Abstract
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.
Highlights
Introduction and preliminariesThroughout this paper, X is assumed to be a real Banach space.An operator A with domain D(A) ⊂ X and range R(A) ⊂ X is said to be accretive if and only if we have the inequality x – y ≤ x – y + r(Ax – Ay) ( . )for all x, y ∈ D(A) and r >
The interest and importance of accretive operators stems mainly from the fact that many physically significant problems can be modeled in terms of an initial value problem of the form dx(t)
A is an accretive operator in a Hilbert space or an appropriate Banach space
Summary
Introduction and preliminariesThroughout this paper, X is assumed to be a real Banach space.An operator A with domain D(A) ⊂ X and range R(A) ⊂ X is said to be accretive if and only if we have the inequality x – y ≤ x – y + r(Ax – Ay) ( . )for all x, y ∈ D(A) and r >. It follows that for the equation x + Ax = f there exists a unique solution q ∈ X. It is well known that the approximative solution of operator equation is closely related with the iterative sequence convergence of a fixed point of the mapping.
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