American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints
  • JIMO Home
  • This Issue
  • Next Article
    Does the existence of "talented outliers" help improve team performance? Modeling heterogeneous personalities in teamwork
September  2020, 16(5): 2495-2502. doi: 10.3934/jimo.2019065

Strict feasibility of variational inclusion problems in reflexive Banach spaces

Xue-Ping Luo 1, , Yi-Bin Xiao 2,, and  Wei Li 3,

1. 

The Key Laboratory for Computer Systems of State Ethnic Affairs Commission, Southwest Minzu University, Chengdu 610041, China
2. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
3. 

Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China

*Corresponding author

Received  November 2018 Revised  December 2018 Published  May 2019

Fund Project: The work was supported by National Natural Science Foundation of China (Grant 11771067, 11701480), China Postdoctoral Science Foundation (Grant 2018M631072), Applied Basic Project of Sichuan Province (Grant 2019YJ0204), Fundamental Research Funds for the Central Universities, Southwest Minzu University (Grant 2018HQZZ23), Key Projects of the Education Department of Sichuan Province (Grant 18ZA0511), Innovation Team Funds of Southwest Minzu University (Grant 14CXTD03), Innovative Research Team of the Education Department of Sichuan Province (Grant 15TD0050) and Sichuan Youth Science and Technology Innovation Research Team (Grant 2017TD0028)

In this paper, we are denoted to introducing the strict feasibility of a variational inclusion problem as a novel notion. After proving a new equivalent characterization for the nonemptiness and boundedness of the solution set for the variational inclusion problem under consideration, it is proved that the nonemptiness and boundedness of the solution set for the variational inclusion problem with a maximal monotone mapping is equivalent to its strict feasibility in reflexive Banach spaces.

Keywords: Variational inclusion problems, strict feasibility, maximal monotone mapping.
Mathematics Subject Classification: 47H04, 47H05.
Citation: Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
References:
[1]

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.  Google Scholar

[2]

Y. He and K. Ng, Strict feasibility of generalized complementarity problems, J. Aust. Math. Soc. Ser A., 81 (2006), 15-20.  doi: 10.1017/S1446788700014609.  Google Scholar

[3]

Y. He, X. Mao and M. Zhou, Strict feasibility of variational inequalities in reflexive Banach spaces, Acta Math. Sin. Engl. Ser., 23 (2007), 563-570.  doi: 10.1007/s10114-005-0918-5.  Google Scholar

[4]

Y. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330 (2007), 352-363.  doi: 10.1016/j.jmaa.2006.07.063.  Google Scholar

[5]

R. Hu and Y. Fang, Strict feasibility and stable solvability of bifunction variational inequalities, Nonlinear Anal., 75 (2012), 331-340.  doi: 10.1016/j.na.2011.08.036.  Google Scholar

[6]

R. Hu and Y. Fang, A characterization of nonemptiness and boundedness of the solution sets for equilibrium problems, Positivity, 17 (2013), 431-441.  doi: 10.1007/s11117-012-0178-4.  Google Scholar

[7]

R. Hu and Y. Fang, Feasibility-solvability theorem for a generalized system, J. Optim. Theory Appl., 142 (2009), 493-499.  doi: 10.1007/s10957-009-9510-y.  Google Scholar

[8]

R. Hu and et. al., Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459.   Google Scholar

[9]

W. Li and et. al., Existence and stability for a generalized differential mixed quasi-variational inequality, Carpathian J. Math., 34 (2018), 347-354.   Google Scholar

[10]

J. Lu, Y. Xiao and N. Huang, A stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian J. Math., 34 (2018), 355-362.   Google Scholar

[11]

X. Luo and N. Huang, A new class of variational inclusions with B-monotone operators in Banach spaces, J. Comput. Appl. Math., 233 (2010), 1888-1896.  doi: 10.1016/j.cam.2009.09.025.  Google Scholar

[12]

X. Luo and N. Huang, $(H,\phi)$ -$\eta$ -monotone operators in Banach spaces with an application to variational inclusions, Appl. Math. Comput., 216 (2010), 1131-1139.  doi: 10.1016/j.amc.2010.02.005.  Google Scholar

[13]

X. Luo and N. Huang, Generalized $H$ -$\eta$ -accretive operators in Banach spaces with an application to variational inclusions, Appl. Math. Mech. Engl. Ser., 31 (2010), 501-510.  doi: 10.1007/s10483-010-0410-6.  Google Scholar

[14]

