DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES

doi:10.1016/S0252-9602(16)30112-6

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (1): 26-32.doi: 10.1016/S0252-9602(16)30112-6

• Articles • Previous Articles Next Articles

DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES

Zhenhai LIU¹, Shengda ZENG^1,2

1. Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, and College of Sciences, Guangxi University for Nationalities, Nanning 530006, China;
2. Institute of Computer Science, Faculty of Mathematics and Computer Science Jagiellonian University, ul. Lojasiewicza 6, 30-348 Krakow, Poland

Received:2015-12-31 Revised:2016-04-29 Online:2017-02-25 Published:2017-02-25
Contact: Zhenhai LIU,E-mail:zhhliu@hotmail.com E-mail:zhhliu@hotmail.com
About author:Shengda ZENG,E-mail:zengshengda@163.com
Supported by:
Project supported by NNSF of China (11671101), the National Science Center of Poland Under Maestro Advanced Project (UMO-2012/06/A/ST1/00262), and Special Funds of Guangxi Distinguished Experts Construction Engineering.

Abstract

Abstract:

In this paper, we consider a new differential variational inequality (DVI, for short) which is composed of an evolution equation and a variational inequality in infinite Banach spaces. This kind of problems may be regarded as a special feedback control problem. Based on the Browder's theorem and the optimal control theory, we show the existence of solutions to the mentioned problem.

Key words: differential variational inequality, C₀-semigroup, monotone operator, existence of solutions

Zhenhai LIU, Shengda ZENG. DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES[J].Acta mathematica scientia,Series B, 2017, 37(1): 26-32.

Trendmd

References

[1] Browder F E. Nonlinear monotone operators and convex sets in Banach spaces. Bull Amer Math Soc, 1965, 71:780-785
[2] Chen X J,Wang Z Y. Differential variational inequality approach to dynamic games with shared constraints. Math Program, 2014, 146:379-408
[3] Gwinner J. On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction. Appl Numer Math, 2009, 59:2774-2784
[4] Gwinner J. hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics. J Comput Appl Math, 2013, 254:175-184
[5] Han L, Pang J S. Non-Zenoness of a class of differential quasi-variational inequalities. Math Program, 2010, 121:171-199
[6] Ke T D, Loi N V, Obukhovskii V. Decay solutions for a class of fractional differential variational inequalities. Fract Calc Appl Anal, 2015, 18:531-553
[7] Kamemsloo M, Obukhovskii V, Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space. Berlin:Water de Gruyter, 2001
[8] Li X J, Yong J M. Optimal Control Theory for infinite Dimensional Systems. Boster:Birkhäuser, 1995
[9] Liu Z H, Zeng S D, Motreanu D. Evolutionary problems driven by variational inequalities. J Differ Equ, 2016, 260:6787-6799
[10] Liu Z H, Loi N V, Obukhovskii V. Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Int J Bifurcation Chaos, 2013, 23(7):ID:1350125
[11] Loi N V. On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities. Nonlinear Anal, 2015, 122:83-99
[12] Migórski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26. New York:Springer, 2013
[13] Papageorgiou N S, Kyritsi-Yiallourou S T. Handbook of Applied Analysis. New York:Springer, 2009
[14] Pang J S, Stewart D E. Differential variational inequalities. Math Program, 2008, 113:345-424
[15] Pang J S, Stewart D E. Solution dependence on initial conditions in differential variational variational inequalities. Math Program, 2009, 116:429-460
[16] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York:Springer-Verlag, 1983
[17] Rykaczewski K. Approximate controllability of differential inclusions in Hilbert spaces. Nonlinear Anal, 2012, 75:2701-2712
[18] Wang X, Huang N J. A class of differential vector variational inequalities in finite dimensional spaces. J Optim Theory Appl, 2014, 162:633-648

DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 12

Metrics

Comments

Recommended 10

[1]	Yekini SHEHU, Olaniyi. S. IYIOLA. CONVERGENCE OF HYBRID VISCOSITY AND STEEPEST-DESCENT METHODS FOR PSEUDOCONTRACTIVE MAPPINGS AND NONLINEAR HAMMERSTEIN EQUATIONS [J]. Acta mathematica scientia,Series B, 2018, 38(2): 610-626.
[2]	Kazuhide NAKAJO. IMPROVED GRAOIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES [J]. Acta mathematica scientia,Series B, 2017, 37(2): 342-354.
[3]	Li WEI, Rui CHEN, Ravi P. AGARWAL, Patricia YJ WONG. EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES [J]. Acta mathematica scientia,Series B, 2016, 36(6): 1780-1792.
[4]	Yue GUAN, Hua ZHANG. CONVERGENCE OF INVARIANT MEASURES FOR MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS [J]. Acta mathematica scientia,Series B, 2016, 36(2): 487-498.
[5]	Wei LI, Xing WANG, Nanjing HUANG. DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES [J]. Acta mathematica scientia,Series B, 2015, 35(2): 407-422.
[6]	CHI Hong-Mei. MULTIVALUED STOCHASTIC MCKEAN-VLASOV EQUATION [J]. Acta mathematica scientia,Series B, 2014, 34(6): 1731-1740.
[7]	LIU San-Yang, HE Hui-Min, CHEN Ru-Dong. APPROXIMATING SOLUTION OF 0 ∈T(x) FOR AN H-MONOTONE OPERATOR IN HILBERT SPACES [J]. Acta mathematica scientia,Series B, 2013, 33(5): 1347-1360.
[8]	PENG Zi-Jia, LIU Yi-Liang. EXISTENCE RESULTS FOR A CLASS OF PSEUDOMONOTONE ELLIPTIC-PARABOLIC INCLUSIONS [J]. Acta mathematica scientia,Series B, 2011, 31(5): 1709-1718.
[9]	YUAN Cheng-Jun, WEN Xiang-Dan, JIANG Da-Qing. EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR NONLINEAR SINGULAR 2mth-ORDER CONTINUOUS AND DISCRETE LIDSTONE BOUNDARY VALUE PROBLEMS [J]. Acta mathematica scientia,Series B, 2011, 31(1): 281-291.
[10]	DIAO Dun-Ning, CENG Xiao-Meng. THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS [J]. Acta mathematica scientia,Series B, 2005, 25(4): 577-586.
[11]	Leszek Gasi′nski;Nikolaos S. Papageorgiou. ON NONLINEAR ELLIPTIC HEMIVARIATIONAL INEQUALITIES OF SECOND ORDER [J]. Acta mathematica scientia,Series B, 2004, 24(3): 451-462.
[12]	XIAO Jin-Sheng, SUN Le-Lin. AN ITERATIVE ALGORITHM FOR MAXIMAL MONOTONE MULTIVALUED OPERATOR EQUATIONS [J]. Acta mathematica scientia,Series B, 2001, 21(2): 152-158.