Geodesic
On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 2, pp. 263-268.

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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.
Classification : 47H10, 54H25
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 
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     author = {Park, Sehie},
     title = {On a problem of {Gulevich} on nonexpansive maps in uniformly convex {Banach} spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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     number = {2},
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     zbl = {0852.47029},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_1996__37_2_a5/}
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Park, Sehie. On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 2, pp. 263-268. http://geodesic.mathdoc.fr/item/CMUC_1996__37_2_a5/