Sehie Park
On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces

Comment.Math.Univ.Carolinae 37,2 (1996) 263-268.

Abstract:Let $X$ be a uniformly convex Banach space, $D\subset X$, $f:D\to X$ a nonexpansive map, and $K$ a closed bounded subset such that $\overline {\text {co}} K\subset D$. If (1) $f|_K$ is weakly inward and $K$ is star-shaped or (2) $f|_K$ satisfies the Leray-Schauder boundary condition, then $f$ has a fixed point in $\overline {\text {co}} K$. This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.

Keywords: uniformly convex, Banach space, Hilbert space, contraction, nonexpansive map, weakly inward map, demi-closed, Rothe condition, Leray-Schauder condition, (KR)-bounded, Opial's condition
AMS Subject Classification: Primary 47H10; Secondary 54H25

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