Fixed point theorems and periodic problems for nonlinear Hill’s equation

Two main results of fixed point theory in infinite dimensional space are Schauder’s theorem and the contraction mapping principle. Krasnoselskii combined them into one fixed point result. In this paper, we continue the study of extensions of these theorems investigating a convex modular in a original vector space, not in modular space and without Δ 2 condition, to provide certain extensions of Banach contraction principle and Krasnoselskii fixed point theorem. We applied that theorem to solve the nonlinear periodic problem of Hill’s equation.