Boundedness and Asymptotic Stability for the Solution of Homogeneous Volterra Discrete Equations

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 position cn+1,n+1 = an+1, cn+1,n = 1 + bn+1,n, cn+1,j = bn+1,j, j = 0, . . . , n − 2, (3) which 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, y󸀠(t) = g(t) +A(t)y(t) +∫t 0 K(t, s)y(s)ds, has the form yn+1 − yn = hg (tn+1) + hA (tn+1) yn+1 + h2n+1 ∑

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  +1,+1 =  +1 = 0, by imposing   = 1.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][8][9][10][11][12].Most of the known results for the nonconvolution case are based on the hypothesis of double summability of the coefficients (∑ +∞ =0 ∑  =0 | , | < +∞); see [1,4,[13][14][15][16][17][18][19].Another interesting approach, resembling the study of continuous VIDE (see, e.g., [20,21]), basically requires that the coefficient  +1 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  0 = 0. From now on we assume that the given datum  0 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 ∑  = V  = 0, if  < .

Main Results
Let V , be a double-indexed sequence and define Our main result gives sufficient conditions for (5) to have a solution   vanishing at infinity.
Proof.Set   =  +1  +1 + ∑ +1 =0  +1,   , then and hence The second addendum in the right-hand side of (8) can be written as where Applying the summation by parts rule, we have By adding and subtracting  +1 ∑  =0 Δ 2  +1, in the righthand side and by setting we get Now, taking into account the fact that we have By ( 8) and ( 15), ( 7) becomes Summing up over , for all  >  ≥ 0, we have Now, let us consider the double summation at the righthand side.By inverting the summation order, applying the summation by part rule and recalling that  , = 0, it becomes Taking account of this and applying the summation by part rule also to the third addendum in (17), we get In view of the first group of hypotheses (i)-(iv), this implies As the whole right-hand side does not depend on , (20) assures the boundedness of |  | and the first part of the theorem is proved.
It is well known that one of the most used tools in the stability analysis of VDEs is the Lyapunov approach [22][23][24][25][26][27][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 ∑ −1 =0 ∑ +∞ =+1 | , /(1−  − , )| < +∞).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. 2.2], 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 Δ'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  , = 0 and   =  < 0. In this situation the hypotheses of Theorem 1 proved above guarantee that the solution vanishes also in few cases not covered by Theorem 2.2 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  =  − 1, then (v bis ) assures   ≤ − * , for all  > .Therefore, if we assume   ≤ 0,  ≥ , hypothesis (v bis ) becomes sufficient for (v).So (v bis ) 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.
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 Δ 12  , is of convolution type, hypothesis (iv) in Theorem 1 becomes Δ 2   ≤ 0, so that the advantage of using hypothesis (iv), which allowed Δ 12  , 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.Then, for any  0 ∈ R, lim →+∞   = 0.
Proof.First of all observe that (a) assures 2 +1  , ≥ 0,  ≥  ≥  − 1.From here and ( 13) we derive Now, proceeding as in the proof of Theorem 1, we arrive to which, taking into account (b), (c), and (e), assures or which 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  0 ∈ R, cannot diverge and if it is convergent then its limit is zero.Remark 8. If, in Theorem 6,  = 1, then hypothesis (e) can be removed, and the theorem assumes a simplified form.
To be more specific (31) fulfills both corollaries with  = 0. Equation (33) satisfies only Corollary 3 with  = 3, whereas (32) satisfies Corollary 4 but not 2.1, because   +   → 0. 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 ∑  =0 |  | 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 by Here 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  = 2.
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 1 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  0 ∈ R. Furthermore, if hypothesis (v) or (V  ) of Theorem 10 holds and  is uniformly continuous, (41) then lim →+∞ () = 0.
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]