Aoyama, Koji; Iiduka, Hideaki; Takahashi, Wataru Weak convergence of an iterative sequence for accretive operators in Banach spaces. (English) Zbl 1128.47056 Fixed Point Theory Appl. 2006, No. 3, 35390, 13 p. (2006). The authors study the generalized variational inequality problem of finding \(u \in C\) such that \[ \langle Au, J(v-u) \rangle \geq 0\;\forall v \in C, \] where \(C\) is a nonempty closed convex subset of a smooth Banach space \(E\), \(A\) is an accretive operator of \(C\) into \(E\), \(J\) is the duality mapping of \(E\) into \(E^*\), and \(\langle \cdot, \cdot \rangle\) is the duality paring between \(E\) and \(E^*\). To solve this problem, the authors propose the following iterative scheme: \(x_1 =x \in C\) and \[ x_{n+1}=\alpha_{n}x_n +(1-\alpha_{n})Q_{C}(x_n -\lambda_{n}Ax_n) \] for \(n=1, 2, 3,\dots\), where \(Q_C\) is a sunny nonexpansive retraction from \(E\) onto \(C\), \(\{\alpha_n\}\) is a sequence in \([0,1]\), and \(\{\lambda_n\}\) is a sequence of real numbers. For this iterative scheme, the authors establish a weak convergence result (Theorem 3.1) in a uniformly convex and \(2\)-uniformly smooth Banach space for an \(\alpha\)-inverse strongly accretive operator. Applications to finding a zero point of an inverse strongly accretive operator and to finding a fixed point of a strictly pseudocontractive mapping are given. Reviewer: Jen-Chih Yao (Kaohsiung) Cited in 6 ReviewsCited in 74 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:accretive operators; strictly pseudocontractive mappings; fixed points PDFBibTeX XMLCite \textit{K. Aoyama} et al., Fixed Point Theory Appl. 2006, No. 3, 35390, 13 p. (2006; Zbl 1128.47056) Full Text: DOI EuDML References: [1] Ball K, Carlen EA, Lieb EH: Sharp uniform convexity and smoothness inequalities for trace norms.Inventiones Mathematicae 1994,115(3):463-482. · Zbl 0803.47037 [2] Beauzamy B: Introduction to Banach Spaces and Their Geometry, North-Holland Mathematics Studies. Volume 68. 2nd edition. North-Holland, Amsterdam; 1985:xv+338. [3] Brezis H: Analyse Fonctionnelle. Théorie et Applications, Collection of Applied Mathematics for the Master’s Degree. Masson, Paris; 1983:xiv+234. 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