On probabilistic φ-contractions on Menger spaces

Recently, Ciric [Lj.B. Ciric, 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 Ciric, 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.]. (C) 2010 Elsevier Ltd. All rights reserved.