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 T 1 , T 2 , … , T N be a family of asymptotically nonexpansive self-mappings of H with sequences { 1 + k p ( n ) i ( n ) } , such that k p ( n ) i ( n ) → 0 as n → ∞ where p ( n ) = j + 1  if  j N < n ≤ ( j + 1 ) N , j = 0 , 1 , 2 , … and n = j N + i ( n ) , i ( n ) ∈ { 1 , 2 , … , N } . Let F ≔ ⋂ i = 1 N F i x ( T i ) ≠ 0̸ and let f : H → H be a contraction mapping with coefficient α ∈ ( 0 , 1 ) , also let A be a strongly positive bounded linear operator with coefficient γ ¯ > 0 , and 0 < γ < γ ¯ α . Let { α n } , { β n } 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 T 1 , T 2 , … , T n , which solves the variational inequality 〈 ( A − γ f ) x ¯ , x ¯ − x 〉 ≤ 0 ∀ x ∈ F . Our result generalizes and improves several recent results.