Coupled fixed point theorems with respect to binary relations in metric spaces

In this paper we present a new extension of coupled fixed point theorems in metric spaces endowed with a reflexive binary relation that is not necessarily neither transitive nor antisymmetric. The key feature in this coupled fixed point theorems is that the contractivity condition on the nonlinear map is only assumed to hold on elements that are comparable in the binary relation. Next on the basis of the coupled fixed point theorems, we prove the existence and uniqueness of positive definite solutions of a nonlinear matrix equation.