Existence results for a class of hemivariational inequalities involving the stable $(g,f,\alpha)$-quasimonotonicity

Zhenhai Liu, Biao Zeng

DOI: http://dx.doi.org/10.12775/TMNA.2016.002


In this paper, by introducing a new concept of the stable (g; f; alpha)-quasimonotonicity and applying the properties of Clarke's generalized gradient and KKM technique, we show the existence results of solutions for hemivariational inequalities when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. In the last section, we also give an example to present the our main result.


Existence results; hemivariational inequalities; Clarke's generalized gradient; KKM principle; stable (g; f; alpha)-quasimonotonicity

Full Text:



F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983).

N. Costea and V. Radulescu, Existence results for hemivariational inequalities involving relaxed eta - alpha monotone mappings, Commun. Appl. Anal. 13 (2009), 293-304.

N. Costea and C. Varga, Systems of nonlinear hemivariational inequalities and applications, Topol. Methods Nonlinear Anal. 41(2013), 39-65.

Z. Denkowski, S. Migorski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003).

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537.

Y.R. He, A relationship between pseudomonotone and monotone mappings, Appl. Math. Lett. 17 (2004), 459-461.

N. Hadjisavvas and S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl. 90 (1996), 95-111.

A. Kristaly, V. Radulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encylopedia of Mathematics (No. 136), Cambridge University Press, Cambridge (2010).

A. Kristaly and C. Varga, A set-valued approach to hemivariational inequalities, Topol. Methods Nonlinear Anal. 24 (2004), 297-307.

Z.H. Liu, Some convergence results for evolution hemivariational inequalities, J. Global Optim. 29 (2004), 85-95.

Z.H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations 244 (2008), 1395-1409.

Z.H. Liu, On boundary variational-hemivariational inequalities of elliptic type, Proc. Roy. Soc. Edinburgh Sect. A Mathematics 140 (2010), 419-434.

S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Springer, New York (2013).

Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York (1995).

P.D. Panagiotopoulos, Nonconvex superpotentials in the sense of F.H. Clarke and applications, Mech. Res. Comm. 8 (1981), 335-340.

P.D. Panagiotopoulos, Nonconvex energy functions. Hemivariational inequalities and substationarity principles, Acta Mech. 42 (1983), 160-183.

P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechnics and Engineering, Springer, Berlin (1993).

G.J. Tang and N.J. Huang, Existence theorems of the variational-hemivariational inequalities, J. Global Optim. 56 (2013), 605-622.

R. Wangkeeree and P. Preechasilp, Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces, J. Global Optim. 57 (2013), 1447-1464.

Y.B. Xiao and N.J. Huang, Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities, J. Global Optim. 45 (2009), 371-388.

Y.L. Zhang and Y.R. He, On stably quasimonotone hemivariational inequalities, Nonlinear Anal. 74 (2011), 3324-3332.


  • There are currently no refbacks.

Partnerzy platformy czasopism