Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings

Habtu Zegeye, Naseer Shahzad

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F (T) ≠ ∅ and f : K → K be a contraction. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t) f (yt) + tT (yt). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then {yt} converges strongly to a fixed point p of T such that p is the unique solution in F (T) to a certain variational inequality. Moreover, if {Ti, i = 1, 2, ..., r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {Ti, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family Ti, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.

Original languageEnglish
Pages (from-to)155-163
Number of pages9
JournalApplied Mathematics and Computation
Volume191
Issue number1
DOIs
Publication statusPublished - Aug 1 2007

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Viscosity Approximation Method
Banach spaces
Nonexpansive Mapping
Common Fixed Point
Variational Inequalities
Viscosity
Converge
Reflexive Banach Space
Strictly Convex
Set theory
Unique Solution
Differentiable
Contraction
Fixed point
Banach space
Iteration
Norm
Closed
Subset
Family

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F (T) ≠ ∅ and f : K → K be a contraction. Then for t ∈ (0, 1), there exists a sequence {yt} ⊂ K satisfying yt = (1 - t) f (yt) + tT (yt). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly G{\^a}teaux differentiable norm, then {yt} converges strongly to a fixed point p of T such that p is the unique solution in F (T) to a certain variational inequality. Moreover, if {Ti, i = 1, 2, ..., r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {Ti, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family Ti, i = 1, 2, ..., r need not satisfy the requirment that {Mathematical expression}.",
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Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings. / Zegeye, Habtu; Shahzad, Naseer.

In: Applied Mathematics and Computation, Vol. 191, No. 1, 01.08.2007, p. 155-163.

Research output: Contribution to journalArticle

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