The variational principle of fixed point theorems in certain fuzzy topological spaces

General topology can be regarded as a special case of fuzzy topology where all membership functions in question take values 0 and 1 only. The usual fuzzy metric spaces, fuzzy Hausdorff topological vector spaces, and Menger probabilistic metric spaces are all the special cases of F-type fuzzy topological spaces. Therefore, one would expect weaker results in the case of fuzzy topology. Recently several metric space fixed point theorems were extended to fuzzy topological spaces. Many authors introduced the concept of fuzzy metric spaces in different ways (see [4, 5, 11]). Grabiec [5] proved the contraction principle in the setting of fuzzy metric spaces introduced by Kramosil and Michalek [7]. The famous Ekeland's variational principle and Caristi's fixed point theorem are forceful tools in nonlinear analysis, control theory, economic theory and global analysis for details (see [1, 2, 3]). In this paper, we establish a variational principle and Caristi's fixed point theorem in F-type fuzzy topological spaces and utilize the results to obtain a fixed point theorem for Menger probabilistic metric spaces. Our results generalize the previous results of [1, 2, 3].