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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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