STRONG CONVERGENCE OF COMPOSITE ITERATIVE METHODS FOR NONEXPANSIVE MAPPINGS

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 <TEX>$\rightarrow$</TEX>C a contractive mapping (or a weakly contractive mapping), and T : C <TEX>$\rightarrow$</TEX> C a nonexpansive mapping with the fixed point set F(T) <TEX>${\neq}{\emptyset}$</TEX>. Let {<TEX>$x_n$</TEX>} be generated by a new composite iterative scheme: <TEX>$y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n$</TEX>, <TEX>$x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n$</TEX>, (<TEX>$n{\geq}0$</TEX>). It is proved that {<TEX>$x_n$</TEX>} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {<TEX>$\lambda_n$</TEX>} <TEX>$\subset$</TEX> (0, 1) satisfies <TEX>$lim_{n{\rightarrow}{\infty}}{\lambda}_n$</TEX> = 0 and <TEX>$\sum_{n=0}^{\infty}{\lambda}_n={\infty}$</TEX>, {<TEX>$\beta_n$</TEX>} <TEX>$\subset$</TEX> [0, a) for some 0 < a < 1 and the sequence {<TEX>$x_n$</TEX>} is asymptotically regular.