Modified Fast Inertial‐Type Krasnosel’skii‐Mann Iterative Scheme Involving Asymptotically Nonexpansive Mapping

It is our aim to propose a new iterative algorithm with an inertial term involving asymptotically nonexpansive mapping in the framework of Hilbert spaces. Let T : H ⟶ H be asymptotically nonexpansive mapping with F ( T ) ≠ ∅ ∅ and let be defined by x n +1 = σ n z n + b n ( T n σ n z n − σ n z n ) + ε n , ∀ n ≥ 1. T satisfies an additional mild condition, then the sequence converges strongly to . Our main contribution lies in establishing a strong convergence theorem for this method, without relying on the assumption that . Our strong convergence theorems extend the corresponding convergence theorems in literature for nonexpansive maps to a more general class of asymptotically nonexpansive maps. Furthermore, our proposed algorithm is implemented by finding the fixed point of common solutions to a variational inequality problem and η −inverse‐strongly monotone mapping in Hilbert space. Numerical illustrations showed some improvements over existing results in literature.