X. Luo, Quasi-strict feasibility of generalized mixed variational inequalities in reflexive Banach spaces, J. Optim. Theory Appl., 178 (2018), 439-454.  doi: 10.1007/s10957-018-1278-5.  Google Scholar

[15]

S. Migórski and S. D. Zeng, Penalty and regularization method for variational hemivariational inequalities with application to frictional contact, Z. Angew. Math. Phys., 98 (2018), 1503-1512.  doi: 10.1002/zamm.201700348.  Google Scholar

[16]

A. Nagurney, Network Economics: A Variational Inequality Approach, Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[17]

F. Qiao and Y. He, Strict feasibility of pseudomotone set-valued variational inequality, Optim., 60 (2011), 303-310.  doi: 10.1080/02331934.2010.507985.  Google Scholar

[18]

M. Sofonea and Y. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 95 (2016), 2464-2484.  doi: 10.1080/00036811.2015.1093623.  Google Scholar

[19]

M. Sofonea and Y. Xiao, Boundary optimal control of a nonsmooth frictionless contact problem, Comp. Math. Appl., in Press. doi: 10.1016/j.camwa.2019.02.027.  Google Scholar

[20]

M. Sofonea, Y. Xiao and M. Couderc, Optimization problems for elastic contact models with unilateral constraints, Z. Angew. Math. Phys., 70 (2019), Art. 1, 17 pp. doi: 10.1007/s00033-018-1046-2.  Google Scholar

[21]

M. Sofonea, Y. Xiao and M. Couderc, Optimization problems for a viscoelastic frictional contact problem with unilateral constraints, submitted. Google Scholar

[22]

M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009.  Google Scholar

[23]

Q. Shu, R. Hu and Y. Xiao, Metric characterizations for well-psedness of split hemivariational inequalities, J. Ineq. Appl., 2018 (2018), Paper No. 190, 17 pp. doi: 10.1186/s13660-018-1761-4.  Google Scholar

[24]

Y. M. Wang and et al., Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.  Google Scholar

[25]

Y. Xiao, N. Huang and Y. Cho, A class of generalized evolution variational inequalities in Banach spaces, Appl. Math. Lett., 25 (2012), 914-920.  doi: 10.1016/j.aml.2011.10.035.  Google Scholar

[26]

Y. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl., 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.  Google Scholar

[27]

Y. Xiao and M. Sofonea, Generalized penalty method for elliptic variational- hemivariational inequalities, Appl. Math. Optim., in press. doi: 10.1007/s00245-019-09563-4.  Google Scholar

[28]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[29]

S. D. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072.  Google Scholar

[30]

Y. Zhang, Y. He and Y. Jiang, Existence and boundedness of solutions to maximal monotone inclusion problem, Optim. Lett., 11 (2017), 1565-1570.  doi: 10.1007/s11590-016-1064-y.  Google Scholar

[31]

R. Zhong and N. Huang, Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 152 (2012), 696-709.  doi: 10.1007/s10957-011-9914-3.  Google Scholar

show all references

References:
[1]

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.  Google Scholar

[2]

Y. He and K. Ng, Strict feasibility of generalized complementarity problems, J. Aust. Math. Soc. Ser A., 81 (2006), 15-20.  doi: 10.1017/S1446788700014609.  Google Scholar

[3]

Y. He, X. Mao and M. Zhou, Strict feasibility of variational inequalities in reflexive Banach spaces, Acta Math. Sin. Engl. Ser., 23 (2007), 563-570.  doi: 10.1007/s10114-005-0918-5.  Google Scholar

[4]

Y. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330 (2007), 352-363.  doi: 10.1016/j.jmaa.2006.07.063.  Google Scholar

[5]

R. Hu and Y. Fang, Strict feasibility and stable solvability of bifunction variational inequalities, Nonlinear Anal., 75 (2012), 331-340.  doi: 10.1016/j.na.2011.08.036.  Google Scholar

[6]

R. Hu and Y. Fang, A characterization of nonemptiness and boundedness of the solution sets for equilibrium problems, Positivity, 17 (2013), 431-441.  doi: 10.1007/s11117-012-0178-4.  Google Scholar

[7]

R. Hu and Y. Fang, Feasibility-solvability theorem for a generalized system, J. Optim. Theory Appl., 142 (2009), 493-499.  doi: 10.1007/s10957-009-9510-y.  Google Scholar

[8]

R. Hu and et. al., Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459.   Google Scholar

[9]

