Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces
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Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces Nawab Hussain1 , DR Sahu2 and Arif Rafiq3* Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday * Correspondence: aarafi[email protected] 3 Department of Mathematics, Lahore Leads University, Lahore, Pakistan Full list of author information is available at the end of the article
Abstract The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and φ -hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Keywords: modified iterative scheme; nonexpansive mappings; φ -hemicontractive mappings; Banach spaces
1 Preliminaries Let K be a nonempty subset of an arbitrary Banach space X, and let X ∗ be its dual space. Let T : X → X be an operator. The symbols D(T) and R(T) stand for the domain and the range of T, respectively. We denote F(T) by the set of fixed points of a single-valued mapping ∗ T : K → K . We denote by J the normalized duality mapping from X to X defined by J(x) = f ∗ ∈ X ∗ : x, f ∗ = x = f ∗ . Let T : D(T) ⊆ X → X be an operator. Definition T is called L-Lipschitzian if there exists L ≥ such that Tx – Ty ≤ Lx – y for all x, y ∈ D(T). If L = , then T is called non-expansive, and if ≤ L < , T is called contraction. Definition [–] (i) T is said to be strongly pseudocontractive if there exists a t > such that for each x, y ∈ D(T), there exists j(x – y) ∈ J(x – y) satisfying Re Tx – Ty, j(x – y) ≤ x – y . t ©2013 Hussain et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hussain et al. Fixed Point Theory and Applications 2013, 2013:247 http://www.fixedpointtheoryandapplications.com/content/2013/1/247
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(ii) T is said to be strictly hemicontractive if F(T) = ∅ and if there exists a t > such that for each x ∈ D(T) and q ∈ F(T), there exists j(x – y) ∈ J(x – y) satisfying Re Tx – q, j(x – q) ≤ x – q . t (iii) T is said to be φ-strongly pseudocontractive if there exists a strictly increasing function φ : [, ∞) → [, ∞) with φ() = such that for each x, y ∈ D(T), there exists j(x – y) ∈ J(x – y) satisfying Re Tx – Ty, j(x – y) ≤ x – y – φ x – y x – y. (iv) T is said to be φ-hemicontractive if F(T) = ∅ and if there exists a strictly increasing function φ : [, ∞) → [, ∞) with φ() = such that for each x ∈ D(T) and q ∈ F(T), there exists j(x – y) ∈ J(x – y) satisfying Re Tx – q, j(x – q) ≤ x – q – φ x – q x – q. Clearly, each strictly hemicontractive operator is φ-hemicontractive. For a nonempty convex subset K of a normed space X, S : K → K an
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