Monotone generalized contractions in partially ordered probabilistic metric spaces

In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205–2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments.