Periodicity in nonlinear systems with infinite delay.

1. Introduction

Let Z be the set of integers. In this article we use Schaefer's fixed-point theorem to deduce the existence of periodic solutions of infinite delay nonlinear Volterra difference equations of the form

x(n + 1) = Dx(n) + f (x(n)) + [n.summation over (j=-[infinity]]) K(n, j)g(x(j)) + p(n), (1.1)

with the existence of positive constant Q such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where D is a k x k matrix and p(n) is a given k x 1 vector with p(n + T) = p(n) for integer T. The kernel K (n, j) satisfies K(n + T, j + T) = K(n, j) for all -[infinity] < j [less than or equal to] n < [infinity], where (n, j) [member of] [Z.sup.2] and K(n, j) = 0 for j > n. The period T is taken to be the least positive integer for which these hold. The functions f and g are continuous. In  the author studied the existence of periodic solutions of the Volterra difference system with infinite delay

[DELTA]x(n) = Dx(n) + [n.summation over (j=-[infinity])] C(n - j)x(j) + g(n), n [member of] Z with [[infinity].summation over (u=0)]|C(u)| < [infinity], (1.2)

where D and C are k x k matrices and g(n) is a given k x 1 vector with g(n + T) = g(n) for integer T, by using Schaefer's fixed point theorem, which we will state later in the paper. In  the mapping was constructed by taking direct sum in (1.2). On the other hand, Elaydi in  considered (1.2) and utilized the notion of the resolvent of an equation associated with (1.2) and concluded the existence of a periodic solution of (1.2). In arriving at his results, Elaydi had to show that the zero solution of an homogenous equation associated with (1.2) is uniformly asymptotically stable. Thus, it was assumed that |D| < 1 where | x | is a suitable matrix norm. Later on, for the purpose of relaxing |D| < 1, Elaydi and S. Zhang , used the notion of degree theory, due to Grannas, and obtained the existence of a periodic solution of (1.2).

Once our results are established, we apply them to nonlinear infinite delay Volterra discrete equations of the form

x(n + 1) = ax(n) + f (x(n)) + [n.summation over (j=-[infinity)] K(n, j)g(x(j)) + p(n). (1.3)

In  the author considered (1.3) with the assumptions that the two functions f and g are uniformly bounded and the coefficient a satisfies the stringent condition -1 [less than or equal to] a [less than or equal to] 1. In this research our objective is to relax those conditions. We achieve our objective by displaying nonnegative definite Lyapunov functionals, which in turn give the a-priori bound. Thus, this paper will advance the theory of existence of periodic solutions in the most general form of nonlinear infinite delay volterra difference equations. For more on the existence of periodic and positive periodic solutions in difference equations we refer the reader to [4-7] and the references therein.

2. Existence of Periodic Solutions

We begin this section by stating Schaefer's fixed point theorem which can be found in .

Theorem 2.1. (Schaefer ) Let (B, | x |) be a normed linear space, H a continuous mapping of B into B which is compact on each bounded subset of B. Then either

(i) the equation x = [lambda]Hx has a solution for [lambda] = 1, or

(ii) the set of all solutions x, for 0 < [lambda] < 1, is unbounded.

For (1.1) a homotopy will have to be constructed which we obtain in the following manner. Let m be a real number such that either m > 1 or m < -1. For 0 [less than or equal to] [lambda] [less than or equal to] 1, we rewrite (1.1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.1)

One may easily verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

is a solution of (1.2) and hence of (1.1). Define the space PT by

PT = {x : x(n + T) = x(n), for all n [member of] Z},

where T is the least positive integer so that x(n + T) = x(n). Then (PT , | x |) defines a Banach space of T -periodic k x 1 real vector sequences x(n) with the maximum norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For x [member of] [P.sub.T], using (2.2) we define the mapping H : [P.sub.T] [right arrow] [P.sub.T] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Thus,

x = [lambda]Hx

is equivalent to (2.2).

Next we prove two lemmas that are essential in the application of Schaefer's theorem.

Lemma 2.2. If H is defined by (2.3), then H is continuous and H : [P.sub.T] [right arrow] [P.sub.T].

Proof. For the continuity of H we let [[phi].sub.1], [[phi].sub.2] [member of] [P.sub.T] and use (2.3) to obtain,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By invoking the continuity of f and g and the fact that the infinite series [n-1.summation over (j = -[infinity])] |[m.sup.-(n-j-1)]| is convergent, we deduce that H is continuous. Left to show is that H : [P.sub.T ] [right arrow] [P.sub.T]. Let [phi] [member of] PT and use the substitution v = j - T followed by the substitution r = s - T to obtain (H[phi])(n + T) = (H[phi])(n). This conclude the proof.

