Positive Solutions for Systems of Nonlinear Higher Order Differential Equations with Integral Boundary Conditions

Abstract

By constructing some general type conditions and using fixed point theorem of cone, this paper investigates the existence of at least one and at least two positive solutions for systems of nonlinear higher order differential equations with integral boundary conditions. As application, some examples are given.

1. Introduction

In this paper, we consider the following systems of nonlinear mixed higher order differential equations with integral boundary conditions: where , , , , and is nonnegative, ; .

Boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], and hydrodynamic problems [3]. Such problems include two-, three-, and multipoint boundary value problems as special cases and attracted much attention (see [4–12] and the references therein). In particular, we would like to mention the result of Pang et al. [9]. In [9], by applying fixed point index theory, Pang et al. study the expression and properties of Green’s function and obtained the existence of positive solutions for th-order -point boundary value problems: Yang and Wei [10], Feng and Ge [11], and Li and Wei [12] improved and generalized the results of [9] by using different methods.

On the other hand, much effort has been devoted to the study of the existence of positive solutions for systems of nonlinear differential equations (see [13–16] and the references therein). In [13], by applying Krasnoselskii fixed point theorem in a cone, Hu and Wang obtained multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations. In [14], Henderson and Ntouyas extended the results of [13] to systems of nonlinear th-order three-point boundary value problems: In [15], by using fixed point index theory, Xie and Zhu improved the results of [14]. At the same time, boundary value problems with integral boundary conditions have received attention [16, 17].

Motivated by the work of the abovementioned papers, our aim in this paper is to study the existence of positive solutions associated with systems (1) by applying fixed point theorem in cone. Further, we present some general type conditions ()–() instead of the sublinear or superlinear conditions which are used in [4, 5, 8, 10, 12–14]. Our conditions are applicable for more general functions.

2. Several Lemmas

For convenience, we make the following notations. Let where is defined by Lemma 6 and is some subset of .

List the following assumptions:), do not vanish identically for , ;(), ;();()there exist , and a sufficiently large such that(1), for all ,(2), for all , where ; is defined by (21).()There exist , and a sufficiently small such that(1), for all ,(2), for all , where .()There exist , , and such that(1), for all ,(2), for all , where .()There exist , and a sufficiently small such that(1), for all ,(2), for all , where .() and are increasing on , and there exists such that , for all .

Lemma 1. If , for any , higher order differential equations have a unique solution where

Proof. By Taylor’s formula, we get Letting in (10), we have Substituting and (11) into (10), we obtain Multiplying (12) with and integrating it, we have so Substituting (14) into (12), we have where is defined by (7).

Definition 2. is said to be a positive solution of systems (1) if and only if satisfies systems (1) and , , for any .

Lemma 3 (see [6]). If , the continuous function , has the following properties: (i), for all , where ;(ii), for all , where .

Remark 4. Combining (i) and (ii), we can easily see where .

Lemma 5. If , the continuous function has the following property:

Proof. From the properties of and the definition of , we can prove easily the results of Lemma 5.

Lemma 6. If , the continuous function defined by (7) satisfies (i), for all ,(ii) for each , and , for all ,
where , is defined in Remark 4 and .

Proof. (1) From Lemma 5 and (i) of Lemma 3, we get the proof of (i) immediately.
(2) From Lemma 5 and (i) of Lemma 3, it is obvious that for each .
Now, we show that the form (ii) holds. In fact, from (16) and (9), we have Then, the proof of Lemma 6 is completed.

Remark 7. From the definition of , it is obvious that .
It is easy to prove that is a positive solution of systems (1) if and only if is a positive solution of systems of integral equations where , , are Green’s functions defined by (7).
It follows from (19) that we can obtain the integral equation:
In a real Banach space , the norm is defined by . Set where . Obviously, is a positive cone in .
Define the operator by

Lemma 8. Suppose that ()–() are satisfied; then the operator is completely continuous.

Proof. Let ; consider (22); from Lemma 3 and (21), we have It follows from (23) that we have ; therefore, operator . It is easy to prove that operator is completely continuous since , , , , , and are continuous.

Lemma 9 (see [18]). Suppose is a real Banach space and is cone in , and let , be bounded open sets in such that , . Let operator be completely continuous. Suppose that one of two conditions holds (i), for all ; , for all ;(ii), for all ; , for all .
Then, operator has at least one fixed point in .

Lemma 10 (see [18]). Suppose is a real Banach space and is cone in , and let , , and be bounded open sets in such that , , and . Let operator be completely continuous, such that (1), for all ;(2), , for all ;(3), for all .
Then, operator has at least two fixed points and in with and .

3. Main Results

Theorem 11. Suppose that assumptions ()–() are satisfied; then systems (1) have at least one positive solution satisfying , .

Proof. At first, let , and set and ; then . By Lemma 6 and the assumption (), we have Therefore, we have Further, set , for ; by the assumption (), we have Therefore, we have Thus, from (25), (27), Lemma 8, and Lemma 9, operator has a fixed point in . This means that systems (1) have at least one positive solution satisfying , .

Theorem 12. Suppose that assumptions ()–() and ()-() are satisfied; then systems (1) have at least one positive solution satisfying , .

Proof. At first, it follows from the assumption () that we have By means of simple calculation, we have Then, there exists a sufficiently large such that Set . For , by (28),(30), we obtain that Further, since and is continuous in , there exists such that Set . For , we have It follows from the assumption () and Lemma 6 that we have Hence, we have Thus, from (31), (35), Lemmas 8 and 9, operator has a fixed point in . This means that systems (1) have at least one positive solution satisfying , .

Theorem 13. Suppose that assumptions ()–() and ()-() hold. Then, systems (1) have at least two positive solutions and .

Proof. Set . For , from (), we obtain that Thus, we have By () and (), we can get So, we can choose , , and such that and satisfying the above three inequalities. By Lemma 8 and Lemma 10, we guarantee that operator has two fixed points and . This means that systems (1) have at least two positive solutions and .
In order to illustrate that our assumptions ()–() are suitable for more general functions, we give some examples.

Example 14. In systems (1), let , , , , , and , so the assumptions ()–() are satisfied. Choose , ; then uniformly with respect to and . It is easy to verify that the assumptions ()-() hold. By Theorem 11, systems (1) have at least one position solution.

Example 15. In systems (1), let , , , , , and , so the assumptions ()–() are satisfied. Choose ; then uniformly with respect to and . It is easy to verify that the assumptions ()-() hold. By Theorem 12, systems (1) have at least one position solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the referees for their time and comments. The authors thank the partial support from the Shandong Provincial Natural Science Foundation of China (Grant no. ZR2012AQ007), the Natural Science Foundation of of Anhui Provincial Education Department of China (Grants nos. KJ2012B187 and KJ2013A248), and Professors (Doctors) Scientific Research Foundation of Suzhou University of China (Grant no. 2013jb04).