Positive solutions of singular fourth-order three-point eigenvalue problem

Abstract:

In this paper we investigate the problem of existence of positive solutions for the nonlinear singular fourth-order three-point eigenvalue problem u^{(4)}(t)=\lambda a(t)f(t,u(t)),t\in [0,1], u(0)=u'(\eta)=u''(1)=u'''(0)=0, where \lambda is a positive parameter and \eta\in[\frac{1}{2},1) is a constant. By using the fixed point theorem of cone expansion-compression type duo to Krasnosel'skii, we establish various results on the existence of single and multiple positive solutions to the eigenvalue problem. Under various assumptions on functions f and a, we give explicitly the intervals for parameter \lambda in which the existence of positive solutions is guaranteed. Especially, we allow the function a(t) of nonlinear term to have suittable singularities.