# Article

Full entry | PDF   (0.1 MB)
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
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:
[A] Altman M.: A fixed point theorem for completely continuous operators in Banach spaces. Bull. Acad. Polon. Sci. 3 (1955), 409-413. MR 0076308 | Zbl 0067.40802
[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
[B1] Browder F.E.: Existence of periodic solutions for nonlinear equations of evolution. Proc. Nat. Acad. Sci. USA 53 (1965), 1100-1103. MR 0177295 | Zbl 0135.17601
[B2] Browder F.E.: Semicontractions and semiaccretive nonlinear mappings in Banach spaces. Bull. Amer. Math. Soc. 74 (1968), 660-665. MR 0230179
[CMP] Canetti A., Marino G., Pietramala P.: Fixed point theorems for multivalued mappings in Banach spaces. Nonlinear Anal. TMA 17 (1991), 11-20. MR 1113446 | Zbl 0765.47016
[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. MR 0296778 | Zbl 0229.47047
[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. MR 0331136 | Zbl 0277.47034
[Go] Goebel K., Kuczumow T.: A contribution to the theory of nonexpansive mappings. Bull. Calcutta Math. Soc. 70 (1978), 355-357. MR 0584472 | Zbl 0437.47040
[Gö] Göhde D.: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30 (1965), 251-258. MR 0190718
[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.
[K] Krasnosel'skii M.A.: New existence theorems for solutions of nonlinear integral equations. Dokl. Akad. Nauk SSSR 88 (1953), 949-952. MR 0055578
[M] Martinez-Yanez C.: A remark on weakly inward contractions. Nonlinear Anal. TMA 16 (1991), 847-848. MR 1106372 | Zbl 0735.47032
[O] Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967), 591-597. MR 0211301 | Zbl 0179.19902
[P] Petryshyn W.V.: A new fixed point theorem and its application. Bull. Amer. Math. Soc. 78 (1972), 225-229. MR 0291920 | Zbl 0231.47030
[R] Ray W.O.: Zeros of accretive operators defined on unbounded sets. Houston J. Math. 5 (1979), 133-139. MR 0533647 | Zbl 0412.47032
[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. MR 0094672 | Zbl 0066.09001
[Sh] Shinbrot M.: A fixed point theorem and some applications. Arch. Rational Mech. Anal. 17 (1964), 255-271. MR 0169068 | Zbl 0156.38502
[Z] Zhang S.: Star-shaped sets and fixed points of multivalued mappings. Math. Japonica 36 (1991), 327-334. MR 1095748 | Zbl 0752.47017

Partner of