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

1. Introduction

Linear Volterra Discrete Equations (VDEs) are usually represented according to two types of formulae (see, e.g.,  and references therein, [2, XIII-10], [3, Chap. 7]):

[mathematical expression not reproducible], (1)

n [greater than or equal to] 0, [x.sub.0] given,

[mathematical expression not reproducible]. (2)

Even if each of the equations above can be easily transformed into the other, we read in the literature (see, e.g., ) 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

[mathematical expression not reproducible], (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, [mathematical expression not reproducible], has the form

[mathematical expression not reproducible], (4)

where h is the stepsize, [t.sub.n] = nh, [w.sub.n,j] are given weights, and [y.sub.n] [approximately equal to] y([t.sub.n]).

Using (3X namely [mathematical expression not reproducible], and [mathematical expression not reproducible], (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 h. For such a reason, in this paper we focus on the following homogeneous VDE:

[mathematical expression not reproducible], (5)

where [x.sub.0] is given and [b.sub.n,j] = 0 for j > n, 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])

[mathematical expression not reproducible]. (6)

This is an explicit equation which can be recasted in the form (5) with [b.sub.n+1,n+1] = [a.sub.n+1] = 0, by imposing [A.sub.n] = 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 ([b.sub,n,j] = [b.sub.n-j]); see, for instance,  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 ([mathematical expression not reproducible]); see [1, 4, 13-19]. Another interesting approach, resembling the study of continuous VIDE (see, e.g., [20, 21]), basically requires that the coefficient [a.sub.n+1] of (5), assumed to be negative, in some sense "prevails" on the summation of the remaining coefficients [b.sub.nj]. 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 [x.sub.0] = 0. From now on we assume that the given datum [x.sub.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 [[summation].sup.M.sub.j=m] [v.sub.j] = 0, if M < m.

2. Main Results

Let [v.sub.n,j] be a double-indexed sequence and define [mathematical expression not reproducible]. Our main result gives sufficient conditions for (5) to have a solution [x.sub.n] vanishing at infinity.

Theorem 1. Consider (5) and assume that

(i) [there exists][bar.n] > 0 such that 2[a.sub.n] + [b.sub.n,n] [less than or equal to] 0, n [greater than or equal to] [bar.n],

(ii) [b.sub.n,0] [less than or equal to] 0, n [greater than or equal to] 2, [[DELTA].sub.1][b.sub.n,0] [greater than or equal to] 0, n [greater than or equal to] [bar.n],

(iii) [[DELTA].sub.2][b.sub.nj] [less than or equal to] 0, n [greater than or equal to] 2, j [less than or equal to] n - 2,

(iv) [[DELTA].sub.12][b.sub.nj] [greater than or equal to] 0, n [greater than or equal to] [bar.n] [greater than or equal to] 1, j [less than or equal to] n - 1.

Then, for any [x.sub.0] [member of] R, there exists [x.sup.*] such that [absolute value of [x.sub.n]] [less than or equal to] [x.sup.*], [for all]n > 0. If, in addition,

(v) 2[a.sub.n] + [b.sub.n,n] [less than or equal to] -[a.sup.*] < 0, n > [bar.n] [greater than or equal to] 1,

or

[mathematical expression not reproducible],

then, for any [x.sub.0] [member of] R, [lim.sub.n[right arrow]+[infinity]][x.sub.n] = 0.

Proof. Set [mathematical expression not reproducible], then

[mathematical expression not reproducible], (7)

and hence

[mathematical expression not reproducible]. (8)

The second addendum in the right-hand side of (8) can be written as

[mathematical expression not reproducible], (9)

where

[mathematical expression not reproducible]. (10)

Applying the summation by parts rule, we have

[mathematical expression not reproducible]. (11)

By adding and subtracting [mathematical expression not reproducible] in the right-hand side and by setting

[mathematical expression not reproducible], (12)

we get

[mathematical expression not reproducible]. (13)

Now, taking into account the fact that

[mathematical expression not reproducible], (14)

we have

[mathematical expression not reproducible]. (15)

By (8) and (15), (7) becomes

[mathematical expression not reproducible]. (16)

Summing up over n, for all N > [bar.n] [greater than or equal to] 0, we have

[mathematical expression not reproducible]. (17)

Now, 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 [V.sub.j,j] = 0, it becomes

[mathematical expression not reproducible]. (18)

Taking account of this and applying the summation by part rule also to the third addendum in (17), we get

[mathematical expression not reproducible]. (19)

In view of the first group of hypotheses (i)-(iv), this implies

[mathematical expression not reproducible]. (20)

As the whole right-hand side does not depend on N, (20) assures the boundedness of [absolute value of [x.sub.n]] 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 that

[mathematical expression not reproducible]. (21)

From (19) and (20) we have [mathematical expression not reproducible] and in view of (v) [mathematical expression not reproducible], which leads to an absurd because of (21). So the series in (21) converges and [lim.sub.n[right arrow]+[infinity]][x.sup.2.sub.n] = 0.

Now consider hypothesis ([v.sub.bis]) and once again proceed by contradiction. Assume that the series [[summation].sup.+[infinity].sub.n=0] does not converge, therefore [there exists][epsilon] > 0 and two increasing sequences of integers {[n.sub.i]} and {[l.sub.i]} with [n.sub.i] > [l.sub.i], [l.sub.0] > [bar.n] such that [mathematical expression not reproducible], and hence

[mathematical expression not reproducible]. (22)

Again, using (19) and (20), we write

[mathematical expression not reproducible]. (23)

Because of (iv) we can write

[mathematical expression not reproducible], (24)

with m(N) such that [n.sub.m(N)] < N < [n.sub.m(N)] + 1. This together with (22) and ([v.sub.bis]) leads to [x.sup.2.sub.N+1] < C - [[epsilon].sup.2][b.sup.*]m(N). Since m(N) [right arrow] +[infinity], as N increases, this is absurd. Hence, the series [[summation].sup.+[infinity].sub.n=0][x.sub.n] 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 [mathematical expression not reproducible]. 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 A'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 [b.sub.n,n] = 0 and [a.sub.n] = a < 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  (just to mention one example, the coefficient of [x.sub.n], which should be negative in , is allowed to assume whatever sign here).

Remark 2. It is easy to see that if hypothesis (iii) in Theorem 1 holds also for j = n - 1, then ([v.sub.bis]) assures [b.sub.nn] [less than or equal to] -[b.sup.*], for all n > [bar.n]. Therefore, if we assume [a.sub.n] [less than or equal to] 0, n [greater than or equal to] [bar.n], hypothesis ([v.sub.bis]) becomes sufficient for (v). So ([v.sub.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.

Corollary 3. Consider (5) and assume that [a.sub.n] [less than or equal to] 0, for n [greater than or equal to] [bar.n], [[DELTA].sub.2][b.sub.n,n1] [less than or equal to] 0, and that (ii)-(v) hold. Then, for any [x.sub.0] [member of] R, [lim.sub.n[right arrow]+[infinity]][x.sub.n] = 0

Furthermore, we point out that checking assumption ([v.sub.bis]) of Theorem 1 maybe difficult; hence the following result can be useful.

Corollary 4. Consider (5) and assume that (i)-(iv) hold and

[[DELTA].sub.12][b.sub.n,n-1] [greater than or equal to] [b.sup.*], n [greater than or equal to] [bar.n], (25)

with [b.sup.*] > 0. Then, for any [x.sub.0] [member of] R, [lim.sub.n[right arrow]+[infinity]][x.sub.n] = 0.

Proof. From (iv) and the definition of [V.sub.n,j] in (12),

[mathematical expression not reproducible]. (26)

The desired result is readily obtained by using (23) and (25).

We want to underline that Theorem 1 is strongly inspired by  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 [[DELTA].sub.12][b.sub.n,j] is of convolution type, hypothesis (iv) in Theorem 1 becomes [[DELTA].sub.2][b.sub.n] [less than or equal to] 0, so that the advantage of using hypothesis (iv), which allowed [[DELTA].sub.12][b.sub.n,j] to have a constant sign only definitely with respect to n, is completely lost. This drawback can be overcome if we know that the sign of [x.sub.n] is definitely constant, as it is shown in the following theorem.

