## Banach Journal of Mathematical Analysis

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

Vy Khoi Le

#### Abstract

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

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

Dates
Accepted: 13 October 2018
First available in Project Euclid: 28 January 2019

https://projecteuclid.org/euclid.bjma/1548666054

Digital Object Identifier
doi:10.1215/17358787-2018-0034

Mathematical Reviews number (MathSciNet)
MR3927875

Zentralblatt MATH identifier
07045460

#### Citation

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. https://projecteuclid.org/euclid.bjma/1548666054

#### References

• [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.