Existence and multiplicity of positive solutions for an elastic beam equation

Abstract

This paper investigates the boundary value problem for elastic beam equation of the form $$u''''(t) = q(t)f(t,u(t)u'(t),u''(t),u'''(t)),0 < t < 1,$$ , with the boundary conditions $$u(0) = u'(1) = u''(0) = u'''(1) = 0.$$ . The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. By using Leray-Schauder nonlinear alternate, Leray-Schauder fixed point theorem and a fixed point theorem due to Avery and Peterson, we establish some results on the existence and multiplicity of positive solutions to the boundary value problem. Our results extend and improve some recent work in the literature.