝕋v-k→ℝ is said to be a solution of (2.1) on 𝕋v-k if ω satisfies (2.1) for n∈ℤv∞. A solution ω of (2.1) on 𝕋v-k is asymptotically convergent if the limit limn→∞ω(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.
Letφ∈𝒞 be increasing (nondecreasing, decreasing, nonincreasing) on 𝕋n0-kn0. Then the solution y(n0,φ)(tn) of (1.1), where n∈ℤn0∞ is increasing (nondecreasing, decreasing, nonincreasing) on 𝕋n0, too.Lemma 2.2.
Letφ∈𝒞 be increasing (nondecreasing) and ω:𝕋→ℝ be a solution of inequality (2.1) with ω(tm)=φ(tm), m∈ℤn0-kn0. Then, ω(tn), where n∈ℤn0∞ 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, n∈ℤn0-k∞, k>j≥0.Theorem 2.3.
Letω: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
(2.2)y(tn)≤ω(tn)
holds for every n∈ℤn0-k∞. In particular, a solution y(n0,ϕ) of (1.1) with ϕ∈𝒞, defined by
(2.3)ϕ(tn):=ω(tn),n∈Zn0-kn0,
is such a solution.Proof.
Letω(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,
(2.4)y(n0,ϕ)(tn)≤ω(tn)
for every n∈ℤn0-k∞. Let W:𝕋→ℝ be defined by
(2.5)W(tn):=ω(tn)-y(tn).
Then W(tn)=0 if n∈ℤn0-kn0 and, in addition, W is a solution of (2.1) on 𝕋. Lemma 2.2 implies that W is nondecreasing. Consequently,
(2.6)W(tn)=ω(tn)-y(tn)≥W(tn0)=ω(tn0)-y(tn0)=0,
and y(tn)≤ω(tn) 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 functionɛ:𝕋→ℝ+ such that
(2.7)ɛ(tn+1)≥∑i=n-k+1n-jβ(ti-1)ɛ(ti)
for every n∈ℤn0∞. Then there exists a solution ω=ωɛ of (2.1) defined on 𝕋 and having the form
(2.8)ωɛ(tn):=∑i=n0-k+1nβ(ti-1)ɛ(ti).Proof.
Assuming thatωɛ defined by (2.8) is a solution of (2.1) for n∈ℤn0∞, we will deduce the inequality for ɛ. We get
(2.9)Δωɛ(tn)=ωɛ(tn+1)-ωɛ(tn)=∑i=n0-k+1n+1β(ti-1)ɛ(ti)-∑i=n0-k+1nβ(ti-1)ɛ(ti)=β(tn)ɛ(tn+1),ωɛ(tn-j)-ωɛ(tn-k)=∑i=n0-k+1n-jβ(ti-1)ɛ(ti)-∑i=n0-k+1n-kβ(ti-1)ɛ(ti)=∑i=n-k+1n-jβ(ti-1)ɛ(ti).
We substitute ωɛ for ω in (2.1). Then, using (2.9), (2.1) turns into
(2.10)β(tn)ɛ(tn+1)≥β(tn)∑n-k+1n-jβ(ti-1)ɛ(ti).
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 functionφ∈𝒞 can be decomposed into the difference of two increasing functions φj∈𝒞, j=1,2, that is,
(2.11)φ(tn)=φ1(tn)-φ2(tn),n∈Zn0-kn0.Proof.
Let constantsMn>0, n∈ℤn0-kn0 be such that
(2.12)Mn+1>Mn+max⁡{0,φ(tn)-φ(tn+1)}
is valid for each n∈ℤn0-kn0-1. We set
(2.13)φ1(tn):=φ(tn)+Mn,n∈Zn0-kn0,φ2(tn):=Mn,n∈Zn0-kn0.
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 n∈ℤn0-kn0-1, which means that
(2.14)φ(tn+1)+Mn+1>φ(tn)+Mn
or
(2.15)Mn+1>Mn+φ(tn)-φ(tn+1).
We conclude that the last inequality holds because, due to (2.12), we have
(2.16)Mn+1>Mn+max⁡{0,φ(tn)-φ(tn+1)}≥Mn+φ(tn)-φ(tn+1).
The inequality φ2(tn+1)>φ2(tn) obviously holds for every n∈ℤn0-kn0-1 due to (2.12) as well.


