Abstract
We consider homogeneous linear Volterra Discrete Equations and we study the asymptotic behaviour of their solutions under hypothesis on the sign of the coefficients and of the first- and second-order differences. The results are then used to analyse the numerical stability of some classes of Volterra integrodifferential equations.
1. Introduction
Linear Volterra Discrete Equations (VDEs) are usually represented according to two types of formulae (see, e.g., [1] and references therein, [2, XIII-10], [3, Chap. ]):Even if each of the equations above can be easily transformed into the other, we read in the literature (see, e.g., [4]) that (1) is the discrete analogue of a Volterra Integrodifferential Equation (VIDE), whereas (2) is seen as the discrete version of a second kind Volterra Integral Equation (VIE). This is due to the fact that the simple positionwhich transforms (2) into (1) is not meaningful when we are dealing with numerical analysis of Volterra equations.
To be more specific, a simple numerical method for the VIDE, , has the formwhere is the stepsize, are given weights, and
Using (3), namely, , , and , (4) turns into the form of (2), the analysis of which would be complicated by the fact that the coefficients do not have the same dependence on For such a reason, in this paper we focus on the following homogeneous VDE:where is given and for , and we study its asymptotic properties exploiting its particular form.
For the sake of completeness, there is also another type of VDE widely used in literature (see, e.g., [5, 6])This is an explicit equation which can be recasted in the form (5) with , by imposing
Asymptotic analysis of difference equations of the form (5) or its explicit version often appeared in literature in the last decades. Some of them deal with the convolution case (); see, for instance, [6] and the references therein and [7–12]. Most of the known results for the nonconvolution case are based on the hypothesis of double summability of the coefficients ; see [1, 4, 13–19]. Another interesting approach, resembling the study of continuous VIDE (see, e.g., [20, 21]), basically requires that the coefficient of (5), assumed to be negative, in some sense “prevails” on the summation of the remaining coefficients Here we would like to add another piece to the framework regarding the analysis of VDE behaviour, by considering hypotheses based on the sign of the coefficients and of their first and second differences.
Since (5) is homogeneous, it has always the trivial solution. Therefore, all the results that follow are valid automatically and no assumptions are necessary when From now on we assume that the given datum is different from zero and we want to analyse the behaviour of the corresponding solution with respect to the trivial one. In Section 2 we report our main results on the asymptotic behaviour of the nontrivial solution to (5) which are then used, in Section 3, to prove the boundedness of the solution and the convergence to zero in some cases of interest.
In the whole paper it is assumed the empty sum convention , if
2. Main Results
Let be a double-indexed sequence and define , , and Our main result gives sufficient conditions for (5) to have a solution vanishing at infinity.
Theorem 1. Consider (5) and assume that(i) such that ,(ii),(iii),(iv) Then, for any , there exists such that If, in addition,(v), or(), then, for any ,
Proof. Set , thenand henceThe second addendum in the right-hand side of (8) can be written aswhereApplying the summation by parts rule, we haveBy adding and subtracting in the right-hand side and by settingwe getNow, taking into account the fact thatwe haveBy (8) and (15), (7) becomesSumming up over , for all , we haveNow, let us consider the double summation at the right-hand side. By inverting the summation order, applying the summation by part rule and recalling that , it becomesTaking account of this and applying the summation by part rule also to the third addendum in (17), we getIn view of the first group of hypotheses (i)–(iv), this impliesAs the whole right-hand side does not depend on , (20) assures the boundedness of and the first part of the theorem is proved.
In order to prove the second part of our result, let us proceed by contradiction. Assume thatFrom (19) and (20) we have and in view of (v) , which leads to an absurd because of (21). So the series in (21) converges and
Now consider hypothesis () and once again proceed by contradiction. Assume that the series does not converge, therefore and two increasing sequences of integers and with such that , and henceAgain, using (19) and (20), we writeBecause of (iv) we can writewith such that This together with (22) and () leads to Since , as increases, this is absurd. Hence, the series converges and the desired result follows.
