Triple fixed point theorems for mixed monotone Prešić-Kannan and Prešić-Chatterjea mappings in partially ordered metric spaces

The aim of this paper is to extend the Kannan fixed point theorem from single-valued self mappings T : X → X to mappings F : X3 → X satisfying a Prešić-Kannan type contractive condition: d (F (x, y, z) , F (y, z, u)) ≤ k 8 [d (x, F (x, y, z)) + d (y, F (y, x, y))+ +d (z, F (z, y, x)) + d (y, F (y, z, u)) + d (z, F (z, y, z)) + d (u, F (u, z, y))], or a Prešić-Chatterjea type contractive condition: d (F (x, y, z) , F (y, z, u)) ≤ k 8 [d (x, F (y, z, u)) + d (y, F (z, y, z))+ +d (z, F (u, z, y)) + d (y, F (x, y, z)) + d (z, F (y, x, y)) + d (u, F (z, y, x))]. The obtained tripled fixed point theorems extend and unify several related results in literature. Acknowledgements. The paper has been finalized during the visit of the first two authors to Department of Mathematics, Universita di Roma Tre. They gratefully thank Professor Andrea Laforgia for kind hospitality and excellent work facilities offered. The research was supported by the Grant PN-II-RU-TE-2011-3-239 of the Romanian Ministry of Education and Research. The second author’s research has been partly supported by the Grant PN-II-ID-PCE-2011-3-0087 of the Romanian Ministry of Education and Research.