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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
uniformly convex; Banach space; Hilbert space; contraction; nonexpansive map; weakly inward map; demi-closed; Rothe condition; Leray-Schauder condition; (KR)-bounded; Opial's condition
Summary:
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.
References:
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