Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we investigate the existence of positive solu tions for singular nth-order boundary value problem where n � 2, a 2 C((0, 1), [0,+1)) may be singular at t = 0 and (or) t = 1 and the nonlinear term f is continuous and is allowed to change sign. Our proofs are based on the method of lower solution and topology degree theorem.


Introduction
Boundary value problems for higher order differential equations play a very important role in both theories and applications.Existence of positive solutions for nonlinear higher order has been studied in the literature by using the Krasnosel'skii and Guo fixed point theorem, Leggett-Williams fixed point theorem, Lower-and upper-solutions method and so on.We refer the reader to [2][3][4][5][6][7][8][9] for some recent results.However, to the best of our knowledge, few papers can be found in the literature for nth-order boundary value problem with sign changing nonlinearity, most papers are dealing with the existence of positive solutions when the nonlinear term f is nonnegative.For example, in [3], by using the Krasnosel'skii and Guo fixed point theorem, Eloe and Henderson studied the existence of positive solutions for the following boundary value problem * E-mail address: czbai8@sohu.comEJQTDE, 2008 No. 8, p. 1 where (A 2 ) a : (0, 1) → [0, ∞) is continuous and does not vanish identically on any subinterval; (A 3 ) f is either superlinear or sublinear.
Motivated by the above works, in this paper, we study the existence of positive solutions for singular nth-order boundary value problem with sign changing nonlinearity as follows Throughout this paper, we assume the following conditions hold.
The purpose of this paper is to establish the existence of positive solutions for BVP (1.2) by constructing available operator and combining the method of lower solution with the method of topology degree.
The rest of this paper is organized as follows: in section 2, we present some preliminaries and lemmas.Section 3 is devoted to prove the existence of positive solutions for BVP (1.2).An example is considered in section 4 to illustrate our main results.

Preliminary Lemmas
Lemma 2.1.Suppose that y(t) ∈ C[0, 1], then boundary value problem has a unique solution Proof.The proof follows by direct calculations.
Remark 2.1.By (C 2 ) and Lemma 2.2, we have By the definition of completely continuous operator, we can check that the following lemma holds.Then, A • T : P → P is also a completely continuous operator.