In the setting of uniformly convex Banach spaces equipped with a partially ordered relation, we survey the existence of fixed points for

  monotone orbitally nonexpansive mappings. In this way, we extend and improve the main results of Alfuraidan and Khamsi [M.R. Alfuraidan, M.A. Khamsi, A fixed point theorem for monotone asymptotic nonexpansive mappings, Proc. Amer. Math. Soc., 146 (2018), 2451-2456]. Examples are given to show the usability of our main conclusions. We also study the existence of an optimal solution for cyclic contractions in such spaces.