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http://dx.doi.org/10.4134/JKMS.2009.46.6.1151

STRONG CONVERGENCE OF COMPOSITE ITERATIVE METHODS FOR NONEXPANSIVE MAPPINGS  

Jung, Jong-Soo (DEPARTMENT OF MATHEMATICS DONG-A UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.46, no.6, 2009 , pp. 1151-1164 More about this Journal
Abstract
Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $\rightarrow$C a contractive mapping (or a weakly contractive mapping), and T : C $\rightarrow$ C a nonexpansive mapping with the fixed point set F(T) ${\neq}{\emptyset}$. Let {$x_n$} be generated by a new composite iterative scheme: $y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n$, $x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$, ($n{\geq}0$). It is proved that {$x_n$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$\lambda_n$} $\subset$ (0, 1) satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n$ = 0 and $\sum_{n=0}^{\infty}{\lambda}_n={\infty}$, {$\beta_n$} $\subset$ [0, a) for some 0 < a < 1 and the sequence {$x_n$} is asymptotically regular.
Keywords
viscosity approximation method; nonexpansive mapping; composite iterative scheme; contractive mapping; weakly contractive mapping; weakly sequentially continuous duality mapping; variational inequality;
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