Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

Abstract

This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Mönch's fixed point theorem is used to prove the main result. By the unification property of time scales, our result is valid for Sturm-Liouville differential equations and difference equations, but more interestingly by the extension property, it is also valid for Sturm-Liouville $q$-difference equation.