W. Li and et. al., Existence and stability for a generalized differential mixed quasi-variational inequality, Carpathian J. Math., 34 (2018), 347-354.   Google Scholar

[10]

J. Lu, Y. Xiao and N. Huang, A stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian J. Math., 34 (2018), 355-362.   Google Scholar

[11]

X. Luo and N. Huang, A new class of variational inclusions with B-monotone operators in Banach spaces, J. Comput. Appl. Math., 233 (2010), 1888-1896.  doi: 10.1016/j.cam.2009.09.025.  Google Scholar

[12]

X. Luo and N. Huang, $(H,\phi)$ -$\eta$ -monotone operators in Banach spaces with an application to variational inclusions, Appl. Math. Comput., 216 (2010), 1131-1139.  doi: 10.1016/j.amc.2010.02.005.  Google Scholar

[13]

X. Luo and N. Huang, Generalized $H$ -$\eta$ -accretive operators in Banach spaces with an application to variational inclusions, Appl. Math. Mech. Engl. Ser., 31 (2010), 501-510.  doi: 10.1007/s10483-010-0410-6.  Google Scholar

[14]

X. Luo, Quasi-strict feasibility of generalized mixed variational inequalities in reflexive Banach spaces, J. Optim. Theory Appl., 178 (2018), 439-454.  doi: 10.1007/s10957-018-1278-5.  Google Scholar

[15]

S. Migórski and S. D. Zeng, Penalty and regularization method for variational hemivariational inequalities with application to frictional contact, Z. Angew. Math. Phys., 98 (2018), 1503-1512.  doi: 10.1002/zamm.201700348.  Google Scholar

[16]

A. Nagurney, Network Economics: A Variational Inequality Approach, Advances in Computational Economics, 1, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[17]

F. Qiao and Y. He, Strict feasibility of pseudomotone set-valued variational inequality, Optim., 60 (2011), 303-310.  doi: 10.1080/02331934.2010.507985.  Google Scholar

[18]

M. Sofonea and Y. Xiao, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 95 (2016), 2464-2484.  doi: 10.1080/00036811.2015.1093623.  Google Scholar

[19]

M. Sofonea and Y. Xiao, Boundary optimal control of a nonsmooth frictionless contact problem, Comp. Math. Appl., in Press. doi: 10.1016/j.camwa.2019.02.027.  Google Scholar

[20]

M. Sofonea, Y. Xiao and M. Couderc, Optimization problems for elastic contact models with unilateral constraints, Z. Angew. Math. Phys., 70 (2019), Art. 1, 17 pp. doi: 10.1007/s00033-018-1046-2.  Google Scholar

[21]

M. Sofonea, Y. Xiao and M. Couderc, Optimization problems for a viscoelastic frictional contact problem with unilateral constraints, submitted. Google Scholar

[22]

M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009.  Google Scholar

[23]

Q. Shu, R. Hu and Y. Xiao, Metric characterizations for well-psedness of split hemivariational inequalities, J. Ineq. Appl., 2018 (2018), Paper No. 190, 17 pp. doi: 10.1186/s13660-018-1761-4.  Google Scholar

[24]

Y. M. Wang and et al., Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.  Google Scholar

[25]

Y. Xiao, N. Huang and Y. Cho, A class of generalized evolution variational inequalities in Banach spaces, Appl. Math. Lett., 25 (2012), 914-920.  doi: 10.1016/j.aml.2011.10.035.  Google Scholar

[26]

Y. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl., 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.  Google Scholar

[27]

Y. Xiao and M. Sofonea, Generalized penalty method for elliptic variational- hemivariational inequalities, Appl. Math. Optim., in press. doi: 10.1007/s00245-019-09563-4.  Google Scholar

[28]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[29]

S. D. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072.  Google Scholar

[30]

Y. Zhang, Y. He and Y. Jiang, Existence and boundedness of solutions to maximal monotone inclusion problem, Optim. Lett., 11 (2017), 1565-1570.  doi: 10.1007/s11590-016-1064-y.  Google Scholar

[31]

R. Zhong and N. Huang, Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 152 (2012), 696-709.  doi: 10.1007/s10957-011-9914-3.  Google Scholar

[1]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[2]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[3]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[4]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
[5]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051
[6]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[7]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200
[8]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[9]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
[10]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : i-i. doi: 10.3934/dcdss.2020446
[11]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[12]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[13]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
[14]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[15]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[16]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[17]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263
[18]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452
[19]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
[20]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (110)
  • HTML views (620)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences