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Fixed Point Results for a Class of Nonexpansive Type Mappings in Banach Spaces

Year 2021, Volume: 5 Issue: 3, 368 - 381, 30.09.2021
https://doi.org/10.31197/atnaa.878951

Abstract

Abstract. In this paper, we present some new fixed point results for a well-known class of generalized nonexpansive type mappings and associated Krasnosel'ski type mappings in Banach spaces. Further, we consider Mann type iteration for finding a common fixed point of a nonexpansive type semigroup. We also present a couple of nontrivial examples to illustrate facts and show numerical convergence.

References

  • [1] K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74(2011), no. 13, 4387-4391.
  • [2] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53(1965), 1272-1276.
  • [3] F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72(1966), 571-575.
  • [4] F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100(1967), 201-225.
  • [5] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, volume 1965 of Lecture Notes in Mathematics, Springer-Verlag London, Ltd., London, (2009).
  • [6] J. García-Falset, E. Llorens-Fuster, and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375(2011), 185-195.
  • [7] K. Goebel and M. Japon-Pineda, A new type of nonexpansiveness, In Proceedings of 8-th international conference on ?xed point theory and applications, Chiang Mai, (2007).
  • [8] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35(1972), 171-174.
  • [9] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30(1965), 251-258.
  • [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72(1965), 1004-1006.
  • [11] W. A. Kirk and B. Sims, Handbook of metric fixed point theory, Springer Science & Business Media, (2013).
  • [12] W. A. Kirk and H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal. 69(2008), 4706-4712.
  • [13] E. Llorens-Fuster, Orbitally nonexpansive mappings, Bull. Aust. Math. Soc. 93(2016), 497-503.
  • [14] E. Llorens-Fuster and E. Moreno-Gálvez, The fixed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal. pages Art. ID 435686(2011), 15.
  • [15] M. A. Krasnosel'ski, Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.), 10(1955), 123-127.
  • [16] A. Nicolae, Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits, Fixed Point Theory Appl. pages Art. ID 458265(2010), 19.
  • [17] K. Nakprasit, W. Nilsrakoo, and S. Saejung, Weak and strong convergence theorems of an implicit iteration process for a countable family of nonexpansive mappings, Fixed Point Theory Appl. pages Art. ID 732193(2008), 18.
  • [18] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73(1967), 591-597.
  • [19] R. Pandey, R. Pant, V. Rakocevic, and R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications, Results Math. 74(2019), Paper No. 7, 24.
  • [20] R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 38(2017), no. 2, 248-266.
  • [21] R. Pant, P. Patel, R. Shukla and M. D. l. Sen, Fixed Point Theorems for Nonexpansive Type Mappings in Banach Spaces, Symmetry 13(2021), no. 43, 585.
  • [22] K. L. Singh, Fixed point theorems for quasi-nonexpansive mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 61(1976), no. 5, 354-363.
  • [23] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340(2008), no. 2, 1088-1095.
  • [24] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems, Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, (1986) xxi+897 pp. ISBN: 0-387-90914-1.
  • [25] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, Math. Anal. Appl. 67(1979), no. 2, 274-276.
  • [26] S. Reich and A. J. Zaslavski, Convergence of Krasnoselskii-Mann iterations of nonexpansive operators, Math. Comput. Modelling 321(2000), no. 11-13, 1423-1431.
  • [27] E. Karapinar and K. Tas, Generalized (C)-conditions and related fixed point theorems, Comput. Math. Appl. 61(2011), no. 11, 3370-3380.
  • [28] A. Fulga, Fixed point theorems in rational form via Suzuki approaches, Results in Nonlinear Analysis 1(2018), no. 1, 19-29.
  • [29] A. Fulga, and A. Proca, A new generalization of Wardowski fixed point theorem in complete metric spaces, Advances in the Theory of Nonlinear Analysis and its Application 1(2017), no. 1, 57-63.
Year 2021, Volume: 5 Issue: 3, 368 - 381, 30.09.2021
https://doi.org/10.31197/atnaa.878951

Abstract

References

  • [1] K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal. 74(2011), no. 13, 4387-4391.
  • [2] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53(1965), 1272-1276.
  • [3] F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72(1966), 571-575.
  • [4] F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100(1967), 201-225.
  • [5] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, volume 1965 of Lecture Notes in Mathematics, Springer-Verlag London, Ltd., London, (2009).
  • [6] J. García-Falset, E. Llorens-Fuster, and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375(2011), 185-195.
  • [7] K. Goebel and M. Japon-Pineda, A new type of nonexpansiveness, In Proceedings of 8-th international conference on ?xed point theory and applications, Chiang Mai, (2007).
  • [8] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35(1972), 171-174.
  • [9] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30(1965), 251-258.
  • [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72(1965), 1004-1006.
  • [11] W. A. Kirk and B. Sims, Handbook of metric fixed point theory, Springer Science & Business Media, (2013).
  • [12] W. A. Kirk and H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal. 69(2008), 4706-4712.
  • [13] E. Llorens-Fuster, Orbitally nonexpansive mappings, Bull. Aust. Math. Soc. 93(2016), 497-503.
  • [14] E. Llorens-Fuster and E. Moreno-Gálvez, The fixed point theory for some generalized nonexpansive mappings, Abstr. Appl. Anal. pages Art. ID 435686(2011), 15.
  • [15] M. A. Krasnosel'ski, Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.), 10(1955), 123-127.
  • [16] A. Nicolae, Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits, Fixed Point Theory Appl. pages Art. ID 458265(2010), 19.
  • [17] K. Nakprasit, W. Nilsrakoo, and S. Saejung, Weak and strong convergence theorems of an implicit iteration process for a countable family of nonexpansive mappings, Fixed Point Theory Appl. pages Art. ID 732193(2008), 18.
  • [18] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73(1967), 591-597.
  • [19] R. Pandey, R. Pant, V. Rakocevic, and R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications, Results Math. 74(2019), Paper No. 7, 24.
  • [20] R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 38(2017), no. 2, 248-266.
  • [21] R. Pant, P. Patel, R. Shukla and M. D. l. Sen, Fixed Point Theorems for Nonexpansive Type Mappings in Banach Spaces, Symmetry 13(2021), no. 43, 585.
  • [22] K. L. Singh, Fixed point theorems for quasi-nonexpansive mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 61(1976), no. 5, 354-363.
  • [23] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340(2008), no. 2, 1088-1095.
  • [24] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems, Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, (1986) xxi+897 pp. ISBN: 0-387-90914-1.
  • [25] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, Math. Anal. Appl. 67(1979), no. 2, 274-276.
  • [26] S. Reich and A. J. Zaslavski, Convergence of Krasnoselskii-Mann iterations of nonexpansive operators, Math. Comput. Modelling 321(2000), no. 11-13, 1423-1431.
  • [27] E. Karapinar and K. Tas, Generalized (C)-conditions and related fixed point theorems, Comput. Math. Appl. 61(2011), no. 11, 3370-3380.
  • [28] A. Fulga, Fixed point theorems in rational form via Suzuki approaches, Results in Nonlinear Analysis 1(2018), no. 1, 19-29.
  • [29] A. Fulga, and A. Proca, A new generalization of Wardowski fixed point theorem in complete metric spaces, Advances in the Theory of Nonlinear Analysis and its Application 1(2017), no. 1, 57-63.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Rajendra Pant 0000-0001-9990-2298

Prashant Patel This is me

Rahul Shukla This is me

Publication Date September 30, 2021
Published in Issue Year 2021 Volume: 5 Issue: 3

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