## 2.4. Auxiliary Asymptotic Decomposition

The following lemma can be proved easily by induction. The symbol𝒪 (capital “O”) stands for the Landau order symbol.Lemma 2.6.
For fixedr, σ∈ℝ∖{0}, the asymptotic representation
(2.17)(n-r)σ=nσ[1-σrn+O(1n2)]
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 ℤn0-k∞, then all the solutions of (1.1) defined on ℤn0-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 functionφ∈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(tn), n∈ℤn0-k∞. Without loss of generality, we assume that y(n0,φ)≢Y on ℤn0-k∞ since, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume
(3.1)ΔY(tn)>0,n∈Zn0-kn0-1.
Hence, there is a constant γ>0 such that
(3.2)ΔY(tn)-γΔy(tn)>0,n∈Zn0-kn0-1
or
(3.3)Δ(Y(tn)-γy(tn))>0,n∈Zn0-kn0-1.
This implies that the function Y(tn)-γy(tn) is increasing on ℤn0-kn0-1, and Lemma 2.1 implies that Y(tn)-γy(tn) is increasing on ℤn0-k∞. Thus,
(3.4)Y(tn)-γy(tn)>Y(tn0)-γy(tn0),n∈Zn0∞
or
(3.5)y(tn)<y(tn0)+1γ(Y(tn)-Y(tn0)),n∈Zn0∞
and, consequently, y(tn) is a bounded function on ℤn0-k∞ 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 solutiony(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 Lemma2.5 that every discrete function φ∈𝒞 can be decomposed into the difference of two increasing discrete functions φj∈𝒞, j=1,2. In accordance with the previous part of the proof, every function φj∈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 Theorem3.1, 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.
Letω 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 Theorems2.3, 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 ℤn0-k∞.(b)
All solutions of (1.1) defined on ℤn0-k∞ are asymptotically convergent.(c)
Inequality (2.1) has a strictly monotone and asymptotically convergent solution on ℤn0-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 Lemma2.1, Theorem 2.3, and Lemma 2.4.Theorem 4.1.
Let there exists a functionɛ:𝕋→ℝ+ satisfying
(4.1)∑i=n0-k+1∞β(ti-1)ɛ(ti)<∞,ɛ(tn+1)≥∑i=n-k+1n-jβ(ti-1)ɛ(ti)
for every n∈ℤn0∞. Then the initial function
(4.2)φ(tn):=∑i=n0-k+1nβ(ti-1)ɛ(ti),n∈Zn0-kn0
defines an increasing and asymptotically convergent solution y(tn0,φ)(tn) of (1.1) on 𝕋 satisfying
(4.3)y(tn0,φ)(tn)≤∑i=n0-k+1nβ(ti-1)ɛ(ti)
for every n∈ℤn0∞.Although Theorem4.1 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 functionsβ*:𝕋→ℝ+ and ɛ:𝕋→ℝ+ such that the inequalities
(4.4)β(tn)≤β*(tn),(4.5)ɛ(tn+1)≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)
hold for all n∈ℤn0-k∞, and moreover
(4.6)∑i=n0-k+1∞β*(ti-1)ɛ(ti)<∞.
Then there exists an increasing and asymptotically convergent solution y:𝕋→ℝ of (1.1) satisfying
(4.7)y(tn)≤∑i=n0-k+1nβ(ti-1)ɛ(ti)
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.8)φ(tn):=∑i=n0-k+1nβ(ti-1)ɛ(ti),n∈Zn0-kn0.Proof.
From (4.5) and (4.6), we get
(4.9)ɛ(tn+1)≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)≥∑i=n-k+1n-jβ(ti-1)ɛ(ti),∞>∑i=n0-k+1∞β*(ti-1)ɛ(ti)≥∑i=n0-k+1∞β(ti-1)ɛ(ti).
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 functionβ* mentioned in Theorem 4.2 can have the form
(4.10)β*(tn)=c-γ(tn),
where c is a positive constant and γ:𝕋→ℝ+ is a suitable function such that γ(tn)<c (at least for all sufficiently large n) and (4.11)lim⁡n→∞γ(tn)=0.
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 constantsc>0, p>0 and α>0 such that
(4.12)β(tn)≤c-ptn,(4.13)1tn+1α≥∑i=n-k+1n-j[c-pti-1]1tiα
hold for all n∈ℤn0-k∞, and moreover
(4.14)∑i=n0-k+1∞1tiα<∞.
Then there exists an increasing and asymptotically convergent solution y:𝕋→ℝ+ of (1.1) satisfying
(4.15)y(tn)≤∑i=n0-k+1n[c-pti-1]1tiα
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.16)φ(tn):=∑i=n0-k+1n[c-pti-1]1tiα,n∈Zn0-kn0.Proof.
We will apply Theorem4.2 with
(4.17)β*(tn):=c-ptn,ɛ(tn):=1tnα.
Inequality (4.5) turns into
(4.18)ɛ(tn+1)=1tn+1α≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)=∑i=n-k+1n-j[c-pti-1]1tiα
and is true due to (4.13). Inequality (4.6) holds due to assumption (4.14) as well because limn→∞tn=∞ and
(4.19)∑i=n0-k+1∞β*(ti-1)ɛ(ti)=∑i=n0-k+1∞[c-pti-1]1tiα<∞.
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 constantsc>0, p>0 and α>0 such that the inequalities
(4.20)β(tn)≤c-pln⁡tn,(4.21)1(ln⁡tn+1)α≥∑i=n-k+1n-j[c-pln⁡ti-1]1(ln⁡ti)α
hold for all n∈ℤn0-k∞, and moreover
(4.22)∑i=n0-k+1∞1(ln⁡ti)α<∞.
Then there exists an increasing and asymptotically convergent solution y:𝕋→ℝ+ of (1.1) satisfying
(4.23)y(tn)≤∑i=n0-k+1n[c-pln⁡ti-1]1(ln⁡ti)α
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.24)φ(tn):=∑i=n0-k+1n[c-pln⁡ti-1]1(ln⁡ti)α,n∈Zn0-kn0.Proof.
We will apply Theorem4.2 with
(4.25)β*(tn):=c-pln⁡tn,ɛ(tn):=1(ln⁡tn)α.
Inequality (4.5) turns into
(4.26)ɛ(tn+1)=1(ln⁡tn+1)α≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)=∑i=n-k+1n-j[c-pln⁡ti-1]1(ln⁡ti)α
and is true due to (4.21). Inequality (4.6) holds due to assumption (4.22) as well because limn→∞tn=∞ and
(4.27)∑i=n0-k+1∞β*(ti-1)ɛ(ti)=∑i=n0-k+1∞[c-pln⁡ti-1]1(ln⁡ti)α<∞.
Now, all assumptions of Theorem 4.2 are true, and its statement gives the statement of Theorem 4.4.


### 4.2. Time ScaleT(t):={n(1+δ(n))}

Now, using Theorem4.3, 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)={tn},tn:=n(1+δ(n)), where δ:𝕋→ℝ, |δ(n)|≤δ*, δ*∈(0,1), n∈ℤn0-k∞, and (4.28)δ(n)=O(1n2).Theorem 4.5.
Let (4.12) be true for
(4.29)c:=1k-j,p:=p*(k+j+1)2(k-j),
where p*>1, that is,
(4.30)β(tn)≤1k-j-p*(k+j+1)2(k-j)tn
holds for all n∈ℤn0-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∈ℤn0∞. Such a solution is defined, for example, by the initial function (4.16).Proof.
We use Theorem4.3 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(tn) of (4.13), we have
(4.31)R(tn)=∑i=n-k+1n-j[1k-j-p*(k+j+1)2(k-j)ti-1]1tiα=1k-j∑i=n-k+1n-j1tiα-p*(k+j+1)2(k-j)∑i=n-k+1n-j1ti-1tiα=1k-j∑i=n-k+1n-j1iα(1+δ(i))α-p*(k+j+1)2(k-j)∑i=n-k+1n-j1(i-1)(1+δ(i-1))iα(1+δ(i))α.
Since i∈{n-k+1,n-k+2,…,n-j} and n→∞, we can asymptotically decompose ℛ(tn) 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
(4.32)1iα=1(n-(n-i))α=1nα[1+α(n-i)n+O(1n2)].
In addition to this, we have
(4.33)1(1+δ(i))α=1+O(1i2)=1+O(1n2).
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
(4.34)1i-1=1n-(n-i+1)=1n⋅11+(n-i+1)/n=1n⋅(1+O(1n)),(4.35)11+δ(i-1)=1+O(1(i-1)2)=1+O(1n2).
Then we get
(4.36)R(tn)=1(k-j)nα[1+O(1n2)]∑i=n-k+1n-j[1+α(n-i)n+O(1n2)]-p*(k+j+1)2(k-j)nα+1[1+O(1n)]∑i=n-k+1n-j[1+O(1n)]=1(k-j)nα[1+α(k-1)n+1+α(k-2)n+⋯+1+αjn+O(1n2)]-p*(k+j+1)2(k-j)nα+1[1+1+⋯+1+O(1n)]=1(k-j)nα+1[(k-j)n+α(k-1)+α(k-2)+⋯+αj+O(1n)]-p*(k+j+1)2(k-j)nα+1[(k-j)+O(1n)]=1nα+α(k-j)nα+1(k+j-1)(k-j)2-p*(k+j+1)2(k-j)nα+1(k-j)+O(1nα+2),
and, finally,
(4.37)R(tn)=1nα+α2nα+1(k+j-1)-p*(k+j+1)2nα+1+O(1nα+2).
A similar decomposition of the left-hand side ℒ(tn) in (4.13) leads to (we use the decomposition formula (2.17) in Lemma 2.6 with σ=-α and r=-1)
(4.38)L(tn)=1tn+1α=1(n+1)α(1+δ(n+1))α=1nα[1-αn+O(1n2)][1+O(1n2)]=1nα-αnα+1+O(1nα+2).
Comparing ℒ(tn) and ℛ(tn), we see that, for ℒ(tn)≥ℛ(tn), 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
(4.39)-α>12α(k+j-1)-12p*(k+j+1).
Simplifying this inequality, we get
(4.40)12p*(k+j+1)>α+12α(k+j-1),
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
(4.41)∑i=n0-k+1∞1tiα=∑i=n0-k+1∞1iα(1+δ(i))α
is convergent for α>1.
Thus, all assumptions of Theorem4.3 are fulfilled and, from the conclusions, we deduce that all conclusions of Theorem 4.5 hold.


### 4.3. Time ScaleT(t):={n}

The time scale𝕋=T(t)={tn},tn:=n, where n∈ℤn0-k∞ is a particular case of the previous time scale defined in Section 4.2 if δ(n)=0 for every n∈ℤn0-k∞. Then (1.1) turns into(4.42)Δy(n)=β(n)[y(n-j)-y(n-k)]
and (4.30), which is crucial for the existence of an increasing and asymptotically convergent solution, takes the form(4.43)β(n)≤1k-j-p*(k+j+1)2(k-j)n,n∈Zn0-k∞
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
(4.44)β(n):=1(n+1)∑i=n-k+1n-j1/i.
It is easy to verify that (4.42) has a solution y(n)=∑i=1n1/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
(4.45)β(n)=1(n+1)∑i=n-k+1n-j1/i=11+1/n⋅1∑i=1k-j(1/(1+((i-k)/n)))=11+1/n⋅1∑i=1k-j[1-(i-k)/n+O(1/n2)]=[1-1n+O(1n2)]⋅1k-j⋅11-∑i=1k-j[(i-k)/(k-j)n+O(1/n2)]=1k-j⋅[1-1n+O(1n2)]⋅[1+∑i=1k-ji-k(k-j)n+O(1n2)]=1k-j⋅[1-1n+O(1n2)]⋅[1-kn+k-j+12n+O(1n2)]=1k-j⋅[1-kn+k-j+12n-1n+O(1n2)]=1k-j-k+j+12(k-j)n+O(1n2).
Now, (4.43) requires that
(4.46)β(n)=1k-j-k+j+12(k-j)n+O(1n2)≤1k-j-p*(k+j+1)2(k-j)n.
The last will hold if
(4.47)-k+j+12(k-j)n<-p*(k+j+1)2(k-j)n,
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 ScaleT(t):={qn}, q>1

