Fixed Point Theorems for Contravariant Maps in Bipolar b-Metric Spaces with Integration Application

Abstract

As a natural extension of the metric and the bipolar metric, this article introduces the new abstract bipolar $b-$ metric. The bipolar $b-$metric is a novel technique addressed in this article; it is explained by combining the well-known $b-$metric in the theory of metric spaces, as defined by Mutlu and G\"{u}rdal (2016) \cite{mg1}, with the description of the bipolar metric. In this new definition, well-known mathematical terms such as Cauchy and convergent sequences are utilized. In the bipolar $b-$metric, fundamental topological concepts are also defined to investigate the existence of fixed points implicated in such mappings under different contraction conditions. An example is provided to demonstrate the presented results.

Keywords

• bipolar b−metric space.
• complete b−metric space
• Fixed point theory

References

1. M. Aslantas, H. Sahin and D. Turkoglu, Some Caristi type fixed point theorems, The Journal of Analysis, 29 (2021), 89—103. https://doi.org/10.1007/s41478-020-00248-8
2. M. Aslantas, H. Sahin and U. Sadullah, Some generalizations for mixed multivalued mappings, Appl. Gen. Topol., 23(1) (2022), 169–178. https://doi.org/10.4995/agt.2022.15214
3. I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis, 30 (1989), 26–37.
4. S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae., 3 (1922), 133–181.
5. V. Berinde, Generalized contractions in quasi metric spaces, Seminar on Fixed Point Theory, 3 (1993), 3–9. Preprint
6. S. Czerwik, Contraction mappings in b−metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5-11.
7. S. Cetin and U. Gürdal, Characterization of bipolar ultrametric spaces and some fixed point theorems, Hacettepe Journal of Mathematics and Statistics, 52(1) (2023), 185 -– 196. DOI : 10.15672/hujms.1024696
8. U. Gürdal, A. Mutlu and K. Özkan, Fixed point results for α − ψ−contractive mappings in bipolar metric spaces, Journal of Inequalities and Special Functions, 11(1) (2020), 64–75.

9. T. Kamran, M. Samreen and Q. UL Ain, A generalization of b−metric space and some fixed point theorems, Mathematics, 5(2) (2017), 1–7.
10. A. Mutlu A. and U. Gürdal, Bipolar metric spaces and some fixed point theorems, Journal of Nonlinear Sciences and Applications, 9(9) (2016), 5362–5373.
11. A. Mutlu, K. Özkan and U. Gürdal, Coupled fixed point theorems on bipolar metric spaces. European Journal of Pure And Applied Mathematics, 10(4) (2017), 655–667.
12. A. Mutlu, K. Özkan and U. Gürdal, C, Locally and weakly contractive principle in bipolar metric spaces. TurkishWorld Mathematical Society Journal of Applied and Engineering Mathematics, 10(2) (2020), 379– 388.
13. A. Mutlu, K. Özkan and U. Gürdal, C, Fixed point theorems for multivalued mappings on bipolar metric spaces, Fixed Point Theory, 21(1) (2020), 271–280. doi: 10.24193/fpt-ro.2020.1.19 BIPOLAR b−METRIC SPACES 43
14. K. Özkan, U. Gürdal and A. Mutlu, Generalization of Amini-Harandi’s fixed point theorem with an application to nonlinear mapping theory. Fixed Point Theory, 21(2) (2020), 707–714. doi: 10.24193/fptro. 2020.2.50
15. K. Özkan, U. Gürdal and A. Mutlu Caristi’s and Downing-Kirk’s fixed point theorems on bipolar metric spaces. Fixed Point Theory, 21(2) (2021), 785–794. doi: 10.24193/fpt-ro.2021.2.51
16. A. Mutlu, K. Özkan and U. Gürdal, Some Common Fixed Point Theorems in BipolarMetric Spaces. Turkish Journal of Mathematics and Computer Science, 14(2) (2022), 346–354. doi: 10.47000/tjmcs.1099118
17. H. Sahin, M. Aslantas and A. A. Nasir Nasir, Some Extended Results for Multivalued F-Contraction Mappings, Axioms, 12(2) (2023), 116. https://doi.org/10.3390/axioms12020116