Abstract
Inspired by the well-known result stating that if any iterate of a mapping is a Banach contraction on a complete metric space, then the mapping itself possesses a unique fixed point, we investigate that claim for a Kannan contraction but by retaining the left-hand side of the inequality as per the mapping itself. With an additional assumption of $k$-continuity, the existence and uniqueness of a fixed point is obtained for a new class of contractions, $m$-Kannan contraction, on a complete metric space. Several examples are given in order to substantiate many theoretical claims such as discontinuity at the unique limit point of the iterative sequence or the ones testifying that this class is wider than the class of Kannan mappings.
Keywords
Kannan contraction Kannan contraction Banach contraction iterates k-continuity m-Kannan contraction
Supporting Institution
Ministry of Science, Technological Development and Innovation, Republic of Serbia
Project Number
451-03-65/2024-03/200124
References
- [1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3, 133-181, 1922.
- [2] R. Batra, R. Gupta and P. Sahni, A new extension of Kannan contractions and related fixed point results, J. Anal. 28, 1143-1154, 2020.
- [3] V. Berinde and M. Pacurar, Kannan’s fixed point appoxim.tion for solving split feasibility and variational inequality problems, J. Comput. Appl. Math. 386, Article ID: 113217, 2021.
- [4] V. Berinde, A. Petrusel and I. A. Rus, Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theory, 24 (2), 525-540, 2023.
- [5] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10, 974-979, 1959.
- [6] J. Gornicki, Fixed point theorems for Kannan type mappings, J. Fixed Point Theory Appl. 19, 2145-2152, 2017.
- [7] R. Kannan, Some remarks on fixed points, Bull. Calcutta Math. Soc. 60, 71-76, 1968.
- [8] A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Volume I, Metric and Normed Spaces, Graylock Press, Rochester, New York, 1957.
- [9] H. Lakzian, V. Rakočević and H. Aydi, Extensions of Kannan contraction via wdistances, Aequat. Math. 93, 1231-1244, 2019.
- [10] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math. 80, 325-330, 1975.
- [11] S. Som, A. Petruel, H. Garai and L. K. Dey, Some characterizations of Reich and Chatterjea type nonexpansive mappings, J. Fixed Point Theory Appl. 21 (4), 2019.
Abstract
Project Number
451-03-65/2024-03/200124
References
- [1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3, 133-181, 1922.
- [2] R. Batra, R. Gupta and P. Sahni, A new extension of Kannan contractions and related fixed point results, J. Anal. 28, 1143-1154, 2020.
- [3] V. Berinde and M. Pacurar, Kannan’s fixed point appoxim.tion for solving split feasibility and variational inequality problems, J. Comput. Appl. Math. 386, Article ID: 113217, 2021.
- [4] V. Berinde, A. Petrusel and I. A. Rus, Remarks on the terminology of the mappings in fixed point iterative methods in metric spaces, Fixed Point Theory, 24 (2), 525-540, 2023.
- [5] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10, 974-979, 1959.
- [6] J. Gornicki, Fixed point theorems for Kannan type mappings, J. Fixed Point Theory Appl. 19, 2145-2152, 2017.
- [7] R. Kannan, Some remarks on fixed points, Bull. Calcutta Math. Soc. 60, 71-76, 1968.
- [8] A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Volume I, Metric and Normed Spaces, Graylock Press, Rochester, New York, 1957.
- [9] H. Lakzian, V. Rakočević and H. Aydi, Extensions of Kannan contraction via wdistances, Aequat. Math. 93, 1231-1244, 2019.
- [10] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math. 80, 325-330, 1975.
- [11] S. Som, A. Petruel, H. Garai and L. K. Dey, Some characterizations of Reich and Chatterjea type nonexpansive mappings, J. Fixed Point Theory Appl. 21 (4), 2019.
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Mathematics |
| Authors | |
| Project Number | 451-03-65/2024-03/200124 |
| Early Pub Date | January 27, 2025 |
| Publication Date | June 24, 2025 |
| Submission Date | August 5, 2024 |
| Acceptance Date | October 3, 2024 |
| Published in Issue | Year 2025 Volume: 54 Issue: 3 |