Theorem 6. Consider (5) and assume that

(a) [there exists][bar.n] > 0 such that [x.sub.n] [greater than or equal to] 0 (< 0), for n [greater than or equal to] [bar.n],

(b) [there exists][n.sup.*] > [bar.n] such that [[beta].sub.n] = 2[a.sub.n] + 2[b.sub.n,n] - [b.sub.n,[bar.n]] [less than or equal to] 0, for n [greater than or equal to] [n.sup.*] [greater than or equal to] [bar.n],

(c) [b.sub.n,0] [less than or equal to] 0, n [greater than or equal to] 0, [[DELTA].sub.1] [b.sub.n,0] [greater than or equal to] 0, n [greater than or equal to] [n.sup.*] [greater than or equal to] [bar.n],

(d) [[DELTA].sub.2][b.sub.n,j] [less than or equal to] 0, n [greater than or equal to] 1, j < n,

(e) [there exists]p [greater than or equal to] 0 such that [[DELTA].sub.12][b.sub.n,j] [greater than or equal to] 0, n [greater than or equal to] [bar.n] + p, 0 [less than or equal to] j [less than or equal to] [bar.n] - 2,

(f) [[beta].sub.n] [less than or equal to] -[a.sup.*] < 0, n [greater than or equal to] [n.sup.*] [greater than or equal to] [bar.n],

or

([f.sub.bis]) [[DELTA].sub.12][b.sub.n,j] [greater than or equal to] [b.sup.*] > 0, n [greater than or equal to] [bar.n] + p, 0 [less than or equal to] j [less than or equal to] [bar.n] - 2.

Then, for any [x.sub.0] [member of] R, [lim.sub.n[right arrow]+[infinity]][x.sub.n] = 0.

Proof. First of all observe that (a) assures 2[x.sub.n+1][V.sub.n,j] [greater than or equal to] 0, n [greater than or equal to] j [greater than or equal to] n - 1. From here and (13) we derive

[mathematical expression not reproducible]. (27)

Now, proceeding as in the proof of Theorem 1, we arrive to

[mathematical expression not reproducible], (28)

which, taking into account (b), (c), and (e), assures

[mathematical expression not reproducible], (29)

or

[mathematical expression not reproducible], (30)

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 [x.sub.n], obtained by (5) with [x.sub.0] [member of] R, cannot diverge and if it is convergent then its limit is zero.

Remark 8. If, in Theorem 6, [bar.n] = 1, 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 [a.sub.n] and [b.sub.n,j]

[mathematical expression not reproducible]; (31)

[mathematical expression not reproducible]; (32)

[mathematical expression not reproducible]; (33)

[mathematical expression not reproducible], (34)

satisfies the assumptions of the corollaries in the previous section.

To be more specific (31) fulfills both corollaries with [bar.n] = 0. Equation (33) satisfies only Corollary 3 with [bar.n] = 3, whereas (32) satisfies Corollary 4 but not 2.1, because [b.sub.nn] + [a.sub.n] [right arrow] 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 [[summation].sup.n.sub.j=0] [absolute value of [b.sub.nj]] 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

[mathematical expression not reproducible]. (35)

Here condition (i) for the coefficients [a.sub.n] and [b.sub.n,n] 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 [x.sub.n].

Theorem 6 can be applied to the following example:

[mathematical expression not reproducible], (36)

First of all we need to show that (a) holds with [bar.n] = 2. Starting from the initial condition given in (36), by simple computation, we have [x.sub.1] = -0.003 < 0, 0 < [x.sub.2] < 1. Our aim is to prove that 0< [x.sub.n] < 1, for n [greater than or equal to] 2. Let us proceed by induction on n. Assume 0 [less than or equal to] [x.sub.j] [less than or equal to] 1, j = 2, ... , n - 1 and verify that the same is true for [x.sub.n] given by

[mathematical expression not reproducible]. (37)

From the definition of [b.sub.n,j] in (36), it easily follows that -4 < 1 + [[summation].sup.n-1.sub.j=0] [b.sub.n,j] < 0. Then taking into account the induction hypothesis and that [b.sub.n,n-1] [less than or equal to] -1, we obtain

[mathematical expression not reproducible]. (38)

