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September  2020, 16(5): 2495-2502. doi: 10.3934/jimo.2019065

## Strict feasibility of variational inclusion problems in reflexive Banach spaces

 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  September 2020 Early access  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.

Citation: Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial and 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. [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. [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. [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. [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. [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. [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. [8] R. Hu and et. al., Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459. [9] W. Li and et. al., Existence and stability for a generalized differential mixed quasi-variational inequality, Carpathian J. Math., 34 (2018), 347-354. [10] J. Lu, Y. Xiao and N. Huang, A stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian J. Math., 34 (2018), 355-362. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] M. Sofonea, Y. Xiao and M. Couderc, Optimization problems for a viscoelastic frictional contact problem with unilateral constraints, submitted. [22] M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009. [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. [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. [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. [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. [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. [28] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. [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. [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. [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.

show all references

##### References:
 [1] F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003. [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. [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. [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. [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. [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. [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. [8] R. Hu and et. al., Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459. [9] W. Li and et. al., Existence and stability for a generalized differential mixed quasi-variational inequality, Carpathian J. Math., 34 (2018), 347-354. [10] J. Lu, Y. Xiao and N. Huang, A stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian J. Math., 34 (2018), 355-362. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] M. Sofonea, Y. Xiao and M. Couderc, Optimization problems for a viscoelastic frictional contact problem with unilateral constraints, submitted. [22] M. Sofonea and A. Matei, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009. [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. [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. [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. [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. [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. [28] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. [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. [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. [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.
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