A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces

Lu Chuan Ceng, Sy Ming Guu*, Jen Chih Yao

*Corresponding author for this work

Research output: Contribution to journalJournal Article peer-review

16 Scopus citations

Abstract

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient α∈(0,1), F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0, and A:H→H is a strongly positive bounded linear operator with coefficient γ∈(1,2). Let 0<μ<2η k2,0<γ<μ(η-μk22)α= τα. It is shown that the sequence xn generated by the following general composite iterative method: yn=(I- αnμF)Txn+αnγf( xn),xn+1=(I-βnA)Txn+ βnyn,∀n<0, where αn⊂ [0,1] and βn⊂(0,1], converges strongly to a fixed point x∈Fix(T), which solves the variational inequality 〈(I-A)x,x- x〉≤0,∀x∈Fix(T).

Original languageEnglish
Pages (from-to)2447-2455
Number of pages9
JournalComputers and Mathematics with Applications
Volume61
Issue number9
DOIs
StatePublished - 05 2011
Externally publishedYes

Keywords

  • Composite iterative method
  • Fixed point
  • Nonexpansive mappings
  • Projection
  • Variational inequality
  • Viscosity approximation

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