Existence and multiplicity of solutions for some three-point nonlinear boundary value problems

Abstract

We study the existence and multiplicity of solutions for the three-point nonlinear boundary value problem u '' ( t) +lambda a(t) f (u) = 0, 0 < t < 1; u( 0) = 0 = u(1)-gamma u(eta), where eta is an element of (0,1), gamma is an element of [0,1), a(t) and f (u) are assumed to be positive and have some singularities, and. is a positive parameter. Under certain conditions, we prove that there exists lambda* > 0 such that the three-point nonlinear boundary value problem has at least two positive solutions for 0 <lambda <lambda*, at least one solution for lambda =lambda*, and no solution for lambda >lambda*.