Common fixed point theorems for a family multivalued mapping in Banach spaces

Let K be a nonempty closed convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J_φ for some gauge φ. Let T_i : K → K be a multivalued nonexpansive mappings with F := ∩_i=0^∞F(T_i) ≠ φ which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For a contraction f on K and {α_n}, {β_n} ϵ (0, 1), we consider the algorithm be called multivalued version of the modified Mann iteration x_n+1 = β_nf(x_n) + α_nx_n + (1 - α_n - β_n)y_n, where y_n ϵ T_nx_n such that ∥y_n+1 - y_n∥≤H(T_n+1x_n+1, T_nx_n).