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

Abstract

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 $$ be defined by x_n+1 = σ_nz_n + b_n(Tⁿσ_nz_n − σ_nz_n) + ε_n, ∀n ≥ 1. T satisfies an additional mild condition, then the sequence $$ 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.