Abstract

We consider the partial differential set-valued variational inequality (PDSVI) on real Banach spaces. We give conditions for the existence and uniqueness of mild solution of the PDSVI, and establish closedness and continuity of the mild solution set mapping of the PDSVI. The proofs of the main results are based on the theory of semigroups, set-valued mappings, and variational inequality.

Author information

Contact details are reproduced from the original publication and may be historical.

Wei Li

Geomathematics Key Laboratory, Chengdu University of Technology, Chengdu 610059, P.R. China

lovelylw@126.com

Xianfu Wang

Dept. of Mathematics, University of British Columbia, Kelowna B.C., Canada V1V 1V7

shawn.wang@ubc.ca

Keywords

  • Partial differential set-valued variational inequality
  • stability
  • mild solution
  • continuity

Mathematics Subject Classification

49J40, 47J20, 49J53

References

  1. C. Baiocchi, A. Capelo: Variational and Quasi-Variational Inequalities: Application to Free Boundary Problems, Wiley, London (1984).Google Scholar
  2. H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York (2010).https://doi.org/10.1007/978-0-387-70914-7Google Scholar
  3. X. J. Chen, Z. Y. Wang: Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optimization 23 (2013) 1647–1671.https://doi.org/10.1137/120875223Google Scholar
  4. X. J. Chen, Z. Y. Wang: Differential variational inequality approach to dynamic games with shared constraints, Math. Programming 146 (2014) 379–408.https://doi.org/10.1007/s10107-013-0689-1Google Scholar
  5. F. Facchinei, J. S. Pang: Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I, II, Springer, New York (2003).https://doi.org/10.1007/b97544Google Scholar
  6. J. H. Fan, R. Y. Zhong: Stability analysis for variational inequality in reflexive Banach spaces, Nonlinear Anal. 69 (2008) 2566–2574.https://doi.org/10.1016/j.na.2007.08.031Google Scholar
  7. J. Gwinner: On a new class of differential variational inequalities and a stability result, Math. Programming 139 (2013) 205–221.https://doi.org/10.1007/s10107-013-0669-5Google Scholar
  8. A. D. Ioffe: Variational Analysis of Regular Mappings: Theory and Applications, Springer Monographs in Mathematics, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  9. A. D. Ioffe: On variational inequalities over polyhedral sets, Math. Programming 168 (2018) 261–278.https://doi.org/10.1007/s10107-016-1077-4Google Scholar
  10. A. D. Ioffe, R. E. Lucchetti: Typical convex program is very well posed, Math. Programming 104 (2005) 483–499.https://doi.org/10.1007/s10107-005-0625-0Google Scholar
  11. A. D. Ioffe, R. E. Lucchetti, J. P. Revalski: Almost every convex or quadratic programming problem is well posed, Math. Oper. Res. 29 (2004) 369–382.https://doi.org/10.1287/moor.1030.0080Google Scholar
  12. A. Jofré, R. T. Rockafellar, J. B. Wets: Variational inequalities and economic equilibrium, Math. Oper. Res. 32 (2007) 32–50.https://doi.org/10.1287/moor.1060.0233Google Scholar
  13. M. Kamenskii, V. Obukhovskii, P. Zecca: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin (2001).https://doi.org/10.1515/9783110870893Google Scholar
  14. S. Lang: Real and Functional Analysis, 3rd ed. Springer, Berlin (1993).https://doi.org/10.1007/978-1-4612-0897-6_1Google Scholar
  15. S. J. Li, C. R. Chen: Stability of weak vector variational inequality, Nonlinear Anal. 70 (2009) 1528–1535.https://doi.org/10.1016/j.na.2008.02.032Google Scholar
  16. W. Li, Y. B. Xiao, N. J. Huang, Y. J. Cho: A class of differential inverse quasi-variational inequalities in finite dimensional spaces, J. Nonlinear Sci. Appl. 10 (2017) 4532–4543.https://doi.org/10.22436/jnsa.010.08.45Google Scholar
