Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces (Q549926)

Let \((X,d,\leq)\) be a partially ordered complete metric space, and \(F:X^3\to X\) a continuous mixed monotone map. Assume that \textbf{i)} there exist \(j,k,l\in [0,1)\) with \(j+k+l< 1\) for which \(d(F(x,y,z),F(u,v,w))\leq jd(x,u)+kd(y,v)+ld(z,w)\), for all \((x,y,z),(u,v,w)\in X^3\) with \(x\geq u\), \(y\leq v\), \(z\geq w\), \textbf{ii)} there exists \((x_0,y_0,z_0)\in X^3\) such that \(x_0\leq F(x_0,y_0,z_0)\), \(y_0\geq F(y_0,x_0,y_0)\), \(z_0\leq F(z_0,y_0,x_0)\). Then, there exists \((x,y,z)\in X^3\) with the triple fixed point property: \(x=F(x,y,z)\), \(y=F(y,x,y)\), \(z=F(z,y,x)\). Sufficient conditions guaranteeing the uniqueness of this tripled fixed point or its diagonal properties are also given.