GERAGHTY PROXIMAL CONTRACTION TYPE MAPPINGS AND RJ-PROPERTY IN bv(s) -METRIC SPACES

Abstract

In this article, we will determine the best proximity point results for the Geraghty proximal contraction type mappings in a more general space called the bv(s)-metric space, and prove the existence of the best proximity point for such mappings which satisfy the RJ-Property. We also derive some consequences as a justification for the validity of the main result. The results presented here extend, generalize, and integrate many previous results in the literature.

Keywords

• best proximity point
• Geraghty proximal contraction
• RJ-Property
• bv(s) metric space

References

1. A. Abkar, M. Gabeleh, Global optimal solutions of non-cyclic mappings in metric spaces, J. Optim. Theory Appl. 153 (2012) 298-305.
2. A. Abkar, M. Gabeleh, Best proximity points of non-self mappings. Top 21 (2013) 287-295.
3. M.A. Al-Thaga, N. Shahzad, Convergence and existence results for best proximity points. Nonlinear Anal. 70(2009) 3665-3671.
4. S. Aleksic, Z. D. Mitrovic, and S. Radenovic, A fixed point theorem of Jungck in bv(s)-metric spaces. Period. Math. Hung. 77(2) (2018) 224-231.
5. E. Altiparmak and I. Karahan, Fixed Point Theorems for Geraghty Type Contraction Mappings in Complete Partial bv(s)-Metric Spaces. Sahand Communications in Mathematical Analysis. 18(2) (2021) 45-62.
6. E. Altiparmak and I. Karahan, Fixed point results in partially ordered partial bv(s)-metric spaces. International Journal of Nonlinear Analysis and Applications. 12(2) (2021) 35-52.
7. S. Banach, Sur les operations dans les ensembles abstraits et leur application auxequations integrales. Fundam. Math. 3(1922) 133-181.
8. S.S. Basha, Best proximity points optimal solutions. J. Optim. Theory Appl. 151(2011) 210-216.

9. S. S. Basha, Discrete optimization in partially ordered sets. J. Glob. Optim. 54(2012) 511-517.
10. P. Charoensawan, Common fixed point theorems for Geraghty's type contraction mapping with two generalized metrics endowed with a directed graph in JS-metric spaces. Carpathian Journal of Mathematics. 34(2018) 305-312.
11. S. Czerwik, Contraction mappings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis. 1(1) (1993) 5-11.
12. F. Dong, P. Ji, Sheng, and X. Wang, Pata-Type Fixed Point Results in bv(s)-Metric Spaces. International Journal of Analysis and Applications. 17(3) (2019) 342-360.
13. T. Doˇsenovi´c, Z. Kadelburg, D. Mitrovi´c, and S. Radenovi´c, New xed point results in bv(s)-metric spaces. Mathematica Slovaca. 70(2) (2020) 441-452.
14. D. Dukic, Z. Kadelburg, and S. Radenovic, Fixed point of Geraghty-type mappings in various generalized metric spaces. Abstr Appl Anal. in (2011).
15. R. George, S. Radenovi´c, K. P. Reshma, and S. Shukla, Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 8(6) (2015) 1005-1013. [ M. Geraghty, On contractive mappings, Proc. Am. Math. Soc. 40(1973) 604-608.
16. M. Gordji, M. Ramezani, Y. Cho, and S. Pirbavafa, A generalization of Geraghty's theorem in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory and Applications. 74(1) 2012.
17. J. Hamzehnejadi, and R. Lashkaripour, Best proximity points for generalized α − ϕ−Geraghty proximal contraction mappings and its application. Fixed point theory and applications. (2016) 2016:72 1-13.
18. M. Jleli, and B. Samet, Best proximity points for α − ψ-proximal contractive type mappings and applications. Bull.Sci.Math. 137(8) (2013) 977-995.
19. Z. Kadelburg, P. Kumam, S. Radenovi¢, and W. Sintunavarat, Common coupled fixed point theorems for Geraghty-type contraction mappings using monotone property. Fixed Point Theory and Applications. 27(2015).
20. E. Karapinar, P. Kumam, and P. Salimi, On α-ψ-Meir-Keeler contractive mappings. Fixed Point Theory and Applications. 94(1) (2013) 1-12.
21. P. Kumam, P. Salimi, and C. Vetro, Best proximity point results for modified α-proximal C-contraction mappings. Fixed Point Theory and Applications. (2014) 2014:99.
22. Z. D. Mitrovi´c, and S. Radenovi´c, The Banach and Reich contractions in bv(s)-metric spaces. J. Fixed Point Theory Appl. 19(2017) 3087-3095.
23. B. Samet C. Vetro, and P. Vetro, Fixed point theorems for α − ψ-contractive type mappings. Nonlinear Anal. 75(4) (2012) 2154-2165.
24. J. Zhang, Y. Su, and Q. Cheng, A note on A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. (2013) 2013:99.