Summary: 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 the weak contraction principle in metric spaces [B. E. Rhoades
, Nonlinear Anal., Theory Methods Appl. 47, No. 4, 2683–2693 (2001; Zbl 1042.47521
)] and, subsequently, in partially ordered metric spaces [J. Harjani
and K. Sadarangani
, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 7–8, A, 3403–3410 (2009; Zbl 1221.54058
)]. Two illustrative examples are also given.