Strong and weak convergence theorems for asymptotically nonexpansive mappings

C. E. Chidume, E. U. Ofoedu, Habz Zegeye

Research output: Contribution to journalArticle

110 Citations (Scopus)

Abstract

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xnnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.

Original languageEnglish
Pages (from-to)364-374
Number of pages11
JournalJournal of Mathematical Analysis and Applications
Volume280
Issue number2
DOIs
Publication statusPublished - Apr 15 2003

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Completely Continuous
Asymptotically Nonexpansive Mapping
Banach spaces
Convergence Theorem
Uniformly Convex Banach Space
Retract
Retraction
Weak Convergence
Strong Convergence
Differentiable
Norm
Closed

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fr{\'e}chet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.",
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Strong and weak convergence theorems for asymptotically nonexpansive mappings. / Chidume, C. E.; Ofoedu, E. U.; Zegeye, Habz.

In: Journal of Mathematical Analysis and Applications, Vol. 280, No. 2, 15.04.2003, p. 364-374.

Research output: Contribution to journalArticle

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N2 - Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.

AB - Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂[1, ∞), limkn = 1, F(T):= {x ∈ K: Tx = x)≠ ∅. Suppose {xn}n≥1 is generated iteratively by x1 ∈ K, xn+1 = P((1-αn xn+αnT(PT)n-1xn), n≥1, where {αn}n≥1 ⊂ (0, 1) is such that ∈ < 1 - αn < 1 - ∈ for some ∈ > 0. It is proved that (I - T) is demiclosed at 0. Moreover, if ∑n≥1 (kn2 - 1) < ∞ and T is completely continuous, strong convergence of {xn} to some x* ∈ F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {xn} to some x* ∈ F(T) is obtained.

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