General Periodic Boundary Value Problem for Systems

The paper deals with the existence of nonzero periodic solution of systems             2 , , 0, , 0, 0 , 0 u k u f t u t T T u u T u u T     ,            where π 0, k T       , ,   are real nonsingular matrices, n n    1, , n u u u   ,            1 , , , , , 0, , n n f t u f t u f t u C T           is periodic of period in the t 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 T 1 0 , lim i u f t u u      . In addition, if all   1 0 , lim i u f t u u      ,   1, i  , n where 1  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.


0
, 0 , is periodic of period in the t 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

Introduction
In this paper, we study the existence of nonzero positive periodic solution of systems 0 , are periodic of period in the t , 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 , , , 0,1 , 0 1, 0 where   2 : 0,1 is continuous ( is n-dimensional real Euclidean space) and 0 1 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 I      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][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][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 and   , f t u , 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   , f t u t is a continuous vector valued function, periodic in with period , and T f 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 where i for (see [6]) with    identity, in the boundary conditions.For systems problem see also [7][8][9][10][11][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][14][15][16][17][18][19][20][21][22] and references therein.In particular, in [13][14][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 f has no singularity.In scalar case problem see [16][17][18][19][20][21][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  , 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: are continuous and periodic of period in the variable and There exits , and 1  is the largest characteristic value of the linear system corresponding to (1.1), where  and   5 H 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

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

 
where the positive sign is taken when , and the negative sign when Proof.It is easy to check that , is continuous and positive on It is easy to verify that the operator A is completely continuous.
It is known that  , is a bounded and surjective linear operator and has a unique extension, denoted by  , to It is known that is an interior point of the positive cone in T is a compact linear operator such that   1 P  1 and for each . By Lemma 2.3 and the well-known Krein-Rutman theorem (see [23,Theorem 3.1] or [27], it is easy to see that   1 0,    and there exists   where We use the following maximum norm in : the Banach space of continuous func- . We use the standard positive cone in defined by We can write where    It is easy to verify that (1.1) is equivalent to the following fixed point equation: Note that (2.10) same as   I .Recall that a solution of (1.1) is said to be a nonzero positive solution if and there exists such that

  and let
Copyright © 2012 SciRes.AM   We need some results from the theory of the fixed point index for compact maps defined on cones in a Banach space X (see [23]).
Lemma 2.4.Assume that : A P   P is a compact map.Then the following results hold: 1) If there exists Now, we are in a position to give our main result and proof analogous results were established in [24].
holds.be the same as in (2.5).Assume that the following conditions hold: is compact and satisfies .This, together with the continuity of i f in ( 3) H , implies that : A P P  is compact.Without loss of generalization, we assume that for , where 1  is the same as in (2.5).
We prove that In fact, if not, there exist It follows that This, together with (2.12),     0 0 0 f i  and (2.5), implies that for all Hence, we have   This, together with     exists and is bounded and satisfies 1 1 Indeed, if not, there exist u P   and . By (2.13), we have for each . Taking the maximum in the above inequality implies that , and

Application
Let the systems . Assume that the following conditions hold: 1) For each , and is continuous and let 2) There exists such that Then for   and