International Journal of Applied Mathematical Research, 3 (4) (2014) 529-533 c Science Publishing Corporation www.sciencepubco.com/index.php/IJAMR doi: 10.14419/ijamr.v3i4.3168 Research Paper Positive solutions for one-dimensional p-Laplacian boundary value problems with nonlinear parameter Ahmed Omer Mohammed Abubaker Department of Mathematics, Faculty of Education, University of Khartoum, Omdurman, 406, Sudan E-mail: wadomar877@hotmail.com Copyright c 2014 Ahmed Abubaker. 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 In this paper, we establish existence of positive solutions of the nonlinear problems of one - dimensional p-Laplacian with nonlinear parameter ϕ p (u ′ (t)) ′ + a(t)f (λ, u)=0, t ∈ (0, 1), u(0) = u(1) = 0. where a :Ω → R is continuous and may change sign, λ> 0 is a parameter, f (λ, 0) > 0 for all λ> 0. By applying Leray-Schauder fixed point theorem we obtain the existence of positive solutions. Keywords : p-Laplacian, Positive solutions, Leray-Schauder fixed point theorem, nonlinear parameter. 1. Introduction The boundary value problem for one- dimensional p-Laplacian  ϕ p (u ′ (t)) ′ + λa(t)f (u)=0, t ∈ (0, 1), u(0) = u(1) = 0, (1) where ϕ p (u(t)) = |u| p−2 u, p> 1, has been studied extensively. For details, see for example, Refs [1,2,5], in the case p=2 see [6], and for case λ =1, see [7,8,9]. In a recent paper [4], Hai considered the boundary value problem  Δu + λa(t)f (u)=0, t ∈ Ω, u =0, t ∈ ∂ Ω, (2) where Ω is a bounded domain in R N , a :Ω → R is continuous and changes its sign, f (0) > 0, and λ> 0 is sufficiently small, under the following assumptions (A1) f : [0, ∞) → R is continuous and f (0) > 0. (A2) a : ¯ Ω → R is continuous, a ≡ 0, and there exists a number k> 1 such that