Optimal coincidence best approximation solution in non-Archimedean Fuzzy Metric Spaces

Document Type : Review Paper


1 Department of Mathematics, National University of Computer and Emerging Sciences, Lahore - Pakistan

2 Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa and Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia


In this paper, we introduce the concept of best proximal contraction theorems in non-Archimedean fuzzy metric space for two mappings and prove some proximal theorems. As a consequence, it provides the existence of an optimal approximate solution to some equations which contains no solution. The obtained results extend further the recently development proximal contractions in non-Archimedean fuzzy metric spaces and famous Banach contraction principle.


[1] S. Chauhan, W. Shatanawi, S. Kumar and S. Radenovic, Existence and uniqueness of xed
points in modi ed intuitionistic fuzzy metric spaces, Journal of Nonlinear Sciences and Ap-
plications, 7(1) (2014), 28{41.
[2] K. Fan, Extensions of two xed point theorems of F. E. Browder, Mathematische Zeitschrift,
112(3) (1969), 234{240.
[3] J. G. Garcia and S. Romaguera, Examples of non-strong fuzzy metrics, Fuzzy sets and sys-
tems, 162(1) (2011), 91{93.
[4] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,
64(3) (1994), 395{399.
[5] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets
and Systems, 90(3) (1997), 365{368.
[6] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27(3) (1983),
[7] V. Gregori and A. Sapena, On xed-point theorems in fuzzy metric spaces, Fuzzy Sets and
Systems, 125(2) (2002), 245{252.
[8] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11(5)
(1975), 336{344.
[9] C. Mongkolkeha, Y. J. Cho and P. Kumam, Best proximity points for generalized proxi-
mal contraction mappings in metric spaces with partial orders, Journal of Inequalities and
Applications, 94(1) (2013), 94{105.
[10] S. Sadiq Basha, Best proximity points, optimal solutions, Journal of Optimal Theory and
Applications, 151(1) (2011), 210{216.
[11] S. Sadiq Basha, Common best proximity points: Global minimization of multi-objective func-
tions, Journal of Global Optimization, 54(2) (2012), 367{373.
[12] N. Saleem, B. Ali, M. Abbas and Z. Raza, Fixed points of Suzuki type generalized multivalued
mappings in fuzzy metric spaces with applications, Fixed Point Theory and Applications,
(36)(1) (2015).
[13] M. Sangurlu and D. Turkoglu, Fixed point theorems for ( o') contractions in a fuzzy metric
spaces, Journal of Nonlinear Sciences and Applications. 8(5) (2015), 687{694.
[14] B. Schweizer and A. Sklar, Statistical metric spaces, Paci c Journal of Mathematics, 10(1)
(1960), 313{334.
[15] C. Vetro and P. Salimi, Best proximity point results in non-Archimedean fuzzy metric spaces,
Fuzzy Information and Engineering, 5(4) (2013), 417{429.
[16] L. A. Zadeh, Fuzzy Sets, Informations and Control, 8(3) (1965), 338{353.