Lemma 2.3. If H is defined by (2.3), then H maps bounded subsets into compact subsets.

Proof. Let J > 0 be given and define the two sets S = {x [member of] [P.sub.T] : |x| [less than or equal to] J and W = {Hx : x [member of] [P.sub.T]}. Then W is a subset of [R.sub.Tk], which is closed and bounded and thus compact. As H is continuous in x it maps compact sets into compact sets. We deduce that W = H(S) is compact. This concludes the proof.

Now we are in position to state and prove our main theorem that yields the existence of a periodic solution of (1.1).

Theorem 2.4. If there exists an L > 0 such that for any T -periodic solution of (2.1), 0 < [lambda] < 1 satisfies |x| [less than or equal to] L, then (1.1) has a solution in PT .

Proof. Let H be defined by (2.3). Then, by Lemmas 2.2 and 2.3, H is continuous, compact and T -periodic. The hypothesis |x| [less than or equal to] L rules out part (ii) of Schaefer's fixed point theorem and thus x = [lambda]Hx has a solution for [lambda] = 1, which solves Equation (1.1). This concludes the proof.

We end this section by making the following remark. When it comes to application, the reader shall see that we may have to require m [member of] (-1, 0) [union] (0, 1). Thus, to take care of such situation we note that Eqn. (1.1) is equivalent for [lambda] = 1 to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.4)

Then it follows readily that x is a bounded solution of (2.4) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.5)

Then one may easily prove a theorem similar to Theorem 2.4 for the case m . (-1, 0). (0, 1).

3. Applications to Scalar Equations

Now we apply the results of the previous section to scalar nonlinear Volterra difference equations with infinite delay of the form

x(n + 1) = ax(n) + f (x(n)) + [n.summation over (j = -[infinity])] K(n, j)g(x(j)) + p(n), (3.1)

where the terms f, g, K and p obey the same conditions as before. The highlight of this paper is to prove the existence of periodic solution of Eq. (3.1), where the magnitude of a could be |a| > 1. In most of the literature, it is required that |a| < 1. To relax this condition we resort to nonnegative definite Lyapunov functional to obtain the a-priori bound on all possible T -periodic solutions of Eqn. (3.1) and then conclude the existence of a periodic solution by invoking Theorem 2.4.

We shall assume in addition to those assumptions made in the previous section that there exists F : [Z.sup.+] [right arrow] R and R > 0 such that

|K(n, u + n)| = F(u), with [[infinity].summation over (u = 0)] |F(u)| [less than or equal to] R, (3.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.3)

We note that assumption (3.2) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we state two theorems; one will show the existence of a periodic solution of (3.1) when |a| < 1, and the other when |a| > 1. The proof of the first theorem will be established in three different cases on the coefficient a.

Theorem 3.1. Assume (3.2) and (3.3). Also, we assume that there exist an a > 0 such that

|f (x)| + R|g(x)| [less than or equal to] [alpha]|x|,

and

|[mu]| + [alpha] - 1 [less than or equal to] -[beta], for some positive constant [beta], (3.4)

where [mu] is to be defined in the body of the proof and R is given by (3.2). Then, Eqn. (3.1) has a T-periodic solution.

Proof. Case 1. 0 < a < 1. Set m = a. Then 0 < m < 1. We shall apply Theorem 2.4 with m . (0, 1) to the corresponding family of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

Our aim is to show that there is a priori bound, say L such that all solutions x of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for 0 < [lambda] < 1 satisfy |x| [less than or equal to] L. Once this is accomplished, we can rule out (ii) of Schaefer's theorem and then conclude the above equation has a solution for [lambda] = 1.

We begin by rewriting (3.5) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.6)

where [mu] = m + [lambda](-m + a). Define the Lyapunov functional V by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.7)

It is clear that for x [member of] [P.sub.T], V(n + T, x) = V (n, x) and hence V is periodic. Along the solutions of (3.6) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since V is periodic for x [member of] [P.sub.T], we have by summing the above inequality over one period that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that

[n + T - 1.summation over (s = n)]|x(s)| [less than or equal to] T |p|/[beta].

Thus, |x(n)| is bounded over one period, and hence for any T -periodic solution of (3.5) there is an E > 0 such that |x(n)| [less than or equal to] E, which serves as the a priori bound on every possible T -periodic solution of (3.5). Therefore, by Theorem 2.4, Eqn. (3.1) has a T -periodic solution for 0 < a < 1. This concludes the proof of Case 1.

Note that since 0 < [lambda] < 1, condition (3.4) reduces to |a| + [alpha] - 1 [less than or equal to] -[beta].

