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ON AN IMPLICIT HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES

Published online by Cambridge University Press:  19 June 2009

FILOMENA CIANCIARUSO
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: cianciaruso@unical.it)
VITTORIO COLAO
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: colao@mat.unical.it)
LUIGI MUGLIA
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: muglia@mat.unical.it)
HONG-KUN XU
Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan (email: xuhk@math.nsysu.edu.tw)
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Abstract

Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math. 13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The first three authors were supported in part by Ministero dell’Universitá e della Ricerca of Italy. The fourth author was supported in part by NSC 97-2628-M-110-003-MY3 (Taiwan).

References

[1] Cabot, A., ‘A proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization’, SIAM J. Optim. 15 (2005), 555572.CrossRefGoogle Scholar
[2] Goebel, K. and Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[3] Luo, Z-Q., Pang, J.-S. and Ralph, D., Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
[4] Maingé, P. E. and Moudafi, A., ‘Strong convergence of an iterative method for hierarchical fixed point problems’, Pac. J. Optim. 3(3) (2007), 529538.Google Scholar
[5] Marino, G. and Xu, H. K., ‘A general iterative method for nonexpansive mappings in Hilbert spaces’, J. Math. Anal. Appl. 318(1) (2006), 4352.CrossRefGoogle Scholar
[6] Moudafi, A., ‘Viscosity approximation methods for fixed-points problems’, J. Math. Anal. Appl. 241 (2000), 4655.CrossRefGoogle Scholar
[7] Moudafi, A. and Maingé, P. E., ‘Towards viscosity approximations of hierarchical fixed-point problems’, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp.CrossRefGoogle Scholar
[8] Xu, H. K., ‘An iterative approach to quadratic optimization’, J. Optim. Theory Appl. 116(3) (2003), 659678.CrossRefGoogle Scholar
[9] Xu, H. K., ‘Viscosity approximation methods for nonexpansive mappings’, J. Math. Anal. Appl. 298 (2004), 279291.CrossRefGoogle Scholar
[10] Xu, H. K., ‘Viscosity method for hierarchical fixed point approach to variational inequalities’, Taiwanese J. Math. 13(6) (2009).Google Scholar
[11] Yamada, I., ‘The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings’, in: Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Studies in Computational Mathematics, 8 (eds. D. Butnariu, Y. Censor and S. Reich) (North-Holland, Amsterdam, 2001), pp. 473504.CrossRefGoogle Scholar
[12] Yao, Y. and Liou, Y. C., ‘Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems’, Inverse Problems 24 (2008), 501508.CrossRefGoogle Scholar
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ON AN IMPLICIT HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES
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