Krasnosel’skiĭ-Mann-Opial type iterative solution of m-accretive operator equation and its stability in arbitrary Banach spaces
Yuguang Xu, Zeqin Liu, Zeqin Liu, Zuhua Liu, Zuhua Liu · Fixed Point Theory and Applications · 2014
Abstract Let X be a Banach space. Suppose that A : X → X is a Lipschitz accretive operator. The objective of this note is to discuss simultaneously the existence and uniqueness of solution of the equation x + A x = f for any given f ∈ X , and its convergence, estimate of convergent rate, and stability of Krasnosel’skiĭ-Mann-Opial type iterative solution { x n } ⊆ X . If an iterative parameter is selected suitably then the iterative procedure converges strongly to a unique solution of the equation and the iterative process is stable in arbitrary Banach space without any convexity or reflexivity. In particular, if A is nonexpansive then an estimate of the convergence rate can be written as ∥ x n + 1 − q ∥ ≤ ( 17 18 ) n + 1 ∥ x 0 − q ∥ where q ∈ X is a solution of x + A X = f . MSC:47H06, 47H10, 47H17.