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

DOI: http://dx.doi.org/10.12775/TMNA.2003.035

#### 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.

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.

#### Keywords

Banach space; equicontractions; fixed point theorem

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