Partially ordered cone metric spaces and coupled fixed point results

Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393] studied the coupled coincidence point of a mapping FF from X×XX×X into XX and a mapping gg from XX into XX. E. Karapinar [E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. (2010), doi:10.1016/j.camwa.2010.03.062] proved some results of the coupled coincidence point of a mapping FF from X×XX×X into XX and a mapping gg from XX into XX over normal cones without regularity. In the present paper, we prove that coupled coincidence fixed point theorems over cone metric spaces are not necessarily normal. Our results generalize several well known comparable results in the literature.