Geodesic
The characteristic of noncompact convexity and random fixed point theorem for set-valued operators
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 269-279.

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Let $(\Omega ,\Sigma )$ be a measurable space, $X$ a Banach space whose characteristic of noncompact convexity is less than 1, $C$ a bounded closed convex subset of $X$, $KC(C)$ the family of all compact convex subsets of $C.$ We prove that a set-valued nonexpansive mapping $T\: C\rightarrow KC(C)$ has a fixed point. Furthermore, if $X$ is separable then we also prove that a set-valued nonexpansive operator $T\: \Omega \times C\rightarrow KC(C)$ has a random fixed point.
Classification : 47H09, 47H10, 47H40
Keywords: random fixed point; set-valued random operator; measure of noncompactness 
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     title = {The characteristic of noncompact convexity and random fixed point theorem for set-valued operators},
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     zbl = {1174.47042},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2007__57_1_a21/}
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Kumam, Poom; Plubtieng, Somyot. The characteristic of noncompact convexity and random fixed point theorem for set-valued operators. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 269-279. http://geodesic.mathdoc.fr/item/CMJ_2007__57_1_a21/