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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 $\left(X,\parallel ·\parallel \right)$ be a Banach space and let $A,B:X\to X$ be two continuous mappings. If $A$ maps bounded subsets into compact sets and $B$ is a strict contraction, i.e., there exists $k\in \left[0,1\right)$ such that $\parallel Bx-By\parallel \le k\parallel x-y\parallel$ 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 $\lambda \in \left(0,1\right)$. 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:
 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H30 Particular nonlinear operators