Existence of variational quasi-hemivariational inequalities involving a set-valued operatorand a nonlinear term

详细信息    查看全文

• 作者：Guo-ji Tang (1)
  Xing Wang (2)
  Zhong-bao Wang (3)
  
  1. School of Science ; Guangxi University for Nationalities ; Nanning ; 530006 ; Guangxi ; People鈥檚 Republic of China
  2. School of Information Technology ; Jiangxi University of Finance and Economics ; Nanchang ; 330013 ; People鈥檚 Republic of China
  3. Department of Mathematics ; Southwest Jiaotong University ; Chengdu ; 610031 ; People鈥檚 Republic of China
  
• 关键词：Variational quasi ; hemivariational inequalities ; Generalized monotonicity ; F ; KKM theorem ; Existence
• 刊名：Optimization Letters
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：9
• 期：1
• 页码：75-90
• 全文大小：204 KB
• 参考文献：1. Carl, S.: Existence of extremal solutions of boundary hemivariational inequalities. J. Differ. Equ. 171, 370鈥?96 (2001) CrossRef
  2. Carl, S., Le, V.K., Motreanu, D.: Existence and comparison principles for general quasilinear variational-hemivariational inequalities. J. Math. Anal. Appl. 302, 65鈥?3 (2005) CrossRef
  3. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities, Comparison Principles and Applications. Springer, New York (2007) CrossRef
  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
  5. Costea, N., R膬dulescu, V.: Existence results for hemivariational inequalities involving relaxed \(\eta -\alpha \) monotone mappings. Commun. Appl. Anal. 13, 293鈥?04 (2009)
  6. Costea, N., R膬dulescu, V.: Hartman鈥揝tampacchia results for stably pseudomonotone operators and nonlinear hemivariational inequalities. Appl. Anal. 89, 175鈥?88 (2010) CrossRef
  7. Costea, N., Ion, D.A., Lupu, C.: Variational-like inequality problems involving set-valued maps and generalized monotonicity. J. Optim. Theory Appl. 155, 79鈥?9 (2012) CrossRef
  8. Costea, N., R膬dulescu, V.: Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term. J. Glob. Optim. 52, 743鈥?56 (2012) CrossRef
  9. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)
  10. Fan, K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266, 519鈥?37 (1984) CrossRef
  11. Farajzadeh, A., Karamian, A., Plubtieng, S.: On the translations of quasimonotone maps and monotonicity, J. Inequalities Appl. (2012) ofinequalitiesandapplications.com/content/2012/1/192" class="a-plus-plus">http://www.journalofinequalitiesandapplications.com/content/2012/1/192
  12. Hadjisavvas, N., Schaible, S.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90(1), 95鈥?11 (1996) CrossRef
  13. Hadjisavvas, N.: Translations of quasimonotone maps and monotonicity. Appl. Math. Lett. 19, 913鈥?15 (2006) CrossRef
  14. He, Y.R.: Asymptotic analysis and error bounds in optimization. Ph.D. Thesis, The Chinese University of Hong Kong (2001)
  15. He, Y.R.: A relationship between pseudomonotone and monotone mappings. Appl. Math. Lett. 17, 459鈥?61 (2004) CrossRef
  16. He, Y.R.: Stable pseudomonotone variational inequality in reflexive Banach spaces. J. Math. Anal. Appl. 330, 352鈥?63 (2007) CrossRef
  17. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
  18. Liu, Z.H.: Existence results for quasilinear parabolic hemivariational inequalities. J. Differ. Equ. 244, 1395鈥?409 (2008) CrossRef
  19. Mig贸rski, S., Ochal, A.: Boundary hemivariational inequality of parabolic type. Nonlinear Anal. 57, 579鈥?96 (2004) CrossRef
  20. Motreanu, D., R膬dulescu, V.: Existence results for inequality problems with lack of convexity. Numer. Funct. Anal. Optim. 21, 869鈥?84 (2000) CrossRef
  21. Motreanu, D., R膬dulescu, V.: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems. Kluwer Academic Publishers, Boston, Dordrecht, London (2003) CrossRef
  22. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)
  23. Panagiotopoulos, P.D.: Nonconvex energy functions. Hemivariational inequalities and substationarity principles. Acta Mech. 42, 160鈥?83 (1983)
  24. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications, Convex and Nonconvex Energy Functions. Birkh盲user, Basel (1985) CrossRef
  25. Panagiotopoulos, P.D.: Coercive and semicoercive hemivariational inequalities. Nonlinear Anal. 16, 209鈥?31 (1991) CrossRef
  26. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993) CrossRef
  27. Panagiotopoulos, P.D., Fundo, M., R膬dulescu, V.: Existence theorems of Hartman鈥揝tampacchia type for hemivariational inequalities and applications. J. Glob. Optim. 15, 41鈥?4 (1999) CrossRef
  28. Park, J.Y., Ha, T.G.: Existence of antiperiodic solutions for hemivariational inequalities. Nonlinear Anal. 68, 747鈥?67 (2008) CrossRef
  29. Park, J.Y., Ha, T.G.: Existence of anti-periodic solutions for quasilinear parabolic hemivariational inequalities. Nonlinear Anal. 71, 3203鈥?217 (2009) CrossRef
  30. Tang, G.J., Huang, N.J.: Existence theorems of the variational-hemivariational inequalities. J. Glob. Optim. 56, 605鈥?22 (2013) CrossRef
  31. Xiao, Y.B., Huang, N.J.: Generalized quasi-variational-like hemivariational inequalities. Nonliear Anal. 69, 637鈥?46 (2008) CrossRef
  32. Xiao, Y.B., Huang, N.J.: Sub-super-solution method for a class of higher order evolution hemivariational inequalities. Nonliear Anal. 71, 558鈥?70 (2009) CrossRef
  33. Zhang, Y.L., He, Y.R.: On stably quasimonotone hemivariational inequalities. Nonlinear Anal. 74, 3324鈥?332 (2011) CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Optimization
  Operation Research and Decision Theory
  Numerical and Computational Methods in Engineering
  Numerical and Computational Methods
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1862-4480

文摘

This paper is devoted to the existence of solutions for a class of variational quasi-hemivariational inequalities involving a lower hemicontinuous set-valued operator and a nonlinear term in reflexive Banach spaces. In the case when the constraint set is bounded, under certain generalized monotonicity conditions, we prove an existence result of solutions for the problem by means of F-KKM theorem. In the case when the constraint set is unbounded, under certain coercivity conditions, we construct an existence theorem of solutions and a boundedness theorem of the solution set for the problem, respectively. Moreover, a necessary and sufficient condition to the existence of solutions is also derived. The results presented in this paper generalize and improve some known results.