Abstract
We prove some fixed point theorems for asymptotically regular self-map**s, not necessarily orbitally continuous or k-continuous, on complete metric spaces. Our results extend several recent results in the literature. An extension for multi-valued map**s with \(\delta \)-distance is also presented. Some examples are given to illustrate our results.
Similar content being viewed by others
References
Baillon, J.-B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive map**s and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)
Beg, I., Azam, A.: Fixed points of asymptotically regular multivalued map**s. J. Austral. Math. Soc. (Series A) 53, 313–326 (1992)
Bisht, R.K., Pant, R.P.: A remark on discontinuity at fixed point. J. Math. Anal. Appl. 445, 1239–1242 (2017)
Bisht, R.K.: A note on the fixed point theorem of Górnicki. J. Fixed Point Theory Appl. 21, 54 (2019)
Bisht, R.K., Rakocevic, V.: Fixed points of convex and generalized convex contractions. Rend. Circ. Mat. Palermo, II. Ser 69, 21–28 (2020)
Bollenbacher, A., Hicks, T.: L: A fixed point theorem revisited. Proc. Amer. Math. Soc. 102, 898–900 (1988)
Browder, F.E., Petryshyn, W.V.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–576 (1966)
Ćirić, L.B.: On contraction type map**s. Math. Balkánica 1, 52–57 (1971)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Górnicki, J.: Fixed point theorems for Kannan type map**s. J. Fixed Point Theory Appl. 19, 2145–2152 (2017)
Górnicki, J.: Remarks on asymptotic regularity and fixed points. J. Fixed Point Theory Appl. 21, 29 (2019)
Górnicki, J.: On some map**s with a unique fixed point. J. Fixed Point Theory Appl. 22, 8 (2020)
Hicks, T. L., Rhoades, B. E.: A Banach type fixed-point theorem. Math. Japon. 24, 327–330 (1979/80)
Jachymski, J., Jóźwik, I.: Nonlinear contractive conditions: a comparison and related problems. Fixed point theory and its applications, 123-146, Banach Center Publ., 77, Polish Acad. Sci. Inst. Math., Warsaw, (2007)
Jachymski, J.: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 74, 768–774 (2011)
Kamran, T.: Mizoguchi-Takahashi’s type fixed point theorem. Comput. Math. Appl. 57, 507–511 (2009)
Naidu, S.V.R.: Fixed point theorems for a broad class of multimaps. Nonlinear Anal. 52, 961–969 (2003)
Nicolae, A.: On some generalized contraction type map**s. Appl. Math. Lett. 23, 133–136 (2010)
Pant, A., Pant, R.P.: Fixed points and continuity of contractive maps. Filomat 31(11), 3501–3506 (2017)
Pant, A., Pant, R.P., Rakocevic, V.: Meir-Keeler type and Caristi type fixed point theorems. Appl. Anal. Discr. Math. 13, 849–858 (2019)
Pant, R.P.: Discontinuity and fixed points. J. Math. Anal. Appl. 240, 284–289 (1999)
Reich, S.: Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)
Rhoades, B.E., Sessa, S., Khan, M.S., M. Swaleh, M.: On fixed points of asymptotically regular map**s. J. Austral. Math. Soc. (Series A) 43, 328–346 (1987)
Rhoades, B.E., Singh, S.L., Kulshrestha, C.: Coincidence theorems for some multivalued map**s. Int. J. Math. Math. Sci. 7, 429–434 (1984)
Singh, S.L., Mishra, S.N., Pant, R.: New fixed point theorems for asymptotically regular multi-valued maps. Nonlinear Anal. 71, 3299–3304 (2009)
Acknowledgements
The author would like to thank the handling editor and the three anonymous referees for valuable comments which helped to improve the manuscript. The author gratefully thank to Professor Simeon Reich for sending him the paper [22].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Nguyen, L.V. On fixed points of asymptotically regular map**s. Rend. Circ. Mat. Palermo, II. Ser 70, 709–719 (2021). https://doi.org/10.1007/s12215-020-00527-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-020-00527-0