Generalized probabilistic metric spaces and fixed point theorems. (English) Zbl 0603.54049

Nguyen, Xuan Tan

Math. Nachr. 129, 205-218 (1986).

The author introduces a new class of spaces called generalized probabilistic metric spaces (GPM-spaces) which are more general than a certain class of Menger spaces and the class of topological linear locally convex spaces with a system of neighbourhoods of zero. The author also shows that a collection of generalized pseudometrics can be defined which generates the usual structure for GPM-spaces. Further, the author proves some new fixed point theorems for single-valued mappings on topological and GPM-spaces, and derives several results as corollaries. One of the results generalizing Edelstein’s fixed point theorem [M. Edelstein, Proc. Am. Math. Soc. 12, 7-10 (1961; Zbl 0096.171)] is (Corollary 3.2): Let (X,d) be a metric space, \(T: X\to X\) be a continuous mapping satisfying one of the following conditions: \[ (a)\quad d(Tx,Ty)<\max \{d(x,y),d(y,Ty),d(x,Tx)\} \] for each x,y\(\in X\), \(x\neq y\); \[ (b)\quad d(Tx,Ty)>\min \{d(x,y),\quad d(x,Tx),d(y,Ty)\} \] for each x,y\(\in X\), \(x\neq y\). Then every limit point of the iterates \(\{T^ nx\}_{n=0}^{+\infty}\) with an arbitrary starting point \(x\in X\) is a fixed point of T in X.

Reviewer: S.L.Singh

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems

Keywords:

generalized probabilistic metric spaces; Menger spaces; topological linear locally convex spaces; generalized pseudometrics; Edelstein’s fixed point theorem

Citations:

Zbl 0096.171

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