Stability and Convergence of Solutions to Volterra Integral Equations on Time Scales

where T is a time scale, which is a nonempty, closed subset of R. In (1) t0 ∈ T , the integral sign has to be intended as a deltaintegral (see Definition 4 in Section 2), and we assume that the given real-valued functions f(t) and k(t, s) are defined in [t0, T and [t0, T × [t0, T , respectively. In the following section we will give examples of time scales; here we observe that the most popular examples are T = R and T = Z. When T = R, (1) takes the form


Introduction
In this paper we consider the Volterra integral equations (VIEs) on time scales of the type where T is a time scale, which is a nonempty, closed subset of R. In (1)  0 ∈ T, the integral sign has to be intended as a deltaintegral (see Definition 4 in Section 2), and we assume that the given real-valued functions () and (, ) are defined in [ 0 , +∞) T and [ 0 , +∞) T × [ 0 , +∞) T , respectively.
In the following section we will give examples of time scales; here we observe that the most popular examples are T = R and T = Z.When T = R, (1) takes the form  () =  () + ∫ So, 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 [2][3][4] 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, [6][7][8][9][10] 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 ∫   0 |(, )|Δ < 1, 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 ∫   |(, )|Δ < 1, for all  > , and we study how the freedom before  affects the solution over the entire interval [ 0 , +∞] T .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.

Background Material
In this section we will recall some definitions and theorems that will be useful in the following (see [1][2][3] and the bibliography therein).
As already mentioned in Section 1, a time scale T is any closed subset of R.
We assume that the topology in T is inherited from the standard one in R.
Definition 1.For all  ∈ T and  < sup T, the forward jump operator is and, for  ∈ T and  > inf T, 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 () = () − .
Definition 2 (see [14]).A function  : T → R has a limit  at  0 ∈ T if and only if for every  > 0 there exists If  0 is an isolated point, then  = ( 0 ).If the limit exists, one writes lim Definition 3. Consider  : T → R, for each  < supT, and define  Δ () to be the number (provided it exists) with the property that, given any  > 0, there is a neighborhood for all  ∈ . Δ () is the delta-derivative of ().
Definition 4. If  Δ () = () and ,  0 ∈ T, one defines the delta-integral by If T = R, then ∫ Of course, every continuous function on T is also rdcontinuous and ld-continuous on T. Furthermore, it is possible to prove (see [15]) that every rd-continuous function on T is delta-integrable on T.
Let  ∈ R + ; the exponential function   (,  0 ),  ∈ T, is defined as the unique solution of the initial value problem (see, e.g., [5,16]) The explicit form of   (,  0 ) is given by Observe that since  > 0, we have   (,  0 ) > 0 for all  ∈ [ 0 , +∞] T .Furthermore,   (,  0 ) is the solution of problem (10), so   (,  0 ) Δ =   (,  0 ) > 0; hence   (,  0 ) is a strictly increasing function (see [3,Th. 1.76]) and When T = R, then   (,  0 ) =  (− 0 ) , and if T = Z, then   (,  0 ) = (1 + ) − 0 .In the following it will be useful to define Let the norm associated with   ([ 0 , +∞] T ; R), and, for  < +∞, set As already mentioned in the previous section, classical examples of time sets are T = R and T = Z.Particularly useful from a theoretical point of view are the following time sets (see, e.g., [2]): T =  Z , T = ℎZ, and T = {  :  ∈ Z}, which lead, respectively, to the -difference equation the Volterra discrete equation with constant stepsize and the discrete equation where the life span of a certain species is supposed to be one unit of time and the time scale for simulating electric circuit is where  represents the time units for discharging the capacitor (see [3, Ex. 1.39, 1.40] for details).

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 [ 0 , +∞) T .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.From now on we assume that, in (1), the kernel  : [ 0 , +∞) 2 T → R is continuous with respect to the first variable and rd-continuous in the second variable.Furthermore, we assume that the forcing function  is continuous on [ 0 , +∞) T (observe that, from Definition 2, if  is an isolated point, the definition of continuity is vacuously true).
In these hypotheses, if in addition  ∈   ([ 0 , +∞) T , R), Theorem 4.2 in [6] assures that the delta-integral equation ( 1) has a unique solution  ∈   ([ 0 , +∞) T , ) and where  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 [ 0 , +∞) T .In [6] it is shown that this is true under the additional hypothesis ∫ and we prove that the solution to ( 1) is bounded at [ 0 , +∞) T .

Theorem 7. Assume that (h1)-(h3) hold. Then, there exists a constant
Proof.Choose  > 0 such that  = / < 1, and consider the exponential function   (,  0 ) defined in (11).Let  < ; dividing each member of (1) by   (,  0 ) one gets Thus, since (h1) and (h2) hold, by using the norm ‖⋅‖   defined in (15) and the identity in (12) ( Since  = / < 1, then When  ≥  (1) can be rewritten as ( We know, from (h3), that  < 1; then sup where 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 (T = R) and Volterra summation equations (T = Z), 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 T = ℎZ 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].

Examples
For our examples we consider According to the discussion related to the cases T = R and T = Z, when (, ) is given by (36), it is possible to find  such that for all  >  both ∫   |(, )| and ∑ −1 = |(, )| are less than 1/2.It turns out that this value for  is ≈ 5.8.
Then, the hypotheses of Theorem 7 are fulfilled in each of the time sets considered.

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.