A function \(\varphi :[0,\infty)\rightarrow [0 \infty)\) satisfies the condition (CBW) if \(\varphi(0)=0\), \(\varphi(t)<t\) and \(\lim_{r\rightarrow t+}\inf \varphi (r)<t\) for each \(t>0\). The main theorem of the paper asserts that if \((S,F, \Delta)\) is a complete Menger PM space under a t-norm \(\Delta\) of H-type and \(T:S \rightarrow S\) is a mapping such that, for some \(\varphi\) satisfying (CBW), \(F_{T(p) T(q)}(\varphi(t)) \geq\) \(F_{pq}(t)\) for every \(p, q\in S\) and all \(t>0\), then T has a unique fixed point.
Reviewer’s comment. We quote from author’s abstract: “The problem to prove the probabilistic versions of the very important generalization of the Banach Contraction Principle, obtained in 1969 by D.W.Boyd and J.S.W.Wong [“On nonlinear contractions”, Proc.Am.Math.Soc.20, 458–464 (1969; Zbl 0175.44903)], who proved the fixed point theorem for a self-mapping of a metric space, satisfying the very general nonlinear contractive condition, is unsolved in these days.” Actually, some results in this direction have been obtained in [O.Hadžić and E.Pap, “New classes of probabilistic contractions and applications to random operators”, in:Fixed point theory and applications, Vol.4.Papers from the 7th international conference on nonlinear functional analysis and applications, Gyeongsang National University, Chinju, Korea and at Kyungnam University, Masan, Korea, August 6–10, 2001.Hauppauge, NY:Nova Science Publishers, 97–119 (2003; Zbl 1069.54026)].