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Positive periodic solutions in neutral delay difference equations.

1 Introduction

In this paper we use Krasnoselskii's fixed point theorem to study the existence of positive periodic solutions of a certain type of difference equation with delay which appear in ecological models. The existence of positive periodic solutions of functional differential equations has gained the attention of many researchers in recent times. We are mainly motivated by the work of the first author [5] and the references therein on neutral differential equations.

Let t be a nonnegative integer and consider the neutral delay difference equation

x(n + 1) = a(n)x(n) + c[DELTA]x(n - [tau]) + g(n, x(n - [tau])), (1.1)

where g is continuous in x. The operator [DELTA] is defined as [DELTA]x(n) = x(n + 1) - x(n). In this paper, we denote by E the shift operator defined as Ex(n) = x(n + 1). Also, the product of x(n) from n = a to n = b is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For more on the calculus of difference equation we refer the reader to [1] and [2]. In the continuous case, equations in the form of 1.1 have applications in food-limited populations, see biological [5] and the references therein. In [3], the first author considered a more complicated form of 1.1 and analyzed the existence of periodic solutions. On the other hand, the second author studied the boundedness of solutions and the stability of the zero solution. In [4], using cone theory, the first author obtained sufficient conditions that guaranteed the existence of multiple positive periodic solutions for the nonlinear delay difference equation

x(n + 1) = a(n)x(n) [+ or -] [lambda]h(n)f (x(n - [tau](n))).

2 Preliminaries

We begin this section by introducing some notations. Let [P.sub.T] be the set of all real T-periodic sequences, where T is an integer with T [greater than or equal to] 1. Then [P.sub.T] is a Banach space when it is endowed with the maximum norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is natural to ask for the periodicity condition

a(n + T) = a(T), g(n + T, *) = g(n, *), (2.1)

to hold for all n [member of] Z. In addition to (2.1), we assume that

0 < a(n) < 1. (2.2)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Note that the denominator in G(n, u) is not zero since 0 < a(n) < 1 for n [member of] [0, T - 1]. Also, let

m := min{G(n,u) : n [greater than or equal to] 0, u [less than or equal to] T} = G(n,n) > 0, (2.4)

M := max{G(n, u) : n [greater than or equal to] 0, u [less than or equal to] T} = G(n, n + T - 1) = G(0, T - 1) > 0. (2.5)

Lemma 2.1. Suppose (2.1) and (2.2) hold. If x(n) [member of] [P.sub.T], then x(n) is a solution of (1.1) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

where G(n, u) is defined by (2.3).

Proof. Rewrite (1.1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

Summing (2.7) from n to n + T - 1, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since x(n + T) = x(n), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus (2.7) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Dividing both sides of the above equation by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] completes the proof.

Now for n [less than or equal to] 0, Equation (1.1) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Summing the above expression from n to n + T - 1, we obtain (1.1) by a similar argument.

We next state Krasnoselskii's theorem in the following lemma.

Lemma 2.2 (Krasnoselskii). Let M be a closed convex nonempty subset of a Banach space (B, [parallel] * [parallel]). Suppose that C andB map M into B such that

(i) x, y [member of] M implies Cx + By [member of] M;

(ii) C is continuous and CM is containedin a compact set;

(iii) B is a contraction mapping.

Then there exists z [member of] M with z = Cz + Bz.

3 Main Results

In this section we obtain the existence of positive periodic solution of (1.1). For some nonnegative constant L and a positive constant K we define the set

M = {[phi] [member of] [P.sub.T] : L [less than or equal to] [phi] [less than or equal to] K}, (3.1)

which is a closed convex and bounded subset of the Banach space [P.sub.T]. In addition we assume that for all u [member of] Z and [rho] [member of] M,

[(1 - c)L/mT] [less than or equal to] g(u,[rho]) - c[1 - a(u)][rho] [less than or equal to] [(1 - c)K/MT], (3.2)

where m and M are defined by (2.4) and (2.5), respectively. We will treat separately the cases 0 [less than or equal to] c < 1 and -1 < c [less than or equal to] 0. Thus, for our first theorem we assume

0 [less than or equal to] c < 1. (3.3)

To apply the theorem stated in Lemma 2.2, we will need to construct two mappings; one is contraction and the other is compact. In view of this we define the map B : M [right arrow] [P.sub.T] by

(B[phi])(n) = cx(n - t ).

In a similar way we define the map C : M [right arrow] [P.sub.T] by

(C[phi])(n) = [n+T-1.summation over (u=n)] G(n,u) [g(u,x(u - [tau])) - c(1 - a(u))x(u - [tau])].

