Strong convergence theorems of the general iterative methods for nonexpansive semigroups in Banach spaces (Q554821)

Summary: Let \(E\) be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from \(E\) to \(E^*\). Let \({\mathcal S}=\{T(s) : 0 \leq s < \infty\}\) be a nonexpansive semigroup on \(E\) such that \(\text{Fix}({\mathcal S}):= \bigcap_{t\geq 0}\text{Fix}(T(t))\neq\emptyset\), and \(f\) be a contraction on \(E\) with coefficient \(0<\alpha<1\). Let \(F\) be \(\delta\)-strongly accretive and \(\lambda\)-strictly pseudocontractive with \(\delta+\lambda>1\) and \(\gamma\) a positive real number such that \(\gamma<1/\alpha(1-\sqrt{1-\delta/\lambda})\). When the sequences of real numbers \(\{\alpha_n\}\) and \(\{t_n\}\) satisfy some appropriate conditions, the three iterative processes given as follows: \(x_{n+1}=\alpha_n\gamma f(x_n)+(I-\alpha_nF)T(t_n)x_n\), \(n\geq 0\), \(y_{n+1}=\alpha_n\gamma f(T(t_n)y_n)+(I-\alpha_nF)T(t_n)y_n\), \(n\geq 0\), and \(z_{n+1}=T(t_n)(\alpha_n\gamma f(z_n)+(I-\alpha_nF)z_n)\), \(n\geq 0\), converge strongly to \(\widetilde x\), where \(\widetilde x\) is the unique solution in \(\text{Fix}({\mathcal S})\) of the variational inequality \(\langle(F-\gamma f)\widetilde x,j(x-\widetilde x)\rangle\geq 0\), \(x\in \text{Fix}({\mathcal S})\). Our results extend and improve corresponding ones of Li et al. (2009), Chen and He (2007), and many others.