A Note on Periodic Solutions of Second Order Nonautonomous Singular Coupled Systems

Some classical tools have been used in the literature to study the positive solutions for twopoint boundary value problems of a coupled system of differential equations. These classical tools include some fixed point theorems in cones for completely continuous operators and Leray-Schauder fixed point theorem; for examples, see 1–3 and literatures therein. Recently, Schauder’s fixed point theorem has been used to study the existence of positive solutions of periodic boundary value problems in several papers; see, for example, Torres 4 , Chu et al. 5, 6 , Cao and Jiang 7 , and the references contained therein. However, there are few works on periodic solutions of second-order nonautonomous singular coupled systems. In these papers above, there are the major assumption that their associated Green’s functions are positive. Since Green’s functions are positive, in the paper, we continue to study the existence of periodic solutions to second-order nonautonomous singular coupled systems in the following form:


Introduction
Some classical tools have been used in the literature to study the positive solutions for twopoint boundary value problems of a coupled system of differential equations.These classical tools include some fixed point theorems in cones for completely continuous operators and Leray-Schauder fixed point theorem; for examples, see 1-3 and literatures therein.
Recently, Schauder's fixed point theorem has been used to study the existence of positive solutions of periodic boundary value problems in several papers; see, for example, Torres 4 , Chu et al. 5, 6 , Cao and Jiang 7 , and the references contained therein.However, there are few works on periodic solutions of second-order nonautonomous singular coupled systems.In these papers above, there are the major assumption that their associated Green's functions are positive.Since Green's functions are positive, in the paper, we continue to study the existence of periodic solutions to second-order nonautonomous singular coupled systems in the following form: x a 1 t x f 1 t, y t e 1 t for a.e.t ∈ 0, T , y a 2 t y f 2 t, x t e 2 t for a.e.t ∈ 0, T , with a 1 , a 2 , e 1 , e 2 ∈ L 1 0, T , f 1 , f 2 ∈ Car 0, T × 0, ∞ , 0, ∞ .Here we write f ∈ Car 0, T × 0, ∞ , 0, ∞ if f : 0, T × 0, ∞ → 0, ∞ is an L 1 -caratheodory function, that is, the map x → f t, x is continuous for a.e.t ∈ 0, 1 and the map t → f t, x is measurable for all x ∈ 0, ∞ , and for every 0 < r < s there exists h r,s ∈ L 1 0, T such that |f t, x | ≤ h r,s t for all x ∈ r, s and a.e.t ∈ 0, T ; here "for a.e." means "for almost every".This paper is mainly motivated by the recent papers 4-6, 8, 9 , in which the periodic singular problems have been studied.Some results in 4-6, 9 prove that in some situations weak singularities may help create periodic solutions.In 6 , the authors consider the periodic solutions of second-order nonautonomous singular dynamical systems, in which the scalar periodic singular problems have been studied by Leray-Schauder alternative principle, a well-known fixed point theorem in cones, and Schauder's fixed point theorem, respectively.
The remaining part of the paper is organized as follows.In Section 2, some preliminary results will be given.In Sections 3-5, by employing a basic application of Schauder's fixed point theorem, we state and prove the existence results for 1.1 under the nonnegative of the Green's function associated with 2.1 -2.2 .Our view point sheds some new light on problems with weak force potentials and proves that in some situations weak singularities may stimulate the existence of periodic solutions, just as pointed out in 9 for the scalar case.
To illustrate our results, for example, we can select the system Here we emphasize that in the new results e 1 , e 2 do not need to be positive.
Let us fix some notation to be used in the following: given a ∈ L 1 0, 1 , we write a 0 if a ≥ 0 for a.e.t ∈ 0, 1 and it is positive in a set of positive measures.For a given function p ∈ L 1 0, T , we denote the essential supremum and infimum by p * and p * , if they exist.The usual L p -norm is denoted by • p .The conjugate exponent of p is denoted by p : 1/p 1/ p 1.

Preliminaries
We consider the scalar equation In other words, the strict antimaximum principle holds for 2.1 -2.2 .Under the conditions H 1 , the solution of 2.1 -2.2 is given by x t T 0 G i t, s e i s ds.

