Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

详细信息    查看全文

• 作者：Vy Khoi Le
• 关键词：Quasi ; variational inequality ; maximal monotone mapping ; mapping of class (S)+ ; multivalued mapping
• 刊名：Results in Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：71
• 期：1-2
• 页码：423-453
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-9012
• 卷排序：71

文摘

This paper is about the existence of solutions of the following abstract quasi-variational inequality:$$\left\{\begin{array}{ll}\langle {\mathcal{A}}(u) + {\mathcal F} (u), v-u\rangle + J_u(v) -J_u(u) \ge 0,&\quad \forall v\in X,\\ u \in D(J_u).&\end{array}\right.$$where X is a reflexive Banach space, \({{\mathcal{A}}}\) is a maximal monotone mapping, \({{\mathcal F}}\) is a multivalued mapping which is compact in certain sense, and J_u is a convex and lower semicontinuous functional depending on u. We are interested in solvability conditions for the above inequality that extends those for a class of boundary value problems containing elliptic operators with multivalued coefficients.KeywordsQuasi-variational inequalitymaximal monotone mappingmapping of class (S)₊multivalued mappingMathematics Subject Classification58E3547J2047J2535J87References1.Baiocchi C., Capelo A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)MATHGoogle Scholar2.Bensoussan A., Goursat M., Lions J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires. C. R. Acad. Sci. Paris Sér. A-B 276, A1279–A1284 (1973)MATHGoogle Scholar3.Bensoussan A., Lions J.-L.: Applications of Variational Inequalities in Stochastic Control. Studies in Mathematics and its Applications, vol. 12. North-Holland, Amsterdam (1982)Google Scholar4.Bensoussan A., Lions J.-L.: Impulse Control and Quasivariational Inequalities. Gauthier-Villars, Montrouge (1984)Google Scholar5.Browder F.E., Hess P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)MathSciNetCrossRefMATHGoogle Scholar6.Carl S., Le V.K.: Elliptic inequalities with multi-valued operators: existence, comparison and related variational–hemivariational type inequalities. Nonlinear Anal. 121, 130–152 (2015)MathSciNetCrossRefMATHGoogle Scholar7.Ceng L.-C., Yao J.-C.: Existence theorems for generalized set-valued mixed (quasi-)variational inequalities in Banach spaces. J. Glob. Optim. 55(1), 27–51 (2013)MathSciNetCrossRefMATHGoogle Scholar8.Cosso A.: Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities. SIAM J. Control Optim. 51(3), 2102–2131 (2013)MathSciNetCrossRefMATHGoogle Scholar9.Fukao T., Kenmochi N.: Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete Contin. Dyn. Syst. 35(6), 2523–2538 (2015)MathSciNetCrossRefMATHGoogle Scholar10.Hu, S.C., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I. Mathematics and its Applications, vol. 419. Kluwer Academic Publishers, Dordrecht (1997)11.Jadamba B., Khan A.A., Sama M.: Generalized solutions of quasi variational inequalities. Optim. Lett. 6(7), 1221–1231 (2012)MathSciNetCrossRefMATHGoogle Scholar12.Joly, J.-L., Mosco, U.: À propos de l’existence et de la régularité des solutions de certaines inéquations quasi-variationnelles. J. Funct. Anal. 34, 107–137 (1979)13.Kadoya, A., Kenmochi, N., Niezgódka, M.: Quasi-variational inequalities in economic growth models with technological development. Adv. Math. Sci. Appl. 24(1), 185–214 (2014)14.Kano, R., Kenmochi, N., Murase, Y.: Existence Theorems for Elliptic Quasi-Variational Inequalities in Banach Spaces. Recent Advances in Nonlinear Analysis, pp. 149–169. World Scientific Publishing, Hackensack (2008)15.Kenmochi, N.: Monotonicity and Compactness Methods for Nonlinear Variational Inequalities. Handbook of Differential Equations, vol. IV, pp. 203–298. Elsevier/North-Holland, Amsterdam (2007)16.Kourogenis N.C., Papageorgiou N.S.: Nonlinear hemivariational inequalities of second order using the method of upper–lower solutions. Proc. Am. Math. Soc. 131, 2359–2369 (2003)MathSciNetCrossRefMATHGoogle Scholar17.Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-Variational Inequalities in Mechanics. Solid Mechanics and its Applications, vol. 147. Springer, Dordrecht (2007)18.Le V.K.: Existence of positive solutions of variational inequalities by a subsolution–supersolution approach. J. Math. Anal. Appl. 252, 65–90 (2000)MathSciNetCrossRefMATHGoogle Scholar19.Le V.K.: Subsolution–supersolution method in variational inequalities. Nonlinear Anal. 45, 775–800 (2001)MathSciNetCrossRefMATHGoogle Scholar20.Le V.K.: A range and existence theorem for pseudomonotone perturbations of maximal monotone operators. Proc. Am. Math. Soc. 139, 1645–1658 (2011)MathSciNetCrossRefMATHGoogle Scholar21.Le V.K.: Sub-supersolution method in variational inequalities with multivalued operators given by integrals. Rocky Mt. J. Math. 41, 535–553 (2011)MathSciNetCrossRefMATHGoogle Scholar22.Le V.K.: Variational inequalities with general multivalued lower order terms given by integrals. Adv. Nonlinear Stud. 11, 1–24 (2011)MathSciNetCrossRefGoogle Scholar23.Le V.K.: On variational inequalities with maximal monotone operators and multivalued perturbing terms in Sobolev spaces with variable exponents. J. Math. Anal. Appl. 388, 695–715 (2012)MathSciNetCrossRefMATHGoogle Scholar24.Le V.K.: On variational and quasi-variational inequalities with multivalued lower order terms and convex functionals. Nonlinear Anal. 94, 12–31 (2014)MathSciNetCrossRefMATHGoogle Scholar25.Lenzen, F., Becker, F., Lellmann, J., Petra, S., Schnörr, C.: A class of quasi-variational inequalities for adaptive image denoising and decomposition. Comput. Optim. Appl. 54(2), 371–398 (2013)26.Loc N.H., Schmitt K.: Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities. Nonlinear Anal. 75, 4664–4671 (2012)MathSciNetCrossRefMATHGoogle Scholar27.Loc N.H., Schmitt K.: Nonlinear elliptic Dirichlet and no-flux boundary value problems. Ann. Univ. Buchar. Math. Ser. 3(LXI), 201–217 (2012)MathSciNetMATHGoogle Scholar28.Moreira D.R., Teixeira E.V.: Weak convergence under nonlinearities. An. Acad. Brasil. Ciênc. 75, 9–19 (2003)MathSciNetMATHGoogle Scholar29.Moreira, D.R., Teixeira, E.V.: On the behavior of weak convergence under nonlinearities and applications. Proc. Am. Math. Soc. 133, 1647–1656 (2005) (electronic)30.Murase, Y.: Abstract Quasi-Variational Inequalities of Elliptic Type and Applications. Nonlocal and Abstract Parabolic Equations and Their Applications, vol. 86, pp. 235–246. Banach Center Publications/Polish Academy of Sciences Institute of Mathematics, Warsaw (2009)31.Tartar L.: Inéquations quasi variationnelles abstraites. C. R. Acad. Sci. Paris Sér. A 278, 1193–1196 (1974)MATHGoogle Scholar32.Zeidler E.: Nonlinear Functional Analysis and its Applications, vol. 3: Variational Methods and Optimization. Springer, Berlin (1985)CrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsVy Khoi Le1Email author1.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA About this article CrossMark Publisher Name Springer International Publishing Print ISSN 1422-6383 Online ISSN 1420-9012 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; letter-spacing: .017em; display: inline-block; line-height: 1.2; padding: 0; } .buybox__header .buybox__login:before { position: absolute; top: 50%; -webkit-transform: perspective(1px) translateY(-50%); transform: perspective(1px) translateY(-50%); content: '\A'; width: 34px; height: 34px; left: 10px; } /*---------------------------------*/ .buybox .buybox__body { padding: 0; padding-bottom: 16px; position: relative; text-align: center; background-color: #fcfcfc; border: 1px solid #b3b3b3; } .buybox__body .buybox__section { padding: 16px 12px 0 12px; text-align: left; } .buybox__section .buybox__buttons { text-align: center; width: 100%; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section .buybox__buttons { border-top: 0; padding-top: 0; } /******/ .buybox__section:nth-child(2) .buybox__buttons { border-top: 1px solid #b3b3b3; padding-top: 20px; } .buybox__buttons .buybox__buy-button { display: inline-block; text-align: center; margin-bottom: 5px; padding: 6px 12px; } .buybox__buttons .buybox__price { white-space: nowrap; text-align: center; font-size: larger; padding-top: 6px; } .buybox__section .buybox__meta { letter-spacing: 0; padding-top: 12px; } .buybox__section .buybox__meta:only-of-type { padding-top: 0; position: relative; bottom: 6px; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section .buybox__meta { margin-top: 0; margin-bottom: 0; } /******/ .buybox__meta .buybox__product-title { display: inline; font-weight: bold; } .buybox__meta .buybox__list { line-height: 1.3; } .buybox__meta .buybox__list li { position: relative; padding-left: 1em; list-style: none; margin-bottom: 5px; } .buybox__meta .buybox__list li:before { font-size: 1em; content: '\2022'; float: left; position: relative; top: .1em; font-family: serif; font-weight: 600; text-align: center; line-height: inherit; color: #666; width: auto; margin-left: -1em; } .buybox__meta .buybox__list li:last-child { margin-bottom: 0; } /*---------------------------------*/ .buybox .buybox__footer { border: 1px solid #b3b3b3; border-top: 0; padding: 8px 12px; position: relative; border-style: dashed; } /*-----------------------------------------------------------------*/ @media screen and (min-width: 460px) and (max-width: 1074px) { .buybox__body .buybox__section { display: inline-block; vertical-align: top; padding: 12px 12px; padding-bottom: 0; text-align: left; width: 48%; } .buybox__body .buybox__section { padding-top: 16px; padding-left: 0; } .buybox__section:nth-of-type(2) .buybox__meta { border-left: 1px solid #d3d3d3; padding-left: 28px; } .buybox__section:nth-of-type(2) .buybox__buttons { border-top: 0; padding-top: 0; padding-left: 16px ; } .buybox__buttons .buybox__buy-button { } /********** article buybox specific **********/ .buybox.article__buybox .buybox__section:nth-of-type(2) { margin-top: 16px; padding-top: 0; } .buybox.article__buybox .buybox__section:nth-of-type(2) .buybox__meta { margin-top: 40px; padding-top: 0; padding-bottom: 45px; } .buybox.article__buybox .buybox__section:nth-of-type(2) .buybox__meta:only-of-type { margin-top: 8px; padding-top: 12px; padding-bottom: 12px; } /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__section:first-child { width: 69%; } .buybox.mycopy__buybox .buybox__section:last-child { width: 29%; } /******/ } /*-----------------------------------------------------------------*/ @media screen and (max-width: 459px) { /********** mycopy buybox specific **********/ .buybox.mycopy__buybox .buybox__body { padding-bottom: 5px; } .buybox.mycopy__buybox .buybox__section:last-child { text-align: center; width: 100%; } .buybox.mycopy__buybox .buybox__buttons { display: inline-block; width: 150px ; } /******/ } /*-----------------------------------------------------------------*/ Log in to check access Buy (PDF) EUR 34,95 Unlimited access to the full article Instant download Include local sales tax if applicable Find out about institutional subscriptions (function () { var forEach = function (array, callback, scope) { for (var i = 0; i Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips