Abstract
We are concerned in this article with the existence of solutions to inclusions containing generalized pseudomonotone perturbations of maximal monotone mappings in general Banach spaces. Our approach is based on a truncation–regularization technique and an extension of the Moreau–Yosida–Brezis–Crandall–Pazy regularization for maximal monotone mappings in general Banach spaces. We also consider some applications to multivalued variational inequalities containing elliptic operators with rapidly growing coefficients in Orlicz–Sobolev spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, R.: Sobolev Spaces. Academic Press, New York (1975)
Alves, M.M., Svaiter, B.F.: Moreau–Yosida regularization of maximal monotone operators of type (D). Set Valued Var. Anal. 19(1), 97–106 (2011)
Brezis, H., Crandall, M.G., Pazy, A.: Perturbations of nonlinear maximal monotone sets in Banach space. Commun. Pure Appl. Math. 23, 123–144 (1970)
Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)
Carl, S., Le, V.K.: Multi-Valued Variational Inequalities and Inclusions, Springer Monographs in Mathematics. Springer, Switzerland (2021)
Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities, Comparison principles and applications. Springer Monographs in Mathematics. Springer, New York (2007)
Gossez, J.P.: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs. J. Math. Anal. Appl. 34, 371–395 (1971)
Gossez, J.P.: Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)
Gossez, J.P., Mustonen, V.: Variational inequalities in Orlicz–Sobolev spaces. Nonlinear Anal. 11, 379–392 (1987)
Guan, Z., Kartsatos, A.G., Skrypnik, I.V.: Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators. J. Differ. Equ. 188, 332–351 (2003)
Hu, S.C., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Mathematics and Its Applications, vol. 419. Kluwer Academic Publishers, Dordrecht (1997)
Kartsatos, A.G., Skrypnik, I.V.: A new topological degree theory for densely defined quasibounded \(({{{\tilde{S}}}}_+)\)-perturbations of multivalued maximal monotone operators in reflexive Banach spaces. Abstr. Appl. Anal. 2, 121–158 (2005)
Kenmochi, N.: Monotonicity and Compactness Methods for Nonlinear Variational Inequalities. Handbook of Differential Equations, vol. IV, pp. 203–298. Elsevier, Amsterdam (2007)
Khan, A.A., Motreanu, D.: Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities. J. Optim. Theory Appl. 167, 1136–1161 (2015)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Krasnosel’skii, M.A., Rutic’kii, J.: Convex Functions and Orlicz Spaces. Noordhoff, Groningen (1961)
Kufner, A., John, O., Fučic, S.: Function Spaces. Noordhoff, Leyden (1977)
Le, V.K.: A range and existence theorem for pseudomonotone perturbations of maximal monotone operators. Proc. Am. Math. Soc. 139, 1645–1658 (2011)
Le, V.K.: On second order elliptic equations and variational inequalities with anisotropic principal operators. Topol. Methods Nonlinear Anal. 44, 41–72 (2014)
Le, V.K.: On the convergence of solutions of inclusions containing maximal monotone and generalized pseudomonotone mappings. Nonlinear Anal. 143, 64–88 (2016)
Le, V.K.: On the existence of solutions of variational inequalities in nonreflexive Banach spaces. Banach J. Math. Anal. 13, 293–313 (2019)
Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
Rudd, M., Schmitt, K.: Variational inequalities of elliptic and parabolic type. Taiwan. J. Math. 6(3), 287–322 (2002)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Variational Methods and Optimization, vol. 3. Springer, Berlin (1985)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications Nonlinear Monotone Operators, vol. 2B. Springer, New York (1990)
Acknowledgements
The author would like to thank the reviewers for their insightful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Akhtar A. Khan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Le, V.K. On Inclusions with Monotone-Type Mappings in Nonreflexive Banach Spaces. J Optim Theory Appl 192, 484–509 (2022). https://doi.org/10.1007/s10957-021-01973-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-021-01973-1
Keywords
- Monotone mapping
- Nonreflexive Banach space
- Multivalued mapping
- Variational inequality
- Orlicz–Sobolev space