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

George L. Karakostas

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'skiń≠'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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