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

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