# Article

 Title: On a problem of Gulevich on nonexpansive maps in uniformly convex Banach spaces (English) Author: Park, Sehie Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 37 Issue: 2 Year: 1996 Pages: 263-268 . Category: math . 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. (English) Keyword: uniformly convex Keyword: Banach space Keyword: Hilbert space Keyword: contraction Keyword: nonexpansive map Keyword: weakly inward map Keyword: demi-closed Keyword: Rothe condition Keyword: Leray-Schauder condition Keyword: (KR)-bounded Keyword: Opial's condition MSC: 47H10 MSC: 54H25 idZBL: Zbl 0852.47029 idMR: MR1399001 . Date available: 2009-01-08T18:23:31Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118831 . Reference: [A] Altman M.: A fixed point theorem for completely continuous operators in Banach spaces.Bull. Acad. Polon. Sci. 3 (1955), 409-413. Zbl 0067.40802, MR 0076308 Reference: [AK] Assad N.A., Kirk W.A.: Fixed point theorems for set-valued mappings of contractive type.Pac. J. Math. 43 (1972), 553-562. MR 0341459 Reference: [B1] Browder F.E.: Existence of periodic solutions for nonlinear equations of evolution.Proc. Nat. Acad. Sci. USA 53 (1965), 1100-1103. Zbl 0135.17601, MR 0177295 Reference: [B2] Browder F.E.: Semicontractions and semiaccretive nonlinear mappings in Banach spaces.Bull. Amer. Math. Soc. 74 (1968), 660-665. MR 0230179 Reference: [CMP] Canetti A., Marino G., Pietramala P.: Fixed point theorems for multivalued mappings in Banach spaces.Nonlinear Anal. TMA 17 (1991), 11-20. Zbl 0765.47016, MR 1113446 Reference: [D] Dotson W.G.: Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces.J. London Math. Soc. (2) 4 (1972), 408-410. Zbl 0229.47047, MR 0296778 Reference: [GK] Gatica J.A., Kirk W.A.: Fixed point theorems for contraction mappings with applications to nonexpansive and pseudo-contractive mappings.Rocky Mount. J. Math. 4 (1974), 69-79. Zbl 0277.47034, MR 0331136 Reference: [Go] Goebel K., Kuczumow T.: A contribution to the theory of nonexpansive mappings.Bull. Calcutta Math. Soc. 70 (1978), 355-357. Zbl 0437.47040, MR 0584472 Reference: [Gö] Göhde D.: Zum Prinzip der kontraktiven Abbildung.Math. Nachr. 30 (1965), 251-258. MR 0190718 Reference: [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 Reference: [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. Zbl 0412.47033, MR 0537116 Reference: [KKM] Knaster B., Kuratowski C., Mazurkiewicz S.: Ein Beweis des Fixpunktsatzes für $n$- dimensionale Simplexe.Fund. Math. 14 (1929), 132-137. Reference: [K] Krasnosel'skii M.A.: New existence theorems for solutions of nonlinear integral equations.Dokl. Akad. Nauk SSSR 88 (1953), 949-952. MR 0055578 Reference: [M] Martinez-Yanez C.: A remark on weakly inward contractions.Nonlinear Anal. TMA 16 (1991), 847-848. Zbl 0735.47032, MR 1106372 Reference: [O] Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings.Bull. Amer. Math. Soc. 73 (1967), 591-597. Zbl 0179.19902, MR 0211301 Reference: [P] Petryshyn W.V.: A new fixed point theorem and its application.Bull. Amer. Math. Soc. 78 (1972), 225-229. Zbl 0231.47030, MR 0291920 Reference: [R] Ray W.O.: Zeros of accretive operators defined on unbounded sets.Houston J. Math. 5 (1979), 133-139. Zbl 0412.47032, MR 0533647 Reference: [S] Schaefer H.H.: Neue Existenzsätze in der Theorie nichtlinearer Integralgleichungen.Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Natur. Kl. 101 (1955), no.7, 40pp. Zbl 0066.09001, MR 0094672 Reference: [Sh] Shinbrot M.: A fixed point theorem and some applications.Arch. Rational Mech. Anal. 17 (1964), 255-271. Zbl 0156.38502, MR 0169068 Reference: [Z] Zhang S.: Star-shaped sets and fixed points of multivalued mappings.Math. Japonica 36 (1991), 327-334. Zbl 0752.47017, MR 1095748 .

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