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Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. (English) Zbl 1265.54127
Summary: In this paper a fixed point theorem for a probabilistic \(q\)-contraction \(f\:S\to S,\) where \((S, \mathcal {F},T)\) is a complete Menger space, \(\mathcal {F}\) satisfies a grow condition, and \(T\) is a \(g\)-convergent t-norm (not necessarily \(T \geq T_L\)) is proved. Also a second fixed point theorem is proved for mappings \(f\:S \rightarrow S\), where \((S, \mathcal {F},T)\) is a complete Menger space, \(\mathcal {F}\) satisfies a weaker condition than in [V. Radu, Lectures on probabilistic analysis. Surveys, in: Lecture Notes and Monographs. Series on Probability, Statistics and Applied Mathematics. 2. Timişoara: Universitatea de Vest din Timişoara, Facultatea de Matematicã, 110 p. (1994; Zbl 0927.60003)], and \(T\) belongs to some subclasses of Dombi, Aczél-Alsina, and Sugeno-Weber families of t-norms. An application to random operator equations is obtained.

MSC:
54E70 Probabilistic metric spaces
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:
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