Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

Migda, Janusz; Migda, Małgorzata; Nockowska-Rosiak, Magdalena

doi:https://doi.org/10.1155/2018/2368694

Discrete Dynamics in Nature and Society

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Volume 2018 | Article ID 2368694 | https://doi.org/10.1155/2018/2368694

Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

Janusz Migda,¹Małgorzata Migda,²and Magdalena Nockowska-Rosiak³

Academic Editor: Jorge E. Macias-Diaz

Received26 Apr 2018

Accepted21 Jun 2018

Published09 Jul 2018

Abstract

We consider the discrete Volterra type equation of the form . We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use , for given nonpositive real , as a measure of approximation.

1. Introduction

In this paper we consider the nonlinear Volterra sum-difference equation of nonconvolution type:Here , denote the set of positive integers and the set of real numbers, respectively. By a solution of we mean a sequence satisfying for large .

Discrete Volterra equations of different types are widely used in the process of modeling of some real phenomena or by applying a numerical method to a Volterra integral equation. Let . The general form of a Volterra sum-difference autonomous equation isSuch equations can be regarded as the discrete analogue of Volterra integrodifferential equations of the formThere are relatively few works devoted to the study of equations of type (2); see, for example, [1–4]. In [5], the asymptotic behaviors of nonoscillatory solutions of the higher-order integrodynamic equation on time scales are presented.

In most papers, the following special case of (2) is considered:see, e.g., [6–9], [10–13], [14], [15], or [16]. For some recent results devoted to nonlinear Volterra equations we refer to [5, 17–22] and references therein.

Note, that equation generalizes the second-order discrete Volterra difference equation of type (2):On the other hand, if for , then denoting equation takes the formHence second-order difference equation (6) is a special case of . The results on asymptotic properties and oscillation of equations of type (6) can be found, i.e., in [23–26].

Our main goal is to present sufficient conditions for the existence of a solution to equation such thatwhere and . We give also sufficient conditions for a given solution of equation to have an asymptotic property (7). Moreover, in Section 5 we show applications of the obtained results to linear Volterra equation of type . We present also some results for the case when is a potential sequence.

2. Preliminaries

We will denote by the space of all sequences . If in , then and denote the sequences defined by and , respectively. Moreover,If , , and , then we write . Analogously, denotes the boundedness of the sequence .

The following two lemmas will be useful in the proof of our main results.

Lemma 1. Assume , , andThen

Proof. We have

Lemma 2 ([27, Lemma 4.7]). Assume , and . In the setwe define a metric by the formulaThen any continuous map has a fixed point.

3. Solutions with Prescribed Asymptotic Behavior

In this section we present sufficient conditions for the existence of a solution to equation such thatwhere and .

Theorem 3. Assume , , , , , ,and is continuous and bounded on . Then there exists a solution of such that

Proof. For and letThere exists such thatLetIf and , thenChoose a positive number such that for any . ThenSince , we haveFor letThen we haveHence, using (21) and (22), we getAnalogously, replacing by , we obtainUsing (25) and (26) we getSince , we haveDefine a sequence byDefine byBy (27), . We haveHence and we getHence there exists an index such thatfor . LetWe define a metric on by formula (13). Note that . Let . By (33) and (20) we have for any . Hence, by (18), for . Using (17) and (29) we obtainfor . Therefore . Now we show that the map is continuous. Using (25) and the assumption , we haveHence, by Lemma 1, we getLet . Choose an index and a positive constant such thatLetChoose a positive such that if and , thenChoose such that . Then we haveUsing Lemma 1 we obtainNote that for andHence we obtainTherefore is continuous. By Lemma 2 there exists a point such that . Then, for , we haveNote thatfor any . Hence, for , we getTherefore is a solution of . Since we have .

If the function is continuous, then from Theorem 3 we get the following two results.

Corollary 4. Assume , , is continuous, andThen for any there exists a solution of such that .

Proof. Taking , , and in Theorem 3, we obtain the result.

Corollary 5. Assume , , is continuous, andThen for any there exists a solution of such that

Proof. Assume and a sequence is defined bySince and , we see that is bounded. Now, taking and in Theorem 3, we obtain the result.

