Nonlinear coupled coincidence and coupled fixed point theorems for not necessary commutative contractive mappings in partially ordered probabilistic metric spaces

In this paper we introduce the notion of compatibility of mappings in partially ordered probabilistic metric spaces and use this notion to establish a coupled coincidence point result. Very recently Hu and Ma [Xin-qi Hu, Xiao-yan Ma, Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces, Nonlinear Anal. 74 (2011) 6451–6458] proved coupled coincidence point theorems for commuting mappings in partially ordered probabilistic metric spaces. In this paper we proved results of Hu and Ma under a different set of conditions. Precisely, we establish our results by assuming that two mappings on a partially ordered probabilistic metric spaces are compatible (not necessary commutative) and satisfy a more general contractive condition than the contractive condition in the main theorem of Hu and Ma. Our results improve and extend a coupled coincidence point theorem due to Hu and Ma, as well as a coupled fixed point theorem due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393]. An example is given to support our result.