Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces

In the first part of this paper we generalize results on common fixed points in ordered cone metric spaces obtained by I. Altun and G. Durmaz [I. Altun, G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo, 58 (2009) 319–325] and I. Altun, B. Damnjanović and D. Djorić [I. Altun, B. Damnjanović, D. Djorić, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. (2009) doi:10.1016/j.aml.2009.09.016] by weakening the respective contractive condition. Then, the notions of quasicontraction and gg-quasicontraction are introduced in the setting of ordered cone metric spaces and respective (common) fixed point theorems are proved. In such a way, known results on quasicontractions and gg-quasicontractions in metric spaces and cone metric spaces are extended to the setting of ordered cone metric spaces. Examples show that there are cases when new results can be applied, while old ones cannot.