Note that Corollaries 4 and 5 concern convergent solutions. However, Theorem 3 includes also divergent solutions. For example, if for , , , , andthen, by Theorem 3, for any nonzero and any there exists a solution of such thatNow we present an example that proves the assumptionin Theorem 3, is essential.

Example 6. Assume , ,, and . Then equation takes the formLet andNotice that is continuous and bounded on . Moreover,andAssume is a solution of (56) such thatSince , we haveHencefor large . Therefore, the sequence is eventually increasing and there exists the limitIf , then the sequence is convergent in . Hence the seriesis convergent. On the other handfor large . Hence . Therefore for large and we getBut since , the series is convergent.

4. Asymptotic Behavior of Solutions

In this section we present sufficient conditions for a given solution of equation to have an asymptotic propertywhere and .

Theorem 7. Assume , ,and is a solution of such that the sequence is bounded. Then there exist such that

Proof. We havefor large . Using boundedness of the sequence and (68) we getDefine byChoose a positive such that for any . Since , we haveMoreover,Hence, by (73)Since , we haveLetThenThus . LetThenHenceand we getfor any . Therefore, there exists a real constant such that . ThusHencewhere .

Corollary 8. Assume , , , is locally bounded,and is a bounded solution of . Then there exist such that

Proof. Since is bounded and is locally bounded, the sequence is bounded. Hence the assertion is a consequence of Theorem 7.

Corollary 9. Assume , , is locally bounded, andThen any bounded solution of is convergent.

Proof. Assume is a bounded solution of . Let . By Corollary 8, there exist such thatDefine a sequence , by . Then is increasing and bounded. Hence is convergent. Therefore is convergent.

Corollary 10. Assume , , , is bounded,and is an arbitrary solution of . Then there exist such that

Proof. The assertion is an immediate consequence of Theorem 7.

5. Additional Results

In this section we present some additional results. First, we give some applications of our results to linear discrete Volterra equations of type . From Corollary 4 we get the following result.

Corollary 11. Assume , ,Then for any there exists a solution of equationsuch that .

From Corollary 5 we get the following.

Corollary 12. Assume , , andThen for any there exists a solution of (92) such that

Example 13. Assume , , andThen (92) takes the formIt is easy to check that all assumptions of Corollary 11 hold. Indeed, we haveandSo, for every , there exists a solution of (96) such that . One such solution is .

In our investigations the conditionplays an important role. In practice, this condition can be difficult to verify. In the following remark we present the condition, which is a little stronger but easier to check.

Remark 14. Assume , , , and . Let , , for any . Then and

Applying this remark to Corollaries 4, 5, 8, and 9, respectively, we obtain following results.

Corollary 15. Assume , , , is continuous, andThen for any there exists a solution of such that .

Corollary 16. Assume , , , is continuous, andThen for any there exists a solution of such that

Corollary 17. Assume , , , is locally bounded,and is a bounded solution of . Then there exist such that

Corollary 18. Assume , , is locally bounded, andThen any bounded solution of is convergent.

Now we present some results for the case when the seriesare strongly convergent.

Remark 19. If and , then, by the root test,for any .

Corollary 20. Assume , is continuous, andThen for any and any there exists a solution of such that

Proof. Let . Choose . By Remark 19, we haveBy Corollary 4, there exists a solution of such that

Analogously, using Corollary 5, we get the following.

Corollary 21. Assume , is continuous, andThen for any and any there exists a solution of such that

To the end we consider the case when is a potential sequence.

Lemma 22. If , then

Proof. Define and byBy [28, Theorem 2.2], we haveSince , we getNote that and . Hence, by discrete L’Hospital’s Rule,Therefore

Corollary 23. Assume , , is continuous, andThen for any there exists a solution of such that

Proof. Assume . LetBy Corollary 5, there exists a solution of such thatBy Lemma 22Hence

Corollary 24. Assume , , is locally bounded, andThen for any bounded solution of there exists a real number such that

Proof. By Corollary 8 there exist such thatBy Lemma 22 we obtain

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The second author was supported by the Ministry of Science and Higher Education of Poland (04/43/DSPB/0095).

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Copyright © 2018 Janusz Migda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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