We will focus our attention on the sufficient conditions for the existence of an increasing and asymptotically convergent solutiony:𝕋→ℝ+ of (1.1) if the time scale is defined as 𝕋=T(t)={tn},tn:=qn, where n∈ℤn0-k∞ and q>1. We will apply Theorem 4.4.Theorem 4.7.
Let (4.20) hold for
(4.48)c:=1k-j,p:=p*(k+j+1)ln⁡q2(k-j),
where p*>1, that is, the inequality
(4.49)β(tn)≤1k-j-p*(k+j+1)ln⁡q2(k-j)ln⁡tn=1k-j-p*(k+j+1)2(k-j)n
holds for all n∈ℤn0-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∈ℤn0∞. Such a solution is defined, for example, by the initial function (4.24).Proof.
We use Theorem4.4 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(tn) of (4.21), we have
(4.50)R(tn)=∑i=n-k+1n-j[1k-j-p*(k+j+1)ln⁡⁡q2(k-j)ln⁡⁡ti-1]1(ln⁡ti)α=1k-j∑i=n-k+1n-j1(ln⁡ti)α-p*(k+j+1)ln⁡q2(k-j)∑i=n-k+1n-j1(ln⁡ti-1)(ln⁡ti)α=1k-j∑i=n-k+1n-j1iα(ln⁡q)α-p*(k+j+1)ln⁡q2(k-j)∑i=n-k+1n-j1(i-1)iα(ln⁡q)α+1=1(k-j)(ln⁡q)α∑i=n-k+1n-j1iα-p*(k+j+1)2(k-j)(ln⁡q)α∑i=n-k+1n-j1(i-1)iα=[weapplydecompositions(4.32)and(4.34)]=1(k-j)(ln⁡q)αnα∑i=n-k+1n-j[1+α(n-i)n+O(1n2)]-p*(k+j+1)2(k-j)(ln⁡q)αnα+1∑i=n-k+1n-j[1+O(1n)].
Finally, applying some of the computations from the proof of Theorem 4.5, we get
(4.51)R(tn)=1(ln⁡q)αnα+α2(ln⁡q)αnα+1(k+j-1)-p*(k+j+1)2(ln⁡q)αnα+1+O(1nα+2).
and, for the left-hand side ℒ(tn) of (4.20),
(4.52)L(tn)=1(ln⁡q)α(n+1)α=1(ln⁡q)αnα-α(ln⁡q)αnα+1+O(1nα+2).
Comparing ℒ(tn) and ℛ(tn), we see that, for ℒ(tn)≥ℛ(tn),
(4.53)-α(ln⁡q)α>α(k+j-1)2(ln⁡q)α-p*(k+j+1)2(ln⁡q)α
is sufficient. Simplifying it, we get
(4.54)p*(k+j+1)>α(k+j+1),
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
(4.55)∑i=n0-k+1∞1(ln⁡ti)α=∑i=n0-k+1∞1iα(ln⁡q)α
is convergent for α>1.
Thus, all assumptions of Theorem4.4 are true, and, from its conclusions, we deduce that all conclusions of Theorem 4.7 are true.Example 4.8.
Consider (1.1), where
(4.56)β(tn):=1(n+1)∑i=n-k+1n-j(ln⁡q)/(ln⁡ti)=1(n+1)∑i=n-k+1n-j1/i.
Then it is easy to verify that (1.1) has a solution
(4.57)y(tn)=∑i=1nln⁡qln⁡ti=∑i=1n1i,
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
(4.58)β(tn)=1k-j-k+j+12(k-j)n+O(1n2).
Inequality (4.49) is valid if
(4.59)β(tn)=1k-j-k+j+12(k-j)n+O(1n2)≤1k-j-p*(k+j+1)2(k-j)n,
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 solutiony:𝕋→ℝ+ of (1.1) expressed by (4.12) and (4.20), that is, by inequalities (4.60)β(tn)≤c-ptn,β(tn)≤c-pln⁡tn
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, (4.61)β(tn)≤1k-j-p*(k+j+1)2(k-j)tn=1k-j-p*(k+j+1)2(k-j)n
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, (4.62)β(tn)≤1k-j-p*(k+j+1)ln⁡q2(k-j)ln⁡tn=1k-j-p*(k+j+1)2(k-j)n
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 scaleT(t)={tn} with properties described in introduction.Theorem 4.9.
Let(4.63)β(tn)≤1k-j-p*(k+j+1)2(k-j)n
holds for all n∈ℤn0-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
(4.64)y(tn)≤∑i=n0-k+1n[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.65)φ(tn):=∑i=n0-k+1n[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα,n∈Zn0-kn0.Proof.
We will apply Theorem4.2 with
(4.66)β*(tn):=1k-j-p*(k+j+1)2(k-j)n,ɛ(tn):=1nα.
Inequality (4.5) turns into
(4.67)ɛ(tn+1)=1(n+1)α≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)=∑i=n-k+1n-j[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα.
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 n∈ℤn0-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
(4.68)∑i=n0-k+1∞[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα
is obviously convergent.Remark 4.10.
Although Theorem4.9 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
(4.69)β(tn)≤1k-j-p*(k+j+1)2(k-j)f(tn)=1k-j-p*(k+j+1)2(k-j)n.Remark 4.11.
On the other hand, in a sense, Theorem4.9 gives the best possible result. Indeed, (1.1) with
(4.70)β(tn):=1(n+1)∑i=n-k+1n-j1/i
has an increasing asymptotically divergent solution y(tn)=∑i=1n1/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,
(4.71)β(tn)=1(n+1)∑i=n-k+1n-j1/i≤1k-j-p*(k+j+1)2(k-j)n
holds, or if
(4.72)β(tn)=1k-j-k+j+12(k-j)n+O(1n2)≤1k-j-p*(k+j+1)2(k-j)n.
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.


