Taiwanese Journal of Mathematics


Hong-Kun Xu

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A viscosity method for a hierarchical fixed point approach to variational inequality problems is presented. This method is used to solve variational inequalities where the involving operators are complements of nonexpansive mappigs and the solutions are sought in the set of the fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings.

Article information

Taiwanese J. Math., Volume 14, Number 2 (2010), 463-478.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Secondary: 58E35: Variational inequalities (global problems) 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 65J25

viscosity method variational inequality nonexpansive mapping iterative method hierarchical fixed point projection


Xu, Hong-Kun. VISCOSITY METHOD FOR HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES. Taiwanese J. Math. 14 (2010), no. 2, 463--478. doi:10.11650/twjm/1500405802. https://projecteuclid.org/euclid.twjm/1500405802

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  • H. Attouch, Variational Convergence for Functions and Operators, Applicable Math. Series, Pitman, London, 1984.
  • F. E. Browder, Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces, Arch. Rational Mech. Anal., 24 (1967), 82-90.
  • L. C. Ceng, C. Lee and J. C. Yao, Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities, Taiwanese J. Math., 12 (2008), 227-244.
  • K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, 1990.
  • B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.
  • P. L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Sèr. A-B Paris, 284 (1977), 1357-1359.
  • P.-E. Mainge and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed-points problems, Pacific J. Optim., 3 (2007), 529-538.
  • G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52.
  • A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.
  • A. Moudafi and P.-E. Mainge, Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Vol. 2006, Article ID 95453, pp. 1-10.
  • S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287-292.
  • S. Reich, Approximating fixed points of nonexpansive mappings, Panamerican. Math. J., 4(2) (1994), 23-28.
  • S. Schaible, J. C. Yao and L. C. Zeng, A Proximal Method for Pseudomonotone type variational-like inequalities, Taiwanese J. Math., 10 (2006) 497-513.
  • N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641-3645.
  • R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58 (1992), 486-491.
  • H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.
  • H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.
  • H. K. Xu, Remarks on an iterative method for nonexpansive mappings, Commu. Applied Nonlinear Anal., 10 (2003), no.1, 67-75.
  • H. K. Xu, An iterative approach to quadratic optimization, J. Optimiz. Theory Appl., 116 (2003), 659-678.
  • H. K. Xu, Viscosity Approximation Methods for Nonexpansive Mappings, J. Math. Anal. Appl., 298 (2004), 279-291.
  • I. Yamada, The hybrid steepest descent for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings, in: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Elservier, New York, 2001, pp. 473-504.
  • L. C. Zeng, L. J. Lin and J. C. Yao, Auxiliary Problem Method for Mixed Variational-Like Inequalities, Taiwanese J. Math., 10 (2006), 515-529.
  • L. C. Zeng and J. C. Yao, A hybrid extragradient method for general variational inequalities, Math. Methods Operations Research, 2009. (to appear.)