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The variational principle and fixed point theorems in certain topological spaces. (English) Zbl 0859.54042
The author introduces the concept of $F$-type topological space and gives a characterization of this kind of spaces. He points out that the usual metric spaces, Hausdorff topological vector space and Menger space are all special cases of $F$-type space. Using this concept, some fixed point theorems and a variational principle in $F$-type topological space are established. As an application, the author utilizes the results presented in this paper to obtain a variational principle and a fixed point theorem in Menger probabilistic metric spaces, which generalize the corresponding results of I. Ekeland [ibid. 47, 324-353 (1974; Zbl 0286.49015)], J. Caristi [Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)] and the reviewer with Chen Yuqing and Guo Jinli [Acta Math. Appl. Sin., Engl. Ser. 7, No. 3, 217-228 (1991; Zbl 0743.54017)].

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 49J40 Variational methods including variational inequalities