Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces (Q937582)

Let \(E\) be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of \(E\) has the fixed point property for nonexpansive mappings. Let \(C\) be a nonempty closed convex subset of \(E\), \( f: C \to C\) be a contractive mapping (or a weakly contractive mapping), and \(T: C \to C\) be a nonexpansive mapping with nonempty fixed point set \(F(T)\). Let the sequence \(\{x_n\}\) be generated by the following composite iterative scheme: \[ \left\{\begin{aligned} &y_n=\lambda_n f(x_n)+(1-\lambda_n)Tx_n,\\ &x_{n+1}=(1-\beta_n)y_n+\beta_n Ty_n,\end{aligned}\right.\quad n\geq0. \] It is proved that \(\{x_n\}\) converges strongly to a point in \(F(T)\), which is a solution of a certain variational inequality, provided that the sequence \(\{\lambda_n\}\subset(0,1)\) satisfies \(\lim_{n\rightarrow\infty}\lambda_n=0\) and \(\sum_{n=1}^\infty\lambda_n=\infty\), \(\{\beta_n\}\subset[0,a)\) for some \(0<a<1\), and the sequence \(\{x_n\}\) is asymptotically regular.