A FIXED-POINT THEOREM OF KRASNOSELSKII-SCHAEFER TYPE

In this paper we focus on three fixed point theorems and an integral e quation. Schaefer's fixed point theorem will yield a T-periodic soluti on of (0.1) x(t) = a(t) + integral(t-h)(t) D(t,s)g(s,x(s))ds if D and g satisfy certain sign conditions independent of their magnitude. A co mbination of the contraction mapping theorem and Schauder's theorem (k nown as Krasnoselskii's theorem) will yield a T-periodic solution of ( 0.2) x(t) = f(t,x(t)) + integral(t-h)(t) D(t,s)g(s,x(s))ds if f define s a contraction and if D and g are small enough. We prove a fixed poin t theorem which is a combination of the contraction mapping theorem an d Schaefer's theorem which yields a T-periodic solution of (0.2) when f defines a contraction mapping, while D and g satisfy the aforementio ned sign conditions.