Journal of Function Spaces

Volume 2016, Article ID 6874643, 6 pages

http://dx.doi.org/10.1155/2016/6874643

## Existence of Positive Solution to System of Nonlinear Third-Order Three-Point BVPs

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received 13 April 2016; Accepted 16 June 2016

Academic Editor: Enrique Llorens-Fuster

Copyright © 2016 Jian-Ping Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We are concerned with the following system of third-order three-point boundary value problems: , , , , , , , and , where and . By imposing some suitable conditions on and , we obtain the existence of at least one positive solution to the above system. The main tool used is the theory of the fixed-point index.

#### 1. Introduction

Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves, or gravity driven flows, and so on [1].

Recently, there are a lot of papers concerning the existence of positive solutions to third-order three-point boundary value problems (BVPs for short); see [2–13] and the references therein. Yet, only in a few papers has the problem of existence of positive solutions to systems of third-order three-point BVPs been considered.

It is worth mentioning that there are some excellent works on systems of second-order or higher-order multipoint BVPs; see [14–18].

In this paper, we study the existence of positive solution for the following system of nonlinear third-order three-point BVPs:Throughout this paper, we always assume that the following conditions are fulfilled: and . and , , .

If satisfies the differential equations and boundary conditions in system (1), then is said to be a solution of system (1). If is a solution of system (1) and , , , then is said to be a positive solution of system (1).

To end this section, we state the following results on the theory of the fixed-point index [19].

Let be a real Banach space, a cone, and the zero element in . For , we denote

Theorem 1. *Let be a completely continuous operator. If there exists such that then .*

Theorem 2. *Let be a completely continuous operator which has no fixed point on . If for all , then .*

#### 2. Preliminaries

Let be equipped with the maximum norm. Then is a Banach space.

Lemma 3 (see [7]). *For any , the BVPhas a unique solution where *

For convenience, we denoteObviously, and .

Lemma 4 (see [7]). *For any , .*

Lemma 5. *For any , .*

*Proof. *Since , we only need to consider .

If , then If , then If , then Therefore, for any , .

Lemma 6. *Let and , . Then the unique solution of BVP (4) satisfies *

*Proof. *Since , , it follows from Lemma 4 that for . In view of , , we have , . This shows that , , which indicates thatOn the other hand, it follows from the fact for that is concave down on , which together with implies that . In view of (12), we get that .

Lemma 7. *There exists such that*

*Proof. *Let Then it is obvious that This indicates that there exists such that , which together with the fact that implies that there exists such that ; that is, (13) is satisfied.

In the remainder of this paper, we always assume that is defined as in Lemma 7 and , .

Corollary 8. * is an eigenvalue of the eigenvalue problem and is an eigenfunction corresponding to the eigenvalue .*

*Proof. *By Lemma 7, it is obvious.

Let Then and are cones in . Now, we define a linear operator as follows:

Lemma 9. *Consider .*

*Proof. *In view of Lemmas 3, 4, and 6, it is not difficult to verify that .

Lemma 10. *Consider and*

*Proof. *Obviously, is continuous and , , which indicates that . And it follows from Lemma 3 and Corollary 8 that (20) is satisfied.

#### 3. Main Results

Obviously, is a solution of system (1) if and only if is a solution of the following system:Moreover, system (21) can be written as the integral equation: If we define an operator on by then it is easy to verify that is completely continuous. Moreover, if is a fixed point of and , , then is a solution of system (1).

Theorem 11. *Assume that the following conditions are fulfilled:**There exists a constant such that uniformly on .**There exists a constant such that uniformly on .**Then system (1) has at least one positive solution.*

*Proof. *First, let Then we may assert that is a bounded subset of .

In fact, if , then there exists such that , which together with Lemma 10 implies thatwhere is defined as follows: In view of (27) and Lemma 9, we have . This implies thatOn the other hand, from (1) of , we know that there exist positive constants and such that which together with Jensen inequality and Lemma 4 implies thatwhere .

From (2) of , we know that there exists such thatwhere .

So, it follows from (31), (32), and Lemmas 4 and 5 thatwhere .

