# SOME RESULTS ON COMMON BEST PROXIMITY POINT AND COMMON FIXED POINT THEOREM IN PROBABILISTIC MENGER SPACE

• Shayanpour, Hamid
• Received : 2015.06.19
• Published : 2016.09.01
• 49 7

#### Abstract

In this paper, we define the concepts of commute proximally, dominate proximally, weakly dominate proximally, proximal generalized ${\varphi}$-contraction and common best proximity point in probabilistic Menger space. We prove some common best proximity point and common fixed point theorems for dominate proximally and weakly dominate proximally mappings in probabilistic Menger space under certain conditions. Finally we show that proximal generalized ${\varphi}$-contractions have best proximity point in probabilistic Menger space. Our results generalize many known results in metric space.

#### Keywords

commute proximally;dominate proximally;weakly dominate proximally;common best proximity point;probabilistic Menger space

#### References

1. M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), no. 10, 3665-3671. https://doi.org/10.1016/j.na.2008.07.022
2. A. Amini-Harandi, Best proximity points for proximal generalized contractions in metric spaces, Optim. Lett. 7 (2013), no. 5, 913-921. https://doi.org/10.1007/s11590-012-0470-z
3. A. Amini-Harandi, Best proximity point theorems for cyclic strongly quasi-contraction mappings, J. Global Optim. 56 (2013), no. 4, 1667-1674. https://doi.org/10.1007/s10898-012-9953-9
4. A. Amini-Harandi, Common best proximity points theorems in metric spaces, Optim. Lett. 8 (2014), no. 2, 581-589. https://doi.org/10.1007/s11590-012-0600-7
5. J. Anuradha and P. Veeramani, Proximal pointwise contraction, Topology Appl. 156 (2009), no. 18, 2942-2948. https://doi.org/10.1016/j.topol.2009.01.017
6. G. L. Cain Jr. and R. H. Kasriel, Fixed and periodic points of local contraction mappings on probabilistic metric spaces, Math. Systems Theory 9 (1975/76), no. 4, 289-297. https://doi.org/10.1007/BF01735146
7. S. S. Chang, Y. J. Cho, and S. M. Kang, Nonlinear Operator Theory in Probabilistic Metric Space, Nova Publishers Science Publishers, Inc., New York, 2001.
8. C. Di Bari, T. Suzuki, and C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008), no. 11, 3790-3794. https://doi.org/10.1016/j.na.2007.10.014
9. A. A. Eldred, W. A. Kirk, and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), no. 3, 283-293. https://doi.org/10.4064/sm171-3-5
10. A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), no. 2, 1001-1006. https://doi.org/10.1016/j.jmaa.2005.10.081
11. M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988), no. 3, 385-389. https://doi.org/10.1016/0165-0114(88)90064-4
12. M. Grabiec, Y. J. Cho, and V. Radu, On Nonsymmetric Topological and Probabilistic Structures, Nova Science Publishers, Inc., New York, 2006.
13. O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic, Dordrecht, 2001.
14. T. L. Hicks, Fixed point theory in probabilistic metric spaces, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 13 (1983), 63-72.
15. J. Jachymski, On probabilistic $\varphi$-contractions on Menger spaces, Nonlinear Anal. 73 (2010), no. 7, 2199-2203. https://doi.org/10.1016/j.na.2010.05.046
16. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), no. 4, 771-779. https://doi.org/10.1155/S0161171286000935
17. S. Karpagam and S. Agrawal, Best proximity point theorems for p-cyclic Meir-Keeler contractions, Fixed Point Theory Appl. 2009 (2009), Art. ID 197308, 9 pp.
18. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535
19. D. O'Regan and R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput. 195 (2008), no. 1, 86-93.
20. A. Razani and M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos Solitons Fractals 32 (2007), no. 1, 26-34. https://doi.org/10.1016/j.chaos.2005.10.096
21. S. Sadiq Basha, Best proximity points: global optimal approximate solution, J. Global Optim. 49 (2011), no. 1, 15-21. https://doi.org/10.1007/s10898-009-9521-0
22. S. Sadiq Basha, Best proximity points: optimal solutions, J. Optim. Theory Appl. 151 (2011), no. 1, 210-216. https://doi.org/10.1007/s10957-011-9869-4
23. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313-334. https://doi.org/10.2140/pjm.1960.10.313
24. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983.
25. V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on prob-abilistic metric spaces, Math. Systems Theory 6 (1972), 97-102. https://doi.org/10.1007/BF01706080
26. Y. Su and J. Zhang, Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces, Fixed Point Theory Appl. 2014 (2014), 170, 15pp. https://doi.org/10.1186/1687-1812-2014-170
27. J. S. Ume, Common fixed point theorems for nonlinear contractions in a Menger space, Fixed Point Theory Appl. 2013 (2013), 166, 10 pp. https://doi.org/10.1186/1687-1812-2013-166