  17. W. Li, Y. B. Xiao, X. Wang, J. Feng: Existence and stability for a generalized differential mixed quasi-variational inequality, Carpathian J. Math. 34 (2018) 347–354.https://doi.org/10.37193/CJM.2018.03.09Google Scholar
  18. X. S. Li, N. J. Huang, D. O’Regan: Differential mixed variational inqualities in finite dimensional spaces, Nonlinear Anal. 72 (2010) 3875–3886.https://doi.org/10.1016/j.na.2010.01.025Google Scholar
  19. X. W. Li, Z. H. Liu: Sensitivity analysis of optimal control problems described by differential hemivariational inequalities, SIAM J. Control Optimization 56 (2018) 3569–3597.https://doi.org/10.1137/17M1162275Google Scholar
  20. X. W. Li, Z. H. Liu, M. Sofonea: Unique solvability and exponential stability of differential hemivariational inequalities, Appl. Analysis, doi.org/10.1080/00036811.2019.1569226 (2019).Google Scholar
  21. Z. H. Liu, S. Migórski, S. D. Zeng: Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differ. Equations 263 (2017) 3989–4006.https://doi.org/10.1016/j.jde.2017.05.010Google Scholar
  22. Z. H. Liu, D. Motreanu, S. D. Zeng: Nonlinear evolutionary systems driven by mixed variational inequalities and its applications, Nonlinear Analysis 42 (2018) 409–421.https://doi.org/10.1016/j.nonrwa.2018.01.008Google Scholar
  23. Z. H. Liu, S. D. Zeng, D. Motreanu: Evolutionary problems driven by variational inequalities, J. Differ. Equations 260 (2016) 6787–6799.https://doi.org/10.1016/j.jde.2016.01.012Google Scholar
  24. Z. H. Liu, S. D. Zeng, D. Motreanu: Partial differential hemivariational inequalities, Adv. Nonlinear Analysis 7(4) (2018) 571–586.https://doi.org/10.1515/anona-2016-0102Google Scholar
  25. N. V. Loi: On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal. 122 (2015) 83–99.https://doi.org/10.1016/j.na.2015.03.019Google Scholar
  26. X. Mao: Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics and Applications, Horwood, Chichester (1997).Google Scholar
  27. S. Migórski, A. Ochal, M. Sofonea: Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4232-5Google Scholar
  28. J. S. Pang, D. Stewart: Differential variational inqualities, Math. Programming 113 (2008) 345–424.https://doi.org/10.1007/s10107-006-0052-xGoogle Scholar
  29. J. S. Pang, D. Stewart: Solution dependence on initial conditions in differential variational inequalities, Math. Programming 116 (2009) 429–460.https://doi.org/10.1007/s10107-007-0117-5Google Scholar
  30. A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin (1983).https://doi.org/10.1007/978-1-4612-5561-1Google Scholar
  31. A. U. Raghunathan, J. R. Pérez-Correa, E. Agosin, L. T. Biegler: Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities, Ann. Oper. Res. 148 (2006) 251–270.https://doi.org/10.1007/s10479-006-0086-8Google Scholar
  32. X. Wang, N. J. Huang: Differential vector variational inequalities in finite-dimensional spaces, J. Optim. Theory Appl. 158 (2013) 109–129.https://doi.org/10.1007/s10957-012-0164-9Google Scholar
  33. X. Wang, W. Li, X. S. Li, N. J. Huang: Stability for differential mixed variational inequalities, Optim. Letters 8 (2013) 1–15.https://doi.org/10.1007/s11590-013-0682-xGoogle Scholar
  34. Y. B. Xiao, M. Sofonea: On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl. 475 (2019) 364–384.https://doi.org/10.1016/j.jmaa.2019.02.046Google Scholar
  35. Y. B. Xiao, M. Sofonea: Generalized penalty method for elliptic variational-hemivariational inequalities, Appl. Math. Optimization, doi.org/10.1007/s00245-019-09563-4 (2019).https://doi.org/10.1007/s00245-019-09563-4Google Scholar

Suggested citation

W. Li, X. Wang. “Stability Analysis of Partial Differential Set-Valued Variational Inequalities on Banach Spaces.” Journal of Convex Analysis 27 (2020), No. 2, 423–442. https://doi.org/10.68381/jca27024

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.