In this chapter, we explore monotone asymptotic pointwise contractive and nonexpansive semigroups. We demonstrate the existence of common fixed points for these operator semigroups in Banach spaces with a partial order structure. Additionally, we present an algorithm for constructing a common fixed point and analyse the conditions that enable both weak and strong convergence. We also provide illustrative examples and contextualise our findings within the framework of differential equations and dynamical systems theory.