## 4.1. Some Special Criteria

It will be demonstrated by examples that, in many applications, the functionβ* mentioned in Theorem 4.2 can have the form
(4.10)β*(tn)=c-γ(tn),
where c is a positive constant and γ:𝕋→ℝ+ is a suitable function such that γ(tn)<c (at least for all sufficiently large n) and (4.11)lim⁡n→∞γ(tn)=0.
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 constantsc>0, p>0 and α>0 such that
(4.12)β(tn)≤c-ptn,(4.13)1tn+1α≥∑i=n-k+1n-j[c-pti-1]1tiα
hold for all n∈ℤn0-k∞, and moreover
(4.14)∑i=n0-k+1∞1tiα<∞.
Then there exists an increasing and asymptotically convergent solution y:𝕋→ℝ+ of (1.1) satisfying
(4.15)y(tn)≤∑i=n0-k+1n[c-pti-1]1tiα
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.16)φ(tn):=∑i=n0-k+1n[c-pti-1]1tiα,n∈Zn0-kn0.Proof.
We will apply Theorem4.2 with
(4.17)β*(tn):=c-ptn,ɛ(tn):=1tnα.
Inequality (4.5) turns into
(4.18)ɛ(tn+1)=1tn+1α≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)=∑i=n-k+1n-j[c-pti-1]1tiα
and is true due to (4.13). Inequality (4.6) holds due to assumption (4.14) as well because limn→∞tn=∞ and
(4.19)∑i=n0-k+1∞β*(ti-1)ɛ(ti)=∑i=n0-k+1∞[c-pti-1]1tiα<∞.
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 constantsc>0, p>0 and α>0 such that the inequalities
(4.20)β(tn)≤c-pln⁡tn,(4.21)1(ln⁡tn+1)α≥∑i=n-k+1n-j[c-pln⁡ti-1]1(ln⁡ti)α
hold for all n∈ℤn0-k∞, and moreover
(4.22)∑i=n0-k+1∞1(ln⁡ti)α<∞.
Then there exists an increasing and asymptotically convergent solution y:𝕋→ℝ+ of (1.1) satisfying
(4.23)y(tn)≤∑i=n0-k+1n[c-pln⁡ti-1]1(ln⁡ti)α
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.24)φ(tn):=∑i=n0-k+1n[c-pln⁡ti-1]1(ln⁡ti)α,n∈Zn0-kn0.Proof.
We will apply Theorem4.2 with
(4.25)β*(tn):=c-pln⁡tn,ɛ(tn):=1(ln⁡tn)α.
Inequality (4.5) turns into
(4.26)ɛ(tn+1)=1(ln⁡tn+1)α≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)=∑i=n-k+1n-j[c-pln⁡ti-1]1(ln⁡ti)α
and is true due to (4.21). Inequality (4.6) holds due to assumption (4.22) as well because limn→∞tn=∞ and
(4.27)∑i=n0-k+1∞β*(ti-1)ɛ(ti)=∑i=n0-k+1∞[c-pln⁡ti-1]1(ln⁡ti)α<∞.
Now, all assumptions of Theorem 4.2 are true, and its statement gives the statement of Theorem 4.4.


## 4.2. Time ScaleT(t):={n(1+δ(n))}

Now, using Theorem4.3, 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)={tn},tn:=n(1+δ(n)), where δ:𝕋→ℝ, |δ(n)|≤δ*, δ*∈(0,1), n∈ℤn0-k∞, and (4.28)δ(n)=O(1n2).Theorem 4.5.
Let (4.12) be true for
(4.29)c:=1k-j,p:=p*(k+j+1)2(k-j),
where p*>1, that is,
(4.30)β(tn)≤1k-j-p*(k+j+1)2(k-j)tn
holds for all n∈ℤn0-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∈ℤn0∞. Such a solution is defined, for example, by the initial function (4.16).Proof.
We use Theorem4.3 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(tn) of (4.13), we have
(4.31)R(tn)=∑i=n-k+1n-j[1k-j-p*(k+j+1)2(k-j)ti-1]1tiα=1k-j∑i=n-k+1n-j1tiα-p*(k+j+1)2(k-j)∑i=n-k+1n-j1ti-1tiα=1k-j∑i=n-k+1n-j1iα(1+δ(i))α-p*(k+j+1)2(k-j)∑i=n-k+1n-j1(i-1)(1+δ(i-1))iα(1+δ(i))α.
Since i∈{n-k+1,n-k+2,…,n-j} and n→∞, we can asymptotically decompose ℛ(tn) 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
(4.32)1iα=1(n-(n-i))α=1nα[1+α(n-i)n+O(1n2)].
In addition to this, we have
(4.33)1(1+δ(i))α=1+O(1i2)=1+O(1n2).
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
(4.34)1i-1=1n-(n-i+1)=1n⋅11+(n-i+1)/n=1n⋅(1+O(1n)),(4.35)11+δ(i-1)=1+O(1(i-1)2)=1+O(1n2).
Then we get
(4.36)R(tn)=1(k-j)nα[1+O(1n2)]∑i=n-k+1n-j[1+α(n-i)n+O(1n2)]-p*(k+j+1)2(k-j)nα+1[1+O(1n)]∑i=n-k+1n-j[1+O(1n)]=1(k-j)nα[1+α(k-1)n+1+α(k-2)n+⋯+1+αjn+O(1n2)]-p*(k+j+1)2(k-j)nα+1[1+1+⋯+1+O(1n)]=1(k-j)nα+1[(k-j)n+α(k-1)+α(k-2)+⋯+αj+O(1n)]-p*(k+j+1)2(k-j)nα+1[(k-j)+O(1n)]=1nα+α(k-j)nα+1(k+j-1)(k-j)2-p*(k+j+1)2(k-j)nα+1(k-j)+O(1nα+2),
and, finally,
(4.37)R(tn)=1nα+α2nα+1(k+j-1)-p*(k+j+1)2nα+1+O(1nα+2).
A similar decomposition of the left-hand side ℒ(tn) in (4.13) leads to (we use the decomposition formula (2.17) in Lemma 2.6 with σ=-α and r=-1)
(4.38)L(tn)=1tn+1α=1(n+1)α(1+δ(n+1))α=1nα[1-αn+O(1n2)][1+O(1n2)]=1nα-αnα+1+O(1nα+2).
Comparing ℒ(tn) and ℛ(tn), we see that, for ℒ(tn)≥ℛ(tn), 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
(4.39)-α>12α(k+j-1)-12p*(k+j+1).
Simplifying this inequality, we get
(4.40)12p*(k+j+1)>α+12α(k+j-1),
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
(4.41)∑i=n0-k+1∞1tiα=∑i=n0-k+1∞1iα(1+δ(i))α
is convergent for α>1.
Thus, all assumptions of Theorem4.3 are fulfilled and, from the conclusions, we deduce that all conclusions of Theorem 4.5 hold.


