Viscosity Approximation to Common Fixed Points of Families of Nonexpansive Mappings with Weakly Contractive Mappings
© A. Razani and S. Homaeipour. 2010
Received: 5 June 2010
Accepted: 26 July 2010
Published: 10 August 2010
Let X be a reflexive Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation sequence , where (0, 1), is a uniformly asymptotically regular sequence, and f is a weakly contractive mapping. Strong convergence of the sequence is proved.
They proved the strong convergence of the iterative sequence , where is a contraction mapping and is a uniformly asymptotically regular sequence of nonexpansive mappings in a reflexive Banach space , as follows.
Theorem 1.3 (see [3, Theorem ]).
The following results are well known which can be founded in .
We say that a Banach space has a weakly sequentially continuous duality mapping if there exists a gauge function such that the duality mapping is single-valued and continuous from the weak topology to the wea topology of .
It is known  that any separable Banach space can be equivalently renormed such that it satisfies Opial's condition. A space with a weakly sequentially continuous duality mapping is easily seen to satisfy Opial's condition .
Lemma 2.2 (see [9, Lemma ]).
Let be a Banach space satisfying Opial's condition and a nonempty, closed, and convex subset of . Suppose that is a nonexpansive mapping. Then is demiclosed at zero, that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .
Definition 2.3 (see ).
3. Main Result
In this section, we prove a new version of Theorem 1.3.
We show that the sequence is sequentially compact. Since is reflexive and is bounded, there exists a subsequence of such that is weakly convergent to as . Since for all , by Lemma 2.2, we have for all . Thus .
It is known that [10, Example ] in a uniformly convex Banach space , the Cesàro means for nonexpansive mapping is uniformly asymptotically regular. So we have the following corollary, which is a new version of [10, Theorem ].
A. Razani would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran for supporting this paper (Grant no.89470126).
- Alber YaI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications. Volume 98. Birkhäuser, Basel, Switzerland; 1997:7–22.View ArticleGoogle Scholar
- Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Analysis: Theory, Methods & Applications 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleMATHGoogle Scholar
- Song Y, Chen R: Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,66(3):591–603. 10.1016/j.na.2005.12.004MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
- Xu ZB, Roach GF: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. Journal of Mathematical Analysis and Applications 1991,157(1):189–210. 10.1016/0022-247X(91)90144-OMathSciNetView ArticleMATHGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
- van Dulst D: Equivalent norms and the fixed point property for nonexpansive mappings. Journal of the London Mathematical Society 1982,25(1):139–144. 10.1112/jlms/s2-25.1.139MathSciNetView ArticleMATHGoogle Scholar
- Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Mathematische Zeitschrift 1967, 100: 201–225. 10.1007/BF01109805MathSciNetView ArticleMATHGoogle Scholar
- Górnicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Commentationes Mathematicae Universitatis Carolinae 1989,30(2):249–252.MathSciNetMATHGoogle Scholar
- Song Y, Chen R: Viscosity approximate methods to Cesàro means for non-expansive mappings. Applied Mathematics and Computation 2007,186(2):1120–1128. 10.1016/j.amc.2006.08.054MathSciNetView ArticleMATHGoogle Scholar
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