E. Messina,¹ Y. Muroya,² E. Russo,¹ and A. Vecchio³

Abstract

We consider nonlinear difference equations of unbounded order of the form xi=bi−∑j=0iai,jfi−j(xj),  i=0,1,2,…, where fj(x)  (j=0,…,i) are suitable functions. We establish sufficient conditions for the boundedness and the convergence of xi as i→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.

1. Introduction

We consider the following nonlinear Volterra discrete equation of nonconvolution type: (1.1) The existence problem for solution of Volterra discrete equations arises in the nonlinear implicit case. For linear implicit equations and nonlinear explicit equations, the problem is easily solved. Recently, some local and global existence theorems for Volterra discrete equations in the general case are given in [1, 2].

From now on, we assume that there exists a strictly increasing function such that (1.2) Note that (1.2) implies that (1.3) The above difference equation can be considered as the discrete counterpart of the Volterra integral equation whose importance in the applications is well known (see, e.g., [3, 4]), and arises also in the application of numerical methods to Volterra integral and integrodifferential equations. The theory of the qualitative behavior of this type of nonlinear difference equation is very important, in particular for the study of numerical stability of such methods (see, e.g., [5–11] and the references therein).

In this paper, we study some sufficient conditions for the boundedness of the solutions (if they exist) of (1.1), subject to (1.2), and their asymptotic behavior as . In particular, in Section 2 we investigate the asymptotic behavior when is upper bounded by a linear function. The case of nonnegative coefficients is investigated in Section 3 and, with additional monotonicity assumptions, in Section 4.

2. Case of

Assume that, in (1.1) with (1.2), the following additional hypotheses hold: (2.1) Observe that the second part of (2.1) is true if in (1.2) . The following lemma can be easily proved.

Lemma 2.1. If (2.2) for all .

Here and in the sequel we assume a sum with a negative superscript to be zero. By using (2.1) and Lemma 2.1, from (1.1), we have that (2.3) and we set (2.4) This inequality will be useful in order to find sufficient conditions for the boundedness of and for its convergence to zero as tends to infinity.

Theorem 2.2. Consider (1.1) with (1.2) and (2.1), if there exists a positive constant such that (2.5) for some positive integer , then is bounded and (2.6) Moreover, if (2.7) then .

Proof. Let us consider (2.3), by using (2.5), we have that (2.8) In particular, assume that the third part of (2.5) holds, then (2.9) and thus, (2.10) Hence, the following inequalities hold for each : (2.11) For this reason, (2.12) from which we obtain that (2.13) Thus, is bounded and satisfies (2.6).

Assume that and put and . Then, since is bounded and the third of (2.5) holds, we have that and . Let's take any and consider a continuous function on . Then, by , there exists a constant such that (2.14) For the above , there exists a positive integer such that and , for any . By assumption (2.7), we have that for , there exists a positive integer such that (2.15) Then, for and , we have that (2.16) which is a contradiction with the definition. Hence, and we obtain .

Note that the third part of (2.5) is equivalent to .

The theorem above gives some conditions on the coefficients of (1.1) for the boundedness of which supplement the results in [12, Theorem 2.1]. Moreover, it worths while to compare our result with the ones in [9, Theorem 3.1] and [5, Theorem 4.1]. In order to do that, we assume , and then is given. In this case, following the line of the proof of Theorem 2.2, we can still show that vanishes as provided that (2.5) and the second part of (2.7) hold. Observe that this represents an additional result with respect to [9, Theorem 3.1] and [5, Theorem 4.1] which, involving the sum of the coefficients on the second index, enlarges the set of conditions for to be bounded and convergent to zero. As an example, for equation (2.17) (3.2) in [9] or the sufficient condition in [5] is not satisfied, however (2.5) is fulfilled. Moreover, it is easy to see that, in the convolution case , the third of (2.5) coincides with the known one [5, 10] (2.18) and the second part of (2.7) is implied by (2.5).

Theorem 2.2 turns out to be quite useful in the linear case when (1.1) represents the linearized equation for the global error of a numerical method applied to a Volterra integral equation. In this case, represents the local truncation error of the method at the step . Thus, if is bounded for all and if (2.5) holds, then the error is bounded and the bound is given in (2.6).

