Best proximity point theorems for proximal b-cyclic contractions on b-metric spaces

Year 2021, Volume: 70 Issue: 1, 483 - 496, 30.06.2021

Mustafa Aslantaş

https://doi.org/10.31801/cfsuasmas.780729

Cited By: 17

Abstract

In this paper, we first introduce a new notion of the property (M_{C}) to improve and generalize the property (G_{C}). After that, we present two new concepts, proximal b-cyclic contraction of first type and second type, on b-metric spaces. Then, we obtain two best proximity point results for such mappings in the frameworks of best proximally complete b-metric spaces by using the property (M_{C}). Hence, we generalize some results existing in the literature. Finally, we present some illustrative and interesting examples.

Keywords

Best proximity point, proximal b-cyclic contraction, b-metric space

References

• Altun, I, Aslantas, M., Sahin, H., Best proximity point results for p-proximal contractions, Acta Math.Hunga, 162 (2020), 393-402. https://doi.org/10.1007/s10474-020-01036-3
• Aslantas, M., Sahin, H., Altun, I., Best proximity point theorems for cyclic p-contractions with some consequences and applications, Nonlinear Analysis: Modelling and Control, 26(1) (2021), 113-129. https://doi.org/10.15388/namc.2021.26.21415
• Aslantas, M., Sahin, H., Turkoglu, D., Some Caristi type fixed point theorems, The Journal of Analysis, 29 (2021), 89-103. https://doi.org/10.1007/s41478-020-00248-8
• Basha, S. S., Shahzad, N. ,Vetro, C., Best proximity point theorems for proximal cyclic contractions, Journal of Fixed Point Theory and Applications, 19(4) (2017), 2647-2661. https://doi.org/10.1007/s11784-017-0447-8
• Banach, S., Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fundamenta Mathematicae, 3(1) (1922), 133-181.
• Czerwik, S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena, 46(2) (1998), 263-276.
• Cevik, C., Altun, I., Sahin, H., Ozeken, C. C., Some fixed point theorems for contractive mapping in ordered vector metric spaces, J. Non. Sci. Ap., 10 (2017), 1424-1432. http://dx.doi.org/10.22436/jnsa.010.04.12
• Eldred, A. A., Veeramani, P., Existence and convergence of best proximity points, Journal of Mathematical Analysis and Applications, 323(2) (2006), 1001-1006. https://doi.org/10.1016/j.jmaa.2005.10.081
• Espínola, R., Kosuru, G. S. R., Veeramani, P., Pythagorean property and best-proximity point theorems, Journal of Optimization Theory and Applications, 164(2), (2015), 534-550. https://doi.org/10.1007/s10957-014-0583-x
• Felhi, A., Aydi, H., Best proximity points and stability results for controlled proximal contractive set valued mappings, Fixed Point Theory and Applications, 2016(1), (2016), 22. https://doi.org/10.1186/s13663-016-0510-y
• Kadelburg, Z., Radenovic, S., Fixed point and tripled fixed point theorems under Pata-Type conditions in ordered metric spaces, International Journal of Analysis and Applications, 6(1) (2014), 113-122.
• Karpagam, S., Agrawal, S., Best proximity points for cyclic contractions, (preprint)
• Kirk, W. A., Srinivasan, P.S., Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003) 79-89.
• Reich, S., Fixed points of contractive functions, Boll. Unione Mat. Ital.,5 (1972), 26-42.
• Ozeken, C. C., Cevik, C., Unbounded vectorial Cauchy completion of vector metric spaces, Gazi Uni. J. of Sci., 33-3 (2020), 761-765. https://doi.org/10.35378/gujs.604784
• Sahin, H., Best proximity point theory on vector metric spaces, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1) (2021), 130-142. https://doi.org/10.31801/cfsuasmas.780723
• Sahin, H., Aslantas, M., Altun, I., Feng-Liu type approach to best proximity point results for multivalued mappings, Journal of Fixed Point Theory and Applications, 22(1) (2020),11. https://doi.org/10.1007/s11784-019-0740-9