Abstract
This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi , K)\)) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP\((F,\phi ,K)\)) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces.
Similar content being viewed by others
References
Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Progr. 86, 433–438 (1999)
Pascali, D., Sburlan, D.: Nonlinear Mappings of Monotone Type. Martinus Nijhoff Publishers, The Hague (1978)
Fu, D.M., He, Y.R.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Sichuan Norm. Univ. (Nat. Sci.) 37, 12–17 (2014)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Qiao, F.S., He, Y.R.: Strict feasibility of pseudomonotone set-valued variational inequalities. Optimization 60, 303–310 (2011)
Terkelsen, F.: Some minimax theorems. Math. Scand. 31, 405–413 (1972)
Ravindran, G., Gowda, M.S.: Regularization of \(P_0\)-functions in box variational inequality problems. SIAM J. Optim. 11, 760–784 (2000–2001)
Qi, H.D.: Tikhonov regularization methods for variational inequality problems. J. Optim. Theory Appl. 102, 193–201 (1999)
Konnov, I.V.: On the convergence of a regularization method for variational inequalities. Comput. Math. Math. Phys. 46, 541–547 (2006)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, Toronto (1984)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Ceng, L.C., Cubiotti, P., Yao, J.C.: Existence of vector mixed variational inequalities in Banach spaces. Nonlinear Anal. 70, 1239–1256 (2009)
Bianchi, M., Hadjisavvas, N., Schaible, S.: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 122, 1–17 (2004)
Bianchi, M., Hadjisavvas, N., Schaible, S.: Exceptional families of elements for variational inequalities in Banach spaces. J. Optim. Theory Appl. 129, 23–31 (2006)
Konnov, I.V., Volotskaya, E.O.: Mixed variational inequalities and economic equilibrium problems. J. Appl. Math. 2, 289–314 (2002)
Goeleven, D.: Existence and uniqueness for a linear mixed variational inequality arising in electrical circuits with transistors. J. Optim. Theory Appl. 138, 397–406 (2008)
Noor, M.A., Noor, K.I.: Set-valued resolvent equations and mixed variational inequalities. J. Math. Anal. Appl. 220, 741–759 (1998)
Noor, M.A.: Mixed variational inequalities. Appl. Math. Lett. 3, 73–75 (1990)
Noor, M.A.: Monotone mixed variational inequalities. Appl. Math. Lett. 14, 231–236 (2001)
Fang, S.C., Peterson, E.L.: Generalized variational inequalities. J. Optim. Theory Appl. 38, 363–383 (1982)
He, Y.R., Ng, K.F.: Strict feasibility of generalized complementarity problems. J. Aust. Math. Soc. 81, 15–20 (2006)
He, Y.R.: Stable pseudomonotone variational inequality in reflexive Banach spaces. J. Math. Anal. Appl. 330, 352–363 (2007)
He, Y.R.: The Tikhonov regularization method for set-valued variational inequalities. Abstr. Appl. Anal. (2012)
Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in banach spaces. J. Optim. Theory Appl. 118, 327–338 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, M. The existence results and Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. Anal.Math.Phys. 7, 151–163 (2017). https://doi.org/10.1007/s13324-016-0134-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13324-016-0134-8
Keywords
- Generalized mixed variational inequalities
- Tikhonov regularization
- Mixed variational inequality property
- Coercivity conditions