The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given.
Similar content being viewed by others
References
M. Bianchi and S. Schaible, “Generalized monotone bifunctions and equilibrium problems,” J. Optimiz. Theory Appl., 90, 31–43 (1996).
E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” Math. Stud., 63, 123–145 (1994).
W. Oettli, “A remark on vector-valued equilibria and generalized monotonicity,” Acta Math. Vietnam, 22, 215–221 (1997).
A. Göpfert, H. Riahi, C. Tammer, and C. Zalinescu, Variational Methods in Partially Ordered Spaces, Springer, New York (2003).
O. Chadli, S. Schaible, and J. C. Yao, “Regularized equilibrium problems with an application to noncoercive hemivariational inequalities,” J. Optimiz. Theory Appl., 121, 571–596 (2004).
P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” J. Nonlin. Convex Anal., 6, No. 1, 117–136 (2005).
I. V. Konnov and O. V. Pinyagina, “D-gap functions for a class of monotone equilibrium problems in Banach spaces,” Comput. Methods Appl. Math., 3, No. 2, 274–286 (2003).
A. C. Stukalov, “Regularization extragradient method for solving equilibrium programming problems in Hilbert spaces,” Zh. Vychisl. Mat. Mat. Fiz., 45, No. 9, 1538–1554 (2005).
I. V. Konnov and O. V. Pinyagina, “D-gap functions and descent methods for a class of monotone equilibrium problems,” Lobachevskii J. Math., 13, 57–65 (2003).
G. Mastroeni, “Gap functions for equilibrium problems,” J. Global Optimiz., 27, 411–426 (2003).
A. S. Antipin, “Equilibrium programming: gradient methods,” Automat. Remote Control, 58, No. 8, 1337–1347 (1997).
A. S. Antipin, “Equilibrium programming: proximal methods,” Comput. Math. Math. Phys., 37, No. 11, 1285–1296 (1997).
M. Bounkhel and B. R. Al-Senan, “An iterative method for nonconvex equilibrium problems,” J. Inequal. Pure Appl. Math., 7, Issue 2, Article 75 (2006).
O. Chadli, I. V. Konnov, and J. C. Yao, “Descent methods for equilibrium problems in Banach spaces,” Comput. Math. Appl., 48, 609–616 (2004).
I. V. Konnov, “Application of the proximal point method to nonmonotone equilibrium problems,” J. Optimiz. Theory Appl., 126, 309–322 (2005).
I. V. Konnov, S. Schaible, and J. C. Yao, “Combined relaxation method for mixed equilibrium problems,” J. Optimiz. Theory Appl., 119, 317–333 (2003).
G. Mastroeni, “On auxiliary principle for equilibrium problems,” Techn. Rep. Dep. Math. Pisa Univ., No. 3, 244–258 (2000).
A. Moudafi, “Second-order differential proximal methods for equilibrium problems,” J. Inequal. Pure Appl. Math., 4, Issue 1, Article 18 (2003).
A. Moudafi and M. Théra, “Proximal and dynamical approaches to equilibrium problems,” Lect. Notes Econ. Math. Syst., 477, 187–201 (1999).
M. A. Noor, “Auxiliary principle technique for equilibrium problems,” J. Optimiz. Theory Appl., 122, 371–386 (2004).
M. A. Noor and K. I. Noor, “On equilibrium problems,” Appl. Math. E-Notes, 4, 125–132 (2004).
N. Buong, “Regularization for unconstrained vector optimization of convex functionals in Banach spaces,” Zh. Vychisl. Mat. Mat. Fiz., 46, No. 3, 372–378 (2006).
N. Buong, “Convergence rates and finite-dimensional approximation for a class of ill-posed variational inequalities,” Ukr. Math. J., 48, No. 5, 697–707 (1996).
N. Buong, “On ill-posed problems in Banach spaces,” South. Asian Bull. Math., 21, 95–193 (1997).
N. Buong, “Operator equations of Hammerstein type under monotone perturbations,” Proc. NCST Vietnam, 11, No. 2, 3–7 (1999).
N. Buong, “Convergence rates in regularization for the case of monotone perturbations,” Ukr. Math. J., 52, No. 2, 285–293 (2000).
A. S. Antipin, “Solution methods for variational inequalities with coupled constraints,” Comput. Math. Math Phys., 40, No. 9, 1239–1254 (2000).
A. S. Antipin, “Solving variational inequalities with coupling constraints with the use of differential equations,” Different. Equat., 36, No. 11, 1587–1596 (2000).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 8, pp. 1098–1105, August, 2009.
Rights and permissions
About this article
Cite this article
Buong, N., Thi Hai Ha, D. Tikhonov regularization method for a system of equilibrium problems in Banach spaces. Ukr Math J 61, 1302–1310 (2009). https://doi.org/10.1007/s11253-010-0277-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0277-y