STRONG CONVERGENCE OF COMPOSITE ITERATIVE METHODS FOR NONEXPANSIVE MAPPINGS
Jong-Soo Jung · Journal of the Korean Mathematical Society · 2009
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) ${ eq}{\emptyset}$ . 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.