Well-posedness for a Class of Variational–Hemivariational Inequalities with Perturbations

In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational inequality and the well-posedness of corresponding inclusion problem.

[1]  Nan-Jing Huang,et al.  Metric characterizations of α-well-posedness for symmetric quasi-equilibrium problems , 2009, J. Glob. Optim..

[2]  Panagiotis D. Panagiotopoulos,et al.  Hemivariational Inequalities: Applications in Mechanics and Engineering , 1993 .

[3]  Dumitru Motreanu,et al.  Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities , 1998 .

[4]  Zhenhai Liu,et al.  Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities , 2005 .

[5]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[6]  Anna Ochal,et al.  Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion , 2008 .

[7]  Nan-jing Huang,et al.  Auxiliary Principle and Iterative Algorithms for Generalized Set-Valued Strongly Nonlinear Mixed Variational-like Inequalities☆ , 2001 .

[8]  Matteo Rocca,et al.  Well Posedness in Vector Optimization Problems and Vector Variational Inequalities , 2007 .

[9]  Jean Cea,et al.  Optimization - Theory and Algorithms , 1978 .

[10]  Jen-Chih Yao,et al.  Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems , 2008 .

[11]  P. D. Panagiotopoulos,et al.  Nonconvex energy functions. Hemivariational inequalities and substationarity principles , 1983 .

[12]  Zhenhai Liu,et al.  Existence results for quasilinear parabolic hemivariational inequalities , 2008 .

[13]  M. B. Lignola.  Well-Posedness and L-Well-Posedness for Quasivariational Inequalities , 2006 .

[14]  Fioravante Patrone,et al.  A characterization of tyhonov well-posedness for minimum problems, with applications to variational inequalities (∗) , 1981 .

[15]  Dumitru Motreanu,et al.  Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications , 2010 .

[16]  T. Zolezzi.  Extended well-posedness of optimization problems , 1996 .

[17]  Enrico Miglierina,et al.  Well-posedness and convexity in vector optimization , 2003, Math. Methods Oper. Res..

[18]  Dumitru Motreanu,et al.  A class of variational–hemivariational inequalities of elliptic type , 2010 .

[19]  Jen-Chih Yao,et al.  WELL-POSEDNESS FOR VECTOR QUASIEQUILIBRIA , 2009 .

[20]  A. N. Tikhonov,et al.  On the stability of the functional optimization problem , 1966 .

[21]  Nan-Jing Huang,et al.  Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities , 2009, J. Glob. Optim..

[22]  Georges Zaccour,et al.  Decision and control in management science , 2002 .

[23]  Yi-bin Xiao,et al.  Sub–super-solution method for a class of higher order evolution hemivariational inequalities ☆ , 2009 .

[24]  B. Lemaire,et al.  Well-Posedness by Perturbations of Variational Problems , 2002 .

[25]  R. Lucchetti,et al.  well posed saddle point problems , 2005 .

[26]  Rong Hu,et al.  Parametric well-posedness for variational inequalities defined by bifunctions , 2007, Comput. Math. Appl..

[27]  Jen-Chih Yao,et al.  Well-posedness by perturbations of mixed variational inequalities in Banach spaces , 2010, Eur. J. Oper. Res..

[28]  D. Goeleven,et al.  Well-posed hemivariational inequalities , 1995 .

[29]  Nan-Jing Huang,et al.  A projected subgradient method for solving generalized mixed variational inequalities , 2008, Oper. Res. Lett..

[30]  Rong Hu,et al.  Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints , 2008, Comput. Math. Appl..

[31]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[32]  Jacqueline Morgan,et al.  Approximate Solutions and α-Well-Posedness for Variational Inequalities and Nash Equilibria , 2002 .

[33]  P. D. Panagiotopoulos,et al.  Mathematical Theory of Hemivariational Inequalities and Applications , 1994 .

[34]  Dumitru Motreanu,et al.  Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient , 2003 .

[35]  Yi-bin Xiao,et al.  Sub–supersolution method and extremal solutions for higher order quasi-linear elliptic hemi-variational inequalities☆ , 2007 .

[36]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[37]  Nan-Jing Huang,et al.  Generalized quasi-variational-like hemivariational inequalities , 2008 .

[38]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[39]  P. D. Panagiotopoulos,et al.  Coercive and semicoercive hemivariational inequalities , 1991 .

[40]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[41]  Fioravante Patrone,et al.  On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games , 2002 .

[42]  Franco Giannessi,et al.  Regularization of non-coercive quasi variational inequalities , 2000 .

[43]  X. X. Huang,et al.  Extended and strongly extended well-posedness of set-valued optimization problems , 2001, Math. Methods Oper. Res..

[44]  Jen-Chih Yao,et al.  Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems , 2008, J. Glob. Optim..

[45]  X. X. Huang,et al.  Levitin–Polyak well-posedness of variational inequality problems with functional constraints , 2009, J. Glob. Optim..

[46]  Dumitru Motreanu,et al.  Evolutionary Variational-hemivariational Inequalities: Existence and Comparison Results , 2008 .

[47]  B. Lemaire,et al.  WELL-POSEDNESS, CONDITIONING AND REGULARIZATION OF MINIMIZATION, INCLUSION AND FIXED-POINT PROBLEMS , 2013 .

[48]  Jen-Chih Yao,et al.  Well-Posedness for Mixed Quasivariational-Like Inequalities , 2008 .