On probabilistic φφ-contractions on Menger spaces

Recently, Ćirić [Lj.B. Ćirić, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72 (2010) 2009–2018] obtained a fixed point theorem with the intention to get a probabilistic version of the Boyd–Wong theorem [D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464]. We give a counterexample to the key lemma of Ćirić, and we establish a corrected version of his main theorem. We also discuss the relations between our result and fixed point theorems of Browder [F.E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968) 27–35] and Matkowski [J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.) 127 (1975), 68 pp.].