Uniqueness theorem of differential system with coupled integral boundary conditions

The paper is devoted to study the uniqueness of solutions for a differential system with coupled integral boundary conditions under a Lipschitz condition. Our approach is based on the Banach’s contraction principle. The interesting point is that the Lipschitz constant is related to the spectral radius corresponding to the related linear operators.

Differential system with coupled boundary conditions arise from the study of reactiondiffusion equations and Sturm-Liouville problems, and have extensive applications in various fields of sciences and engineering such as the heat equation and mathematical biology.
The existence of solutions or positive solutions of differential system with coupled boundary conditions has been studied by many researchers, see [1-4, 6-10, 13] for some recent work.For example, by using the Guo-Krasnosel'skii fixed-point theorem, the existence of positive solution of the following singular system with coupled four-point boundary value conditions are obtained [1] In [8], Infante, Minh ós and Pietramala, by means of classical fixed point index theory, provided a general theory for existence of positive solutions for coupled systems.
The uniqueness of solutions can be an important problem for boundary value problems of differential equation or differential system.This problem has been investigated by many authors by use of techniques of nonlinear analysis.We refer the reader to [3,4] for some recent uniqueness results for differential system, to [5,12,14] for differential equation.In [3], by means of the Guo-Krasnosel'skii fixed-point theorem and mixed monotone method, Cui, Liu and Zhang investigated the uniqueness of positive solutions of singular system (1.1) in the case that the nonlinearities f and g may be singular at t = 0, 1.
However, to our best knowledge, there are fewer results concerned the uniqueness of solutions for differential systems with coupled integral boundary conditions.So, we consider the uniqueness of solutions for differential system (1.1) under a Lipschitz condition on f and g.By using Banach's contraction principle, a new result on the uniqueness of solutions for differential system (1.1) is obtained.It is worthwhile to mention that the Lipschitz constant is related to the spectral radius corresponding to the related linear operators.
Throughout the paper, we assume that the following conditions hold.

Preliminaries
Let C[0, 1] be the Banach space of continuous functions endowed with the norm x = max t∈[0,1] |x(t)| and let P 1 be the cone of nonnegative functions in C[0, 1] given by ] is a Banach space with the norm defined by (x, y) E = max{ x , y }, and P = P 1 × P 1 is a cone in E.

Lemma 2.1 ([2]
).Let u, v ∈ C[0, 1], then the system of BVPs where Employing Lemma 2.1, we can reformulate BVP (1.1) as a fixed point for the following integral equations: Define an operator S by where operators S 1 , S 2 : E → C[0, 1] are defined by Then the existence of a solution of differential system (1.1) is equivalent to the existence of a fixed point of S on E.
It is well known that the function k(t, s) has the following properties: From this and (H 1 ), for t, s ∈ [0, 1], we have and Therefore we have and where where operators T a,1 , T a,2 : E → C[0, 1] are defined by It is not difficult to verify that T a : E → E is a completely continuous linear operator.

Definition 2.2 ([11]
).Let E be a Banach space, P ⊂ E be a cone in E. Let e ∈ P\{θ}, a mapping T : P → P is called e−positive if for every nonzero x ∈ P a natural number n = n(x) and two positive number c x , d x can be found such that Recall that a real number λ is an eigenvalue of the operator T if there exists a non-zero element x ∈ E such that Tx = λx.Lemma 2.3 ([11, Theorem 2.5, Lemma 2.1, Theorem 2.10]).Suppose that T : E → E is a e−positive, completely continuous linear operator.If there exist ψ ∈ E\(−P) and a constant c > 0 such that cTψ ≥ ψ, then the spectral radius r(T) = 0, and r(T) is the unique positive eigenvalue with its eigenfunction in P.
Lemma 2.4.Suppose that (H 1 ) holds.Then for the operator T a defined by (2.4), there is a unique positive eigenvalue r(T a ) with its eigenfunction in P.

Main results
Theorem 3.1.Suppose that there exists a = (a, b, c, d and If r(T a ) < 1, then differential system (1.1) has a unique solution in E.
Proof.It is clear that the fixed points of operator S coincide with the solutions to differential system (1.1).For (x, y) ∈ E, by (2.2), (2.7), (3.1) and(3.2) we have In the same way, we can prove that Therefore, S maps all of E into the following vector subspace Evidently, E 1 is a subspace of E and E 1 is an Banach space with the norm So it suffices to consider the fixed point of S in E 1 .Note that In the same way, we can prove that The above two inequalities imply that Notice that r(T a ) < 1, the operator S is a contraction.Hence, it follows from the well known Banach's contraction principle that S has a unique fixed point (x, y) ∈ E 1 , which is obviously a unique solution of differential system (1.1).It ends the proof.
From the above argument, we know that the basic space used in the proof of Theorem 3.1 is E 1 , not in E. If we consider differential system (1.1) in E by use of Banach's contraction principle, the result of Theorem 3.1 remains true except that the condition r(T a ) < 1 is replaced by T a < 1, where It follows from the well-known Gelfand's Formula that r(T a ) = lim n→∞ n T n a ≤ T a which concludes that it may be favorable to consider the uniqueness of differential system (1.1) in E 1 .
In the following, we give two examples to illustrate our main result.Obviously, it is rather difficult to determine the value of r(T a ) in general.In the two examples, we determine the spectral radius r(T a ) for certain four-point coupled boundary conditions which can be seen as a special cases of coupled integral boundary conditions.

Thus we have
By ordinary method, we conclude that (ϕ(t), ψ(t)) = (c 1 , c 2 ) sin √ λt for some c 1 , c 2 ∈ R.This together with the four-point coupled boundary conditions yields So, λ is the unique positive solution of the equation We can obtain λ ≈ 1.9585 2 ≈ 3.83584 by MATLAB.Therefore, if |a| < 3.83584, the problems (3.3) has a unique solution.
Example 3.3.Consider the differential system where a ∈ R, h