On the well-posedness of differential mixed quasi-variational-inequalities

Zhenhai Liu, Dumitru Motreanu, Shengda Zeng


We discuss the well-posedness and the well-posedness in the generalized sense of differential mixed quasi-variational inequalities ((DMQVIs), for short) in Hilbert spaces. This gives us an outlook to the convergence analysis of approximating sequences of solutions for (DMQVIs). Using these concepts we point out the relation between metric characterizations and well-posedness of (DMQVIs). We also prove that the solution set of (DMQVIs) is compact, if problem (DMQVIs) is well-posed in the generalized sense.


Differential mixed quasi-variational inequalities; well-posedness; approximating sequence; relaxed $\alpha$-monotonicity

Full Text:



E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123–145.

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.

X. Chen and Z. Wang, Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optim. 23 (2013), 1647–1671.

X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program. 146 (2014), 379–408.

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

R. Glowinski, J.L. Lions and R. Trèmoliéres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

S.M. Guu and J. Li, Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets, Nonlinear Anal. 71 (2009), 2847–2855.

J. Gwinner, On differential variational inequalities and projected dynamical systemsequivalence and a stability result, Discret. Contin. Dyn. Syst. (2007), 467–476.

J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program. 139 (2013), 205–221.

J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics, J. Comput. Appl. Math. 254 (2013), 175–184.

L. Han and J.S. Pang, Non-Zenoness of a class of differential quasi-variational inequalities, Math. Program. 121 (2010), 171–199.

L. Han, A. Tiwari, K. Camlibel and J.S. Pang, Convergence of time-stepping schemes for passive and extended linear complementarity systems, SIAM J. Numer. Anal. 47 (2009), 3768–3796.

N.J. Huang, J. Li and B.H. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling 43 (2006), 1267–1274.

M. Kamemskiı̆, V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Water de Gruyter, Berlin, 2001.

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.

C. Kuratowski, Topology, vol. I and II, Academic Press, New York, 1966.

B. Lemaire, Well-posedness, conditioning and regularization of minimization, inclusion and fixed point problems, Pliska Stud. Math. Bulg. 12 (1998), 71–84.

X.S. Li, N.J. Huang and D. O’Regan, Differential mixed variational inequalities in finite dimensional spaces, Nonlinear Anal. 72 (2010), 3875–3886.

Z.H. Liu, N.V. Loi and V. Obukhovskiı̆, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Int. J. Bifurcation Chaos 23 (2013), ID # 1350125, 10 pages.

Z.H. Liu, S. Migórski and S.D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations 263 (2017), 3989–4006.

Z.H. Liu and S.D. Zeng, Differential variational inequalities in infinite Banach spaces, Acta Mathematica Scientia 37 (2017), 26–32.

Z.H. Liu, S.D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations 260 (2016), 6787–6799.

Z.H. Liu, S.D. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal., DOI: 10.1515/anona-2016-0102.

Z.H. Liu, S.D. Zeng and B. Zeng, Well-posedness for mixed quasi-variationalhemivariational inequalities, Topol. Methods in Nonlinear Anal. 47 (2016), 561–578.

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.

J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2008), 345–424.

J.S. Pang and D.E. Stewart, Solution dependence on initial conditions in differential variational variational inequalities, Math. Program. 116 (2009), 429–460.

D.E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints, SIAM, Philadelphia, 2011.

A.N. Tykhonov, On the stability of the functional optimization problem, U.S.S.R. Comput. Math. Math. Phys. 6 (1966), 28–33.

X. Wang and N.J. Huang, A class of differential vector variational inequalities in finite dimensional spaces, J. Optim. Theory Appl. 162 (2014), 633–648.

E. Zeidler, Nonlinear Functional Analysis and its Applications II/B. Nonlinear Monotone Operators, Springer, New York, 1990.

S.D. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl. 455 (2017), 619–637.


  • There are currently no refbacks.

Partnerzy platformy czasopism