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:
For any belong to , and , we define the following integral operator:
Kernel is given by
Lemma 2.2 Let () and () hold and , , , , , , , , as well as x are all belong to as well as . Then is T-periodic function and satisfies the following differential equation:
Proof Appealing to presses of the proof of Lemma 2.1 for operators , and one obtains
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 () and () hold, assume further that () for
Then the integral operator maps into and has at least one fixed point.
Proof Let , then
On the other hand,
In addition, since and () are positive functions, we have
Therefore, for , belong to and any , 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
Let . For any , one obtains
This implies that is Lipschitzan with Lipschit constance M. In this way, for given , if we consider , then
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).
Let and (), (), and () hold. Based on Theorem 2.4 for , operator has at least a fixed point in . Let 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 .
Proof It is obvious that is bounded on B. Also, according to inequality (2.10), we have
For given , if we consider , then
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 ,
Wherein, . For each , the partial derivative exist and straightforward calculation shows
Also, in the case ,
We set , wherein,
Based on property (2.15) and by induction, one can obtain
where is a specific polynomial in terms of σ.
Let be a sequence belong to and as . Let . According to relative compactness of in B, there exist a subsequence of and such that , uniformly, in B as . It is obvious that for any q, we have
Based on property (2.16), we have
uniformly, as for all and .
Thus, derivative exist on and
for all and .
Consequently, based on definition of map we have . Also, . Thus, we obtain
Hence, , which it shows that is continuous on .
Therefore, applying Schauder’s fixed-point theorem, map has a fixed point , i.e.,
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
Let . We consider the system (1.4), (1.5) together with the following initial conditions:
Lemma 2.6 (see )
If , and , 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, i.e., there exist and , , , and () such that
Proof Since for any , appealing to the proof of the Lemma 2.3 given in  we can immediately demonstrate that there exist a such that
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 such that
Since ϵ is arbitrary small, one may assume that
Remark 2.8 For the system without the external source, the sets must be replaced by
Similar calculation shows that condition () is reduced to the following one: