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
,
for
where
. 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. 