Abstract

We study an iterative process induced by a set-valued quasi-contraction in a complete metric space and show it generates approximate fixed points under the presence of small computational errors.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Complete metric space
  • fixed point
  • iterate
  • set-valued mapping

Mathematics Subject Classification

47H09, 47H10, 54E50

References

  1. S. Banach: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922) 133–181.https://doi.org/10.4064/fm-3-1-133-181Google Scholar
  2. C. Bargetz, E. Medjic: On the rate of convergence of iterated Bregman projections and of the alternating algorithm, J. Math. Anal. Appl. 481 (2020), 23 pp.https://doi.org/10.1016/j.jmaa.2019.123482Google Scholar
  3. A. Betiuk-Pilarska, T. Domínguez Benavides: Fixed points for nonexpansive mappings and generalized nonexpansive mappings on Banach lattices, Pure Appl. Funct. Analysis 1 (2016) 343–359.Google Scholar
  4. F. S. de Blasi, J. Myjak, S. Reich, A. J. Zaslavski: Generic existence and approximation of fixed points for nonexpansive set-valued maps, Set-Valued Var. Analysis 17 (2009) 97–112.https://doi.org/10.1007/s11228-009-0104-5Google Scholar
  5. B. Djafari-Rouhani, K. R. Kazmi, S. Moradi, R. Ali, S. A. Khan: Solving the split equality hierarchical fixed point problem, Fixed Point Theory 23 (2022) 351–369.https://doi.org/10.24193/fpt-ro.2022.1.22Google Scholar
  6. W.-S. Du: Some generalizations of fixed point theorems of Caristi type and Mizoguchi-Takahashi type under relaxed conditions, Bull. Braz. Math. Soc. (N.S.) 50 (2019) 603–624.https://doi.org/10.1007/s00574-018-0117-5Google Scholar
  7. K. Goebel, W. A. Kirk: Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511526152Google Scholar
  8. K. Goebel, S. Reich: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York (1984).Google Scholar
  9. E. Karapinar, R. P. Agarwal, S. S. Yesilkaya: Perov type mappings with a contractive iterate, J. Nonlinear Convex Analysis 22 (2021) 2531–2541.Google Scholar
  10. E. Karapinar, Z. Mitrovic, A. Ozturk, S. Radenovic: On a theorem of Ciric in b-metric spaces, Rend. Circ. Mat. Palermo 70 (2021) 217–225.https://doi.org/10.1007/s12215-020-00491-9Google Scholar
  11. A. A. Khan, J. Li, S. Reich: Generalized projections on general Banach spaces, J. Nonlinear Convex Analysis 24 (2023) 1079–1112.Google Scholar
  12. W. A. Kirk, P. S. Srinivasan, P. Veeramani: Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003) 79–89.Google Scholar
  13. W. M. Kozlowski: An introduction to fixed point theory in modular function spaces, in: Topics in Fixed Point Theory, S. Almezel et al. (eds.), Springer, Cham (2014) 159–222.https://doi.org/10.1007/978-3-319-01586-6_5Google Scholar
  14. S. B. Nadler Jr.: Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–488.https://doi.org/10.2140/pjm.1969.30.475Google Scholar
  15. A. Nicolae, D. O’Regan, A. Petruşel: Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J. 18 (2011) 307–327.https://doi.org/10.1515/gmj.2011.0019Google Scholar
  16. A. Petruşel, G. Petruşel, J.-C. Yao: Multi-valued graph contraction principle with applications, Optimization 69 (2020) 1541–1556.https://doi.org/10.1080/02331934.2019.1633652Google Scholar
  17. E. Rakotch: A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962) 459–465.https://doi.org/10.1090/S0002-9939-1962-0148046-1Google Scholar
  18. S. Reich, A. J. Zaslavski: Genericity in Nonlinear Analysis, Developments in Mathematics 34, Springer, New York (2014).https://doi.org/10.1007/978-1-4614-9533-8Google Scholar
  19. S. Reich, A. J. Zaslavski: Contractive mappings on unbounded sets, Set-Valued Var. Analysis 26 (2018) 27–47.https://doi.org/10.1007/s11228-017-0430-yGoogle Scholar
  20. R. Suparatulatorn, W. Cholamjiak, S. Suantai: A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs, Numerical Algorithms 77 (2018) 479–490.https://doi.org/10.1007/s11075-017-0324-yGoogle Scholar
  21. W. Takahashi, H. K. Xu, J.-C. Yao: Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Analysis 23 (2015) 205–221.https://doi.org/10.1007/s11228-014-0285-4Google Scholar
  22. A. J. Zaslavski: Approximate Solutions of Common Fixed Point Problems, Springer Optimization and Its Applications 112, Springer, Cham (2016).https://doi.org/10.1007/978-3-319-33255-0Google Scholar
  23. A. J. Zaslavski: Algorithms for Solving Common Fixed Point Problems, Springer Optimization and Its Applications 132, Springer, Cham (2018).https://doi.org/10.1007/978-3-319-77437-4Google Scholar

Suggested citation

A. J. Zaslavski. “Approximate Fixed Points for a Set-Valued Quasi-Contraction.” Journal of Convex Analysis 32 (2025), No. 3, 927–936. https://doi.org/10.68381/jca32048

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.