﻿General Periodic Boundary Value Problem for Systems

Applied Mathematics
Vol.3 No.8(2012), Article ID:21481,6 pages DOI:10.4236/am.2012.38130

General Periodic Boundary Value Problem for Systems

Mohammed Elnagi1,2

1Department of Mathematics, Faulty of Educations, Khartoum University, Omdurman, Sudan

2Department of Mathematics, Northwest Normal University, Lanzhou, China

Email: mohdnajy@hotmail.com

Received May 20, 2012; revised July 2, 2012; accepted July 9, 2012

Keywords: Systems; Principle Eigenvalues; Positive Solutions; Green’s Function; Fixed Point Index Theorem in Cones

ABSTRACT

The paper deals with the existence of nonzero periodic solution of systems where, are real nonsingular matrices, ,

is periodic of period in the variable are continuous and nonnegative functions. We determine the Green’s function and prove that the existence of nonzero periodic positive solutions if one of. In addition, if all, where is the principle eigenvalues of the corresponding linear systems. The proof based on the fixed point index theorem in cones. Application of our result is given to such systems with specific nonlinearities.

1. Introduction

In this paper, we study the existence of nonzero positive periodic solution of systems

(1.1)

where, are real nonsingular matrices, , and

are periodic of period in the, are continuous and nonnegative functions and.

Beginning with the paper of Erbe and Palamides [1], obtained the sufficient conditions for existence solution of the systems of nonlinear boundary value problem

(1.2)

where is continuous (is n-dimensional real Euclidean space) and are nonsingular matrices, with orthogonal matrix, Erbe and Palamides generalize earlier conditions of Bebernes and Schmitt [2] for periodic case. Erbe and Schmitt [3] extend the results in [1] established the sufficient conditions for existence solution of the systems (1.2). The results in [1,3] were obtained via a modifications of a degree-theoretic approach and Leray-Schuader degree and eliminates the modified function approach respectively. None of these earlier results use Green’s function and the first eigenvalues of the corresponding to the linear systems of (1.1).

There has been progress in the study of the existence of positive solutions of system problem. If identity, then (1,1) reduces to the usual periodic boundary value problem for which the literature in both the scalar and systems versions is very extensive (We refer to [4-22] and references therein). For instance a recent paper, Wang [4] obtained the existence of periodic solution of a class of non-autonomous second-order systems

where

,

are periodic of period in the, and is a constantif, and, is bounded below or above for appropriate ranges of, via fixed point theorem in cones. Franco and Webb [5] established the existence of -periodic solutions for systems of (1.1) with identity, in the boundary conditions, where and is a continuous vector valued function, periodic in with period, and is allowed to have a singularity when. Non-singular systems, which are included in the same framework that we study here (i.e., they can be reduced to a Hammerstein integral system with positive kernel), have been considered using some other approaches based on fixed point theorems in conical shells, but previously it has always been assumed that the nonlinearity has a constant sign behaviour: where for (see [6]) with identity, in the boundary conditions. For systems problem see also [7-12] and references therein.

Even in the scalar case the existence of periodic solutions for problems with nonsingular and singular case has commanded much attention in recent years (see [13-22] and references therein. In particular, in [13-15] fixed point theorems in conical shells are used to obtain existence and multiplicity results, some of these are improved in this paper. In this notes, we prove result in the case where has no singularity. In scalar case problem see [16-22] and references therein.

Motivated by these problems mentioned above, we study the existence of nonzero positive solution of (1.1) while we assume that if one components satisfy

, , and all components of nonlinearity are, , where

, is the largest characteristic value of the linear system corresponding to (1.1). The approach is to use the theory of fixed point index for compact maps defined on cones [23]. To apply this theory one needs to find the Green’s function. Our purpose here is prove that (1.1) has nontrivial nonnegative solution, assuming the following conditions:

,

and

,

are continuous and periodic of period in the variable and, on any subinterval of.

There exits such that

where and, and

is the largest characteristic value of the linear system corresponding to (1.1),

For all,

where and, and

is the largest characteristic value of the linear system corresponding to (1.1).

Remark 1.1. The assumptions and appeared in Lan [24].

