LOCALLY AND WEAKLY CONTRACTIVE PRINCIPLE IN BIPOLAR METRIC SPACES

Abstract

In this article, we introduce concepts of , λ -uniformly locally contractive and weakly contractive mappings, which are generalizations of Banach contraction mapping, in bipolar metric spaces. Also, we express the results showing the existence and uniqueness of fixed point for these mappings. bipolar metric space, -chainable, , λ -uniformly locally contractive, weakly contractive, fixed point.

References

1. Azam, A. and Arshad, M., (2009), Fixed points of a sequence of locally contractive multivalued maps, Comput. Math. Appl., 57, pp. 96–100.
2. Boyd, D. W. and Wong, J. S., (1969), On nonlinear contractions, Proc. Amer. Math. Soc. 20, pp. 458–464.
3. C´ır´ıc, L. J., (1971), On contraction type mappings, Math. Balk., 1, pp. 52–57.
4. Dey, D. and Saha, M., (2013), Partial cone metric space some fixed point theorems, M.TWMS J. App. Eng. Math., 3 (1), pp. 1–9.
5. Edelstein, M., (1961), An extension of Banach’s contraction principle, Proc. Amer. Math. Soc., 12, pp. 7–10.
6. Kılın¸c, E. and Alaca, C., (2014), A Fixed point theorem in modular metric spaces, Adv. Fixed Point Theory, 4, pp. 199–206,.
7. Raja, P. and Vaezpour, S. M., (2008), Some extensions of Banach’s contraction principle in complete cone metric spaces, Fixed Point Theory Appl., 2008, 11 pages, Article ID 768294.
8. Rakoch, E., (1962), A note on contractive mappings, Proc. Amer. Math. Soc., 10F, pp. 459–465.

9. Rakoch, E., (1962), A note on α-locally contractive mappings, Bull. Res. Counc. Israel, 10F, pp. 188– 191.
10. Rakoch, E., (1962), On -contractive mappings, Bull. Res. Counc. Israel, 10F, pp. 53–58.
11. Reich, S. and Zaslavski, A. J., (2008), A note on Rakotch contraction, Fixed Point Theory, 9, pp. 267–273.
12. Meir, A. and Keeler, E. (1969), A theorem on contraction mappings, J. Math. Anal. Appl., 28, pp. 326–329.
13. Mutlu, A. and G¨urdal, U., (2015), An infinite dimensional fixed point theorem on function spaces of ordered metric spaces, Kuwait J. Sci., 42 (3), pp. 36-49.
14. Mutlu, A. and G¨urdal, U., (2016), Bipolar metric spaces and some fixed point theorems, J. Nonlinear Sci. Appl., 9(9), pp. 5362–5373.
15. Mutlu, A., ¨Ozkan, K. and G¨urdal, U., (2017), Coupled Fixed Point Theorems on Bipolar Metric Spaces, European Journal of Pure and Applied Mathematics, 10 (4), pp. 655–667.
16. Mutlu, A., ¨Ozkan, K. and G¨urdal, U., (2018), Coupled fixed point theorem in partially ordered modular metric spaces and its an application, J. Comput. Anal. Appl., 25 (2), pp. 1–10.
17. Mutlu, A., ¨Ozkan, K., G¨urdal, U., Fixed point theorems for multivalued mappings on bipolar metric spaces, Fixed Point Theory, in press.
18. Shatanawi, W., Karapinar, E. and Aydi, H., (2012), Coupled coincidence points in partially ordered cone metric spaces with a c-distance, Journal of Applied Mathematics, 2012, Article ID 312078.
19. Shatanawi, W. and Pitea, A., (2013), Some coupled fixed point theorems in quasi-partial metric spaces, Fixed Point Theory Appl., 2013 (153), pp. 1-–15.
20. Tahat, N., Aydi, H., Karapinar, E. and Shatanawi, W., (2012), Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces, Fixed Point Theory Appl., 2012 (48), doi:10.1186/1687-1812-2012-48.