Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces
Abstract
:1. Introduction
- (d1)
- (d2)
- iff
- (d3)
- (d4)
- (d5)
- iff
- left (right) -sequentially complete if each left (right) -Cauchy sequence converges in the topology
- d-sequentially complete if any Cauchy sequence in converges in topology ,
- bicomplete if the metric space is complete, and
- Smyth is complete if every left -Cauchy sequence in converges in the topology .
2. Fixed Points of Suzuki-Type d -Contractions of Quasi-Metric Spaces
- 1.
- if there is a such thatfor all then Ψ is called d -contraction and -contraction on respectively (see [16]),
- 2.
- if there is a such thatfor all then Ψ is called Suzuki-type d-contraction on ,
- 3.
- if there is a such thatfor all then Ψ is called Suzuki-type -contraction on ,
- 4.
- if there is a such thatfor all then Ψ is called Suzuki type contraction on ,
- 5.
- if there is a such thatfor all then Ψ is called basic contraction of Suzuki type (see [18]).
- If and with we have
- If and we have
- If and we have
- (a)
- Ψ is a Suzuki-type contraction on .
- (b)
- For any , the sequence is a Cauchy sequence in .
3. Fixed Points and -Symmetric Quasi-Metric Spaces
- That is, for all This implies is a -Cauchy sequence in Since is d-sequential complete, there exists a such that
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Younis, M.; Ahmad, H.; Chen, L.; Han, M. Computation and convergence of fixed points in graphical spaces with an application to elastic beam deformations. J. Geom. Phys. 2023, 192, 104955. [Google Scholar] [CrossRef]
- Suzuki, T. A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. 2008, 136, 1861–1869. [Google Scholar] [CrossRef]
- Kikkawa, M.; Suzuki, T. Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. Theory, Methods Appl. 2008, 69, 2942–2949. [Google Scholar] [CrossRef]
- Ali, B.; Abbas, M. Existence and stability of fixed point set of Suzuki-type contractive multivalued operators in b-metric spaces with applications in delay differential equations. J. Fixed Point Theory Appl. 2017, 19, 2327–2347. [Google Scholar] [CrossRef]
- Eroglu, I.; Güner, E.; Aygün, H.; Valero, O. A fixed point principle in ordered metric spaces and applications to rational type contractions. AIMS Math. 2022, 7, 13573–13594. [Google Scholar] [CrossRef]
- Romaguera, S. Generalized Ćirić’s contraction in quasi-metric spaces. Lett. Nonlinear Anal. Appl. 2023, 1, 30–38. [Google Scholar]
- Ghasab, E.L.; Majani, H.; Karapinar, E.; Rad, G.S. New fixed point results in F-quasi-metric spaces and an application. Adv. Math. Phys. 2020, 2020, 9452350. [Google Scholar] [CrossRef]
- Kannan, R. Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60, 71–76. [Google Scholar]
- Subrahmanyam, P.V. Completeness and fixed-points. Monatsh. Math. 1975, 80, 325–330. [Google Scholar] [CrossRef]
- Connell, E.H. Properties of fixed point spaces. Proc. Amer. Math. Soc. 1959, 10, 974–979. [Google Scholar] [CrossRef]
- Jarosław, G. Fixed point theorems for Kannan type mappings. J. Fixed Point Theory Appl. 2017, 19, 2145–2152. [Google Scholar]
- Vasile, B.; Păcurar, M. Fixed point theorems for Kannan type mappings with applications to split feasibility and variational inequality problems. arXiv 2019, arXiv:1909.02379. [Google Scholar]
- Secelean, S.A.; Mathew, S.; Wardowski. D. New fixed point results in quasi-metric spaces and applications in fractals theory. Adv. Differ. Equations 2019, 1, 1–23. [Google Scholar] [CrossRef]
- Shahzad, N.; Valero, O.; Alghamdi, M.A.; Alghamdi, M.A. A fixed point theorem in partial quasi-metric spaces and an application to software engineering. Appl. Math. Comput. 2015, 268, 1292–1301. [Google Scholar] [CrossRef]
- Schellekens, M. The Smyth completion: A common foundation for denotational semantics and complexity analysis. Electron. Notes Theoretical Comput. Sci. 1995, 1, 535–556. [Google Scholar] [CrossRef]
- Dağ, H.; Romaguera, S.; Tirado, P. The Banach contraction principle in quasi-metric spaces revisited. In Proceedings of the Workshop on Applied Topological Structures, WATS’15, Valencia, Spain, 3–4 September 2015; pp. 25–31. [Google Scholar]
- Alegre, C.; Dağ, H.; Romaguera, S.; Tirado, P. Characterizations of quasi-metric completeness in terms of Kannan-type fixed point theorems. Hacet. J. Math. Stat. 2017, 46, 67–76. [Google Scholar] [CrossRef]
- Romaguera, S. Basic contractions of Suzuki-type on quasi-metric spaces and fixed point results. Mathematics 2022, 10, 3931. [Google Scholar] [CrossRef]
- Badr, A.; Fulga, A.; Karapınar, E. Fixed point results on Δ-symmetric quasi-metric space via simulation function with an application to Ulam stability. Mathematics 2018, 6, 10208. [Google Scholar]
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Ali, B.; Ali, H.; Nazir, T.; Ali, Z. Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces. Mathematics 2023, 11, 4445. https://doi.org/10.3390/math11214445
Ali B, Ali H, Nazir T, Ali Z. Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces. Mathematics. 2023; 11(21):4445. https://doi.org/10.3390/math11214445
Chicago/Turabian StyleAli, Basit, Hammad Ali, Talat Nazir, and Zakaria Ali. 2023. "Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces" Mathematics 11, no. 21: 4445. https://doi.org/10.3390/math11214445
APA StyleAli, B., Ali, H., Nazir, T., & Ali, Z. (2023). Existence of Fixed Points of Suzuki-Type Contractions of Quasi-Metric Spaces. Mathematics, 11(21), 4445. https://doi.org/10.3390/math11214445