The Uniqueness Theorem of the Solution for a Class of Differential Systems with Coupled Integral Boundary Conditions

We discuss the uniqueness of the solution to a class of differential systems with coupled integral boundary conditions under a Lipschitz condition. Our main method is the linear operator theory and the solvability for a system of inequalities. Finally, an example is given to demonstrate the validity of our main results.

The question of existence and uniqueness of solution of differential equations and differential systems is an ageold problem and it has a great importance, as much in theory as in applications.This problem has been investigated by use of a variety of nonlinear analyses such as fixed point theorem for mixed monotone operator [7,15,[22][23][24][25], maximal principle [6], Banach's contraction mapping principle [26][27][28][29], and the linear operator theory [27,30,31].
For example, the authors [31] introduced a Banach space using the positive eigenfunction of linear operator related to differential system (1).They established the uniqueness results for differential system (1) under a Lipschitz condition.It should be noted that the Lipschitz constant is related to the spectral radius corresponding to the related linear operators.The obtained results are optimal from the viewpoint of theory.However, it is very difficult to determine the spectral radius for differential system (1) with general functions (), ().
Motivated by the above works, we investigate the uniqueness of solutions for differential system (1) by using a system of inequalities and the linear operator theory.The main features of this paper are as follows: (1) The main results are mostly implemented to the uniqueness result for coupled boundary value problems.(2) An easy criterion to determine the uniqueness result is obtained by using a system of inequalities.(3) An example shows that the main result provides the same results with weaker conditions.
Throughout the paper, we assume that the following condition hold:

Main Result
For notational convenience, let where Take () = .By (7), we get By use of ( 21), ( 22), (23), and (24), we present the main result of this paper.Theorem 4. Suppose that there exist four nonnegative constants  1 ,  1 ,  1 ,  1 such that the following conditions hold: Then differential system (1) has a unique solution in .
Proof.We divide the proof into several main steps to show that the operator  has a unique point in  under the conditions of Theorem 4.
In the following, we give an example to illustrate our theory.( then where Therefore, according to Theorem 4, the problem (42) has a unique solution.
When the nonlinearity of differential equation and differential system satisfies Lipschitz condition, the usual method to obtain the uniqueness is the well-known Banach's contraction principle.For this purpose, we should add some restriction on the Lipschitz constants to guarantee the norm of a linear operator related to differential equation and differential system less than 1.Next, we discuss the estimate of the norm of a linear operator related to differential system (42).