The general iterative methods for nonexpansive semigroups in Banach spaces

Abstract

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let $${\mathcal{S} = \{T(s): 0 \leq s 1 and $${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$$ . When the sequences of real numbers {? n } and {t n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to $${\tilde{x}}$$ , where $${\tilde{x}}$$ is the unique solution in $${Fix(\mathcal{S})}$$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065---3071, 2009) and Chen and He (Appl Math Lett 20:751---757, 2007) and many others.