Well-posedness for mixed quasi-variational-hemivariational inequalities
DOI:
https://doi.org/10.12775/TMNA.2016.016Keywords
Mixed quasi-variational-hemivariational inequality, well-posedness, $L$-$\alpha$-well-posedness, lower semi-Mosco convergence, $\alpha$-$\eta$-monotonicityAbstract
In this paper, we consider the well-posedness of mixed quasi-variational-hemivariational inequalities ((MQVHVI) for short). By introducing a new concept of the $\alpha$-$\eta$-monotone mappings, we establish several metric characterizations and equivalent conditions of well-posedness for the (MQVHVI).References
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123–145.
F.E. Browder, Nonlinear maximal monotone operators in Banach space, Math. Ann. 175 (1968), 89–113.
S. Carl and D. Motreanu, Extremal solutions of quasilinear parabolic inclusions with generalized Clarke’s gradient, J. Differential Equations 191 (2003), 206–233.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983).
N. Costea, Existence and uniqueness results for a class of quasi-hemivariational inequalities, J. Math. Anal. Appl. 373 (2011) 305-315.
N. Costea and C. Lupu, On a class of variational-hemivariational inequalities involving set valued mappings, Adv. Pure Appl. Math. 1 (2010), 233–246.
N. Costea and C. Varga, Systems of nonlinear hemivariational inequalities and applications, Topol. Methods Nonlinear Anal. 41 (2013), 39–65.
L.C. Ceng and J.C. Yao, Well-posedness of generailized mixed variational inequalities inclusion problems and fixed point problem, Nonlinear Anal. 69 (2008), 4585–4603.
Y.P. Fang and N.J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl. 118 (2003), 327–338.
K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519–537.
S.M. Guu and J. Li, Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets, Nonlinear Anal. 71 (2009), 2847–2855.
R. Glowinski, J.L. Lions and R. Trèmoliéres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam (1981).
D. Goeleven and D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim. 16 (1995), 909–921.
N.J. Huang, J. Li and B.H. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model. 43 (2006), 1267–1274.
C. Kuratowski, Topology, Vol. I and II, Academic Press, New York, NY, USA (1966).
K. Kimura, Y.C. Liou, S.Y. Wu and J.C. Yao, Well-posedness for parametric vector equilibrium problems with applications, J. Ind. Manag. Optim. 4 (2008), 313–327.
B. Lemaire, Well-posedness conditioning and regularization of minimization inclusion and fixed point problems, Pliska Stud. Math. Bulgar. 12 (1998), 71–84.
Z.H. Liu, A class of evolution hemivariational inequalities, Nonlinear Anal. 36 (1999), 91–100.
Z.H. Liu, Ellipic variational hemivariational inequalities, App. Math. Lett. 16 (2003), 871–876.
Z.H. Liu, Generalized quasi-variational hemivariational inequalities, Appl. Math. Lett. 17 (2004), 741–745.
Z.H. Liu, Some convergence results for evolution hemivariational inequalities, J. Global. Optim. 29 (2004), 85–95.
Z.H. Liu, Browder–Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems 21 (2005), 13–20.
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 140 (2010), 419–434.
Z.H. Liu and X.W. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Anal. Real World Appl. 22 (2015), 581–591.
Z.H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim. 72 (2015), 305–323.
X.J. Long and N.J. Huang, Metric characterizations of α-well-posedness for symmetric quasi-equilibrium problems, J. Global. Optim. 45 (2009), 459–471.
N.K. Mahato and C. Nahak, Weakly relaxed α-pseudomonotonicity and equilibrium, Appl. Math. Comput. 40 (2012), 499–509.
S. Migórski and A. Ochal, Boundary hemivariational inequality of parabolic type, Nonlinear Anal. 57 (2004), 579–596.
S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013.
U. Mosco, Convergence of convex sets and of solutions of variational ineuqalities, Adv. Math. 3 (1969), 510–585.
D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer Academic, Dordrecht, 1999.
P.D. Panagiotopoulos, Nonconvex superpotentials in sense of F.H. Clarke and applications, Mech. Res. Comm. 8 (1981), 335–340.
P.D. Panagiotopoulos, Hemivariational inequalities, Applications in Mechanics and Engineering, Springer, Berlin, 1993.
Z. Peng, Z.H. Liu and X. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann. 356 (2013), 1339–1358.
J.W. Peng and J. Tang, α-well-posedness for mixed quasi variational-like inequality problems, Abstr. Appl. Anal. (2011), doi:10.1155/2011/683140.
A.N. Tykhonov, On the stability of the functional optimization problem, U.S.S.R. Comput. Math. Math. Phys. 6 (1966), 631–634.
M.A. Noor, K.I. Noor and V. Gupta, On equilibrium-like problems, Appl. Anal. 86 (2007), 807–818.
M.A. Noor, K.I. Noor and S. Zainab, On a predictor-corrector method for solving invex equilibrium problems , Nonlinear Anal. 71 (2009), 3333–3338.
Y.B. Xiao and N.J. Huang, Well-posedness for a class of variational-hemivariational ineuqalities with perturbations, J. Global. Optim. 151 (2011), 33–51.
Y.B. Xiao, X.M. Yang and N.J. Huang, Some equivalence results for well-posedness of hemivariational inequalities, J. Global. Optim., doi: 10.1007/s10898-014-0198-7.
Y.L. Zhang and Y.R. He, The hemivariational inequalities for an upper semicontinuous set-valued mapping, J. Optim. Theory Appl. 156 (2013), 716–725.
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