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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References:

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