As [x.sub.0] = -10 and [x.sub.1] < 0, it turns out that the right-hand side of (37) is positive, then [x.sub.n] > 0. On the other hand (38) implies [x.sub.n] < ([b.sub.n,0][x.sub.0] + [b.sub.n,1][x.sub.1])/21, n [greater than or equal to] 2, with [b.sub.n,0] [less than or equal to] -1/2, [b.sub.n,1] [less than or equal to] -1, which assures 0 < [x.sub.n] < 1. We conclude that hypothesis (a) of Theorem 6 is satisfied. Since (d) is true and (c), (f) are obvious with [n.sup.*] = 0, it remains to prove (e). In our case (e) corresponds to [there exists]p [greater than or equal to] 0 such that [[DELTA].sub.12][b.sub.n,j] [greater than or equal to] 0, for n [greater than or equal to] 2 + p and j = 0. Observe that [[DELTA].sub.12][b.sub.n,0] can be written as [mathematical expression not reproducible in ascii], since (c) holds and [mathematical expression not reproducible]. In conclusion, all the hypotheses of Theorem 6 are fulfilled and [x.sub.n] [right arrow] 0, as can be seen in Figure 2.

Remark 9. We want explicitly to mention that Theorem 1 cannot be applied to (36) because [mathematical expression not reproducible], and hypothesis (iv) is not satisfied.

Finally, we observe that, if in (36) we choose [b.sub.n,0] according to the remaining coefficients for j [not equal to] 0, that is, b[n.sub.,0] = -((n + 1)[e.sup.-n] + 1/[(n+ 1).sup.2]), then Theorem 6 is still valid with [bar.n] = 1. 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 problem

[mathematical expression not reproducible], (39)

and a simple method of family (4), the Backward Euler method (see , [12, (3.8)])

[mathematical expression not reproducible], (40)

where h>0 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) [there exists][bar.t] such that A(t) [less than or equal to] 0, t > [bar.t];

(ii) K(t, 0) [less than or equal to] 0, t [greater than or equal to] 0, [partial derivative]K(t, 0)/[partial derivative]t [greater than or equal to] 0, t > [bar.t];

(iii) [partial derivative]K(t, s)/ [partial derivative]s [less than or equal to] 0, t > 0;

(iv) [[partial derivative].sup.2] K(t, s)/[partial derivative]t[partial derivative]s [greater than or equal to] 0, t > [bar.t].

Then the solution [y.sub.n] of (40) is bounded [for all][y.sub.0] [member of] R. If in addition

(v) A(t) [less than or equal to] -[A.sup.*] < 0, t > [bar.t],

or

[mathematical expression not reproducible],

then [lim.sub.n[right arrow]+[infinity]][y.sub.n] = 0

Proof. Note that (5) coincides with (40) whenever [a.sub.n] = hA([t.sub.n]), n [greater than or equal to] 1, and [b.sub.nj] = [h.sup.2]K([t.sub.n], [t.sub.j]), n [greater than or equal to] 1, j [less than or equal to] n. Now, assumptions (i)-(iv) immediately assure (i)-(iv) of Theorem 1 for any fixed h and [bar.n](h) such that [bar.n](h)h = [bar.t]. Moreover, (v) implies (v) of Theorem 1 with [a.sup.*] = h[A.sup.*], so that Corollary 3 holds. In order to exploit ([v.sub.bis]) note that it is equivalent to [[partial derivative].sup.2](K(t, s) - [K.sup.*]ts)/[partial derivative]t[partial derivative]s > 0, t > [bar.t], s < [bar.t], which in turn implies that, for any fixed h, the function [GAMMA](i, s, h) = K(t + h, s) - K(t, s) - [K.sup.*]ts is increasing with respect to s, so that ([v.sub.bis]) of Theorem 1 is fulfilled with [b.sup.*] = [h.sup.4][K.sup.*] and [bar.n] 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 y(t) 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 . 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 y(t) of (39) is bounded, for any choice of [y.sub.0] [member of] R. Furthermore, if hypothesis (v) or ([v.sub.bis]) of Theorem 10 holds and

y is uniformly continuous, (41)

then [lim.sub.t[right arrow]+[infinity]]y(t) = 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 [b.sub.nj] represents the samples K([t.sub.n], [t.sub.j]), and in the numerical solution obtained by the Backward Euler method (40).

Moreover, it is worth noting that if in (39)

[mathematical expression not reproducible], (42)

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., ):

[mathematical expression not reproducible]. (43)

This kind of equation can be easily recast in the form (39), with [mathematical expression not reproducible], and the method (40) for it will read

[mathematical expression not reproducible]. (44)

Theorem 10 immediately becomes as follows.

