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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)

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