The variational principle and fixed point theorems in certain topological spaces (Q1922934)

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 \textit{I. Ekeland} [ibid. 47, 324-353 (1974; Zbl 0286.49015)], \textit{J. Caristi} [Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)] and the reviewer with \textit{Chen Yuqing} and \textit{Guo Jinli} [Acta Math. Appl. Sin., Engl. Ser. 7, No. 3, 217-228 (1991; Zbl 0743.54017)].