On coupled common fixed points for mixed weakly monotone maps in partially ordered $S$-metric spaces

In this paper, we use the notion of a mixed weakly monotone pair of maps of Gordji et al. [Gordji ME, Ramezani M, Cho YJ, and Akbartabar E, \emph{Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces}, Fixed Point Theory Appl. \textbf{2012:95} (2012)] to state a coupled common fixed point theorem for maps on partially ordered $S$-metric spaces. This result generalizes the main results of Gordji et al. [Gordji ME, Ramezani M, Cho YJ, and Akbartabar E, \emph{Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces}, Fixed Point Theory Appl. \textbf{2012:95} (2012)], Bhaskar, Lakshmikantham [Bhaskar TG and Lakshmikantham V, \emph{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. \textbf{65} (2006), no.~7, 1379 -- 1393], Kadelburg et al. [Kadelburg Z, Pavlovi{\'c} M, and Radenovi{\'c} S, \emph{Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces}, Comput. Math. Appl. \textbf{59} (2010), 3148 -- 3159] into the structure of $S$-metric spaces.