Singular higher-order semipositone nonlinear eigenvalue problems

Abstract

In this paper, we consider the existence of positive solutions to the following singular higher-order semipositone eigenvalue problem (HSEP): {(-1)(n-k)u(n)(t) = λ[f(t, u(t)) + q(t)], 0 < t < 1, u(i)(0) = 0, 0 ≤ i ≤ k - 1, u(i)(1) = 0, 0 ≤ i ≤ n-k-1, where n ≥ 2, 1 ≤ k ≤ n - 1, and λ > 0 is a parameter. The functions f and q may have singularity at t = 0 and (or) 1, and furthermore, the nonlinear function may change sign for 0 < t < 1. Without making any monotone-type assumptions, we obtain the positive solution of the problem for λ lying in some interval, based on the Krasnaselskii's fixedpoint theorem in a cone. Copyright © 2007 Watam Press.