Convergence Comparison of Several Iteration Algorithms for the Common Fixed Point Problems
© Y. Song and X. Liu. 2009
Received: 20 January 2009
Accepted: 2 May 2009
Published: 9 June 2009
We discuss the following viscosity approximations with the weak contraction for a non-expansive mapping sequence , , . We prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Mouda's viscosity approximations with the weak contraction.
The following famous theorem is referred to as the Banach Contraction Principle.
Theorem 1.1 (Banach ).
In 2001, Rhoades  proved the following very interesting fixed point theorem which is one of generalizations of Theorem 1.1 because the weakly contractions contains contractions as the special cases .
Theorem 1.2 (Rhoades, Theorem 2).
and proved that converges to a fixed point of in a Hilbert space. They are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. Xu  extended Moudafi's results to a uniformly smooth Banach space. Recently, Song and Chen [12, 13, 16–18] obtained a number of strong convergence results about viscosity approximations (1.8). Very recently, Petrusel and Yao , Wong, et al.  also studied the convergence of viscosity approximations, respectively.
We will prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Moudafi's viscosity approximations with the weak contraction.
2. Preliminaries and Basic Results
Let denote the set of all fixed points for a mapping , that is, , and let denote the set of all positive integers. We write (resp. ) to indicate that the sequence weakly (resp. wea ) converges to ; as usual will symbolize strong convergence.
exists for each ; (ii) a uniformly Gâteaux differentiable norm, if for each in , the limit (2.2) is uniformly attained for ; (iii) a Fréchet differentiable norm, if for each , the limit (2.2) is attained uniformly for ; (iv) a uniformly Fréchet differentiable norm (we also say that is uniformly smooth), if the limit (2.2) is attained uniformly for . A Banach space is said to be (v) strictly convex if (vi) uniformly convex if for all , such that For more details on geometry of Banach spaces, see [21, 22].
If is a nonempty convex subset of a Banach space and is a nonempty subset of , then a mapping is called a retraction if is continuous with . A mapping is called sunny if whenever and . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . We note that if is closed and convex of a Hilbert space , then the metric projection coincides with the sunny nonexpansive retraction from onto . The following lemma is well known which is given in [22, 23].
Lemma 2.1 (see [22, Lemma 5.1.6]).
In order to showing our main outcomes, we also need the following results. For completeness, we give a proof.
Let be a convex subset of a smooth Banach space . Let be a subset of and let be the unique sunny nonexpansive retraction from onto . Suppose is a weak contraction with a function on and is a nonexpansive mapping. Then
Namely, is a weakly contractive mapping with a function . Thus, Theorem 1.2 guarantees that has a unique fixed point in , that is, satisfying (2.4) is uniquely defined for each . (i) and (ii) are proved.
We know that if is a uniformly smooth Banach space or a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, is bounded, is a constant sequence , then has both Browder's and Halpern's property (see [7, 10, 11, 23], resp.).
Lemma 2.3 (see [24, Proposition 4]).
Lemma 2.4 (see [24, Proposition 5]).
3. Main Results
We first discuss Browder's type convergence.
We next discuss Halpern's type convergence.
Let be a Banach space whose norm is uniformly Gâteaux differentiable, and satisfies the condition (C2). Assume that have Browder's property and for every , where is a bounded sequence in defined by (2.10). then have Halpern's property.
4. Deduced Theorems
Using Theorems 3.1, 3.2, and 3.3, we can obtain many convergence theorems. We state some of them.
We now discuss convergence theorems for families of nonexpansive mappings. Let be a nonempty closed convex subset of a Banach space . A (one parameter) nonexpansive semigroups is a family of selfmappings of such that
Recently, Song and Xu  showed that have both Browder's and Halpern's property in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm whenever . As a direct consequence of Theorems 3.1, 3.2, and 3.3, we obtain the following.
Recently, Chen and Song  showed that have both Browder's and Halpern's property in a uniformly convex Banach space with a uniformly Gâeaux differentiable norm whenever . Then we also have the following.
The authors would like to thank the editors and the anonymous referee for his or her valuable suggestions which helped to improve this manuscript.
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