2.3
For a nonconstant function a t , there is an L p -criterion proved in 9 , which is given in the following lemma for the sake of completeness.Let K q denote the best Sobolev constant in the following inequality: The explicit formula for K q is where Γ is the Gamma function.See 10 .
Lemma 2.1.For each i 1, 2, assume that a i t 0 and a i ∈ L p 0, T for some 1 ≤ p ≤ ∞.If then the standing hypothesis (H 1 ) holds.
We define the function γ i : R → R by which is the unique T -periodic solution of x a i t x e i t .

3.2
By a direct application of Schauder's fixed point theorem, the proof is finished if we prove that A maps the closed convex set defined as > 0 are positive constants to be fixed properly.For convenience, we introduce the following notations: Given x, y ∈ K, by the nonnegative sign of G i and f i , i 1, 2, we have and note for every x, y ∈ K that

3.6
Also, follow the same strategy,

3.7
Thus Ax, Ay ∈ K if r 1 , r 2 , R 1 , and R 2 are chosen so that

3.8
Note that 3.9 and these inequalities hold for R being big enough because α i < 1.

The Case
and where 0 < r 10 < ∞ is a unique positive solution of the equation and 0 < r 20 < ∞ is a unique positive solution of the equation then there exists a positive T -periodic solution of 1.1 .
Proof.We follow the same strategy and notation as in the proof of ahead theorem.In this case, to prove that γ * 2 , then the first inequality of 4.5 holds if r 2 satisfies According to we have g 0 −∞, g ∞ 1; then there exists r 20 such that g r 20 0, and

4.8
Then the function g r 2 possesses a minimum at r 20 , that is, g r 20 min r 2 0, ∞ g r 2 .Note g r 20 0; then we have or equivalently, Similarly, 4.11 g r 10 min r 1 0, ∞ g r 1 , and

4.12
Taking r 1 r 10 and r 2 r 20 , then the first inequality in 4.4 and 4.5 holds if γ 1 * ≥ g r 10 , γ 2 * ≥ g r 20 , which are just condition 4.1 .The second inequalities hold directly by the choice of R 1 and R 2 , and it would remain to prove that r 10 < R 1 and r 20 < R 2 .This is easily verified through elementary computations .

4.13
The proof is the same as that in .

Mathematical Problems in Engineering
Next,we will prove r 10 < R 1 , r 20 < R 2 , or equivalently,

4.14
Namely, On the other hand,

The Case γ
and

5.1
where 0 < r 11 < ∞ is a unique positive solution of the equation then there exists a positive T -periodic solution of 1.1 .
Proof.In this case, to prove that

5.3
If we fix γ * 2 , then the first inequality of 6.4 holds if r 2 satisfies or equivalently

5.5
Then the function f r 2 possesses a minimum at On the analogy of 5.4 , we obtain or equivalently,

5.8
According to we have h 0 −∞, h ∞ 1; then there exists r 11 such that h r 11 0, and

5.10
Then the function h r 1 possesses a minimum at r 11 , that is, h r 11 min r 1 0, ∞ f r 1 .Note h r 11 0; then we have Namely, Taking r 2 r 21 and r 1 r 11 , then the first inequality in 5.3 hold if γ 2 * ≥ h r 21 and γ 1 * ≥ h r 11 which are just condition 5.1 .The second inequalities hold directly by the choice of R 2 and R 1 , so it would remain to prove that r α 2 11 .

5.16
In what follows, we will verify that R 2 > r 21 .In fact, Thus On the other hand,

5.19
Thus one can see easily that

5.23
Recall 5.16 , we obtain r 21 < R 2 immediately.The proof is complete.
Similarly, we have the following theorem.

5.24
where 0 < r 21 < ∞ is a unique positive solution of the equation then there exists a positive T -periodic solution of 1.1 .

The Case γ
and where 0 < r 22 < ∞ is a unique positive solution of the equation then there exists a positive T -periodic solution of 1.1 .
Proof.The following proof is the same as the proof of ahead theorem.In this case, to prove that A 6.9 and these inequalities hold for R 2 being big enough because α i < 1.The proof is completed.
Similarly, we have the following theorem.