The following theorem provides some sufficient conditions on the coefficients of (1.1) for the summability of , which turn out to be less restrictive of those stated by [13, Theorem 2.8].

Theorem 2.3. For (1.1) with (1.2), assume (2.1). If (2.19) then , and consequently, .

Proof. By (2.3), (2.20) Therefore, by (2.19), we have that (2.21) and then, .

In the case (1.1) is linear, (2.22) the following theorem is easily proved.

Theorem 2.4. For the linear equation (2.22), assume , and for (2.23)

(i) Suppose that (2.24) Then, . In particular, if there exists a positive integersuch that(2.25)thenis bounded and(2.26)Moreover, if(2.27)then. (ii) If (2.28) then , and consequently, .

Proof. By (2.22), we obtain that (2.29) Then, we have that (2.30) Thus, analogously to the proofs of Theorems 2.2 and 2.3, we obtain the conclusion of this theorem.

3. Nonnegative Coefficients

In this section, we focus on the solutions of (1.1) with (1.2) and (3.1) Such discrete equations are useful, above all, in the investigations on the behavior of the solution of some numerical methods when used to solve nonlinear heat flow in a material with memory (see [14] and the bibliography therein). Let us start with the following lemma, which describes some aspects of the solution of (1.1)-(1.2) with (3.1) when has a sign eventually constant for all . The utility of this lemma is not in itself, but as an instrument to prove some of the next theorems (see Theorems 3.4, 4.1 and 4.3).

Lemma 3.1. Let be the solution of (1.1) and assume that

(i) and for each ; (ii) there exists such that (resp., ) for any,

then (3.2) where and is a positive constant.

Moreover, assume that one of the following conditions holds:

(iii₁) , (iii₂) and there exists a strictly increasing function on such thatand (resp., ), then .

Furthermore, if, in addition to , there exists a positive constant such that (resp., ), then .

Proof. Since and is a continuous function, then is bounded. Assume that there exists a nonnegative integer such that for any (the analysis of the case for all is analogous). Then, by the fact that, for the main hypothesis (1.2), whenever , we have (3.3) Hence, the first part of the lemma is proved. Consider now the two cases and separately.

Case (iii₁): of course implies .

Case (iii₂): put . Assume that , and let be a subsequence of such that . Then, one can prove that . By (1.1) and assumptions, we have (3.4) Therefore, (3.5) which is a contradiction because and . Hence, we have .

In addition, suppose that there exists a positive constant such that . Then, we have that (3.6) Thus, from , we conclude that .

The proof is completely analogous when there exists a nonnegative integer such that for any .

Remark 3.2. Observe that in the linear case (2.22), the last conditions of Lemma 3.1 are satisfied whenever and .

Hereafter, we investigate on the boundedness of the solution of (1.1)-(1.2) when (3.7)

Lemma 3.3. Let be the solution of (1.1) with (1.2) and (3.7), and assume that (3.8) then is bounded.

Proof. Let be the bound for and . Let us write as the sum of the following two contributions: (3.9) where and . Therefore, since (1.2), (3.1), (3.7), and (3.8) hold, we have that (3.10) Thus, is bounded and the proof is complete.

As an example we consider the equation (3.11) in this case and . Hence, (3.12) Another example is given by the explicit equation (3.13) Here and , hence (3.14) From Figure 1 it is clear that the bounds established by Lemma 3.3 (represented by dotted lines) may be quite sharp. We are able to prove the following result.

Figure 1: Plot of (3.13) and its bounds given by (3.14).

Theorem 3.4. Assume that is continuous on , (3.15) (3.16) for all . Then .

Proof. Let and assume . Since we are in the hypotheses of Lemma 3.3, is bounded and then . For any fixed , consider a continuous function on . Then, by , there exists a constant such that (3.17) For the above , there exists a positive integer such that and , for any . By Assumption (3.16), we have that for , there exists a positive integer such that (3.18) Then, for and , we have that (3.19) Let us rewrite (1.1) in the following form: (3.20) where , for and , for and . Thus, (3.21) and, since is an increasing function, we have that, for all , (3.22) Since we are in the hypothesis that the coefficients are nonnegative, it follows that (3.23) In conclusion, from (3.24) and by using (3.20), (3.19), and (3.17), the following inequality holds: (3.25) This result contradicts the definition. Hence, , so are eventually nonnegative. Since it is easy to see that we are in the hypotheses of Lemma 3.1 (case (iii₁)), then .