Corollary 14. Consider (44) and assume that

(a1) [there exists][bar.t] > 0: A(t) [less than or equal to] 0, t > [bar.t],

(a2) f(t) [less than or equal to] 0, t [greater than or equal to] 0,

(a3) f'(t) [greater than or equal to] 0, t [greater than or equal to] 0,

(a4) A(t) [less than or equal to] [A.sup.*] < 0, t [greater than or equal to] [bar.t],

or

(a[4.sub.bis]) f'(t) [greater than or equal to] [K.sup.*] > 0, t > 0.

Then, for any [y.sub.0] [member of] R, [lim.sub.n[right arrow]+[infinity]][y.sub.n] = 0.

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)) [[partial derivative].sup.2]K(t - s)/[partial derivative]t[partial derivative]s is of convolution type. The application of Theorem 6 leads to the following result.

Theorem 15. Consider (44) and assume that

(HA) [there exists][bar.n] [greater than or equal to] 0: [y.sub.n] [greater than or equal to] 0 (<0), n > [bar.n];

(HB) [there exists][??] > 0: A(t) [less than or equal to] 0, t > [??];

(HC) f(t) [less than or equal to] 0, t > 0;

(HD) f' (t) [greater than or equal to] 0, t > [??].

Then, for any [y.sub.0] [member of] R, [lim.sub.n[right arrow]+[infinity]][y.sub.n] = 0.

Proof. We want to prove that all the hypotheses of Theorem 6 are fulfilled with [a.sub.n] = hA([t.sub.n]) and [b.sub.nj] = [h.sup.2] [[summation].sup.n.sub.i=n-j] f([t.sub.i]). Let h > 0 be fixed and let [mathematical expression not reproducible] be such that [mathematical expression not reproducible in ascii]. 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 n [greater than or equal to] n = max{[??], [bar.n]},

[mathematical expression not reproducible]. (45)

As [b.sub.n,0] = [h.sup.2]f([t.sub.n]), (HC) implies the first condition in (c), and since [[DELTA].sub.1][b.sub.n,0] = [h.sup.2](f([t.sub.n+1]) - f([t.sub.n])), (HD) immediately assures the second condition of (c). Furthermore, since [[DELTA].sub.2][b.sub.n,j] = [h.sup.2] f([t.sub.n-j-1]), (HC) also implies (d) and, taking into account the fact that [[DELTA].sub.1,2][b.sub.n+1,j] = [h.sup.2](f([t.sub.n+1-j]) - f ([t.sub.n-j])), we have that (HD) implies (e) with p = max{0, [??] - 2}. Finally, whenever f is not identically zero, we can always assume, with no loss of generality, that [??] is such that f([??]) < 0, so that (45) assures (f) with [A.sup.*] = -[h.sup.2]f([??]), and the proof is complete.

As already mentioned above, in  equation (3.32) represents the velocity of the centre of mass of a system of particles in collective motion under alignment and chemotaxis effect. Here A(t) = 0 and

[mathematical expression not reproducible], (46)

with [c.sub.1], [c.sub.2], and [c.sub.3] depending on the number of particles, their dimension, and the dynamic of the motion. For significative values of [c.sub.1], [c.sub.2], and [c.sub.3] hypotheses (a1), (a2), (a3), and (a[4.sub.bis]) 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 [c.sub.1] = [10.sup.5], [c.sub.2] = [10.sup.-2], and [c.sub.3] = [10.sup.-4].

https://doi.org/10.1155/2018/6935069

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.

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E. Messina (iD) (1) and A. Vecchio (2)

(1) Dipartimento di Matematica e Applicazioni, Universita degli Studi di Napoli "Federico II", Via Cintia, 80126 Napoli, Italy

(2) C.N.R. National Research Council of Italy, Institute for Computational Application "Mauro Picone", Via P. Castellino 111, 80131 Napoli, Italy

Correspondence should be addressed to E. Messina; eleonora.messina@unina.it

Received 17 November 2017; Revised 9 January 2018; Accepted 15 January 2018; Published 15 February 2018

Academic Editor: Mustafa R. S. Kulenovic

Caption: Figure 1: Problem (35): unbounded [x.sub.n].

Caption: Figure 2: Problem (36): vanishing [x.sub.n].

Caption: Figure 3: Problem (43)-(46): vanishing [y.sub.n].
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