Convergence of implicit and explicit schemes for common fixed points for finite families of asymptotically nonexpansive mappings

Let H be a real Hilbert space and T1,T2,…,TN be a family of asymptotically nonexpansive self-mappings of H with sequences {1+kp(n)i(n)}, such that kp(n)i(n)→0 as n→∞ where p(n)=j+1 if jN0, and 0<γ<γ¯α. Let {αn},{βn} be sequences in (0,1) satisfying some conditions. Strong convergence of the implicit and explicit schemes are proved for a common fixed point of the family T1,T2,…,Tn, which solves the variational inequality 〈(A−γf)x¯,x¯−x〉≤0∀x∈F. Our result generalizes and improves several recent results.