Coupled fixed point theorems in partially ordered cone metric space (Q2853231)

One of the main results of the paper is the following. Let \((X,\sqsubseteq,d)\) be a complete ordered cone metric space in the sense of \textit{L.-G. Huang} and \textit{X. Zhang} [Math. Anal. Appl. 332, No. 2, 1468--1476 (2007; Zbl 1118.54022)] and let \(F:X\times X\to X\) be a continuous mapping. Suppose that: (a)~\(F\) has the mixed monotone property, i.e., \(F(x_1,y_1)\sqsubseteq F(x_2,y_2)\) whenever \(x_1\sqsubseteq x_2\) and \(y_2\sqsubseteq y_1\); (b)~there exist \(\alpha,\beta,\gamma\geq0\) with \(2\alpha+3\beta+3\gamma<2\) such that \(d(F(x,y),F(u,v))\leq\frac\alpha2(d(x,u)+d(y,v)) +\frac\beta2(d(x,F(x,y))+d(u,F(u,v))+d(y,v))+\frac\gamma2(d(x,F(u,v))+d(u,F(x,y))+d(y,v))\) for all \(u\sqsubseteq x\), \(y\sqsubseteq v\); (c)~there exist \(x_0,y_0\in X\) such that \(x_0\sqsubseteq F(x_0,y_0)\) and \(F(y_0,x_0)\sqsubseteq y_0\). The authors prove that under these assumptions, \(F\) has a coupled fixed point. Also, a result for a class of quasicontractions is proved. No example is given.