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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 764351, 12 pages
http://dx.doi.org/10.1155/2012/764351
Research Article

Asymptotic Upper and Lower Estimates of a Class of Positive Solutions of a Discrete Linear Equation with a Single Delay

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

Received 20 February 2012; Accepted 1 March 2012

Academic Editor: Elena Braverman

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

Abstract

We study a frequently investigated class of linear difference equations with a positive coefficient and a single delay k. Recently, it was proved that if the function is bounded above by a certain function, then there exists a positive vanishing solution of the considered equation, and the upper bound was found. Here we improve this result by finding even the lower bound for the positive solution, supposing the function is bounded above and below by certain functions.

1. Introduction

Throughout this paper, we use the following notation: for an integer , we define We investigate the asymptotic behavior as of the solutions of the discrete delayed equation of the -th order where is the independent variable assuming values from the set with a fixed . The number is the fixed delay, , and .

Along with (1.2), we consider initial conditions Initial problem (1.2), (1.3) obviously has a unique solution, defined for every . Moreover, the solution of (1.2) continuously depends on initial conditions (1.3).

Equation (1.2) is investigated very frequently. It was analyzed, for example, in papers [13] (where the comparison method [4, 5] was used) and [6]. Similar problems for differential and dynamic equations are studied, for example, in [710].

In a recent work of the authors [6], it is proved that if the function is bounded above by a certain function, then there exists a positive vanishing (i.e., tending to 0 as ) solution of the considered equation. Moreover, its upper bound was found. Our aim is to improve this result and to show that if the coefficient is between two functions and (see (2.3), (2.6), and (2.7) below) then (1.2) has a positive vanishing solution which is bounded from below by the function (see (2.5)) and from above by the function (see (2.4)). Due to the linearity of equation considered it becomes clear that a similar result holds for a one-parametric family of positive vanishing solutions of (1.2).

To prove this, we will use Theorem 1.1 which is one of the main results of [6]. This theorem is valid for any delayed difference equation of the form:

Theorem 1.1. Let , , be real functions defined on . Further, let be a continuous function and let the inequalities: hold for every and every such that Then there exists a solution of (1.4) satisfying the inequalities
for every .

For related comparison theorems for solutions of difference equations as well as related methods and their applications, see, for example, [1, 1121] and the related references therein. Investigation of positive solutions (and connected problems of oscillating solutions) attracted recently large attention. Except the references given above, one refers as well to [11, 2233], and to the references therein. Existence of positive solutions of some classes of difference equations has been also studied in papers [1216]. The existence of unbounded solutions by some comparison methods can be found, for example, in [17, 18].

2. Auxiliary Functions and Lemmas

Define the expression , , as where . We will write only instead of . Further, for a fixed integer define auxiliary functions: where , is a constant. Notice that if is sufficiently large, all these functions are well defined for .

Finally, let functions satisfy for the inequalities: for fixed and .

In [3], it was proved that if in (1.2) is a positive function bounded above by for some , then there exists a positive solution of (1.2) bounded above by the function for sufficiently large. Since , such solution will vanish as . This result was further improved in [6], where it was shown that (1.2) has a positive solution bounded above by even if the coefficient satisfies a less restrictive inequality, namely, . Here we will prove that function provides the lower estimate of the solution, supposing . The proof of this statement will be based on the following four lemmas. The symbols “” and “” stand for the Landau order symbols and are used as .

Lemma 2.1. For fixed and fixed , the asymptotic representation: holds as .

Proof. Relation (2.8) can be proved by induction with respect to , for details, see [6].

Lemma 2.2. For fixed and fixed , the asymptotic representations: hold as .

Proof. Both these relations are simple consequences of the asymptotic formula: and of Lemma 2.1 (for formula (2.9)).
In the case of relation (2.10), we put and .
To prove relation (2.9), first notice that dividing (2.8) by , we get Thus, putting and using (2.11), we get (2.9).

The following lemma is proved in [6].

Lemma 2.3. For fixed and fixed , the asymptotic representation: holds as .

Lemma 2.4. For fixed , and , the asymptotic representation: holds as .

Proof. Using Lemma 2.2 with and instead of , we get for Multiplying the asymptotic representations (2.14) and (2.16), we get (2.15).

3. Main Result

Now we are ready to prove that there exists a positive solution of (1.2) which is bounded below and above. Remind the functions , , , , and were defined by (2.3)–(2.6) and (2.7), respectively.

Theorem 3.1. Suppose that there exist numbers , and , such that the function in (1.2) satisfies the inequalities for every . Then there exists a solution of (1.2) such that for sufficiently large the inequalities: hold.

Proof. Show that all the assumptions of Theorem 1.1 are fulfilled. For (1.2), This is a continuous function. Put We have to prove that for every such that the inequalities (1.5) and (1.6) hold for sufficiently large. Start with (1.5). That gives that for it has to be which is equivalent to the inequality Denote the left-hand side of (3.8) as . As and as by (2.3), (2.6), and (3.1) we have Further, we can easily see that Thus, to prove (3.8), it suffices to show that for sufficiently large, the following inequality holds: Denote the left-hand side of inequality (3.12) as and the right-hand side as . In the following computation we will use the fact that and and we will omit all the terms which are of order . Applying Lemma 2.4 with and , we can write Using Lemma 2.4 with and , we get for It is easy to see that the inequality (3.12) reduces to This inequality is equivalent to The last inequality holds for sufficiently large because , , , and as , tend to zero faster than does.
Thus, we have proved that inequality (1.5) holds.
Next, according to (1.6), we have to prove that which is equivalent to the inequality: Denote the left-hand side of (3.21) as . As and as by (2.3), (3.1), and (2.7) we have Further, we can easily see that Thus, to prove (3.21), it suffices to show that for sufficiently large, the following inequality holds: Denote the left-hand side of inequality (3.25) as and the right-hand side as . Using Lemma 2.3 with and , we can write Using Lemma 2.3 with and , we get for It is easy to see that the inequality (3.25) reduces to This inequality is equivalent to The last inequality holds for sufficiently large because and . We have proved that all the assumptions of Theorem 1.1 are fulfilled and hence there exists a solution of (1.2) satisfying conditions (1.8), that is, in our case, conditions (3.2).

Acknowledgments

The research was supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30519 and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology.

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