Song, Yisheng; Cho, Yeol Je Iterative approximations for multivalued nonexpansive mappings in reflexive banach spaces. (English) Zbl 1183.47073 Math. Inequal. Appl. 12, No. 3, Article ID 47, 611-624 (2009). Let \(K\) be a nonempty closed convex subset of a Banach space \(E\) and let \(T:K\rightarrow C(E)\) be a nonexpansive weakly inward mapping on \(K\), where \(C(E)\) denotes the family of nonempty closed and bounded subsets of \(E\). In order to approximate the fixed points of \(T\), the authors consider two methods, namely, an implicit method, given by \[ x_t\in (1-t) Tx_t+tu,\,t\in (0,1),\,u\in K, \]and an explicit method of Halpern type, defined by\[ x_{n+1}\in (1-\alpha_n) Tx_n+\alpha_n u,\,\alpha_n\in (0,1),\,u\in K. \]Some strong convergence theorems for \(\{x_t\}\) and \(\{x_n\}\) in particular Banach spaces are obtained, which improve and extend several results in literature. Reviewer: Vasile Berinde (Baia Mare) Cited in 6 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H04 Set-valued operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H10 Fixed-point theorems Keywords:Banach space; multivalued nonexpansive mapping; fixed point; explicit scheme; implicit scheme; convergence theorem; strong convergence; reflexive and strictly convex Banach space; weakly sequentially continuous duality mapping; uniformly Gâteaux differentiable norm PDFBibTeX XMLCite \textit{Y. Song} and \textit{Y. J. Cho}, Math. Inequal. Appl. 12, No. 3, Article ID 47, 611--624 (2009; Zbl 1183.47073) Full Text: DOI