In this section, we shall study the existence of periodic solutions of the multi-species system (1.4)-(1.5). To do this, we transform this system of couple equations into one integral equation. For each
, we introduce the following integral operator
on the Banach space
The kernel of the integral operator (2.1) is in fact the Green’s function of Eq. (1
.2) and is given by
where ; see .
Lemma 2.1 Let and , , , and are belong to as well as . Suppose that is a continuous real function such that for some , . Then is a T-periodic solution of Eq. (1.5).
The proof of Lemma 2.1 is similar to the proof of Lemma 3.1 of .
According to Lemma 2.1, it may be deduced that the existence problem of T
-periodic solution of system (1.4), (1.5) is equivalent to that of the T
-periodic solution of the following equation:
, we define the following integral operator:
Kernel is given by
Lemma 2.2 Let
) hold and
, as well as x are all belong to as well as
. Then is T
-periodic function and satisfies the following differential equation
Appealing to presses of the proof of Lemma 2.1 for operators
The sum of terms above and taking the equality (2.3) into account leads to (2.4). □
Corollary 2.3 Let , , , , , , , , and x all belong to as well as . Let that be a fixed point of the operator , i.e., , then is a solution of Eq. (2.2).
, assume further that
Then the integral operator maps into and has at least one fixed point.
On the other hand,
In addition, since
) are positive functions, we have
, we obtain
Also, we have
In these regards, based on (2.6) and (2.8) one obtains
This shows that is belong to () and, therefore, the integral operator maps into .
In addition, based on inequality (2.9), we have
which shows that
is bounded by
. For any
, one obtains
This implies that
is Lipschitzan with Lipschit constance M
. In this way, for given
, if we consider
Consequently, for any the family is equicontinuous on .
Suppose is a sequence on , . Thus, as a sequence of functions on is equicontinuous. Appealing to the Arzela-Ascoli theorem, there exist a subsequent denoted by , which is uniformly convergence on . This means that is convergent on and consequently, is compact. Appealing to Schauder’s fixed-point theorem, has at least a fixed point on . □
We emphasize that according to Corollary 2.3, for any , and belong to fixed point of is the positive T-periodic solutions of Eq. (2.2) or equivalently, T-periodic solution of the nonlinear population system (1.3) and (1.4).
), and (
) hold. Based on Theorem 2.4 for
has at least a fixed point in
denote the set of fixed points of
. Applying the Axiom of Choice, we chose a representative point, say
, in i.e.
We introduce the following operator on
Theorem 2.5 Let (), (), and () hold. Then operator defined by (2.11) has at least a fixed point on .
It is obvious that
is bounded on B
. Also, according to inequality (2.10), we have
, if we consider
which shows that is equicontinuous on [0,T]. In this regard, any sequence, say in satisfies all the conditions of the Arzela-Ascoli theorem on [0,T]. Hence, has a subsequence such that is uniformly convergent on [0,T]. This shows that is relatively compact in B.
In the sequel, we show that
is continuous. We define the following map on
. For each
, the partial derivative
exist and straightforward calculation shows
Also, in the case
Based on property (2.15) and by induction, one can obtain
where is a specific polynomial in terms of σ.
be a sequence belong to
. According to relative compactness of
, there exist a subsequence
, uniformly, in B
. It is obvious that for any q
, we have
Based on property (2.16), we have
uniformly, as for all and .
for all and .
Consequently, based on definition of map
. Thus, we obtain
Hence, , which it shows that is continuous on .
Therefore, applying Schauder’s fixed-point theorem, map
has a fixed point
Finally, this indicate that is a positive periodic solution of system of Eq. (2.2) or equivalently, T-periodic solution of the multi-species cooperation system (1.3) and (1.4). This completes the proof of the theorem. □
2.1 Permanence of system
. We consider the system (1.4), (1.5) together with the following initial conditions:
Lemma 2.6 (see )
, when and
, we have
Theorem 2.7 Let
) hold and be any solution differential equation
(2.2) and with
be the solution of system
(1.5) with initial condition
(2.19). Then solution of system
(1.4), (1.5) is permanent
., there exist and
) such that
, appealing to the proof of the Lemma 2.3 given in [6
] we can immediately demonstrate that there exist a
while and .
On the other hand, from Lemma 2.1 and inequality (2.5), we have
then by Lemma 2.6, for arbitrary
, there exist
is arbitrary small, one may assume that
For the system without the external source, the sets
must be replaced by
Similar calculation shows that condition (
) is reduced to the following one: