## Advances in Operator Theory

### A variational inequality theory for constrained problems in reflexive Banach spaces

T. M. ‎‎Asfaw

#### Abstract

‎‎Let $X$ be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space $X^*$‎, ‎and let $K$ be a nonempty‎, ‎closed‎, ‎and convex subset of $X$‎. ‎Let $T‎: ‎X\supseteq D(T)\to 2^{X^*}$ be maximal monotone‎, ‎let $S‎: ‎K\to 2^{X^*}$ be bounded and of type $(S_+)$‎, ‎and let $C‎: ‎X\supseteq D(C)\to X^*$ with $D(T)\cap D(\partial \phi)\cap K\subseteq D(C)$‎. ‎Let $\phi‎ : ‎X\to (-\infty‎, ‎\infty]$ be a proper‎, ‎convex‎, ‎and lower semicontinuous function‎. ‎New existence theorems are proved for solvability of variational inequality problems of the type $\rm{VIP}(T+S+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $C$ is compact and $\rm{VIP}(T+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $T$ is of compact resolvent and $C$ is bounded and continuous‎. ‎Various improvements and generalizations of the existing results for $T+S$ and $\phi$ are obtained‎. ‎The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems‎.

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 2 (2019), 462-480.

Dates
Received: 26 September 2018
Accepted: 14 October 2018
First available in Project Euclid: 1 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1543633238

Digital Object Identifier
doi:10.15352/aot.1809-1423

Mathematical Reviews number (MathSciNet)
MR3883147

Zentralblatt MATH identifier
07009320

#### Citation

‎‎Asfaw, T. M. A variational inequality theory for constrained problems in reflexive Banach spaces. Adv. Oper. Theory 4 (2019), no. 2, 462--480. doi:10.15352/aot.1809-1423. https://projecteuclid.org/euclid.aot/1543633238

#### References

• T. M. Asfaw, New variational inequality and surjectivity theories for perturbed noncoercive operators and application to nonlinear problems, Adv. Math. Sci. Appl. 24 (2014), 611–668.
• T. M. Asfaw, New surjectivity results for perturbed weakly coercive operators of monotone type in reflexive Banach space, Nonlinear Anal. 113 (2015), 209–229.
• T. M. Asfaw, Maximality theorems on the sum of two maximal monotone operators and application to variational inequality problems, Abstr. Appl. Anal. 2016, Art. ID 7826475, 10pp.
• T. M. Asfaw, A degree theory for compact perturbations of monotone type operators and application to nonlinear parabolic problem, Abstr. Appl. Anal. 2017, Art. ID 7236103, 13pp.
• T. M. Asfaw and A. G. Kartsatos, A Browder topological degree theory for multivalued pseudomonotone perturbations of maximal monotone operators in reflexive Banach spaces, Adv. Math. Sci. Appl. 22 (2012), 91–148.
• T. M. Asfaw and A. G. Kartsatos, Variational inequalities for perturbations of maximal monotone operators in reflexive Banach spaces, Tohoku Math. J. 66 (2014), 171–203.
• V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer Monographs in Mathematics, New York, 2010.
• H. Brézis, M. G. Crandall, and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. Pure Appl. Math. 23 (1970), 123–144.
• H. Brèzis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa 5 (1978), 225–326.
• F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math. 18 (1976), 1–308.
• F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal. 11 (1972), 251–294.
• D. Goeleven and D. Motreanu, Variational and hemivariational inequalities: Theory, methods and applications Vol. II. Unilateral problems. Nonconvex optimization and its applications, 70.}, Kluwer Academic Publishers, Boston, 2003.
• A. G. Kartsatos, New results in the perturbation theory of maximal monotone operators in Banach spaces, Trans. Amer. Math. Soc. 348 (1996), 1663–1707.
• N. Kenmochi, Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations, Hiroshima Math. J. 4 (1974), 229–263.
• N. Kenmochi, Pseudomonotone operators and nonlinear ellpitic boundary value problems, J. Math. Soc. Japan 27 (1975), 121–149.
• N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of differential equations, IV, Elsevier/North-Holland, Amsterdam, 2007, 203–298.
• A. A. Khan and D. Motreanu, Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities, J. Optim. Theory Appl. 167 (2015), 1136–1161.
• D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Pure Appl. Math. 88, Academic Press, New York, 1980.
• J. Kobayashi and M. Otani, Topological degree for $({\rm S})\sb +$-mappings with maximal monotone perturbations and its applications to variational inequalities, Nonlinear Anal. 59 (2004), no. 1-2, 147–172.
• V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc. 139 (2011), 1645–1658.
• Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995.
• D. Pascali and S. Sburlan, Nonlinear mappings of monotone type, Sijthoff and Noordhoff, Bucharest, 1978.
• R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75–88.
• R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997.
• S. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1971), 173–180.
• E. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990.