Some fixed point results for enriched nonexpansive type mappings in Banach spaces

• Shukla, Rahul ^[1] ; Pant, Rajendra ^[1]
  1. [1] University of Johannesburg

     University of Johannesburg

     City of Johannesburg, Sudáfrica

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 23, Nº. 1, 2022, págs. 31-43
• Idioma: inglés
• DOI: 10.4995/agt.2022.16165
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in Banach spaces. More precisely, we employ the Krasnosel'skii iterative method to obtain fixed points of Suzuki-enriched nonexpansive mappings under different conditions. Moreover, we approximate the fixed point of enriched-quasinonexpansive mappings via Ishikawa iterative method. 

• Referencias bibliográficas
  • V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math....
  • https://doi.org/10.37193/CJM.2019.03.04
  • V. Berinde, Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition, Carpathian...
  • https://doi.org/10.37193/CJM.2020.01.03
  • F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044.
  • https://doi.org/10.1073/pnas.54.4.1041
  • 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
  • R. E. Bruck, Asymptotic behavior of nonexpansive mappings, Fixed points and nonexpansive mappings (Cincinnati, Ohio, 1982), vol. 18, Contemp....
  • https://doi.org/10.1090/conm/018/728592
  • C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, vol. 1965 Lecture Notes in Mathematics, Springer-Verlag London,...
  • https://doi.org/10.1007/978-1-84882-190-3
  • J. B. Diaz and F. T. Metcalf, On the set of subsequential limit points of successive approximations, Trans. Amer. Math. Soc. 135 (1969), 459-485.
  • https://doi.org/10.1090/S0002-9947-1969-0234327-0
  • J. García-Falset, E. Llorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl....
  • https://doi.org/10.1016/j.jmaa.2010.08.069
  • M. K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 207 (1997), 96-103.
  • https://doi.org/10.1006/jmaa.1997.5268
  • K. Goebel and W. Kirk, Topics in metric fixed point theory, vol. 28, Cambridge Studies in Advanced Mathematics, Cambridge University Press,...
  • https://doi.org/10.1017/CBO9780511526152
  • 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
  • D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258.
  • https://doi.org/10.1002/mana.19650300312
  • S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
  • https://doi.org/10.1090/S0002-9939-1974-0336469-5
  • S. H. Khan and T. Suzuki, A Reich-type convergence theorem for generalized nonexpansive mappings in uniformly convex Banach spaces, Nonlinear...
  • https://doi.org/10.1016/j.na.2012.09.005
  • 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
  • M. A. Krasnosel'skiĭ, Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.) 10 (1955), 123-127.
  • E. Llorens-Fuster and E. Moreno Gálvez, The fixed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal. 2011, Art. ID...
  • https://doi.org/10.1155/2011/435686
  • M. Maiti and M. K. Ghosh, Approximating fixed points by Ishikawa iterates, Bull. Austral. Math. Soc. 40 (1989), 113-117.
  • https://doi.org/10.1017/S0004972700003555
  • Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.
  • https://doi.org/10.1090/S0002-9904-1967-11761-0
  • R. Pandey, R. Pant, V. Rakočević and R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with...
  • https://doi.org/10.1007/s00025-018-0930-6
  • R. Pant and R. Shukla, Some new fixed point results for nonexpansive type mappings in Banach and Hilbert spaces. Indian J. Math. 62 (2020),...
  • https://doi.org/10.23952/jnfa.2020.36
  • H. F. Senter and W. G. Dotson, Jr., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380.
  • https://doi.org/10.1090/S0002-9939-1974-0346608-8
  • R. Shukla and R. Pant, Some new fixed point results for monotone enriched nonexpansive mappings in ordered Banach spaces, Adv. Theory Nonlinear...
  • https://doi.org/10.31197/atnaa.954446
  • R. Shukla and A. Wiśnicki, Iterative methods for monotone nonexpansive mappings in uniformly convex spaces, Adv. Nonlinear Anal. 10 (2021),...
  • https://doi.org/10.1515/anona-2020-0170
  • T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088-1095.
  • https://doi.org/10.1016/j.jmaa.2007.09.023
  • K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993),...
  • https://doi.org/10.1006/jmaa.1993.1309
  • E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems, Springer-Verlag, New York, 1986.
  • https://doi.org/10.1007/978-1-4612-4838-5