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Existence of fixed points and measures of weak noncompactness. (English) Zbl 1194.47060

The author proves the existence of fixed points of an operator \(A\) which is defined on a closed convex subset \(M\) of a Banach space \(X\) into itself and satisfies the following properties: (a) the measure of weak noncompactness of \(A(C)\) where \(C\subset M\) is not relatively weakly compact is strictly less than the measure of weak compactness of \(C\); (b) the sequence \(\{Ax_n\}\) has a strongly convergent subsequence whenever \(\{x_n\}\) is a weakly convergent sequence in \(M\); and (c) there exist \(x_0\in M\) and \(R>0\) such that \(Ax-x_0\neq\lambda(x-x_0)\) for all \(\lambda>1\) and all \(x\in\{z\in M:\|z-x_0\|=R\}\). As a result, the existence of fixed points for the sum of two operators and the solvability of a variant of Hammerstein’s integral equation are obtained.

MSC:

47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations
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