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Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness. (English) Zbl 1252.47047
In [“A fixed point theorem of Krasnoselskii-Schaefer type”, Math. Nachr. 189, 23–31 (1998; Zbl 0896.47042)], T. A. Burton and C. Kirk proved the following theorem of Krasnoselskii-Schaefer type. Let X,· be a Banach space and let A,B:XX be two continuous mappings. If A maps bounded subsets into compact sets and B is a strict contraction, i.e., there exists k[0,1) such that Bx-Bykx-y for every x,yX, then either A+B has a fixed point or the set xX:x=λBx λ+λAx is unbounded for each λ(0,1). In the paper under review, the authors establish some variants of the above result. They use their results to study the existence of solutions of a nonlinear integral equation in the context of L 1 -spaces.

MSC:
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H30Particular nonlinear operators
References:
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