## 4.3. Time ScaleT(t):={n}

The time scale𝕋=T(t)={tn},tn:=n, where n∈ℤn0-k∞ is a particular case of the previous time scale defined in Section 4.2 if δ(n)=0 for every n∈ℤn0-k∞. Then (1.1) turns into(4.42)Δy(n)=β(n)[y(n-j)-y(n-k)]
and (4.30), which is crucial for the existence of an increasing and asymptotically convergent solution, takes the form(4.43)β(n)≤1k-j-p*(k+j+1)2(k-j)n,n∈Zn0-k∞
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
(4.44)β(n):=1(n+1)∑i=n-k+1n-j1/i.
It is easy to verify that (4.42) has a solution y(n)=∑i=1n1/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
(4.45)β(n)=1(n+1)∑i=n-k+1n-j1/i=11+1/n⋅1∑i=1k-j(1/(1+((i-k)/n)))=11+1/n⋅1∑i=1k-j[1-(i-k)/n+O(1/n2)]=[1-1n+O(1n2)]⋅1k-j⋅11-∑i=1k-j[(i-k)/(k-j)n+O(1/n2)]=1k-j⋅[1-1n+O(1n2)]⋅[1+∑i=1k-ji-k(k-j)n+O(1n2)]=1k-j⋅[1-1n+O(1n2)]⋅[1-kn+k-j+12n+O(1n2)]=1k-j⋅[1-kn+k-j+12n-1n+O(1n2)]=1k-j-k+j+12(k-j)n+O(1n2).
Now, (4.43) requires that
(4.46)β(n)=1k-j-k+j+12(k-j)n+O(1n2)≤1k-j-p*(k+j+1)2(k-j)n.
The last will hold if
(4.47)-k+j+12(k-j)n<-p*(k+j+1)2(k-j)n,
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 ScaleT(t):={qn}, q>1

