Ceng, Lu-Chuan; Yao, Jen-Chih Relaxed viscosity approximation methods for fixed point problems and variational inequality problems. (English) Zbl 1163.47052 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 10, 3299-3309 (2008). Let \(X\) be a strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and \(C\) a nonempty closed convex subset of \(X\). Let \(\{T_n\}_{n=1}^\infty\) be a sequence of nonexpansive self-mappings on \(C\) with the nonempty common fixed point set \(F\) and \(f:C\to C\) a given contractive map. The authors present an iterative scheme which converges strongly to some \(p\in F\) which is the unique solution to the variational inequality \[ \langle(I - f)p, J(p-x^*)\rangle\leq 0, \quad \forall x^* \in F, \] where \(J\) is the duality map of \(X\). Reviewer: Valerii V. Obukhovskij (Voronezh) Cited in 29 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47J20 Variational and other types of inequalities involving nonlinear operators (general) Keywords:relaxed viscosity approximation method; nonexpansive mapping; strong convergence; common fixed point; uniformly Gâteaux differentiable norm PDFBibTeX XMLCite \textit{L.-C. Ceng} and \textit{J.-C. Yao}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 10, 3299--3309 (2008; Zbl 1163.47052) Full Text: DOI References: [1] Bauschke, H. H., The approximation of fixed points of compositions of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 202, 150-159 (1996) · Zbl 0956.47024 [2] Chang, S. S., Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 323, 2, 1402-1416 (2006) · Zbl 1111.47057 [3] Chang, S. 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