Since , we haveIn view of (33) and (34), we get So, and so, which together with (29) indicates thatThis shows that is a bounded subset of . Therefore, there exists a sufficiently larger such that So, it follows from Theorem 1 thatNext, from (1) of , we have which together with , , implies thatLet where . Then from (2) of and , , we know that there exists such thatBy (44), for any , we have which together with (42) shows that This indicates that , . So, it follows from Theorem 2 thatIn view of (40) and (47), we get Therefore, has at least one fixed point and . Let Then is a solution of system (1).

Thirdly, we prove for .

Suppose on the contrary that there exists such that . Since , , we know that , . So, is concave down on , which together with implies that . At the same time, it follows from , , that , . So, , ; that is, , , which contradicts with the fact that . Therefore, , .

Finally, we verify for .

First, we may assert that . Indeed, if , , in view of , , then we get which is a contradiction. This shows that . So, by using the similar method to prove for , we may obtain , .

To sum it up, is a positive solution of system (1).

*Competing Interests*

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

*References*

- M. Greguš,
*Third Order Linear Differential Equations*, vol. 22 of*Mathematics and Its Applications (East European Series)*, D. Reidel, Dordrecht, The Netherlands, 1987. - D. R. Anderson, “Green's function for a third-order generalized right focal problem,”
*Journal of Mathematical Analysis and Applications*, vol. 288, no. 1, pp. 1–14, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Sun, “Positive solutions of singular third-order three-point boundary value problem,”
*Journal of Mathematical Analysis and Applications*, vol. 306, no. 2, pp. 589–603, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Bai and X. Fei, “Existence of triple positive solutions for a third order generalized right focal problem,”
*Mathematical Inequalities and Applications*, vol. 9, no. 3, pp. 437–444, 2006. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. Yang, “Positive solutions of a third-order three-point boundary-value problem,”
*Electronic Journal of Differential Equations*, vol. 2008, no. 99, pp. 1–10, 2008. View at Google Scholar - L.-J. Guo, J.-P. Sun, and Y.-H. Zhao, “Existence of positive solutions for nonlinear third-order three-point boundary value problems,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 68, no. 10, pp. 3151–3158, 2008. View at Publisher · View at Google Scholar · View at Scopus - J.-P. Sun, L.-J. Guo, and J.-G. Peng, “Multiple nondecreasing positive solutions for a singular third-order three-point BVP,”
*Communications in Applied Analysis*, vol. 12, no. 1, pp. 91–100, 2008. View at Google Scholar · View at MathSciNet - Q. Yao, “Positive solutions of singular third-order three-point boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 354, no. 1, pp. 207–212, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Sun, “Positive solutions for third-order three-point nonhomogeneous boundary value problems,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 45–51, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Feng, H. Feng, and D. Bai, “Eigenvalue for a singular third-order three-point boundary value problem,”
*Applied Mathematics and Computation*, vol. 219, no. 18, pp. 9783–9790, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X.-L. Li, J.-P. Sun, and F.-D. Kong, “Existence of positive solution for a third-order three-point BVP with sign-changing Green's function,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 30, pp. 1–11, 2013. View at Google Scholar · View at MathSciNet - F. J. Torres, “Positive solutions for a third-order three-point boundary-value problem,”
*Electronic Journal of Differential Equations*, vol. 2013, no. 147, pp. 1–11, 2013. View at Google Scholar - J.-P. Sun and J. Zhao, “Iterative technique for a third-order three-point BVP with sign-changing Green's function,”
*Electronic Journal of Differential Equations*, vol. 2013, no. 215, pp. 1–9, 2013. View at Google Scholar · View at MathSciNet - Y. Zhou and Y. Xu, “Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 320, no. 2, pp. 578–590, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Henderson and R. Luca, “Positive solutions for a system of second-order multi-point boundary value problems,”
*Applied Mathematics and Computation*, vol. 218, no. 10, pp. 6083–6094, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Henderson and R. Luca, “Positive solutions for a system of second-order nonlinear multi-point eigenvalue problems,”
*Applied Mathematics and Computation*, vol. 223, pp. 197–208, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Henderson and R. Luca, “Positive solutions for systems of nonlinear second-order multipoint boundary value problems,”
*Mathematical Methods in the Applied Sciences*, vol. 37, no. 16, pp. 2502–2516, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Henderson and R. Luca, “Positive solutions for a system of higher-order multi-point boundary value problems,”
*Computers & Mathematics with Applications*, vol. 62, no. 10, pp. 3920–3932, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,”
*SIAM Review*, vol. 18, no. 4, pp. 620–709, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus

*
*