Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Ma
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Research Article Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings L. C. Ceng,1 David S. Shyu,2 and J. C. Yao3 1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China Department of Finance, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan 3 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan 2
Correspondence should be addressed to J. C. Yao, [email protected] Received 26 November 2008; Accepted 28 May 2009 Recommended by Lai-Jiu Lin We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established. Copyright q 2009 L. C. Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and Preliminaries Let E be a real Banach space, and let E∗ be its dual space. Denote by J the normalized duality ∗ mapping from E into 2E defined by Jx ϕ ∈ E∗ : x, ϕ x2 ϕ2 ,
∀x ∈ E,
1.1
where ·, · is the generalized duality pairing between E and E∗ . If E is smooth, then J is single valued and continuous from the norm topology of E to the weak∗ topology of E∗ . A mapping T with domain DT and range RT in E is called λ-strictly pseudocontractive in the terminology of Browder and Petryshyn 1, if there exists a constant λ > 0 such that
T x − T y, j x − y ≤ x − y2 − λx − y − T x − T y2
1.2
2
Fixed Point Theory and Applications
for all x, y ∈ DT and all jx − y ∈ Jx − y. Without loss of generality, we may assume λ ∈ 0, 1. If I denotes the identity operator, then 1.2 can be written in the form I − T x − I − T y, j x − y ≥ λI − T x − I − T y2
1.3
for all x, y ∈ DT and all jx − y ∈ Jx − y. In 1.2 and 1.3, the positive number λ > 0 is said to be a strictly pseudocontractive constant. The class of strictly pseudocontractive mappings has been studied by several authors see, e.g., 1–10. It is shown in 4 that a strictly pseudocontractive map is L-Lipschitzian i.e., T x − T y ≤ Lx − y, ∀x, y ∈ DT for some L > 0. Indeed, it follows immediately from 1.3 that x − y ≥ λI − T x − I − T y ≥ λ T x − T y − x − y ,
1.4
and hence T x − T y ≤ Lx − y, ∀x, y ∈ DT where L 1 1/λ. It is clear that in Hilbert spaces the important class of nonexpansive mappings mappings T for which T x − T y ≤ x − y, ∀x, y ∈ DT is a subclass of the class of strictly pseudocontractive maps. Let K be a nonempty convex subset of E, and let {Ti }N i1 be a finite family of nonexpansive self-maps of K. In 11, Xu and Ori introduced the following implicit iteration ∞ process; for any initial x0 ∈ K and {αn }∞ n1 ⊂ 0, 1, the sequence {xn }
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