Abstract
We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.
Highlights
Introduction and preliminariesThroughout this paper, we assume that E is a real Banach space with the normalized duality mapping J from E into 2E∗ give byJ x f ∗ ∈ E∗ : x, f ∗ x 2, f x, ∀x ∈ E, 1.1 where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing
Recall that T is nonexpansive if T x − T y ≤ x − y, for all x, y ∈ C
If T is a nonexpansive mapping with a fixed point and the control sequence {αn} is chosen so that 1 − αn
Summary
Introduction and preliminariesThroughout this paper, we assume that E is a real Banach space with the normalized duality mapping J from E into 2E∗ give byJ x f ∗ ∈ E∗ : x, f ∗ x 2, f x , ∀x ∈ E, 1.1 where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing. We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Reich 2 extended Broweder’s result to the setting of Banach spaces and proved that if X is a uniformly smooth Banach space, xt converges strongly to a fixed point of T and the limit defines the unique sunny nonexpansive retraction from C onto F T .
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