Fixed point theorems in probabilistic metric spaces. (English) Zbl 0589.54059

This paper contains some fixed point theorems for a continuous mapping f, \(f: S\to S\), S a complete probabilistic metric space, satisfying one of the following kinds of conditions: \[ D_{O_ f(f(x))}(\phi (\epsilon))\geq D_{O_ f(x)}(\epsilon), \] for all \(x\in S\) and all \(\epsilon >0,\) \(F_{f(x),f(y)}(\phi (\epsilon))\geq F_{x,y}(\epsilon),\) for all x,y\(\in S\) and all \(\epsilon >0\), where \(\phi\) is a suitable nonnegative function, and \[ O_ f(x)=\{x,f(x),f^ 2(x),...\};\quad D_ M(\epsilon)=\sup_{\delta <\epsilon}\inf_{x,y\in M}(\delta). \] Then under certain hypotheses on \(\phi\) and S four results on existence and uniqueness of fixed point are shown. Moreover, each result guarantees the convergence of \(\{f^ nx\}\) to the fixed point of f.
{Reviewer’s remark: The main results in this paper have been published by the reviewer himself [cf. Math. Jap. 26, 121-129 (1981; Zbl 0475.54032); the reviewer and S. Kang, J. Chengdu Univ. Sci. Technol. 1983, No.1, 103-109 (1983; Zbl 0519.60068)].}
Reviewer: Zhang Shisheng


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