Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces

Volume 9, Issue 6, pp 4478--4488 http://dx.doi.org/10.22436/jnsa.009.06.86

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)

• 1642 Downloads
• 2690 Views

Authors

Liang-Cai Zhao - College of Mathematics, Yibin University, Yibin, Sichuan, 644007, P. R. China. Shih-Sen Chang - Center for General Education, China Medical University, Taichung, 40402, Taiwan. Ching-Feng Wen - Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 80708, Taiwan.

Abstract

The purpose of this paper is to introduce the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces. The strong convergence of this viscosity method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to nonlinear variational inclusion problem, nonlinear Volterra integral equations, variational inequality problem and hierarchical minimization problems are included. The results presented in the paper extend and improve some recent results announced in the current literature.

Share and Cite

Keywords

• Viscosity
• implicit midpoint rule
• asymptotically nonexpansive mapping
• projection
• variational inequality.

MSC

•  47J25
•  47H09

References

• [1] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad, H. K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), 9 pages.


• [2] H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769-806.


• [3] 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.


• [4] G. Bader, P. Deu hard , A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), 373-398.


• [5] I. D. Berg, I. G. Nikolaev , Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195-218.


• [6] S. S. Chang, Y. J. Cho, H. Zhou, Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc., 38 (2001), 1245-1260.


• [7] P. Deu hard , Recent progress in extrapolation methods for ordinary differential equations , SIAM Rev., 27 (1985), 505-535.


• [8] K. Geobel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge (1990)


• [9] A. Moudafi , Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., 241 (2000), 46-55.


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


• [11] S. Somali, Implicit midpoint rule to the nonlinear degenerate boundary value problems, Int. J. Comput. Math., 79 (2002), 327-332.


• [12] 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.


• [13] M. Van Veldhuxzen, Asymptotic expansions of the global error for the implicit midpoint rule (stiff case), Computing, 33 (1984), 185-192.


• [14] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.


• [15] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.


• [16] 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.