Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S -iteration process recently introduced by Sahu in  [14] converges strongly to a unique fixed point of a mappi...

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Bibliographic Details
Published in:Nonlinear analysis 2011-12, Vol.74 (17), p.6012-6023
Main Authors: Sahu, D.R., Petruşel, Adrian
Format: Article
Language:English
Subjects:
[formula omitted]-iteration process
[formula omitted]-strictly pseudocontractive
Banach space
Convergence
Exact sciences and technology
Functional analysis
Inequalities
Mapping
Mathematical analysis
Mathematics
Metric projection mapping
Nonlinearity
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Steepest descent method
Strongly pseudocontractive
Theorems
Uniformly Gâteaux differentiable norm
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Summary:In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S -iteration process recently introduced by Sahu in  [14] converges strongly to a unique fixed point of a mapping T , where T is κ -strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ i -strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980)  [1] and Yamada (2001)  [15] to a general class of λ -strictly pseudocontractive mappings in uniformly smooth Banach spaces.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2011.05.078