We investigated the existence and uniqueness of coupled best proximity points for some cyclic and semi-cyclic maps in a reflexive Banach space. We found sufficient conditions, ensuring the existence of coupled best proximity points in reflexive Banach spaces and some convexity types of conditions, ensuring uniqueness of the coupled best proximity points in strictly convex Banach spaces. We illustrate the results with examples and we present an application of one of the theorems in the modeling of duopoly markets, to have an existence of market equilibrium. We show that, in general, the iterative sequences can have chaotic behavior.

Sufficient conditions have been obtained for the existence and uniqueness of coupled fixed points for Hardy-Rodgers maps with the mixed monotone property in partially ordered metric spaces using a variational technique. The obtained result generalizes and enriches already known results for other types of maps with mixed monotone property. It is shown that in partially ordered metric spaces the maps of Hardy-Rodgers and Reich with mixed monotonic properties do not coincide.