Remark 1.2. The nonzero positive solution has been studied by Lan [24] and Hai and Wang [25].

Throughout this paper, we will use the notation, , and denote by

the usual norm of for, and.

2. Preliminaries

In this section, we shall introduce some basic lemmas which are used throughout this paper.

Lemma 2.1. Let and holds. Let then for, the periodic boundary value problems problem

(2.1)

has a unique solution

where

(2.1*)

where.

Proof. Consider the scalar periodic boundary value problems of (2.1) and let and be linearly distinct solutions of the scalar equation of (2.1) and consider the function

where the positive sign is taken when, and the negative sign when we can obtain this result by routine substitutions of scalar boundary conditions, we do not state it here.

Lemma 2.2. Let conditions hold, then, , is continuous and positive on, and we can find it’s positive minimum value and maximum value of, by

, and

,.

Proof. It is easy to check that, is continuous and positive on,.

It is clear that the problem (1.1) has a solution if and only if solves the operator equation

(I)

It is easy to verify that the operator is completely continuous.

We define corresponding to linear equation of (1.1) by

(2.2)

where and is Green’s function define in (2.1*), and define

and

where is completely continuous.

Remark 2.1. Equations (2.2) appeared in [26].

It is known that, is a bounded and surjective linear operator and has a unique extension, denoted by, to. We write

(2.3)

It is known that is an interior point of the positive cone in, where

(2.4)

Lemma 2.3. [24] is a compact linear operator such that and for each there exists such that.

By Lemma 2.3 and the well-known Krein-Rutman theorem (see [23, Theorem 3.1] or [27], it is easy to see that and there exists such that

(2.5)

where and is the spectral radius of.

We use the following maximum norm in:

(2.6)

where. We denote by

the Banach space of continuous functions from into with norm

where for.

We use the standard positive cone in defined by

(2.7)

We can write defined in (2.2) as operator equations

(2.8)

where and are define above and define a Nemytskii operator

(2.9)

It is easy to verify that (1.1) is equivalent to the following fixed point equation:

(2.10)

Note that (2.10) same as.

Recall that a solution of (1.1) is said to be a nonzero positive solution if; that is, and satisfies for and and there exists such that on.

Let and let

,

and

.

We need some results from the theory of the fixed point index for compact maps defined on cones in a Banach space (see [23]).

Lemma 2.4. Assume that is a compact map. Then the following results hold:

1) If there exists such that for and, then

2) If for and, then

3) If and for some, then has a fixed point in.

Now, we are in a position to give our main result and proof analogous results were established in [24].

Theorem 2.1. Assume that - holds. be the same as in (2.5). Assume that the following conditions hold:

. There exist, and such that for and all with.

. There exist and such that for, for and all with. Then (1.1) has a nonzero positive solution in.

Proof. By Lemma 2.1, Lemma 2.2 and Lemma 2.3, is compact and satisfies.

This, together with the continuity of in, implies that is compact. Without loss of generalization, we assume that for. Let,  where is the same as in (2.5). We prove that

(2.11)

In fact, if not, there exist and such that. Then

(2.12)

It follows that for. Let

.

Then and,. This, together with (2.12), and (2.5), implies that for all

Hence, we have, a contradiction. It follows from (2.11) and Lemma 2.4 (1)

For each, by the continuity of, there exists such that for, with.

This, together with implies that, for each for and all

(2.13)

Since

,

exists and is bounded and satisfies

.

Let for,

and

where for. Let. We prove

Indeed, if not, there exist and such that. By (2.13), we have for each,

forwhere. Taking the maximum in the above inequality implies that

for, and

for.

Since

,

for.

Hence, we have

.

a contradiction. By (2.14) and Lemma 2.3 (2), By Lemma 2.4 (3), (1.1) has a solution in.

3. Application

Let the systems

(3.1)

where, and

. Assume that the following conditions hold:

1) For each, and is continuous and let

.

2) There exists such that and.

Then equation (3.1) have a nonzero positive solution in.

Proof. For each, we define a function by

Let and

.

Then for and with and,

Hence, holds. Let,

.

Then for and with,

for it follows that holds. The result follows from Theorem 2.1.

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