Asymptotic behavior of inexact orbits of nonexpansive mappings
DOI:
https://doi.org/10.12775/TMNA.2020.043Keywords
Complete metric space, fixed point, inexact iteration, nonexpansive mappingAbstract
We study the convergence of inexact iterates of nonexpansive mappings which take a nonempty closed subset of a complete metric space into the space in the case where the errors are sufficiently small.References
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.
A. Betiuk-Pilarska and T. Domı́nguez Benavides, Fixed points for nonexpansive mappings and generalized nonexpansive mappings on Banach lattices, Pure Appl. Func. Anal. 1 (2016), 343–359.
R.E. Bruck, T. Kuczumow and S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloquium Math. 65 (1993), 169–179.
D. Butnariu, S. Reich and A.J. Zaslavski, Convergence to fixed points of inexact orbits of Bregman-monotone and of nonexpansive operators in Banach spaces, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2006, pp. 11–32.
A. Cegielski, S. Reich and R. Zalas, Regular sequences of quasi-nonexpansive operators and their applications, SIAM J. Optim. 28 (2018), 1508–1532.
Y. Censor and M. Zaknoon, Algorithms and convergence results of projection methods for inconsistent feasibility problems: a review, Pure Appl. Func. Anal. 3 (2018), 565–586.
F.S. de Blasi, J. Myjak, S. Reich and A.J. Zaslavski, Generic existence and approximation of fixed points for nonexpansive set-valued maps, Set-Valued Var. Anal. 17 (2009), 97–112.
A. Gibali, A new split inverse problem and an application to least intensity feasible solutions, Pure Appl. Funct. Anal. 2 (2017), 243–258.
K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
K. Goebel and W.A. Kirk, Classical Theory of Nonexpansive Mappings, Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, pp. 49–91.
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
J. Jachymski, Extensions of the Dugundji–Granas and Nadler’s theorems on the continuity of fixed points, Pure Appl. Funct. Anal. 2 (2017), 657–666.
W.A. Kirk, Contraction mappings and extensions, Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, pp. 1–34.
R. Kubota, W. Takahashi and Y. Takeuchi, Extensions of Browder’s demiclosedness principle and Reich’s lemma and their applications, Pure Appl. Func. Anal. 1 (2016), 63–84.
E. Pustylnyk, S. Reich and A.J. Zaslavski, Convergence to compact sets of inexact orbits of nonexpansive mappings in Banach and metric spaces, Fixed Point Theory Appl. 2008 (2008), 1–10.
S. Reich and A.J. Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci., Special Volume (Functional Analysis and Its Applications), Part III (2001), 393–401.
S. Reich and A.J. Zaslavski, Generic aspects of metric fixed point theory, Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, pp. 557–575.
S. Reich and A.J. Zaslavski, Convergence to attractors under perturbations, Commun. Math. Anal. 10 (2011), 57–63.
S. Reich and A.J. Zaslavski, Genericity in Nonlinear Analysis, Developments in Mathematics, vol. 34, Springer, New York, 2014.
S. Reich and A.J. Zaslavski, Convergence to attractors of nonexpansive set-valued mappings, Commun. Math. Anal. 22 (2019), 51–60.
W. Takahashi, The split common fixed point problem and the shrinking projection method for new nonlinear mappings in two Banach spaces, Pure Appl. Funct. Anal. 2 (2017), 685–699.
W. Takahashi, A general iterative method for split common fixed point problems in Hilbert spaces and applications, Pure Appl. Funct. Anal. 3 (2018), 349–369.
A.J. Zaslavski, Approximate Solutions of Common Fixed Point Problems, Springer Optimization and Its Applications, Springer, Cham, 2016.
A.J. Zaslavski, Algorithms for Solving Common Fixed Point Problems, Springer Optimization and Its Applications, Springer, Cham, 2018.
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 Topological Methods in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