We will focus our attention on the sufficient conditions for the existence of an increasing and asymptotically convergent solutiony:𝕋→ℝ+ of (1.1) if the time scale is defined as 𝕋=T(t)={tn},tn:=qn, where n∈ℤn0-k∞ and q>1. We will apply Theorem 4.4.Theorem 4.7.
Let (4.20) hold for
(4.48)c:=1k-j,p:=p*(k+j+1)ln⁡q2(k-j),
where p*>1, that is, the inequality
(4.49)β(tn)≤1k-j-p*(k+j+1)ln⁡q2(k-j)ln⁡tn=1k-j-p*(k+j+1)2(k-j)n
holds for all n∈ℤn0-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∈ℤn0∞. Such a solution is defined, for example, by the initial function (4.24).Proof.
We use Theorem4.4 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(tn) of (4.21), we have
(4.50)R(tn)=∑i=n-k+1n-j[1k-j-p*(k+j+1)ln⁡⁡q2(k-j)ln⁡⁡ti-1]1(ln⁡ti)α=1k-j∑i=n-k+1n-j1(ln⁡ti)α-p*(k+j+1)ln⁡q2(k-j)∑i=n-k+1n-j1(ln⁡ti-1)(ln⁡ti)α=1k-j∑i=n-k+1n-j1iα(ln⁡q)α-p*(k+j+1)ln⁡q2(k-j)∑i=n-k+1n-j1(i-1)iα(ln⁡q)α+1=1(k-j)(ln⁡q)α∑i=n-k+1n-j1iα-p*(k+j+1)2(k-j)(ln⁡q)α∑i=n-k+1n-j1(i-1)iα=[weapplydecompositions(4.32)and(4.34)]=1(k-j)(ln⁡q)αnα∑i=n-k+1n-j[1+α(n-i)n+O(1n2)]-p*(k+j+1)2(k-j)(ln⁡q)αnα+1∑i=n-k+1n-j[1+O(1n)].
Finally, applying some of the computations from the proof of Theorem 4.5, we get
(4.51)R(tn)=1(ln⁡q)αnα+α2(ln⁡q)αnα+1(k+j-1)-p*(k+j+1)2(ln⁡q)αnα+1+O(1nα+2).
and, for the left-hand side ℒ(tn) of (4.20),
(4.52)L(tn)=1(ln⁡q)α(n+1)α=1(ln⁡q)αnα-α(ln⁡q)αnα+1+O(1nα+2).
Comparing ℒ(tn) and ℛ(tn), we see that, for ℒ(tn)≥ℛ(tn),
(4.53)-α(ln⁡q)α>α(k+j-1)2(ln⁡q)α-p*(k+j+1)2(ln⁡q)α
is sufficient. Simplifying it, we get
(4.54)p*(k+j+1)>α(k+j+1),
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
(4.55)∑i=n0-k+1∞1(ln⁡ti)α=∑i=n0-k+1∞1iα(ln⁡q)α
is convergent for α>1.
Thus, all assumptions of Theorem4.4 are true, and, from its conclusions, we deduce that all conclusions of Theorem 4.7 are true.Example 4.8.
Consider (1.1), where
(4.56)β(tn):=1(n+1)∑i=n-k+1n-j(ln⁡q)/(ln⁡ti)=1(n+1)∑i=n-k+1n-j1/i.
Then it is easy to verify that (1.1) has a solution
(4.57)y(tn)=∑i=1nln⁡qln⁡ti=∑i=1n1i,
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
(4.58)β(tn)=1k-j-k+j+12(k-j)n+O(1n2).
Inequality (4.49) is valid if
(4.59)β(tn)=1k-j-k+j+12(k-j)n+O(1n2)≤1k-j-p*(k+j+1)2(k-j)n,
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 solutiony:𝕋→ℝ+ of (1.1) expressed by (4.12) and (4.20), that is, by inequalities (4.60)β(tn)≤c-ptn,β(tn)≤c-pln⁡tn
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, (4.61)β(tn)≤1k-j-p*(k+j+1)2(k-j)tn=1k-j-p*(k+j+1)2(k-j)n
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, (4.62)β(tn)≤1k-j-p*(k+j+1)ln⁡q2(k-j)ln⁡tn=1k-j-p*(k+j+1)2(k-j)n
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 scaleT(t)={tn} with properties described in introduction.Theorem 4.9.
Let(4.63)β(tn)≤1k-j-p*(k+j+1)2(k-j)n
holds for all n∈ℤn0-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
(4.64)y(tn)≤∑i=n0-k+1n[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα
for every n∈ℤn0∞. Such a solution is defined, for example, by the initial function
(4.65)φ(tn):=∑i=n0-k+1n[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα,n∈Zn0-kn0.Proof.
We will apply Theorem4.2 with
(4.66)β*(tn):=1k-j-p*(k+j+1)2(k-j)n,ɛ(tn):=1nα.
Inequality (4.5) turns into
(4.67)ɛ(tn+1)=1(n+1)α≥∑i=n-k+1n-jβ*(ti-1)ɛ(ti)=∑i=n-k+1n-j[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα.
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 n∈ℤn0-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
(4.68)∑i=n0-k+1∞[1k-j-p*(k+j+1)2(k-j)(i-1)]1iα
is obviously convergent.Remark 4.10.
Although Theorem4.9 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
(4.69)β(tn)≤1k-j-p*(k+j+1)2(k-j)f(tn)=1k-j-p*(k+j+1)2(k-j)n.Remark 4.11.
On the other hand, in a sense, Theorem4.9 gives the best possible result. Indeed, (1.1) with
(4.70)β(tn):=1(n+1)∑i=n-k+1n-j1/i
has an increasing asymptotically divergent solution y(tn)=∑i=1n1/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,
(4.71)β(tn)=1(n+1)∑i=n-k+1n-j1/i≤1k-j-p*(k+j+1)2(k-j)n
holds, or if
(4.72)β(tn)=1k-j-k+j+12(k-j)n+O(1n2)≤1k-j-p*(k+j+1)2(k-j)n.
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.



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*Source: 580750-2012-01-03.xml*