Nonlinear contractions and semigroups in general complete probabilistic metric spaces. (English) Zbl 1033.54024

El Amrani, M.; Mbarki, A. B.; Mehdaoui, B.

Panam. Math. J. 11, No. 4, 79-87 (2001).

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

Keywords:

nonlinear contractions; self map; complete probabilistic metric space; fixed point