On Fixed-Point Equations Involving Geraghty-Type Contractions with Solution to Integral Equation
Abstract
:1. Introduction
2. Preliminaries
- (i)
- , for all with ;
- (ii)
- , if and only if ;
- (iii)
- , for all .
- (i)
- Convergent to the point , if for any given , there exists such that or , for all , and it is denoted by .
- (ii)
- A Cauchy sequence if, for some given , we can find some such that , for each .
3. Main Results
4. Application
- (1)
- there exists for all
- (2)
- there exists withwhere
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Banach, S. Sur les opérations dans les ensembles abstraits et leur applications aux èquations intégrales. Fund. Math. 1922, 3, 133–181. [Google Scholar] [CrossRef]
- Bakhtin, I.A. The contraction mapping principle in quasimetric spaces. Funct. Anal. 1989, 30, 26–37. [Google Scholar]
- Sedghi, S.; Shobe, N.; Aliouche, A. A generalization of fixed point theorems in S-metric spaces. Mat. Vesn. 2012, 64, 258–266. [Google Scholar]
- Rohen, Y.; Došenović, T.; Radenović, S. A Note on the Paper “A Fixed Point Theorems in Sb-Metric Spaces”. Filomat 2017, 31, 3335–3346. [Google Scholar] [CrossRef]
- Souayah, N.; Mlaiki, N. A fixed point theorem in Sb-metric spaces. J. Math. Comput. Sci. 2016, 16, 131–139. [Google Scholar] [CrossRef]
- Sedghi, S.; Gholidahneh, A.; Došenović, T.; Esfahani, J.; Radenović, S. Common fixed point of four maps in Sb-metric spaces. J. Linear Topol. Algebra 2016, 5, 93–104. [Google Scholar]
- Geraghty, M. On contractive mappings. Proc. Am. Math. Soc. 1973, 40, 604–608. [Google Scholar] [CrossRef]
- Aydi, H.; Felhi, A.; Afshari, H. New Geraghty type contractions on metric-like spaces. J. Nonlinear Sci. Appl. 2017, 10, 780–788. [Google Scholar] [CrossRef]
- Karapinar, E.; Alsulami, H.H.; Noorwali, M. Some extensions for Geraghty type contractive mappings. J. Inequal. Appl. 2015, 2015, 303. [Google Scholar] [CrossRef]
- Wang, Y.; Chen, C. Two new Geraghty type contractions in Gb-metric spaces. J. Funct. Spaces 2019, 2019, 7916486. [Google Scholar] [CrossRef]
- Alam, K.H.; Rohen, Y.; Saleem, N. Fixed points of (α, β, F*) and (α, β, F**)-weak Geraghty contractions with an application. Symmetry 2023, 15, 243. [Google Scholar] [CrossRef]
- Chidume, C.E.; Adamu, A.; Nnakwe, M.O. An Inertial Algorithm for Solving Hammerstein Equations. Symmetry 2021, 13, 376. [Google Scholar] [CrossRef]
- Chidume, C.E.; Adamu, A.; Minjibir, M.S.; Nnyaba, U.V. On the strong convergence of the proximal point algorithm with an application to Hammerstein euations. J. Fixed Point Theory Appl. 2020, 22, 61. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Singh, M.P.; Rohen, Y.; Saleem, N.; Alam, K.H.; Singh, K.A.; Razzaque, A. On Fixed-Point Equations Involving Geraghty-Type Contractions with Solution to Integral Equation. Mathematics 2023, 11, 4882. https://doi.org/10.3390/math11244882
Singh MP, Rohen Y, Saleem N, Alam KH, Singh KA, Razzaque A. On Fixed-Point Equations Involving Geraghty-Type Contractions with Solution to Integral Equation. Mathematics. 2023; 11(24):4882. https://doi.org/10.3390/math11244882
Chicago/Turabian StyleSingh, Moirangthem Pradeep, Yumnam Rohen, Naeem Saleem, Khairul Habib Alam, Kumam Anthony Singh, and Asima Razzaque. 2023. "On Fixed-Point Equations Involving Geraghty-Type Contractions with Solution to Integral Equation" Mathematics 11, no. 24: 4882. https://doi.org/10.3390/math11244882
APA StyleSingh, M. P., Rohen, Y., Saleem, N., Alam, K. H., Singh, K. A., & Razzaque, A. (2023). On Fixed-Point Equations Involving Geraghty-Type Contractions with Solution to Integral Equation. Mathematics, 11(24), 4882. https://doi.org/10.3390/math11244882