It is clear from condition (3.3) that B defines a contraction map under the supremum norm.

Lemma 3.1. If (2.1), (2.2), (3.2) and (3.3) hold, then the operator C is completely continuous on M.

Proof. For n [member of] [0, T - 1] and for [phi] [member of] M, we have by (3.2) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the estimation of [absolute value of C[phi](n)] it follows that

[parallel]C[phi][parallel] [less than or equal to] (1 - c)K [less than or equal to] K.

This shows that C(M) is uniformly bounded. Due to the continuity of all terms, we have that C is continuous. Next, we show that A maps bounded subsets into compact sets. Let J be given, S = {[phi] [member of] [P.sub.T] : [parallel][phi][parallel] [less than or equal to] J} and Q = {(C[phi])(t) : [phi] [member of] S}. Then S is a subset of [R.sup.T] which is closed and bounded and thus compact. As C is continuous in [phi], it maps compact sets into compact sets. Therefore Q = C(S) is compact. This completes the proof. []

Theorem 3.2. Suppose that (2.1), (2.2), (3.2) and (3.3) hold. Then equation (1.1) has a positive periodic solution z satisfying L [less than or equal to] [parallel]z[parallel] [less than or equal to] K.

Proof. Let [phi], [psi] [member of] M. Then, by (3.2), we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This shows that A[phi] + C[psi][member of] M. All the hypotheses of the theorem stated in Lemma 2.2 are satisfied and therefore equation (1.1) has a positive periodic solution, say z, residing in M. This completes the proof. []

For the next theorem we substitute conditions (3.2) and (3.3) with

-1 < c [less than or equal to] 0 (3.4)

and for all u [member of] R and [rho] [member of] M

[L - cK/mT] [less than or equal to] g(u[rho]) - c[1 - a(u)][rho] [less than or equal to] [K - cL/MT], (3.5)

where M and m are defined by (2.4) and (2.5), respectively.

Theorem 3.3. If (2.1), (2.2), (3.2), (3.4) and (3.5) hold, then Equation (1.1) has a positive periodic solution z satisfying L [less than or equal to] [parallel]z[parallel] [less than or equal to] K.

Proof. The proof follows along the lines of Theorem 3.2, and hence we omit it. []

4 Example

The neutral difference equation

x(n + 1) = [1/8]x(n) + [1/10][DELTA]x(n - 4) + [1/[x.sup.2](n - 4) + 100] + [7/80]x(n - 4) + [1/20] (4.1)

has a positive periodic solution x of period 4 satisfying [1/18428] [less than or equal to] [parallel][phi][parallel] [less than or equal to] 2. To see this, we have

g(u, [rho]) = [1/[[rho].sup.2] + 100] + [7/80][rho] + [1/20],

a(n) = [1/8], c = [1/10], and T = 4.

A simple calculation yields M = [4096/4095] and m = [8/4095]. Let K = 2, L = [1/18428], and define the set

M = {[phi] [member of] [P.sub.4] : [1/18428] [less than or equal to] [phi] [less than or equal to] 2}.

Then for p [member of] [[1/18428], 2] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Theorem 3.2, equation (4.1) has a positive periodic solution x with period 4 such that [1/18428] [less than or equal to] [parallel]x[parallel] [less than or equal to] 2.

References

[1] S. Elaydi, An Introduction to Difference Equations, Second edition. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999.

[2] W. Kelley and A. Peterson, Difference Equations: An Introduction with Applications, Harcourt Academic Press, San Diego, 2001.

[3] Y. Raffoul, Periodic solutions for scaler and vector nonlinear difference equations, Panamer. J. Math. 9 (1999), 97-111.

[4] Y. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electr. J. Diff. Eq. 55 (2002), 1-8.

[5] Y. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, Electr. J. Qual. Th. Diff. Eq. 16 (2007), 1-10.

Youssef N. Raffoul

University of Dayton

Department of Mathematics

Dayton, OH 45469-2316, U.S.A.

youssef.raffoul@notes.udayton.edu

Ernest Yankson

University of Cape Coast

Department of Mathematics and Statistics

Cape Coast, Ghana

ernestoyank@yahoo.com

Received December 7, 2009; Accepted May 20, 2010 Communicated by Martin Bohner
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Author:Raffoul, Youssef N.; Yankson, Ernest
Publication:International Journal of Difference Equations
Article Type:Report
Geographic Code:6GHAN
Date:Jun 1, 2010
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