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Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. (English) Zbl 1193.54035
Summary: 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 [Rend. Circ. Mat. Palermo (2) 58, No. 2, 319–325 (2009; Zbl 1184.54038)] and I. Altun, B. Damnjanović and D. Djorić [Appl. Math. Lett. 23, No. 3, 310–316 (2010; Zbl 1197.54052)] by weakening the respective contractive condition. Then, the notions of quasicontraction and $g$-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 $g$-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.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 65J15 Equations with nonlinear operators (numerical methods)