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uniformly convex; Banach space; Hilbert space; contraction; nonexpansive map; weakly inward map; demi-closed; Rothe condition; Leray-Schauder condition; (KR)-bounded; Opial's condition
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.
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[Gu] Gulevich N.M.: Existence of fixed points of nonexpansive mappings satisfying the Rothe condition. J. Soviet Math. 26 (1984), 1607-1611. Zbl 0538.47032
[KR] Kirk W.A., Ray W.O.: Fixed-point theorems for mappings defined on unbounded sets in Banach spaces. Studia Math. 64 (1979), 127-138. MR 0537116 | Zbl 0412.47033
[KKM] Knaster B., Kuratowski C., Mazurkiewicz S.: Ein Beweis des Fixpunktsatzes für $n$- dimensionale Simplexe. Fund. Math. 14 (1929), 132-137.
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[M] Martinez-Yanez C.: A remark on weakly inward contractions. Nonlinear Anal. TMA 16 (1991), 847-848. MR 1106372 | Zbl 0735.47032
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[Z] Zhang S.: Star-shaped sets and fixed points of multivalued mappings. Math. Japonica 36 (1991), 327-334. MR 1095748 | Zbl 0752.47017
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