An extension of Krasnosel'skii's fixed point theorem for contractions and compact mappings
Let \(X\) be a Banach space, \(Y\) a metric space, \(A\subseteq X\), \(C:A\to Y\) 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 (i.e., there is a \(k\in[0,1[\) such that \(\| T(x_1,y)-T(x_2,y) \|\leq k\| x_1-x_2\|\) for all \((x_1, y)\), \((x_2,y)\in A\times C(A))\), the author presents two fixed point theorems (Theorems 2.2 and 2.5) for the operator \(Ex:=T(x,C (x))\), extending the following fixed point theorem of \textit{M. A. Krasnosel'skij} [Am. Math. Soc., Transl., II. Ser. 10, 345--409 (1958; Zbl 0080.10403)]. Theorem 1.1 (Krasnovel'skij). Suppose that \(A\) is a closed bounded convex subset of a Banach space \(X\). If \(T:A\to X\) is a contraction, \(C:A\to X\) is compact and \(T(A)+C(A):=\{z=T(x)+ C(y):x,y \in A\} \subseteq A\), then \(T+C\) has a fixed point in \(A\). Theorem 2.5 also extends the fixed point theorem proved by \textit{B. C. Dhage}, \textit{S. K. Ntouyas} and \textit{P. Ch. Tsamatos} [Bull. Greek Math. Soc. 46, 119--127 (2002; Zbl 1049.47050)]. The two theorems are used to prove the existence of (global) solutions of integral and integrodifferential equations.
scientific article; zbMATH DE number 2096724