Common fixed point theorems under implicit relations on ordered metric spaces and application to integral equations (Q2844607)

Let \((X,d,\preceq)\) be an ordered metric space and let \(R,G,F:X\to X\) be three mappings satisfying a contractive condition of the form NEWLINE\[NEWLINET(d(Fx,Gy),d(Rx,Ry),d(Rx,Fx),d(Ry,Gy),d(Rx,Gy),d(Ry,Fx))\leq0NEWLINE\]NEWLINE for a function \(T:\mathbb{R}_+^6\to \mathbb{R}\) satisfying suitable conditions and all comparable \(x,y\) belonging to the orbit of \((G,F,R)\) corresponding to a certain point~\(x_0\). Under some additional conditions, involving asymptotical regularity of \((G,F)\), orbital completeness of \(X\) and certain monotonicity and compatibility conditions on the given mappings, the author proves the existence of their common fixed point. As an application, an existence result for solutions of a class of nonlinear integral equations is presented.