Case 2. -1 < a < 0. Set m = a. Then -1 < m < 0 and we apply Theorem 2.4 with m [member of] (-1, 0) to the corresponding family of equations (3.5) with [mu] = m + [lambda](-m + a) = a. Define the Lyapunov functional V by (3.7) and proceed with the proof as in Case 1.

Note that since 0 < [lambda] < 1, [mu] = a and hence condition (3.4) reduces to |a| + [alpha]-1 [less than or equal to] -[beta].

Case 3. a = 0. Let m be any fixed number strictly between 0 and 1. Then, [mu] = m-[lambda]m < m. Choose m small enough so that (3.4) is satisfied. Then apply Theorem 2.4 with m [member of] (0, 1) to the corresponding family of equations (3.5). Define the Lyapunov functional V by (3.7) and proceed with the proof as in Case 1.

The next theorem handles the case |a| > 1.

Theorem 3.2. Assume (3.2) and (3.3). Also, we assume that there exist an a > 0 such that

|f (x)| + R|g(x)| [less than or equal to] [alpha]|x|,

and

|[mu]| - [alph] - 1 [greater than or equal to] [beta], for some positive constant [beta],

where [mu] is to be defined in the body of the proof. Then, Eqn. (3.1) has a T -periodic solution.

Proof. Case 1. a > 1. Set m = a. We shall apply Theorem 2.4 with m > 1 to the corresponding family of equations (3.5). Then, [mu] = m + [lambda](-m + a) = a.

Define the Lyapunov functional V by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.8)

It is clear that for x [member of] [P.sub.T], then V (n + T, x) = V (n, x) and hence V is periodic. Along the solutions of (3.6) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since V is periodic for x [member of] [P.sub.T], we have by summing the above inequality over one period that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that

[n + T-1.summation over (s=n)] |x(s)| [less than or equal to] T p/[beta].

Thus, |x(n)| is bounded over one period, and hence for any T -periodic solution of (3.5) there is an E > 0 such that |x(n)| [less than or equal to] E, which serves as the a-priori bound on every possible T -periodic solution of (3.5). Therefore, by Theorem 2.4 Eqn. (3.1) has a T-periodic solution for a > 1. This concludes the proof of Case 1.

Again, we remark that the condition |[mu]| - [alpha] - 1 [less than or equal to] [beta], for some positive constant [beta], reduces to |a| - [alpha] - 1 = [beta].

Case 2. a < -1. Set m = a. Then m < -1 and we apply Theorem 2.4 to the corresponding family of equations (3.5) with [mu] = m + [lambda](-m + a) = a. Thus, |[mu]| = |a|. Define the Lyapunov functional V by (3.8) and then the proof is the same as in Case 2. This concludes the proof of the theorem.

Remark 3.3.

1) By relaxing the condition |a| < 1, we point point out that Theorem 3.2 significantly improves the literature that is related to the existence of periodic solutions in Volterra difference equations.

2) In , for |a| = 1, the author was able to show the existence of a periodic solution under the stringent condition that the functions f and g are uniform bounded by certain positive constants. However, we could not do the same here under the condition

|f (x)| + R|g(x)| [less than or equal to] [alpha]|x|.

Received September 5, 2007; Accepted January 22, 2008 Communicated by Andreas Ruffing

References

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 S. Elaydi, Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl., 181(2):483-492, 1994.

 Saber N. Elaydi and Shunian Zhang, Periodic solutions of Volterra difference equations with infinite delay. I. The linear case, In Proceedings of the First International Conference on Difference Equations (San Antonio, TX, 1994), pages 163-174, Luxembourg, 1995. Gordon and Breach.

 Y.N. Raffoul, Periodic solutions for scalar and vector nonlinear difference equations, Panamer. Math. J., 9(1):97-111, 1999.

 Y.N. Raffoul, T -periodic solutions and a priori bounds, Math. Comput. Modelling, 32(5-6):643-652, 2000. Boundary value problems and related topics.

 Youssef N. Raffoul, Periodic solutions for nonlinear volterra difference equations with infinite delay, Int. J. Nonlinear Differential Equations, 5(1-2):25-33, 1999.

 Youssef N. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electron. J. Differential Equations, pages No. 55, 8 pp. (electronic), 2002.

 Helmut Schaefer, Uber die Methode der a priori-Schranken, Math. Ann., 129:415-416, 1955.

(1) Dedicated to Professor Allan Peterson, on his 65th birthday and Euler Prize in Time Scale Research.

Youssef N. Raffoul

Department of Mathematics, University of Dayton,

Dayton, OH 45469-2316

E-mail: youssef.raffoul@notes.udayton.edu