Existence of Solutions for Noncoercive Hemivariational Inequalities by an Equilibrium Approach Under Pseudomonotone Perturbation

In this paper, we study the existence of a solution for a hemivariational inequality problem in a noncoercive framework. The approach adopted is an equilibrium problem formulation associated with a maximal monotone bifunction with pseudomonotone perturbation. We proceed by introducing auxiliary problems that will be studied using a new existence result for equilibrium problems. An example to illustrate the use of the theory is given.

[1]  Monica Bianchi,et al.  Coercivity Conditions for Equilibrium Problems , 2005 .

[2]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .

[3]  E. Boschi.  Recensioni: J. L. Lions - Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Vi;;ars, Paris, 1969; , 1971 .

[4]  D. Motreanu,et al.  Topological Approach to Hemivariational Inequalities with Unilateral Growth Condition , 2001 .

[5]  J. Mawhin,et al.  Nonlinear Operators and the Calculus of Variations , 1976 .

[6]  Monica Bianchi,et al.  Generalized monotone bifunctions and equilibrium problems , 1996 .

[7]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[8]  G. Stampacchia,et al.  A remark on Ky fan's minimax principle , 2008 .

[9]  Daniel Goeleven,et al.  Variational and Hemivariational Inequalities , 2003 .

[10]  P. Panagiotopoulos.  Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions , 1985 .

[11]  Z. Chbani,et al.  Some existence results for coercive and noncoercive hemivariational inequalities , 1998 .

[12]  P. Panagiotopoulos.  Inequality problems in mechanics and applications , 1985 .

[13]  Jen-Chih Yao,et al.  Regularized Equilibrium Problems with Application to Noncoercive Hemivariational Inequalities , 2004 .

[14]  Daniel Goeleven,et al.  Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets , 1996, J. Glob. Optim..

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

[16]  C. Baiocchi,et al.  Variational and quasivariational inequalities: Applications to free boundary problems , 1983 .

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

[18]  Hadi Khatibzadeh,et al.  Maximal monotonicity of bifunctions , 2010 .

[19]  Haim Brezis,et al.  Characterizations of the ranges of some nonlinear operators and applications to boundary value problems , 1978 .

[20]  H. Brezis,et al.  Image d’une somme d’operateurs monotones et applications , 1976 .

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

[22]  S. Karamardian.  Complementarity problems over cones with monotone and pseudomonotone maps , 1976 .

[23]  U. Mosco.  Implicit variational problems and quasi variational inequalities , 1976 .

[24]  Giuseppe Buttazzo,et al.  Compatibility conditions for nonlinear Neumann problems , 1991 .

[25]  Nikolaos S. Papageorgiou,et al.  Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance , 2007 .

[26]  Eberhard Zeidler,et al.  Nonlinear monotone operators , 1990 .

[27]  Kung-Ching Chang,et al.  Variational methods for non-differentiable functionals and their applications to partial differential equations , 1981 .

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

[29]  P. D. Panagiotopoulos,et al.  Non-convex superpotentials in the sense of F.H. Clarke and applications , 1981 .

[30]  Zhen-Hai Liu,et al.  Elliptic variational hemivariational inequalities , 2003, Appl. Math. Lett..

[31]  Samir Adly,et al.  Recession mappings and noncoercive variational inequalities , 1996 .

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

[33]  Daniel Goeleven,et al.  Variational and Hemivariational Inequalities : Theory, Methods and Applications , 2003 .

[34]  Z. Chbani,et al.  Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities , 2000 .

[35]  F. Tomarelli.  Noncoercive variational inequalities for pseudomonotone operators , 1991 .

[36]  H. Nikaidô,et al.  Note on non-cooperative convex game , 1955 .

[37]  Giuseppe Buttazzo,et al.  General existence theorems for unilateral problems in continuum mechanics , 1988 .

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