scholarly journals Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces

2002 ◽  
Vol 272 (2) ◽  
pp. 435-447 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Yeol Je Cho ◽  
Jun Li ◽  
Nan Jing Huang
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Common Fixed Points ◽  
Contractive Mappings ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces ◽  
Iterative Procedures

scholarly journals Strong convergence theorems for fixed points of pseudo-contractive semigroup

2007 ◽  
Vol 76 (3) ◽  
pp. 441-452 ◽  
Author(s):  
Xue-Song Li ◽  
Nan-Jing Huang
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Common Fixed Points ◽  
Nonexpansive Semigroup ◽  
Uniformly Smooth ◽  
Uniformly Convex Banach Spaces ◽  
Special Cases ◽  
Uniformly Smooth Banach Spaces ◽  
Iteration Processes ◽  
Implicit Iteration

We study some convergence of two kinds of implicit iteration processes for approximating common fixed points of a pseudo-contractive semigroup in uniformly convex Banach spaces with uniformly Gateaux differential norms. As special cases, we get some convergence of the implicit iteration processes for approximating common fixed points of a nonexpansive semigroup in uniformly smooth Banach spaces and give a positive answer to an open problem proposed by Xu in Bull. Austral. Math. Soc. (2005). The results presented in this paper generalise some corresponding results from Osilike in Panamer. Math. J. (2004), Suzuki in Proc. Amer. Math. Soc. (2002) and Xu in Bull. Austral. Math. Soc. (2005).

MODIFIED HALPERN ITERATIVE ALGORITHM FOR NONEXPANSIVE MAPPINGS II

2011 ◽  
Vol 04 (04) ◽  
pp. 683-694
Author(s):  
Mengistu Goa Sangago
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Convergence Conditions ◽  
Uniformly Smooth ◽  
Additional Control ◽  
Uniformly Smooth Banach Spaces

Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Yao et al. [16] modification of Halpern iterative algorithm for nonexpansive mappings in uniformly smooth Banach spaces. Unfortunately, some deficiencies are found in the Yao et al. [16] control conditions imposed on the modified iteration to obtain strong convergence. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Yao et al. [16] are not sufficient and it is also proved that with some additional control conditions on the parameters strong convergence of the iteration is obtained.

scholarly journals Construction of sunny nonexpansive retractions in Banach spaces

2002 ◽  
Vol 66 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Tomas Dominguez Benavides ◽  
Genaro López Acedo ◽  
Hong-Kun Xu
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Banach Spaces ◽  
Convex Subset ◽  
Regularity Condition ◽  
Common Fixed Points ◽  
Sunny Nonexpansive Retraction ◽  
Uniformly Smooth Banach Space ◽  
Uniformly Smooth ◽  
Commutative Family

Let  be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.

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