In this paper, we study the existence of nonzero positive periodic solution of systems
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 [
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 [
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 [
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 [
-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 [
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 [
,
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 [
Remark 1.2. The nonzero positive solution has been studied by Lan [
Throughout this paper, we will use the notation
, 
, and denote by 
the usual norm of 
for
, 
and
.
and 
holds. Let 
then for
, the periodic boundary value problems problem
where
.
and 
be linearly distinct solutions of the scalar equation of (2.1) and consider the function
, and the negative sign when 
we can obtain this result by routine substitutions of scalar boundary conditions, we do not state it here. 
hold, then
, 
, is continuous and positive on
, and we can find it’s positive minimum value 
and maximum value 
of
, 
by
, 
and
,
.
, is continuous and positive on
,
.
if and only if 
solves the operator equation
is completely continuous.
corresponding to linear equation of (1.1) by
and 
is Green’s function define in (2.1*), and define
where 
is completely continuous.
, is a bounded and surjective linear operator and has a unique extension, denoted by
, to
. We write
is an interior point of the positive cone 
in
, where
is a compact linear operator such that 
and for each 
there exists 
such that
.
and there exists 
such that
and 
is the spectral radius of
.
:
. We denote by
the Banach space of continuous functions from 
into 
with norm
where 
for
.
defined by
defined in (2.2) as operator equations
and 
are define above and define a Nemytskii operator
.
of (1.1) is said to be a nonzero positive solution if
; that is, 
and 
satisfies 
for 
and 
and there exists such that 
on
.
and let
, 
.
(see [
is a compact map. Then the following results hold:
such that 
for 
and
, then 
for 
and
, then 
and 
for some
, then 
has a fixed point in
.
-
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
.
is compact and satisfies
.
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
and 
such that
. Then
for
. Let
.
and
,
. This, together with (2.12), 
and (2.5), implies that for all 
, a contradiction. It follows from (2.11) and Lemma 2.4 (1) 
, by the continuity of
, there exists 
such that 
for
, 
with
.
implies that, for each 
for 
and all
,
exists and is bounded and satisfies
.
for
,
where 
for
. Let
. We prove
and 
such that
. By (2.13), we have for each
,
for
where
. Taking the maximum in the above inequality implies that
, and
.
,
.
.
By Lemma 2.4 (3), (1.1) has a solution in
.
, 
and
. Assume that the following conditions hold:
, 
and 
is continuous and let
.
such that 
and
.
.
, we define a function 
by
and
.
and 
with 
and
,
holds. Let
,
.
and 
with
,
it follows that 
holds. The result follows from Theorem 2.1. 