Discrete Dynamics of Nonlinear Systems in Nature and SocietyView this Special Issue
Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type
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.
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 , 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], , , or . 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.
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
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.
Proof. Taking , , and in Theorem 3, we obtain the result.
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
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 .
Proof. Since is bounded and is locally bounded, the sequence is bounded. Hence the assertion is a consequence of Theorem 7.
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.
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
Now we present some results for the case when the seriesare strongly convergent.
Remark 19. If and , then, by the root test,for any .
Analogously, using Corollary 5, we get the following.
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 ,