On modified enriched versions of the Browder-Göhde-Kirk fixed point theorem
Abstract
In this paper, we propose a modified enriched version of the classical Browder-Göhde-Kirk fixed point theorem in the setting of uniformly convex Banach spaces. Furthermore, we define modified enriched asymptotically nonexpansive mappings and proved some results extending the Goebel-Kirk fixed point theorem for these types of mappings. These findings contribute to the ongoing development of fixed point theory and its applications in nonlinear analysis.
References
bibitem[Anjum & Abbas, 2024]{riz} Anjum, R., & Abbas, M. 2024. Remarks on $b$-enriched nonexpansive mappings, arXiv preprint arXiv:2405.07999. Available at: https://doi.org/10.48550/arXiv.2405.07999
bibitem[Berinde, 2019]{ber} Berinde, V. 2019. Approximating fixed points of enriched nonexpansive mappings by Krasnoselskii iteration in Hilbert spaces, textit{Carpathian Journal of Mathematics}, 35(3), pp.293-304. Available at: https://www.jstor.org/stable/26905206
bibitem[Berinde & Pacurar, 2020]{bv1} Berinde, V., & Pacurar, M. 2020. Approximating fixed points of enriched contractions in Banach spaces, textit{Journal of Fixed Point Theory and Applications}, 22, 10. Available at: https://doi.org/10.1007/s11784-020-0769-9
bibitem[Berinde, 2020]{18} Berinde, V. 2020. Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition, textit{Carpathian Journal of Mathematics}, 36, pp.27-34. Available at: https://www.jstor.org/stable/26898779
bibitem[Browder, 1965]{Brow} Browder, F.E. 1965. Nonexpansive nonlinear operators in a Banach space, textit{Proceedings of the National Academy of Sciences of the USA}, 54, pp.1041-1044. Available at: https://doi.org/10.1073/pnas.54.4.104
bibitem[Dugundji & Granas, 1982]{DJ} Dugundji, J., & Granas, A. 1982. Fixed Point Theory, textit{Monografie Mathematyczne, Polska Akademia Nauk, War-szawa}, 1. ISBN: 0387001735, 9780387001739
bibitem[Goebel & Kirk, 1972]{Kirk} Goebel, K., & Kirk, W. A. 1972. A fixed point theorem for asymptotically nonexpansive mappings, textit{Proceedings of the American Mathematical Society}, 35(1), pp.171-174. Available at: https://doi.org/10.1090/S0002-9939-1972-0298500-3
bibitem[Goebel & Reich, 1984]{Goeb} Goebel, K., & Reich, S. 1984. Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, textit{Series of Monographs and Textbooks in Pure and Applied Mathematics}, Dekker, New York. ISBN: 0824772237.
bibitem[Göhde, 1965]{Ghode} Göhde, D. 1965. Zum Prinzip der kontraktiven Abbildung, textit{Mathematische Nachrichten}, 28, pp.251-258. Available at: https://doi.org/10.1002/mana.19650300312
bibitem[Kirk, 1965]{Kirk} Kirk, W. A. 1965. A fixed point theorem for mappings which do not increase distances, textit{American Mathematical Monthly}, 72, pp.1004-1006. Available at: https://doi.org/10.2307/2313345
bibitem[Matkowski, 2007]{JM2} Matkowski, J. 2007. Remarks on Lipschitzian mappings and some fixed point theorems, textit{Banach Journal of Mathematical Analysis}, 2, pp.237-244. Available at: 10.15352/bjma/1240336222
bibitem[Matkowski, 2022]{JM} Matkowski, J. 2022. A refinement of the Browder–Göhde–Kirk fixed point theorem and some applications, textit{Journal of Fixed Point Theory and Applications}, 24, 70. Available at: https://doi.org/10.1007/s11784-022-00985-2
bibitem[Rawat et al., 2023]{R} Rawat, S., Bartwal, A. & Dimri, R.C. 2023. Approximation and existence of fixed points via interpolative enriched contractions, textit{Filomat}, 37(16), pp.5455-5467. Available at: https://doiserbia.nb.rs/img/doi/0354-5180/2023/0354-51802316455R.pdf
bibitem[Reich, 1976]{RS} Reich, S. 1976. The fixed point property for nonexpansive mappings, I, textit{American Mathematical Monthly}, 83(4), pp.266-268. Available at: https://www.tandfonline.com/doi/pdf/10.1080/00029890.1976.11994096
bibitem[Reich, 1980]{RS1} Reich, S. 1980. The fixed point property for nonexpansive mappings, II, textit{American Mathematical Monthly}, 87, pp.292-294. Available at: https://www.tandfonline.com/doi/pdf/10.1080/00029890.1980.11995019
Copyright (c) 2025 Divyanshu Chamoli, Shivam Rawat, Monika Bisht, R.C. Dimri
This work is licensed under a Creative Commons Attribution 4.0 International License.
Proposed Creative Commons Copyright Notices
Proposed Policy for Military Technical Courier (Journals That Offer Open Access)
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).