Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings

• Autores: Mohanasundaram Radhakrishnan, S. Rajesh
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 20, Nº. 1, 2019, págs. 119-133
• Idioma: inglés
• DOI: 10.4995/agt.2019.10360
• Enlaces
  • Texto completo
• Resumen

  • Kirk introduced the notion of pointwise eventually asymptotically nonexpansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically nonexpansive maps. Further, Kirk raised the following question: ``Does a Banach space $X$ have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever $X$ has the fixed point property for nonexpansive mappings?". In this paper, we prove that a Banach space $X$ has the fixed point property for pointwise eventually asymptotically nonexpansive maps if $X$ has uniform normal sturcture or $X$ is uniformly convex in every direction with the Maluta constant $D(X)<1.$ Also, we study the asymptotic behavior of the sequence $\{T^nx\}$ for a pointwise eventually asymptotically nonexpansive map $T$ defined on a nonempty weakly compact convex subset $K$ of a Banach space $X$ whenever $X$ satisfies the uniform Opial condition or $X$ has a weakly continuous duality map.

• Referencias bibliográficas
  • A. G. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-1-4612-3444-9
  • J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math....
  • M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948), 837-840.
  • F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967), 201-225. https://doi.org/10.1007/BF01109805
  • W. L. Bynum, Normal structure coefficients for Banach spaces, Pac. J. Math. 86 (1980), 427-436. https://doi.org/10.2140/pjm.1980.86.427
  • E. Casini and E. Maluta, Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure, Nonlinear Anal. 9 (1985),...
  • G. Emmanuele, Asymptotic behavior of iterates of nonexpansive mappings in Banach spaces with Opial's condition, Proc. Amer. Math. Soc....
  • G. Li and B. Sims, Fixed point theorems for mappings of asymptotically nonexpansive type, Nonlinear Anal. 50 (2002), 1085-1091. https://doi.org/10.1016/S0362-546X(01)00744-1
  • K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511526152
  • J. P. Gossez and E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pac. J. Math. 40 (1972),...
  • T. H. Kim and H. K. Xu, Remarks on asymptotically nonexpansive mappings, Nonlinear Anal. 41 (2000), 405-415. https://doi.org/10.1016/S0362-546X(98)00284-3
  • W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345
  • W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339-346. https://doi.org/10.1007/BF02757136
  • W. A. Kirk, Remarks on nonexpansive mappings and related asymptotic conditions, J. Nonlinear Convex Anal. 18 (2017), 1-15.
  • W. A. Kirk and H. K. Xu, Asymptotic pointwise contraction, Nonlinear Anal. 68 (2008), 4706-4712. https://doi.org/10.1016/j.na.2007.11.023
  • T. C. Lim and H. K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994), 1345-1355. https://doi.org/10.1016/0362-546X(94)90116-3
  • P. K. Lin, K. K. Tan and H. K. Xu, Demiclosed principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal....
  • E. Maluta, Uniformly normal structure and related coefficients, Pac. J. Math. 111 (1984), 357-369. https://doi.org/10.2140/pjm.1984.111.357
  • Z. Opial, Weak convergence of the sequences of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597....
  • S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992), 697-704. https://doi.org/10.1016/0362-546X(92)90165-B
  • H. K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), 1139-1146....