Convergence analysis of generalized viscosity implicit rules for a nonexpansive semigroup with gauge functions

Volume 11, Issue 9, pp 1031--1044 http://dx.doi.org/10.22436/jnsa.011.09.02

Publication Date: June 19, 2018 Submission Date: December 08, 2017 Revision Date: April 01, 2018 Accteptance Date: April 06, 2018

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2115 Downloads
• 4026 Views

Authors

Pongsakorn Sunthrayuth - Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani, 12110, Thailand. Nuttapol Pakkaranang - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. Poom Kumam - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Facuty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan.

Abstract

In this paper, we introduce an iterative algorithm for finding the set of common fixed points of nonexpansive semigroups by the generalized viscosity implicit rule in certain Banach spaces which has a uniformly Gateaux differentiable norm and admits the duality mapping \(j_\varphi\), where \(\varphi\) is a gauge function. We prove strong convergence theorems of proposed algorithm under appropriate conditions. As applications, we apply main result to solving the fixed point problems of countable family of nonexpansive mappings and the problems of zeros of accretive operators. Furthermore, we give some numerical examples for supporting our main results.

Share and Cite

Keywords

• Nonexpansive semigroup
• Banach spaces
• strong convergence
• fixed point problem
• iterative method

MSC

•  47H09
•  47H10
•  47J05
•  47J25

References

• [1] A. Aleyner, Y. Censor, Best approximation to common Fixed points of a semigroup of nonexpansive operator, J. Nonlinear Convex Anal., 6 (2005), 137–151.
  • MathSciNet
  • Google Scholar


• [2] A. Aleyner, S. Reich , An explicit construction of sunny nonexpansive retractions in Banach spaces, Fixed Point Theory Appl., 2005 (2005), 295–305.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mapping in a Banach space, Nonlinear Anal., 67 (2007), 2350–2360.
  • View Article
  • MathSciNet
  • Google Scholar


• [4] H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769–806.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math., 56 (1989), 469–499.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] G. Bader, P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373–398.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-Verlag, London (2006)
  • View Article
  • MathSciNet
  • Google Scholar


• [8] T. D. Benavides, G. L. Acedo, H.-K. Xu , Construction of sunny nonexpansive retractions in Banach spaces, Bull. Austral. Math. Soc., 66 (2002), 9–16.
  • View Article
  • Google Scholar
  • MATH


• [9] F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z., 100 (1967), 201–225.
  • View Article
  • Google Scholar


• [10] S.-S. Chang, Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 323 (2006), 1402–1416.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] R. D. Chen, H. M. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space, Appl. Math. Lett., 20 (2007), 751–757.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] R. D. Chen, Y. Y. Song, Convergence to common fixed point of nonexpansive semigroup, J. Comput. Anal. Appl., 200 (2007), 566–575.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] P. Cholamjiak, S. Suantai, Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions, J. Glob. Optim., 52 (2012), 185–197.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27 (1985), 505–53.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] K.-J. Engel, R. Nagel , One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York (2000)
  • View Article
  • MathSciNet
  • Google Scholar


• [16] K. Eshita, W. Takahashi, Approximating zero points of accretive operators in general Banach spaces , JP J. Fixed Point Theory Appl., 2 (2007), 105–116.
  • MathSciNet
  • Google Scholar
  • MATH


• [17] Y. F. Ke, C. F. Ma, The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 21 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] T.-C. Lim, H.-K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal., 22 (1994), 1345– 1355.
  • View Article
  • Google Scholar


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


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


• [21] C. Schneider, Analysis of the linearly implicit mid-point rule for differential-algebra equations, Electron. Trans. Numer. Anal., 1 (1993), 1–10.
  • MathSciNet
  • Google Scholar


• [22] S. Somalia , Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327–332.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] Y. Song, R. Chen, Viscosity approximative methods to Cesàro means for non-expansive mappings, Appl. Math. Comput., 186 (2007), 1120–1128.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [25] P. Sunthrayuth, Y. J. Cho, P. Kumam, Viscosity Approximation Methods for Zeros of Accretive Operators and Fixed Point Problems in Banach Spaces, Bull. Malays. Math. Sci. Soc., 39 (2016), 773–793.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequence for one–parameter nonexpansive semigroup without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239.
  • View Article
  • MathSciNet
  • Google Scholar


• [27] W. Takahashi, Nonlinear Functional Analysis: Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)
  • Google Scholar


• [28] M. Van Veldhuxzen , Asymptotic expansions of the global error for the implicit midpoint rule (stiff case), Computing, 33 (1984), 185–192.
  • View Article
  • MathSciNet
  • Google Scholar


• [29] H.-K. Xu , Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109–113.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [31] H.-K. Xu, M. A. Alghamdi, 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


• [32] C. Schneider, Analysis of the linearly implicit mid-point rule for differential–algebra equations , Electron. Trans. Numer. Anal., 1 (1993), 1–10.
  • MathSciNet
  • Google Scholar