A new iterative technique for solving fixed point problem involving quasi-nonexpansive and firmly nonexpansive mappings

In this paper,  we introduce a modified Halpern algorithm  to approximate a common fixed points of quasi-nonexpansive and firmly nonexpansive mappings in real Hilbert spaces. We start by showing that  $Fix(T_1) \intersection Fix(T_2)= Fix( T_1oT_2)$ without commuting assumption and etablish strong converge theorems for the proposed iterative process. Our convergence theorems extend and improve some known corresponding results in the contemporary  literature  for a wider class of nonexpansive type-mappings in HIlbert spaces. Finally, applications of our theorems to equilibrium problems and inclusion problems are given.