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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)], {\it T. A. Burton} and {\it C. Kirk} proved the following theorem of Krasnoselskii-Schaefer type. Let $\left( X,\| \cdot \| \right) $ be a Banach space and let $A,B: X\rightarrow X$ be two continuous mappings. If $A$ maps bounded subsets into compact sets and $B$ is a strict contraction, i.e., there exists $k\in [0,1)$ such that $\|Bx-By\| \leq k\| x-y\| $ for every $ x,y\in X,$ then either $A+B$ has a fixed point or the set $\left\{ x\in X:x=\lambda B\left( \frac{x}{\lambda }\right) +\lambda Ax\right\} $ is unbounded for each $\lambda \in (0,1)$. 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.

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