We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.

1. Introduction

In this paper we consider the Volterra integral equations (VIEs) on time scales of the typewhere is a time scale, which is a nonempty, closed subset of . In (1) , the integral sign has to be intended as a delta-integral (see Definition 4 in Section 2), and we assume that the given real-valued functions and are defined in and , respectively.

In the following section we will give examples of time scales; here we observe that the most popular examples are and . When , (1) takes the formof the classical VIE and when , we get the explicit Volterra discrete equationSo, all the results proved on the general time scale include results for both integral and explicit discrete Volterra equations.

A generalized differential and integral calculus on time scales was developed for the first time by Hilger in [1], where he put the basis for establishing the theory of dynamical equations (delta-derivative equations) over very general time scales. This theory has received great attention [24] in order to address many realistic continuous-discrete models in biological and economic applications and to furnish a theoretical framework for developing a unifying analysis. In particular, in [5], a qualitative study of the solutions to nonlinear dynamic equations is described as well as an application to an economic model.

More recently, there has been a growing interest in Volterra integral equations on time scales as they represent a powerful instrument for the mathematical representation of memory dependent phenomena in population dynamic, economy, and so forth. Therefore, this theory has been extended to the integral operator. A pioneering research on this subject is [6], where the main results concerning the existence, uniqueness, and boundedness on compact intervals are presented. After that, a Volterra theory on time scales has been developed and it is still evolving; see, for example, [610] and the bibliography therein.

In [6] a very accurate analysis of the qualitative behavior of the solutions of both linear and nonlinear problems on noncompact intervals is given. In case of linear problems it has been proved that if is bounded and , the solution of (1) is bounded, which means that it is stable with respect to bounded perturbations. In this paper we prove the stability of solutions to VIEs on time scales under more general hypotheses on the delta-integral of the kernel . Hence, we assume that there exists a such that , for all , and we study how the freedom before affects the solution over the entire interval . The investigation carried out here represents an extension of a result already known both for continuous VIEs (see, e.g., [11, Ch. 9] and for discrete implicit Volterra equations (see [12]). However, the technique used in the proof is different and takes inspiration from [6, 13].

Moreover, when the kernel vanishes at infinity, we study the asymptotic behavior of the solution . In the particular case of discrete equation this result has already been proved by analogous techniques by Győri and Reynolds in [12].

The paper is organized as follows. In Section 2 we introduce some basic material needed in the paper. In Section 3 we define the linear model problem and obtain a bound for its solution. Furthermore, in case of vanishing kernel, we prove that the solution of (1) tends to a finite limit if the forcing function tends to as . In Section 4 an extension of the previous results to a Hammerstein type nonlinear equation is shown; in Section 5 some examples are given and Section 6 contains our concluding remarks.

2. Background Material

In this section we will recall some definitions and theorems that will be useful in the following (see [13] and the bibliography therein).

As already mentioned in Section 1, a time scale is any closed subset of .

We assume that the topology in is inherited from the standard one in .

Definition 1. For all and , the forward jump operator isand, for and , the backward jump operator is

If , the point is said to be right-scattered (, left-scattered). If , the point is said to be right-dense (, left-dense). Points that are simultaneously right-scattered and left-scattered are called isolated. The graininess function is defined by .

When , then , and ; when , then , and .

Definition 2 (see [14]). A function has a limit at if and only if for every there exists such that if , thenIf is an isolated point, then . If the limit exists, one writes

Definition 3. Consider , for each , and define to be the number (provided it exists) with the property that, given any , there is a neighborhood of such thatfor all . is the delta-derivative of .

If , then , the usual derivative, and if , then , the forward difference operator.

Definition 4. If and , one defines the delta-integral by

If , then corresponds to the Cauchy integral and if , then .

Definition 5. A function is right-dense continuous if it is continuous at every right-dense point and exists for every left-dense point . Similarly, a function is left-dense continuous if it is continuous at every left-dense point and exists for every right-dense point .

Of course, every continuous function on is also -continuous and -continuous on . Furthermore, it is possible to prove (see [15]) that every -continuous function on is delta-integrable on .

Let ; the exponential function , , is defined as the unique solution of the initial value problem (see, e.g., [5, 16])The explicit form of is given byObserve that since , we have for all . Furthermore, is the solution of problem (10), so ; hence is a strictly increasing function (see [3, Th. 1.76]) andWhen , then , and if , then .

In the following it will be useful to define as the space of continuous functions such thatLetthe norm associated with , and, for , setAs already mentioned in the previous section, classical examples of time sets are and . Particularly useful from a theoretical point of view are the following time sets (see, e.g., [2]): , , and , which lead, respectively, to the -difference equationthe Volterra discrete equation with constant stepsizeand the discrete equationIn addition to the previous ones, examples are , , , and the biological relevant time scalewhere the life span of a certain species is supposed to be one unit of time and the time scale for simulating electric circuit iswhere represents the time units for discharging the capacitor (see [3, Ex. 1.39, 1.40] for details).

3. Stability and Convergence for Linear Equations

In this section we investigate the boundedness of the solution to (1) when the forcing term is bounded on . Since (1) is linear, it may be regarded as the error equation. Hence, our purpose here is to prove stability results for (1) under bounded perturbations, according to the following definition.