It is well known that one of the most used tools in the stability analysis of VDEs is the Lyapunov approach [22–28]. As already mentioned in the introduction, among the results that can be obtained by Lyapunov techniques, the most popular are based on the hypothesis that the coefficients are summable (e.g., the result in [28, Th. 2] applied to (5) requires, among other hypotheses, that ). Our attempt to construct new functionals for the form (5) leads inevitably to this type of hypothesis. Very few are the cases where no summability requirements are made. One of these can be found in [26, Th. ], where a Lyapunov functional is constructed which allows the stability analysis of an explicit equation, provided that some conditions on the sign of the coefficients and of their ’s are satisfied. In order to compare the technique developed in this paper with the Lyapunov one, we refer precisely to this theorem and consider (5) with and In this situation the hypotheses of Theorem 1 proved above guarantee that the solution vanishes also in few cases not covered by Theorem in [26] (just to mention one example, the coefficient of , which should be negative in [26], is allowed to assume whatever sign here).
Remark 2. It is easy to see that if hypothesis (iii) in Theorem 1 holds also for , then () assures , for all Therefore, if we assume , , hypothesis () becomes sufficient for (v). So does no more represent an alternative with respect to (v) and can be dropped out. In this case Theorem 1 can be stated as follows.
Corollary 3. Consider (5) and assume that , for , and that (ii)–(v) hold. Then, for any ,
Furthermore, we point out that checking assumption () of Theorem 1 may be difficult; hence the following result can be useful.
Corollary 4. Consider (5) and assume that (i)–(iv) hold andwith Then, for any ,
Proof. From (iv) and the definition of in (12),The desired result is readily obtained by using (23) and (25).
We want to underline that Theorem 1 is strongly inspired by [29] where the asymptotic behaviour of a nonlinear VIDE is studied and that “in some sense” our result can be viewed as its discrete analogue. This will be illustrated in the following section.
Remark 5. Observe that, when is of convolution type, hypothesis (iv) in Theorem 1 becomes , so that the advantage of using hypothesis (iv), which allowed to have a constant sign only definitely with respect to , is completely lost. This drawback can be overcome if we know that the sign of is definitely constant, as it is shown in the following theorem.
Theorem 6. Consider (5) and assume that(a) such that , for ,(b) such that , for ,(c),(d),(e) such that ,(f), or Then, for any
Proof. First of all observe that (a) assures From here and (13) we deriveNow, proceeding as in the proof of Theorem 1, we arrive towhich, taking into account (b), (c), and (e), assuresorwhich corresponds to (14) and (23) of Theorem 1, respectively. The desired result follows as in the proof of Theorem 1.
As a consequence of this result, the following can be easily proved.
Corollary 7. Under assumptions (b)–(f) of Theorem 6 a sequence , obtained by (5) with , cannot diverge and if it is convergent then its limit is zero.
Remark 8. If, in Theorem 6, , then hypothesis (e) can be removed, and the theorem assumes a simplified form.
3. Examples of Applications
Consider the following theoretical examples of application of Corollaries 3 and 4.
It can be easily seen that (5), with the choices for and satisfies the assumptions of the corollaries in the previous section.
To be more specific (31) fulfills both corollaries with Equation (33) satisfies only Corollary 3 with , whereas (32) satisfies Corollary 4 but not 2.1, because Equation (34) is only a slight modification of (33) and, like (33), it fulfills all the hypotheses of Corollary 3; furthermore it can be easily seen that is an unbounded sequence. So (5) with coefficients as in (34) is an example of VDE with vanishing solution and nonsummable coefficients.
As a counterexample, consider (5) with coefficients given byHere condition (i) for the coefficients and in Theorem 1 is violated and the boundedness of the solution of (5) is not guaranteed any more. In fact, this is clear in Figure 1, which shows the actual behaviour of
Theorem 6 can be applied to the following example:First of all we need to show that (a) holds with Starting from the initial condition given in (36), by simple computation, we have Our aim is to prove that , for Let us proceed by induction on Assume and verify that the same is true for given byFrom the definition of in (36), it easily follows that Then taking into account the induction hypothesis and that , we obtainAs and , it turns out that the right-hand side of (37) is positive, then On the other hand (38) implies , with , which assures We conclude that hypothesis (a) of Theorem 6 is satisfied. Since (d) is true and (c), (f) are obvious with , it remains to prove (e). In our case (e) corresponds to such that , for and Observe that can be written as So , since (c) holds and In conclusion, all the hypotheses of Theorem 6 are fulfilled and , as can be seen in Figure 2.
