In this paper, we introduce new classes of subsemigroups of Order Preserving Full Contraction (OCTη) and Order Preserving Full Contractive (OC*Tη) mappings respectively in metric spaces. The relationship between the fixed elements of these subsemigroups were thoroughly examined in line with approximate fixed points. We show that, every fixed elements of subsemigroups OCTη and OC*Tη has a comparable deterministic fixed point and every subsemigroup OC*Tη belong to a class of nonexpansive mapping. The existence and uniqueness results of these subsemigroups were also given in a complete metric space (Ξ, ρ) under weakly contractivity conditions which was justified with classical examples.