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Nonlinear contractions and semigroups in general complete probabilistic metric spaces. (English) Zbl 1033.54024
The authors have proved the following probabilistic version of a theorem of D. W. Boyd and J. S. W. Wong [Proc. Am. Math. Soc. 20, 458–464 (1969; Zbl 0175.44903)].
Theorem: If $$T$$ is a self map on a complete probabilistic metric space $$(M,F,\tau)$$ satisfying: (i) there exists $$x\in M$$, its orbits $$O_T(x)$$ is bounded; (ii) there is a lower semi-continuous from the left, nondecreasing function $$\varphi: [0,\infty] \to [0, \infty]$$ (with $$\varphi(0)=0$$ and $$\varphi(t)>t$$ for $$t>0)$$ such that $$T$$ is $$\varphi$$-contractive. Then $$T$$ has a unique fixed point.
Subsequently several consequences of the result above are also obtained.
Reviewer: Ismat Beg (Kuwait)

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems 54E70 Probabilistic metric spaces 47S50 Operator theory in probabilistic metric linear spaces