In this paper, we study the iterations of strongly (asymptotic) quasi \phi-nonexpansive mappings in Banach spaces. First, we prove weak convergence of the generated sequence to a common fixed point of an infinite family of strongly asymptotic quasi \phi-nonexpansive mappings. Next we prove strong convergence of the generated sequence by an additional assumption. In the sequel, invoke of Halpern regularization method, we prove strong convergence of the generated sequence to a common fixed point of the family of mappings without any extra conditions. Finally, we give some applications of our main results in convex minimization and equilibrium problems and present numerical examples to illustrate and support them.
