Multivalued and singlevalued fixed point results in partially ordered metric spaces

Fixed point theory in partially ordered metric spaces has greatly developed in recent times. In this paper we prove certain fixed point theorems for multivalued and singlevalued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities in the case where the arguments of the functions are related by partial order. In one of our theorems we assume a weak contractive inequality. It is in the line with the research following the establishing of weak contraction principle in metric spaces [Rhoades BE. Some theorems on weakly contractive maps. Nonlinear Anal 2001;47(4):2683–93] and subsequently in partially ordered metric spaces [Harjani J, Sadarangani K. Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal 2009;71:3403–10]. Two illustrative examples are also given.