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 ⊆ X, C: A → Y a compact operator and T an operator defined at least on the set A × C(A) with values in X. By assuming that the family {T (· , y) : y ∈ 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'ski˘ ı's fixed point theorem for contractions and compact mappings. The results are used to prove the existence of (global) solutions of integral and inte-grodifferential equations.