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

Abstract

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.

1. Introduction

In this paper, we study the following differential system with coupled integral boundary conditions: where are bounded linear functionals on given by involving Riemann-Stieltjes integrals defined via positive Stieltjes measures of .

Differential systems with coupled boundary conditions have some applications in various fields of sciences and engineering, for example, the heat equation [1], reaction-diffusion phenomena [2], and interaction problems [3]. The existence of solutions for differential system with coupled boundary conditions has received a growing attention in the literature; for details, see [4–21]. For example, Asif and Khan in [4] obtained the existence of positive solution for singular sublinear system with coupled four-point boundary value conditions by using the Guo-Krasnosel’skii fixed point theorem. In [5], Cui and Sun discuss the existence of positive solutions of singular superlinear coupled integral boundary value problems by constructing a special cone and using fixed point index theory. In [7], Cui and Zou proved the existence of extremal solutions of coupled integral boundary value problems by monotone iterative method. In [10], Infante, Minhós, and Pietramala presented a general theory for existence of positive solutions for coupled systems by use of fixed point index theory.

The question of existence and uniqueness of solution of differential equations and differential systems is an age-old 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, 22–25], maximal principle [6], Banach’s contraction mapping principle [26–29], 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:, , . are continuous.

2. Preliminaries

Let be the Banach space with the maximal norm given by . Let , . Then is a Banach space.

Lemma 1 (see [5]). Let , then the system of BVPs has integral representation where

Lemma 2 (see [5]). The functions satisfy the following properties: where

With the help of Lemma 1, BVP (1) can be viewed as a fixed point in for the completely continuous operator where are defined by

In order to prove our main result, the following criterion for solving system of inequalities is needed.

Lemma 3. Let with . Then the inequality system has a solution with if and only if satisfy

Proof. Necessity. The proof is obviously true for the case: . So we consider the remaining case . From the first inequality in (11), we get Substituting it into the second inequality in (11), we have Thus, Sufficiency. For the case , we can take . So we consider the last case . Let From the derivative of , we conclude that is increasing on . This together with the locally sign-preserving property of implies that there exists such that The above inequality can be rewritten as Hence (11) holds for .

3. 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 , , 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.
Step 1. It follows from (25), (26), and Lemma 3 that there exist such that Let us introduce a linear operator on as where is given by Take . Now, (21)-(24) and (27) show that i.e., Then for , by induction, we obtain Step 2. For all with and , there exists such that Indeed, by Lemma 2, we have So, we can take such that (34) holds.
Step 3. For any given , , let . By Step 2, there exists such that Notice for that Thus, by (33) and (36), we obtain that Thus for , we conclude thatThe above two inequalities ensure that is a Cauchy sequence in . Since is complete, there exists such that . Therefore, is a fixed point of that follows from the continuity of operator .
Step 4. We show that has a unique fixed point. Suppose there exist two elements with and . By Step 2, there exists such that Applying the method used in Step 3 again, for , we get Hence we get the desired results.

In the following, we give an example to illustrate our theory.

Example 5. Consider the differential system where . We have Let then where . Hence, there exists a solution of the following inequality system: 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).

Take , . After standard computation, we get Then So it follows from the definition of the norm for linear operator that Thus Example 5 shows that Theorem 4 provides the same results with weaker conditions.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (11371221, 11571207), Shandong Natural Science Foundation (ZR2018MA011), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.