Strong convergence of a new iterative algorithm for fixed points of asymptotically nonexpansive mappings

Volume 11, Issue 4, pp 529--540 http://dx.doi.org/10.22436/jnsa.011.04.09

Publication Date: March 19, 2018 Submission Date: November 10, 2017 Revision Date: December 01, 2017 Accteptance Date: January 26, 2018

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Authors

Yuanheng Wang - Department of Mathematics, Zhejiang Normal University, Jinhua, China. Jialei Feng - Department of Mathematics, Zhejiang Normal University, Jinhua, China.

Abstract

In this paper, we investigate a new iterative implicit algorithm for fixed points of asymptotically nonexpansive mapping in Hilbert spaces. We also prove its strong convergence theorem under certain assumptions imposed on the parameters and extend some well-known results. As an application, we apply our main result to \(\mu\)-inverse strongly monotone mapping.

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Keywords

• Asymptotically nonexpansive
• strong convergence
• \(\mu\)-inverse strongly monotone mapping
• Hilbert space

MSC

•  47H09
•  47H10
•  54H25

References

• [1] E. Blum, W. Oettli , From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.
  • MathSciNet


• [2] L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 67 (2008), 375–390.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] Q. Fan, Z. Yao , Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem, J. Nonlinear Sci. Appl., 10 (2017), 1470–1477.
  • View Article
  • MathSciNet


• [4] S. D. Flåm, A. S. Antipin , Equilibrium programming using proximal-like algorithms, Math. Programming, 78 (1996), 29–41.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] T.-H. Kim, H.-K. Xu , Strong convergence of modified Mann iterations , Nonlinear Anal., 61 (2005), 51–60.
  • View Article
  • Google Scholar


• [6] G. López, V. Martín-Márquez, F. Wang, H.-K. Xu , Forward-backword splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012 ), 25 pages.
  • View Article
  • Google Scholar


• [7] J. Lou, L.-J. Zhang, Z. He, Viscosity approximation methods for asymptotically nonexpansive mappings, Appl. Math. Comput., 203 (2008), 171–177.
  • View Article
  • Google Scholar


• [8] A. Moudafi , Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
  • View Article
  • Google Scholar


• [9] K. Nakajo, W. Takahashi , Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , J. Math. Anal. Appl., 279 (2003), 372–379.
  • View Article
  • Google Scholar


• [10] J.-W. Peng, J.-C. Yao , A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings , Nonlinear Anal., 71 (2009), 6001–6010.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] J.-W. Peng, J.-C. Yao, Strong convergence theorems of iterative scheme based on the extra gradient method for mixed equilibrium problems and fixed point problems, Math. Comput. Modelling, 49 (2009), 1816–1828.
  • View Article
  • Google Scholar


• [12] X. Qin, Y. J. Cho, S. M. Kang, Viscosity approximation methods for generalized equlibrium problems and fixed point problems with applications , Nonlinear Anal., 72 (2010), 99–112.
  • View Article
  • Google Scholar


• [13] Y. Song, R. Chen, H. Zhou , Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces, Nonlinear Anal., 66 (2007), 1016–1024.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] P. Sunthrayuth, P. Kumam, Viscosity approximation methods base on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces , Math. Comput. Modelling, 58 (2013), 1814–1828.
  • View Article
  • Google Scholar


• [15] Y.-H.Wang, Y.-H. Xia , Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces , Fixed Point Theory Appl., 2012 (2012 ), 8 pages.
  • View Article
  • Google Scholar
  • MATH


• [16] H.-K. Xu , Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
  • View Article
  • Google Scholar


• [17] H.-K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659–678.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] H.-K. Xu , Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
  • View Article
  • Google Scholar


• [19] H.-K. Xu, M. A. Aoghamdi, N. Shahzad , The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015 ), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Q. Yan, G. Cai , Convergence analysis of modified viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces, Revista de la Real Academia de Ciencias Exactas, Fasicas y Naturales., Serie A. Matematicas (RACSAM), 2017 (2017 ), 16 pages.
  • View Article
  • Google Scholar


• [21] Y. Yao, Y.-C. Liou, R. Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007), 3311–3317.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] Y. Yao, Y.-C. Liou, S. M. Kang , Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method , Comput. Math. Appl., 59 (2010), 3472–3480.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] Y. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Some new algorithms for solving mixed equilibrium problems, Comput. Math. Appl., 60 (2010), 1351–1359.
  • View Article
  • MathSciNet
  • Google Scholar


• [24] Y. Yao, N. Shahzad, Y.-C. Liou, Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015 ), 15 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH