A Fixed Point Theorem of Krasnoselskii—Schaefer Type

Abstract

Abstract In this paper we focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem will yield a T‐periodic solution of (0.1) x ( t ) = a (t) + t t‐h D(t,s)g(s,x(s))ds if D and g satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) will yield a T‐periodic solution of (0.2) x ( t ) = f(t,x(t)) + t t‐h D(t,s)g(s,x(s))ds if f defines a contraction and if D and g are small enough. We prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T‐periodic solution of (0.2) when / defines a contraction mapping, while D and g satisfy the aforementioned sign conditions.