Nonlinear Analysis 68 (2008) 216–225 www.elsevier.com/locate/na Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary conditions ✩ Zhilin Yang Department of Mathematics, Qingdao Technological University, No 11 Fushun Road, 266033, Qingdao, Shandong Province, PR China Received 23 July 2006; accepted 30 October 2006 Abstract This paper deals with the existence of nontrivial solutions for a nonlinear singular Sturm–Liouville problem with integral boundary conditions. c 2006 Elsevier Ltd. All rights reserved. MSC: 34B10; 34B15; 34B16; 47H11 Keywords: Singular integral boundary value problem; Nontrivial solution; Spectral radius; Topological degree; Fixed point 1. Introduction Consider the Sturm–Liouville problem with integral boundary value conditions -(au ) + bu = g(t ) f (t , u ), t ∈ (0, 1), (cos γ 0 )u (0) - (sin γ 0 )u (0) = 1 0 u (τ)dα(τ), (cos γ 1 )u (1) + (sin γ 1 )u (1) = 1 0 u (τ)dβ(τ), (1.1) where a ∈ C 1 ([0, 1],(0, +∞)) and b ∈ C ([0, 1], R + )(R + := [0, +∞)); f ∈ C ([0, 1]× R, R) and g ∈ C ((0, 1), R + ) ∩ L (0, 1) with 1 0 g(t )dt > 0; α and β are right continuous on [0, 1), left continuous at t = 1, and nondecreasing on [0, 1], with α(0) = β(0) = 0; γ 0 ∈ [0,π/2] and γ 1 ∈ [0,π/2]; 1 0 u (τ)dα(τ) and 1 0 u (τ)dβ(τ) denote the Riemann–Stieltjes integrals of u with respect to α and β , respectively. Notice that g, under the aforementioned hypothesis, may be singular at t = 0 or/and at t = 0. Thus (1.1) is a singular Sturm–Liouville problem with integral boundary conditions. We are concerned with the existence of nontrivial solutions for (1.1), where by a nontrivial solution of (1.1) we mean a function u ∈ C 2 (0, 1) ∩ C [0, 1] that solves (1.1) and satisfies u (t ) ≡ 0. ✩ Supported by the Science and Technology Project of the Education Department of Shandong Province, China, No. J06P05. E-mail addresses: zhilinyang@qtech.edu.cn, zhilinyang@sina.com. 0362-546X/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.10.044