Banach Journal of Mathematical Analysis

On the existence of solutions of variational inequalities in nonreflexive Banach spaces

Vy Khoi Le

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.

Article information

Banach J. Math. Anal., Volume 13, Number 2 (2019), 293-313.

Received: 20 July 2018
Accepted: 13 October 2018
First available in Project Euclid: 28 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40]
Secondary: 46B10: Duality and reflexivity [See also 46A25] 35J87: Nonlinear elliptic unilateral problems and nonlinear elliptic variational inequalities [See also 35R35, 49J40] 58E35: Variational inequalities (global problems)

variational inequality nonreflexive Banach space Orlicz–Sobolev space multivalued mapping


Le, Vy Khoi. On the existence of solutions of variational inequalities in nonreflexive Banach spaces. Banach J. Math. Anal. 13 (2019), no. 2, 293--313. doi:10.1215/17358787-2018-0034.

Export citation


  • [1] R. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975.
  • [2] F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal. 11 (1972), 251–294.
  • [3] S. Carl and V. K. Le, Elliptic inequalities with multi-valued operators: Existence, comparison and related variational-hemivariational type inequalities, Nonlinear Anal. 121 (2015), 130–152.
  • [4] M. L. Carvalho and J. V. Goncalves, Multivalued equations on a bounded domain via minimization on Orlicz-Sobolev spaces, J. Convex Anal. 21 (2014), no. 1, 201–218.
  • [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin, 1977.
  • [6] T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Differential Equations 10 (1971), 507–528.
  • [7] J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205.
  • [8] J.-P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math. 74 (1982), no. 1, 17–24.
  • [9] J.-P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Analysis 11 (1987), no. 3, 379–392.
  • [10] S. C. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Math. Appl. 419, Kluwer Academic, Dordrecht, 1997.
  • [11] A. G. Kartsatos and I. V. Skrypnik, A new topological degree theory for densely defined quasibounded $({\widetilde{S}}_{+})$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces, Abstr. Appl. Anal. 2005, no. 2, 121–158.
  • [12] N. Kenmochi, “Monotonicity and compactness methods for nonlinear variational inequalities” in Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. IV, Elsevier/North-Holland, Amsterdam, 2007, 203–298.
  • [13] M. A. Krasnosel’skij and J. Rutiskij, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
  • [14] A. Kufner, O. John, and S. Fučik, Function Spaces, Noordhoff, Leyden, 1977.
  • [15] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1645–1658.
  • [16] V. K. Le, Variational inequalities with general multivalued lower order terms given by integrals, Adv. Nonlinear Stud. 11 (2011), 1–24.
  • [17] V. K. Le, On variational inequalities with maximal monotone operators and multivalued perturbing terms in Sobolev spaces with variable exponents, J. Math. Anal. Appl. 388 (2012), no. 2, 695–715.
  • [18] V. K. Le and D. Motreanu, On nontrivial solutions of variational-hemivariational inequalities with slowly growing principal parts, Z. Anal. Anwend. 28 (2009), no. 3, 277–293.
  • [19] R. C. M. Nemer and J. A. Santos, Multiple solutions for an inclusion quasilinear problem with nonhomogeneous boundary condition through Orlicz-Sobolev spaces, Commun. Contemp. Math. 19 (2017), no. 3, art. ID 1650031.