Well-posedness for generalized variational-hemivariational inequalities with perturbations in reflexive banach spaces
AbstractIn this paper, we consider an extension of well-posedness for a minimization problem to a class of generalized variational-hemivariational inequalities with perturbations in reflexive Banach spaces. We establish some metric characterizations for the $\alpha$-well-posed generalized variational-hemivariational inequality and give some conditions under which the generalized variational-hemivariational inequality is strongly $\alpha$-well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the $\alpha$-well-posedness of the generalized variational-hemivariational inequality and the $\alpha$-well-posedness of the corresponding inclusion problem.
S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications, Springer, Berlin, 2005.
S. Carl and D. Motreanu, General comparison principle for quasilinear elliptic inclusions, Nonlinear Anal., 70(2009), 1105--1112.
L. C. Ceng, N. C. Wong and J. C. Yao, Well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations, J. Appl. Math., (2012) Art.ID 712306, 21 pp.
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
Y. P. Fang, N. J. Huang and J. C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed-point problems, J. Glob. Optim., 41(2008), 117--133.
Y. P. Fang, N. J. Huang and J. C. Yao, Well-posedness by perturbations of mixed variational inequalities in Banach spaces, European J. Oper. Res., 201(2010), 682--692.
F. Giannessi and A. Khan, Regularization of non-coercive quasi variational inequalities, Control Cybern., 29(2000), 91--110.
D. Goeleven and D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16(1995), 909--921.
D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume II: Unilateral Problems, Kluwer, Dordrecht, 2003.
R. Hu and Y. P. Fang, Levitin-Polyak well-posedness by perturbations of inverse variational inequalities, Optim. Lett.,7(2013), 343--359.
X. X. Huang nad X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim., 17(2006), 243--258.
X. X. Huang, X. Q. Yang and D. L. Zhu, Levitin-Polyak well-posedness of variational inequality problems with functional constraints, J. Glob. Optim., 44(2009), 159--174.
M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16(2000), 57--67.
L. J. Lin and C. S. Chuang, Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint, Nonlinear Anal., 70(2009), 3609--3617.
Z. H. Liu and J. Z. Zou, Strong convergence results for hemivariational inequalities, Sci. China Ser. A, 49(2006), 893--901.
Z. H. Liu and D. Motreanu, A class of variational-hemivariational inequalities of elliptic type, Nonlinearity, 23(2010), 1741-1752.
R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3(1981), 461--476.
S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Springer, New York, 2013.
D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer, Dordrecht, 1999.
Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.
P. D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationarity principles, Acta Mech., 48(1983), 111--130.
D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer, Dordrecht, 1999.
J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optim. Lett., 4(2010), 501--512.
A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Phys., 6(1966), 631--634.
S. B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math., 30(1969), 475--488.
Y. B. Xiao and N. J. Huang, Well-posedness for a class of variational-hemivariational inequalities with perturbations, J. Optim. Theory Appl., 151 (2011), 33--51.
Y. B. Xiao, N. J. Huang and M. M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math., 15(2011), 1261--1276.
E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. II, Springer, Berlin, 1990.
J. Zeng, S. J. Li, W. Y. Zhang and X. W. Xue, Hadamard well-posedness for a set-valued optimization problem, Optim. Lett. 7 (2013), 559--573.
T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91(1996), 257--266.
X. B. Li and F. Q. Xia, Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces, Nonlinear Anal. TMA, 75(2012), 2139--2153.
L. C. Ceng and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems, Nonlinear Anal. TMA, 69(2008), 4585--4603.
L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139(2008), 109--125.
L. C. Ceng, H. Gupta and C. F. Wen, Well-posedness by perturbations of variational-hemivariational inequalities with perturbations, Filomat, 26(2012), 881--895.
K. Kuratowski, Topology, vols. 1-2, Academic, New York, NY, 1968.
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1972.