Remark 3.5. Once again, in the convolution case, the first part of (3.15) implies the second one of (3.16).

For the special case , we establish the following sufficient condition from Theorem 3.4.

Theorem 3.6. Suppose that and assume that (3.26) hold, then the solution of (1.1) tends to zero as tends to infinity.

Proof. Put . Since for , hence is a strictly monotone decreasing function in . Now, we will prove that , for .

Let for . Then we have that (3.27) By recalling that and , we have . Since the function is increasing for , there results . Thus, . Hence, we have that , for . Thus, for with , the second of (3.15) is true and, by Theorem 3.4, we have .

Remark 3.7. From the proof of Theorem 3.6 it is clear that the second part of (3.15) is satisfied by . By playing with , this allows us to consider a wide variety of functions which satisfy (1.2). For instance, in the cases and the stained areas in Figure 2 represent the admissible regions for the functions respectively (the solid lines show, as an example, the graphs of and ).

Figure 2: Plots of some admissible regions for according to Theorem 3.6.

4. Monotonic Nonnegative Coefficients

In this section, for (1.1), first, we consider the case that (4.1) We provide the following theorem which generalizes [15, Theorem 2.1] to the nonlinear case.

Theorem 4.1. In addition to condition (4.1), suppose that (4.2) Then, any solution of (1.1) satisfies (resp., ), . Moreover, if , for all and there exists a strictly increasing function on such that and (resp., ), then .

In addition, if there exists a positive constant such that (resp., ), then .

Proof. We prove the theorem in the case , the proof for is perfectly symmetric. Then by (1.1), we have that (4.3) hence, for the properties of described in (1.2), it has to be .

Proceeding by induction, suppose that . By (1.1), (4.4) and hence, by adding the two relations and taking into account that, for the second part of (4.1), , we have that (4.5) So we have that and, from (1.1), .

Thus, we are in the hypotheses of Lemma 3.1 part and, hence, we get the thesis.

Observe that when (1.1) is linear, the last condition of Theorem 4.1 is satisfied by choosing and . In this case, the hypotheses of Theorem 4.1 include, as particular cases, those of [15, Theorem 2.1]. In particular we note that, as Theorem 4.1 prove the summability of the solution , it is interesting when applied to the equation satisfied by the fundamental matrix of a Volterra difference equation (see, e.g., [15, equation (1.4)]). Namely, in [15] it is underlined that such a result can be employed in the study of the stability of some numerical methods.

A simple application of Theorem 4.1 in the linear case is given by the following example.

Example 4.2. Let us consider the difference equation (4.6) whose solution is given by (4.7) Then, for and , all the conditions in Theorem 4.1 are satisfied with , which implies and . Observe that in this case the bound coincides with the exact value of the sum of the series.

Next, we provide the following theorem whose proof is a direct extension of the proof of Crisci et al. [6, Theorem 2.1], which gives a priori bound for the solution of (1.1) depending on the forcing terms .

Theorem 4.3. In addition to the conditions (4.1), assume that (4.8) Then, any solution of (1.1) is bounded and satisfies (4.9) Moreover, suppose that and there exists a strictly increasing function on such that and (resp., ), then .

In addition, if there exists a positive constant such that (resp., ), then .

Proof. Consider the two possible subcases: (a) , and (b) .

(a) Assume .