Remark 9. We want explicitly to mention that Theorem 1 cannot be applied to (36) because , and hypothesis (iv) is not satisfied.
Finally, we observe that, if in (36) we choose according to the remaining coefficients for , that is, , then Theorem 6 is still valid with So we are in the case of Remark 8 and the assumption (e) of Theorem 6 can be ignored.
A more practical application of our results is the study of the longtime behaviour of the numerical solution to VIDEs. Let us consider the homogeneous problemand a simple method of family (4), the Backward Euler method (see [30], [12, (3.8)])where is the stepsize. With the help of the results of the previous section we can prove the following.
Theorem 10. Consider (40) and assume that(i) such that ;(ii);(iii);(iv) Then the solution of (40) is bounded . If in addition(v), or(), then
Proof. Note that (5) coincides with (40) whenever , and Now, assumptions (i)–(iv) immediately assure (i)–(iv) of Theorem 1 for any fixed and such that Moreover, (v) implies (v) of Theorem 1 with , so that Corollary 3 holds. In order to exploit () note that it is equivalent to , which in turn implies that, for any fixed , the function is increasing with respect to , so that () of Theorem 1 is fulfilled with and given above. Once again all the hypotheses of Theorem 1 are satisfied and the desired result follows.
Remark 11. Theorem 10 would be particularly interesting whenever it is known that, under the same hypotheses, also the analytical solution of (39) goes to zero as tends to infinity.
As we mentioned in the previous section, the analogue of Theorem 1 in the continuous case can be obtained following the line of the proof of Theorem in [29]. The following is a reformulation of such a theorem suited to our case.
Theorem 12. Assume that hypotheses (i)–(iv) of Theorem 10 are valid, then the solution of (39) is bounded, for any choice of Furthermore, if hypothesis (v) or () of Theorem 10 holds andthen
Remark 13. From Theorems 1 and 10 it is clear that the asymptotic behaviour of the solution to (39) is preserved both in a generic discretization of the kind (5), where represents the samples , and in the numerical solution obtained by the Backward Euler method (40).
Moreover, it is worth noting that if in (39)then assumption (41) is automatically verified; nevertheless in the discrete case, the summability of the coefficients, which is the analogous of (42), is not required as showed in example (34).
In the literature we sometimes encounter VIDEs with the following structure (see, e.g., [31]):This kind of equation can be easily recast in the form (39), with , and the method (40) for it will readTheorem 10 immediately becomes as follows.
Corollary 14. Consider (44) and assume that(a1),(a2),(a3),(a4), or() Then, for any
A comparison to Theorem 12 reveals that, contrarily to (iv), hypothesis (a3) is required to hold in the whole integration range. This is due to the fact that (see Remark 5 in case of (43)) is of convolution type. The application of Theorem 6 leads to the following result.
Theorem 15. Consider (44) and assume that(HA);(HB);(HC);(HD) Then, for any ,
Proof. We want to prove that all the hypotheses of Theorem 6 are fulfilled with and Let be fixed and let be such that Hypothesis (a) of Theorem 6 is obviously true because of (HA). In order to prove (b), observe that (HB) together with (HC) assure that, for ,As , (HC) implies the first condition in (c), and since , (HD) immediately assures the second condition of (c). Furthermore, since , (HC) also implies (d) and, taking into account the fact that , we have that (HD) implies (e) with Finally, whenever is not identically zero, we can always assume, with no loss of generality, that is such that , so that (45) assures (f) with , and the proof is complete.
As already mentioned above, in [31] equation represents the velocity of the centre of mass of a system of particles in collective motion under alignment and chemotaxis effect. Here andwith , and depending on the number of particles, their dimension, and the dynamic of the motion. For significative values of , and hypotheses (a1), (a2), (a3), and of Theorem 15 are satisfied. Hence, we expect that in the numerical simulation of (43) obtained by using the Backward Euler method (44), any convergent numerical solution vanishes at infinity. Figure 3 shows exactly this behaviour when integrating (43)–(46) with , and
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research was supported by GNCS-INdAM.