A CLASS OF DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES

Jun Feng; Wei Li; Hui Chen; Yuanchun Chen

doi:10.11948/2018.1664

2018 Volume 8 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(6): 1664-1678

Next Article Previous Article

Jun Feng, Wei Li, Hui Chen, Yuanchun Chen. A CLASS OF DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1664-1678. doi: 10.11948/2018.1664

Citation:

Jun Feng, Wei Li, Hui Chen, Yuanchun Chen. A CLASS OF DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1664-1678. doi: 10.11948/2018.1664

A CLASS OF DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES

1 Geomathematics Key Lab of Sichuan Province, Chengdu University of Technology, Chengdu, 610059, China;
2 State Key Laboratory of Geohazard Prevention and Geoenvironment, Chengdu University of Technology, Chengdu, 610059, China

Fund Project:

Abstract

Abstract

In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Carathéodory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.
- Differential inverse variational inequality /
- Carathéodory weak solution /
- Euler time-stepping procedure /
- algorithm
MSC: 49J40;47J20

FullText(HTML)

References

[1]	A. Barbagallo and P. Mauro, Inverse variational inequality approach and applications, Numerical Functional Analysis and Optimization, 2014, 35, 851-867. Google Scholar
[2]	M. K. Camlibel, W. P. M. H. Heemels and J. M. Schumacher, Consistency of a time-stepping method for a class of pieceise linear networks, IEEE Trans. Circuit System, 2002, 49, 349-357. Google Scholar
[3]	M. K. Camlibel, J. S. Pang and J. Shen, Lyapunov stability of complementarity and extended systems, SIAM J. Optim., 2006, 17, 1056-1101. Google Scholar
[4]	X. J. Chen and Z. Y. Wang, Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optim., 2013, 23, 1647-1671. Google Scholar
[5]	X. J. Chen and Z. Y. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 2014, 146, 379-408. Google Scholar
[6]	T. L. Friesz, Differential variational inequalities and differential Nash games, Dyn. Optim. Differ. Games, 2010, 135, 267-312. Google Scholar
[7]	J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program. Ser. B, 2013, 139, 205-221. Google Scholar
[8]	L. S. Han, et al., Convergence of time-stepping schemes for passive and extended linear complementrity systems, SIAM J. Numer. Anal., 2009, 47, 3768-3796. Google Scholar
[9]	L. S. Han and J. S. Pang, Non-zenoness of a class of differential quasivariational inequalities, Math. Program. Ser. A, 2010, 121, 171-199. Google Scholar
[10]	B. S. He, et al., PPA-based methods for monotone inverse variational inequalities, Sciencepaper Online, 2006. Google Scholar
[11]	B. S. He, X. Z. He and H. X. Liu, Sloving a class of constrained ‘blak-box’ inverse variational inequalities, Eur. J. Oper. Res., 2010, 204(3), 391-401. Google Scholar
[12]	B. S. He and H. X. Liu, Inverse variational inequalities in economicsapplications and algorithms, Sciencepaper Online, 2006. Google Scholar
[13]	X. Z. He and H. X. Liu, Inverse variational inequalities with projection-based solution methods, Eur. J. Oper. Res., 2011, 208, 12-18. Google Scholar
[14]	R. Hu, et al., Equivalence results of well-posedness for split variationalhemivariational inequalities, J. Nonlinear Convex Anal., to appear. Google Scholar
[15]	R. Hu and Y. P. Fang, Well-Posedness of the split inverse variational inequality problem, Bull. Malays. Math. Sci. Soc., 2015. DOI:10.1007/s40840-015-0213-2. Google Scholar
[16]	W. Li, et al., A class of differential inverse quasi-variational inequalities in finite dimensional spaces, J. Nonlinear Sci. Appl., 2017, 10, 4532-4543. Google Scholar
[17]	W. Li, et al., Existence and stability for a generalized differential mixed quasivariational inequality, Carpathian J. Math., 2018, 34, 347-354. Google Scholar
[18]	W. Li, X. Wang and N. J. Huang, Differential inverse variational inequalities in finite dimensional spaces, Acta Math. Sci., 2015, 35B, 407-422. Google Scholar
[19]	X. S. Li, N. J. Huang and D. O'Regan, Differential mixed variational inqualities in finite dimensional spaces, Nonlinear Anal., 2010, 72, 3875-3886. Google Scholar
[20]	J. L. Lions, Quelques Mthodes de Rsolution des Problems aux Limites Non Linaires, Dunod, Gauthier-Villars, Paris, 1969. Google Scholar
[21]	Z. H. Liu, S. D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 2016, 260, 6787-6799. Google Scholar
[22]	J. Lu, Y. B. Xiao and N. J. Huang, A Stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian J. Math., 2018, 34, 355-362. Google Scholar
[23]	D. Melanz, P. Jayakumar and D. Negrut, Experimental validation of a differential variational inequality-based approach for handling friction and contact in vehicle/granular-terrain interaction, Journal of Terramechanics, 2016, 65, 1-13. Google Scholar
[24]	J. S. Pang and J. Shen, Strongly regular differential variational systems, IEEE Trans. Automat. Control, 2007, 52, 242-255. Google Scholar
[25]	J. S. Pang and D. Stewart, Differential variational inqualities, Math. Program. Ser. A, 2008,113, 345-424. Google Scholar
[26]	A. U. Raghunathan, et al., Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities, Ann. Oper. Res., 2006, 148, 251-270. Google Scholar
[27]	L. Scrimali, An inverse variational inequality approach to the evolutionary spatial price equilibrium problem, Optim. Eng., 2012, 13, 375-387. Google Scholar
[28]	Q. Y. Shu, R. Hu and Y. B. Xiao, Metric characterizations for well-psedness of split hemivariational inequalities, J. Ineq. Appl., 2018, 190. https://doi.org/10.1186/s13660-018-1761-4. Google Scholar
[29]	M. Sofonea and Y. B. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 2016, 95, 2464-2484. Google Scholar
[30]	D. E. Stewart, Uniqueness for index-one differential variational inequalities, Nonlinear Anal. Hybrid Syst., 2008, 2, 812-818. Google Scholar
[31]	X. Wang, et al., A class of delay differential variational inequalities, J. Optim. Theory Appl., 2017, 172, 56-69. Google Scholar
[32]	X. Wang and N. J. Huang, Differential vector variational inequalities in finitedimensional spaces, J. Optim. Theory Appl., 2013, 158, 109-129. Google Scholar
[33]	Y. M. Wang, et al., Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 2016, 9, 1178-1192. Google Scholar
[34]	Y. B. Xiao, N. J. Huang and M. M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese Journal of Mathematics, 2011, 15(3), 1261-1276. Google Scholar
[35]	Y. B. Xiao and M. Sofonea, On the optimal control of variationalhemivariational inequalities, J. Math. Anal. Appl., to appear. Google Scholar
[36]	J. Yang, Dynamic power price problem:an inverse variational inequality approach, J. Ind. Manag. Optim., 2008, 4, 673-684. Google Scholar