If , then by (1.1), we get and . Hence, (4.9) holds if is oscillatory about . Let and denote by the time moment of the first passage of the solution through the zero, that is, (4.10) The time moment of the following passage through the zero of the solution after is denoted by , that is, (4.11) In a similar way, we introduce the indexes as follows: (4.12)

(1) Consider that , hence . Then, from (1.1), we have that (4.13) where every summation of the type involves only positive , while the negative ones. Now observe that, for , by using (4.1) and the fact that , we have , furthermore , because of (4.12), hence both and are less than or equal to zero, thus . By using these considerations it is easy to see that the following inequality holds: (4.14) With analogues considerations we get (4.15) By adding each side of (4.13) and taking into account (4.14), (4.15), it comes out that (4.16) By using the monotonicity of stated by (4.1) and the main hypothesis (1.2), taking into account (4.12), we have that (4.17)

(2) Consider that , hence . Proceeding as above, we have (4.18) Hence, from (1) and (2), we obtain (4.9). Part (b) of the proof is essentially mirror-like of part (a) and leads once again to (4.9). Thus, any solution of (1.1) is bounded and satisfies (4.9).

Moreover, suppose that and there exists a strictly increasing function on such that and (resp., ).

If there exists a nonnegative integer such that (resp., ) for any , since and for all , we are in the hypotheses of Lemma 3.1 part (iii₂) and we obtain . On the contrary, if such an index does not exist, let and consider the extract subsequence of all the positive values in . Assume that , then (4.19) Taking into account (4.12), there exists an index such that , then plays the role of in (4.13) and, analogously to (4.16), we have (4.20) Hence, since , for all , we have that (4.21) and so, since only positive quantities are involved, we get (4.22)Passing to the as , we have that(4.23) Taking into account that all the involved in the summations above form the extract of the positive values in , we get , which is a contradiction with (4.19), so . An analogous proof on the extract subsequence of all negative values of leads to . The same happens when . Hence, in conclusion, we obtain that .

In addition, suppose that there exists a positive constant such that . Then, by (4.22) and the fact that is strictly positive, we conclude that . Similarly, we obtain that . Hence, .

Acknowledgment

This research was partially supported by Waseda University grant for special research Projects 2006B–167 and Scientific Research (c), no. 19540229 of Japan Society for the Promotion of Science.

References

1. Y. Song and C. T. H. Baker, “Perturbation theory for discrete Volterra equations,” Journal of Difference Equations and Applications, vol. 9, no. 10, pp. 969–987, 2003.
2. C. T. H. Baker and Y. Song, “Discrete Volterra operators, fixed point theorems and their application,” Nonlinear Studies, vol. 10, no. 1, pp. 79–101, 2003.
3. G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, vol. 34 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.
4. M. S. Klamki, Ed., Problems in Applied Mathematics: Selections from SIAM Review, M. S. Klamki, Ed., SIAM, Philadelphia, Pa, USA, 1990.
5. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Stability of difference Volterra equations: direct Liapunov method and numerical procedure,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 77–97, 1998.
6. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Stability of discrete Volterra equations of Hammerstein type,” Journal of Difference Equations and Applications, vol. 6, no. 2, pp. 127–145, 2000.
7. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Asymptotic properties of solutions of Volterra difference equations with finite linear part,” Dynamic Systems and Applications, vol. 8, no. 1, pp. 1–20, 1999.
8. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “A priori bounds on the solution of a nonlinear Volterra discrete equation,” Stability and Control: Theory and Applications, vol. 3, no. 1, pp. 38–47, 2000.
9. S. N. Elaydi, V. L. Kocic, and J. Li, “Global stability of nonlinear delay difference equations,” Journal of Difference Equations and Applications, vol. 2, no. 1, pp. 87–96, 1996.
10. S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.
11. Ch. Lubich, “On the numerical solution of Volterra equations with unbounded nonlinearity,” Journal of Integral Equations, vol. 10, no. 1–3, pp. 175–183, 1985.
12. A. Vecchio, “Boundedness of the global error of some linear and nonlinear methods for Volterra integral equations,” Journal of Integral Equations and Applications, vol. 12, no. 4, pp. 449–465, 2000.
13. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Boundedness of discrete Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 106–130, 1997.
14. Ph. Clément and J. A. Nohel, “Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels,” SIAM Journal on Mathematical Analysis, vol. 12, no. 4, pp. 514–535, 1981.
15. A. Vecchio, “Volterra discrete equations: summability of the fundamental matrix,” Numerische Mathematik, vol. 89, no. 4, pp. 783–794, 2001.