An extension of Krasnoselskii's fixed point theorem for contractions and compact mappings

Abstract

Let $X$ be a Banach space, $Y$ a metric space, $A\subseteq X$, $C\colon A\to Y$
a compact
operator and $T$ an operator defined at least on the set $A\times C(A)$
with values in $X$. By assuming
that the family
$\{T(\cdot,y):y\in C(A)\}$ is equicontractive we present two fixed point
theorems for the operator of the form
$Ex:=T(x,C(x))$. Our results extend the well known Krasnosel'skiĭ's fixed
point theorem for contractions and compact mappings. The results are used
to prove the existence of (global) solutions of integral and integrodifferential equations.