Cianciaruso, Filomena; Marino, Giuseppe; Muglia, Luigi; Zhou, Haiyun Strong convergence of viscosity methods for continuous pseudocontractions in Banach spaces. (English) Zbl 1219.47106 J. Appl. Math. Stochastic Anal. 2008, Article ID 149483, 11 p. (2008). Summary: We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes, improving the main theorems in [Y.-H. Yao, Y.-C. Liou and R.-D. Chen, Nonlinear Anal., Theory Methods Appl. 67, No. 12, A, 3311–3317 (2007; Zbl 1129.47059)] and [H.-Y. Zhou, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 11, A, 4039–4046 (2009; Zbl 1218.47131)]. Cited in 1 Document MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:pseudocontractive mapping; fixed point; viscosity method; family of mappings; strong convergence Citations:Zbl 1129.47059; Zbl 1218.47131 PDFBibTeX XMLCite \textit{F. Cianciaruso} et al., J. Appl. Math. Stochastic Anal. 2008, Article ID 149483, 11 p. (2008; Zbl 1219.47106) Full Text: DOI EuDML References: [1] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46-55, 2000. · Zbl 0957.47039 [2] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279-291, 2004. · Zbl 1061.47060 [3] C. H. Morales, “Strong convergence of path for continuous pseudo-contractive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2831-2838, 2007. · Zbl 1121.47055 [4] H. Zegeye, N. Shahzad, and T. Mekonen, “Viscosity approximation methods for pseudocontractive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 538-546, 2007. · Zbl 1178.47049 [5] Y. Yao, Y.-C. Liou, and R. Chen, “Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3311-3317, 2007. · Zbl 1129.47059 [6] H. Zhou, “Viscosity approximation methods for pseudocontractive mappings in Banach spaces,” Nonlinear Analysis, 2008. · Zbl 1162.47054 [7] Y. Song and R. Chen, “Convergence theorems of iterative algorithms for continuous pseudocontractive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 2, pp. 486-497, 2007. · Zbl 1126.47054 [8] L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114-125, 1995. · Zbl 0872.47031 [9] K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, no. 4, pp. 365-374, 1974. · Zbl 0288.47047 [10] W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.