Existence of Positive Solutions for Nonlinear Third-Order Boundary Value Problem

Abstract

We are concerned with the existence of positive solutions for the nonlinear third-order three-point boundary value problem , , , , where , , is a positive parameter, is continuous. We construct Green’s function for the associated linear boundary value problem and obtain some useful properties of Green’s function. Finally, by using fixed-point index theorem in cones, we establish the existence results of positive solutions for the boundary value problem an example illustrates the application of the results obtained.

1. Introduction

The existence of positive solutions for third-order three-point boundary value problems has attracted considerable attention by a number of authors. For example, Anderson [1], by using the well-known Guo-Krasnoselskii fixed point theorems [2, 3], obtained some existence results for positive solutions for the following boundary value problem (BVP for short): In [4], Sun considered the existence of multiple positive solutions (at least three) to the BVP where , are constants and and are continuous. The methods used in the paper were based on an application of a fixed point theorem due to Graef and Yang [5]. In [6], the authors considered the existence of positive solution to the third-order three-point BVP where and . The proof relies on Guo-Krasnoselskii fixed point theorem.

Motivated by the abovementioned works, in this paper, the fixed-point index theorem in cones was used to discuss the third-order three-point BVP where , , and is a positive parameter. Throughout this paper, we assume that the following conditions are satisfied: is continuous; and is not identically zero on any subinterval of ;, .

2. Preliminaries

In this section, we will state some important preliminary lemmas.

Let , ; then is a Banach space with norm .

Lemma 1. Assume that is continuous, , , ; then the BVP has a unique solution where is corresponding Green’s function, and

Proof. We can reduce equation to an equivalent integral equation By , it follows that ; from , we get that , thanks to ; we have .
Thus, substituting into (9), we have where

Lemma 2. Let , , for be given as (7); then we obtain the following results:(i), , ,(ii), , .

Proof. If , for all , , it follows from (8) that and therefore If , it follows from (8) that By (8), we have and thus So we obtain that From (14) and (16), it follows that Hence, by (7), (9), and (16) we get and by (7), (14), (16), and (20) we have

Remark 3. Let  , ; then , , and it follows that

Lemma 4. Assume that , , is continuous, ; then the unique solution of the BVP (4) is nonnegative and satisfies where . By , it is easy to know .

Proof. From (7), (14), (16), and (20), for , we have and thus, is nonnegative.
Let ; by (12) and (19), we get and therefore On the other hand, for , from Lemma 2 (ii), we have and thus where .

Choose a cone in as follows: Define an operator by By the definition of operator , a positive solution of BVP (4) is equivalent to a nonzero fixed point of .

Lemma 5. Assume that (C1)–(C3) hold; then the operator is completely continuous.

Proof. By the definition of operator and Lemma 5, for , it is easy to prove that . According to the Ascoli-Arzela theorem, we can easily obtain that is a completely continuous mapping.

Lemma 6 (see [7]). Let be a bounded open subset of with , and let be a completely continuous mapping. If for every and , then .

Lemma 7 (see [7]). Let be a bounded open subset of , and let be a completely continuous mapping. If there exists an , such that for every and , then .

3. Main Results

In this section, we state and prove our main results.

To be convenient, we introduce the following notations: Our main results are as follows.

Theorem 8. Assume that (C1)–(C3) hold and , , ; then for any , BVP (4) has at least one positive solution.

Proof. For every , according to the condition of and the definition of , there exist constants , , when ; we have ; namely, Let , for ; from (31), we get Therefore, for every , , when , we have . In fact, if there exist and , such that , then ; this is a contradiction. Hence satisfies the condition of Lemma 6 in . By Lemma 6, we have On the other hand, by the condition of and the definition of , there exist , , and for , we get . Let , ; then, , for all .
Choose , . We prove that satisfies the condition of Lemma 7 in ; namely, for every and . If it is not true, there exist and , such that . Let , , for ; we have and this is a contradiction. Thus satisfies the condition of Lemma 7 in . By Lemma 7 we get Now, from (35) and (37) it follows that Therefore, has a fixed point in , which is a positive solution of BVP (4).

Remark 9. If , let ; if , let .

Theorem 10. Assume that (C1)–(C3) hold and , , ; then for any , BVP (4) has at least one positive solution.

Proof. Let the operator be defined as (31).
For every , according to the condition of and the definition of , there exist constants , , and when , we have . Let ; then Choose ; let . Then for every , we have Thus, for and , when , we have . In fact, if there exist and , such that , then ; this is a contradiction. Hence satisfies the condition of Lemma 6 in . By Lemma 6, we have On the other hand, by the condition of and the definition of , there exist , , such that , , and when , we have . Let . Choose . The process of the proof is similar to Theorem 8; we can get for every and . By Lemma 7 we get Combining (41), (42), and , it follows that Therefore, has a fixed point in , which is a positive solution of BVP (4).

Remark 11. If , let ; if , let .

Example 12. We consider the following third-order three-point BVP: where is a positive parameter; clearly It is easy to obtain that and thus, hold.
By calculating, we get By Theorem 8, for any , the BVP (44) has at least one positive solution.

4. Conclusions

In this work, Green’s function for the associated linear boundary value problem was constructed, and some useful properties of the Green’s function were obtained. On these bases, the existence results of positive solutions for the boundary value problem were established by using fixed-point index theorem in cones. To demonstrate the applications of our results, an example is given.

Conflict of Interests

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

Acknowledgments

This research is supported by NNSF of China (11261053) and NSF of Gansu province (1208R-JZA129).