Definition 6. The zero solution of the VIE on time scales (1) corresponding to is called stable on if for each there exists a such that and , implying that each solution of (1) exists and satisfies , for all .

From now on we assume that, in (1), the kernel is continuous with respect to the first variable and -continuous in the second variable. Furthermore, we assume that the forcing function is continuous on (observe that, from Definition 2, if is an isolated point, the definition of continuity is vacuously true).

In these hypotheses, if in addition , Theorem 4.2 in [6] assures that the delta-integral equation (1) has a unique solution andwhere is a positive constant. This bound, which is useful in applications (see [17, p. 37]), does not address the problem of the boundedness of the solution to (1) on . In [6] it is shown that this is true under the additional hypothesis , for all . Here, we consider more general assumptions; that is,(h1);(h2), for each ;(h3) such that .and we prove that the solution to (1) is bounded at .

Theorem 7. Assume that (h1)–(h3) hold. Then, there exists a constant such that

Proof. Choose such that , and consider the exponential function defined in (11). Let ; dividing each member of (1) by one getsThus, since (h1) and (h2) hold, by using the norm defined in (15) and the identity in (12), we getSince , thenWhen (1) can be rewritten asObserving that , then result (25) can be used to obtain a bound and hence, since (h3) holds,We know, from (h3), that ; thenwhere .

Remark 8. Boundedness results under hypothesis (h3) can be found, for example, in [11, Sec. 9, Th. 9.1] and in [12, 13] for nonconvolution VIEs and Volterra summation equations , respectively. The novelty here is that the different approach used in the proof of Theorem 7 allows the generalization to other kinds of time scales, as, for example, the ones in (19) and (20) motivated by the applications or motivated by numerical schemes. An analysis completely devoted to the stability of parameter-dependent Volterra summation equations has been carried out by the authors in [18].

Consider . If , is said to be left-dense (see [15, Sec. 4]). So we can consider for any function defined in . For the definition of on time scales we refer to [15, Sec. 4].

The following theorem is a generalization of Theorem in [12] to Volterra equations on time scales.

Theorem 9. Assume that (h1)–(h3) hold. Furthermore, let(h4),(h5) ,  ,(h6) , for some .Then , where .

Proof. First of all observe that, from (h3), . Manipulating (1) and subtracting , one getsLet us take the limit superior of each side of (29) as . Since is sufficiently smooth on and hypotheses (h2) and (h5) hold, the limit can be passed under the integral sign in to obtain zero (we refer to [19] for the Lebesgue bounded convergence theorem on time scales). Thus,By taking the for , in view of (h4), (h6), and (h3), we get the result.

4. Extension to Nonlinear Equations

For nonlinear equations of the formwe assume that is a differentiable map on and . This condition is not restrictive; as a matter of fact, if this is not the case, we replace with and with .

Theorem 10. Assume that, for (31), , with , (h1) and (h2) hold, and(h3′) such that .Then the solution to (31) is bounded.

Proof. By Taylor’s theorem we write , . Since and , for all , from (31) one getsThen, for , by dividing each member of (32) by it turns out thatChoosing such that , thenWhen , the following result is easily obtained:where .

5. Examples

For our examples we consider (1) with bounded on andwhich does not satisfy the hypothesis , considered in [6]. We study the stability of the solution for different choices of the time set .

. Here . With defined as in (36) it is and , the last term in the inequality being a function in which tends to zero as . Hence for .

. Here . With defined as in (36) it is and, by simple manipulations, for , one gets , the last term in the inequality being a function in which tends to zero as . Hence for .

. Here . By simple manipulations, for , one gets , the last term in the inequality being a function in which tends to zero as . Hence for depending on . Of course, when we reduce to the case . For we get .

= (19). Let and ; thenAccording to the discussion related to the cases and , when is given by (36), it is possible to find such that for all both and are less than . It turns out that this value for is .

Then, the hypotheses of Theorem 7 are fulfilled in each of the time sets considered.

Observe that the kernel (36) satisfies also (h5) and (h6) for all the time sets considered above. So, for Theorem 9, , if .

6. Concluding Remarks

The research reported in this paper deals with the stability properties of Volterra equations on time scales. After examining the importance and the potential impact of this operator on the applications, we have surveyed the literature related to the calculus on time scales. As already mentioned in Section 1 and in Remark 8, in this paper, we extend some aspects of the stability theory, already developed in the continuous [11] and discrete [12] cases, to VIEs on time scales. Among the existing results for Volterra equations on time scales, the one in [6], concerning the existence and boundedness of the solutions to (1), has been our starting point for investigating their long time behavior. With respect to the results contained in [6], the boundedness of the solution is obtained here under more general sufficient conditions. This allows us to enlarge the class of problems under consideration, as shown in the example reported in Section 5 where kernel (36) is stable according to the current analysis, but does not satisfy the sufficient condition stated in [6].

The technique used in the proof of the main theorem put the basis for an analogous investigation about the numerical stability of Volterra integral equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors thank the referee for his helpful suggestions. This work was supported by GNCS-INDAM.