Existence theory for nonlinear Sturm-Liouville problems with non-local boundary conditions

Abstract

In this work we provide conditions for the existence of solutions to nonlinear SturmLiouville problems of the form (p(t)x′(t))′ +q(t)x(t)+λx(t) = f (x(t)) subject to non-local boundary conditions ax(0)+bx′(0) = η1(x) and cx(1)+dx′(1) = η2(x). Our approach will be topological, utilizing Schaefer’s fixed point theorem and the LyapunovSchmidt procedure.