#### Abstract

The paper investigates a dynamic equation for , where and are integers such that , on an arbitrary discrete time scale with , , , , , and . We assume . 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 . The results are presented as inequalities for the function . Examples demonstrate that the criteria obtained are sharp in a sense.

#### 1. Introduction

We use the following notation: for an integer , we define that , and if an integer , we define .

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 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 as an arbitrary unbounded increasing sequence of real numbers, that is, , where , , , is an integer, , and . For a fixed , we define a time scale , where . Obviously, . In addition, for integers , , , we define the set .

In the paper we study a dynamic equation as . The difference is defined as usual: , integers and in (1.1) satisfy the inequality , and . Without loss of generality, we assume that (this is a technical detail, necessary for some expressions to be well defined). Throughout the paper, we adopt the notation where is an integer and denotes the function under consideration.

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

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

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 be the space of discrete functions mapping the discrete interval into . Let be given. The function is said to be a *solution of* (1.1) *on * if it satisfies (1.1) for every . A solution of (1.1) on is *asymptotically convergent* if the limit exists and is finite. For a given and , we say that is a *solution of* (1.1) defined by the initial conditions if is a solution of (1.1) on and for .

##### 2.1. Auxiliary Inequality

The inequality
is a helpful tool in the analysis of solutions of (1.1). Let . The function is said to be a *solution* of (2.1) on if satisfies (2.1) for . A solution of (2.1) on is *asymptotically convergent* if the limit 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 be increasing (nondecreasing, decreasing, nonincreasing) on . Then the solution of (1.1), where is increasing (nondecreasing, decreasing, nonincreasing) on , too.*

Lemma 2.2. *Let be increasing (nondecreasing) and be a solution of inequality (2.1) with , . Then, , where is increasing (nondecreasing).*

The proofs of both lemmas above follow directly from the form of (1.1), (2.1), and from the properties , , .

Theorem 2.3. *Let be a solution of (2.1) on . Then there exists a solution of (1.1) on such that
**
holds for every . In particular, a solution of (1.1) with , defined by
**
is such a solution.*

* Proof. *Let be a solution of (2.1) defined on . We will show that the solution of (1.1) with defined by (2.3) satisfies (2.2), that is,
for every . Let be defined by
Then if and, in addition, is a solution of (2.1) on . Lemma 2.2 implies that is nondecreasing. Consequently,
and for all .

##### 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 such that
**
for every . Then there exists a solution of (2.1) defined on and having the form
*

* Proof. * Assuming that defined by (2.8) is a solution of (2.1) for , we will deduce the inequality for . We get
We substitute for in (2.1). Then, using (2.9), (2.1) turns into
Reducing the last inequality by , 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 can be decomposed into the difference of two increasing functions , , that is,
*

*Proof. *Let constants , be such that
is valid for each . We set
It is obvious that (2.11) holds. Now we verify that both functions , are increasing. The first one should satisfy for , which means that
or
We conclude that the last inequality holds because, due to (2.12), we have
The inequality obviously holds for every due to (2.12) as well.

##### 2.4. Auxiliary Asymptotic Decomposition

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

Lemma 2.6. *For fixed , , the asymptotic representation
**
holds for .*

#### 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 , then all the solutions of (1.1) defined on 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 is given. For definiteness, let 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 is monotone, that is, it is either increasing or nondecreasing. We prove that is convergent.

Denote the assumed increasing and asymptotically convergent solution of (1.1) as , . Without loss of generality, we assume that on since, in the opposite case, we can choose another initial function. Similarly, without loss of generality, we can assume
Hence, there is a constant such that
or
This implies that the function is increasing on , and Lemma 2.1 implies that is increasing on . Thus,
or
and, consequently, is a bounded function on because of the boundedness of . Obviously, in such a case, 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 generated by a nonmonotone initial function , there are two possibilities: is either eventually monotone and, consequently, convergent, or is eventually nonmonotone.

Now we use the statement of Lemma 2.5 that every discrete function can be decomposed into the difference of two increasing discrete functions , . In accordance with the previous part of the proof, every function , defines an increasing and asymptotically convergent solution . Now it is clear that the solution 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 be an increasing and bounded solution of (2.1) on . Then there exists an increasing and asymptotically convergent solution of (1.1) on .*

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 .*(b)*All solutions of (1.1) defined on are asymptotically convergent.*(c)*Inequality (2.1) has a strictly monotone and asymptotically convergent solution on . *

#### 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 satisfying
**
for every . Then the initial function
**
defines an increasing and asymptotically convergent solution of (1.1) on satisfying
**
for every .*

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 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 and such that the inequalities
**
hold for all , and moreover
**
Then there exists an increasing and asymptotically convergent solution of (1.1) satisfying
**
for every . Such a solution is defined, for example, by the initial function
*

*Proof. * From (4.5) and (4.6), we get
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 mentioned in Theorem 4.2 can have the form where is a positive constant and is a suitable function such that (at least for all sufficiently large ) and 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 , and such that
**
hold for all , and moreover
**
Then there exists an increasing and asymptotically convergent solution of (1.1) satisfying
**
for every . Such a solution is defined, for example, by the initial function
*

*Proof. *We will apply Theorem 4.2 with
Inequality (4.5) turns into
and is true due to (4.13). Inequality (4.6) holds due to assumption (4.14) as well because and
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 , and such that the inequalities
**
hold for all , and moreover
**
Then there exists an increasing and asymptotically convergent solution of (1.1) satisfying
**
for every . Such a solution is defined, for example, by the initial function
*

* Proof. * We will apply Theorem 4.2 with
Inequality (4.5) turns into
and is true due to (4.21). Inequality (4.6) holds due to assumption (4.22) as well because and
Now, all assumptions of Theorem 4.2 are true, and its statement gives the statement of Theorem 4.4.

##### 4.2. Time Scale

Now, using Theorem 4.3, we derive sufficient conditions for the existence of an increasing and asymptotically convergent solution of (1.1) in the case when the time scale is defined as , where , , , , and

Theorem 4.5. *Let (4.12) be true for
**
where , that is,
**
holds for all . Let, moreover, . Then there exists an increasing and asymptotically convergent solution of (1.1) satisfying (4.15) for . 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 is sufficiently large for the asymptotic computations performed below to be correct. Let us verify that (4.13) holds. For the right-hand side of (4.13), we have
Since and , we can asymptotically decompose as using decomposition formula (2.17) in Lemma 2.6. Applying this formula to the term in the first sum with and with , we get
In addition to this, we have
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
Then we get
and, finally,
A similar decomposition of the left-hand side in (4.13) leads to (we use the decomposition formula (2.17) in Lemma 2.6 with and )
Comparing and , we see that, for , it is necessary to match the coefficients of the terms because the coefficients of the terms are equal. It means that we need
Simplifying this inequality, we get
and, finally, . This inequality is assumed, and therefore (4.13) that holds is sufficiently large.

It remains to prove that (4.14) holds for . But it is a well-known fact that the series
is convergent for .

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

The time scale , where is a particular case of the previous time scale defined in Section 4.2 if for every . Then (1.1) turns into and (4.30), which is crucial for the existence of an increasing and asymptotically convergent solution, takes the form with a . 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
It is easy to verify that (4.42) has a solution , which is the th partial sum of harmonic series and, therefore, is divergent as . Now we asymptotically compare the function with the right-hand side of (4.43). First we develop an asymptotic decomposition of when . We get
Now, (4.43) requires that
The last will hold if
that is, if . This inequality is the opposite to 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 and was considered.

##### 4.4. Time Scale ,

We will focus our attention on the sufficient conditions for the existence of an increasing and asymptotically convergent solution of (1.1) if the time scale is defined as , where and . We will apply Theorem 4.4.

Theorem 4.7. *Let (4.20) hold for
**
where , that is, the inequality
**
holds for all . Let, moreover, . Then there exists an increasing and asymptotically convergent solution of (1.1) satisfying (4.23) for . 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 is sufficiently large for the asymptotic computations performed below to be correct. Let us verify (4.21). For the right-hand side of (4.21), we have
Finally, applying some of the computations from the proof of Theorem 4.5, we get
and, for the left-hand side of (4.20),
Comparing and , we see that, for ,
is sufficient. Simplifying it, we get
and, finally, . This inequality is assumed, and therefore (4.21) is valid if is sufficiently large.

It remains to prove that (4.22) holds for . But it is a well-known fact that the series
is convergent for .

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
Then it is easy to verify that (1.1) has a solution
which is the th partial sum of harmonic series and, as such, is divergent as . Now we asymptotically compare the function with the right-hand side of (4.49). Proceeding as in Example 4.6, we get
Inequality (4.49) is valid if
that is if . This inequality is the opposite to 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 of (1.1) expressed by (4.12) and (4.20), that is, by inequalities with suitable constants and , we can state the following. The first criterion (4.12) can successfully be used, for example, for the time scale , where . In this case, as stated in Theorem 4.5, (4.30), that is, is assumed with a .

The second criterion (4.20) can successfully be used, for example, for the time scale where and . Then, as stated in Theorem 4.7, (4.49), that is, is assumed with a . Comparing (4.61) and (4.62), we see that, although their left-hand sides are different due to different meaning of in every case, their right-hand sides are identical.

The following result gives a criterion for every discrete time scale with properties described in introduction.

Theorem 4.9. *Let
**
holds for all and for a fixed . Let, moreover, . Then there exists an increasing and asymptotically convergent solution of (1.1) satisfying
**
for every . Such a solution is defined, for example, by the initial function
*

* Proof. *We will apply Theorem 4.2 with
Inequality (4.5) turns into
Asymptotic decompositions of the left-hand and right-hand sides were used in the proof of Theorem 4.5 (if , i.e., for every ) 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 . (4.6) holds as well because the series
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 , the right-hand side of the crucial inequality (4.63) cannot, in a general situation of arbitrary time scale , be explicitly expressed using only the terms. This is only possible if, for a given time scale, a function is explicitly known such that . Then, (4.63) can be written in the form

*Remark 4.11. *On the other hand, in a sense, Theorem 4.9 gives the best possible result. Indeed, (1.1) with
has an increasing asymptotically divergent solution . 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,
holds, or if
The last holds for . This inequality is the opposite to 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).