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Differential Vector Variational Inequalities in Finite-Dimensional Spaces

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Abstract

In this paper, a differential vector variational inequality is introduced and studied in finite-dimensional Euclidean spaces. The existence of a Carathéodory weak solution for the differential vector variational inequality is presented under some suitable conditions. Furthermore, the upper semicontinuity and the lower semicontinuity of the solution sets for the differential variational inequality are established when both the mapping and the constraint set are perturbed by two different parameters.

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References

  1. Pang, J.S., Stewart, D.E.: Differential variational inequalities. Math. Program., Ser. A 113, 345–424 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Han, L.S., Pang, J.S.: Non-zenoness of a class of differential quasi-variational inequalities. Math. Program., Ser. A 121, 171–199 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Pang, J.S., Shen, J.: Strongly regular differential variational systems. IEEE Trans. Autom. Control 52, 242–255 (2007)

    Article  MathSciNet  Google Scholar 

  4. Li, X.S., Huang, N.J., O’Regan, D.: Differential mixed variational inequalities in finite dimensional spaces. Nonlinear Anal. 72, 3875–3886 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Stewart, D.E.: Uniqueness for index-one differential variational inequalities. Nonlinear Anal. Hybrid Syst. 2, 812–818 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Mandelbaum, A.: The dynamic complementarity problem. Unpublished manuscript (1989)

  7. Çamlıbel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: On linear passive complementarity systems. Eur. J. Control 8, 220–237 (2002)

    Article  Google Scholar 

  8. Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Linear complementarity systems. SIAM J. Appl. Math. 60, 1234–1269 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Anitescu, M., Hart, G.D.: A constraint-stabilized time-stepping for multi-body dynamics with contact and friction. Int. J. Numer. Methods Eng. 60, 2335–2371 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for three-dimensional rigid-body dynamics. Comput. Methods Appl. Mech. Eng. 177, 183–197 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brogliato, B.: Nonsmooth Mechanics. Models, Dynamics and Control. Springer, London (1999)

    Book  MATH  Google Scholar 

  12. Cojocaru, M.G., Daniele, P., Nagurney, A.: Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications. J. Optim. Theory Appl. 127, 549–563 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Projected dynamical systems in a complementarity formalism. Oper. Res. Lett. 27, 83–91 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Song, P., Krauss, P., Kumar, V., Dupont, P.: Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. J. Appl. Mech. 68, 118–128 (2001)

    Article  MATH  Google Scholar 

  15. Song, P., Pang, J.S., Kumar, V.: Semi-implicit time-stepping models for frictional compliant contact problems. Int. J. Numer. Methods Eng. 60, 2231–2261 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Stewart, D.E.: Convergence of a time-stepping scheme for rigid body dynamics and resolution of Painlevé’s problem. Arch. Ration. Mech. Anal. 145, 215–260 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Trinkle, J.C., Tzitzouris, J.A., Pang, J.S.: Dynamic multi-rigid-systems with concurrent distributed contacts. R. Soc. Philos. Trans. Math. Phys. Eng. Sci. 359, 2575–2593 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tzitzouris, J., Pang, J.S.: A time-stepping complementarity approach for frictionless systems of rigid bodies. SIAM J. Optim. 12, 834–860 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Çamlıbel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: Consistency of a time-stepping method for a class of piecewise-linear networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49, 349–357 (2002)

    Article  Google Scholar 

  20. Giannessi, F.: Theorems of alterative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)

    Google Scholar 

  21. Ceng, L.C., Schaible, S., Yao, J.C.: Existence of solutions for generalized vector variational-like inequalities. J. Optim. Theory Appl. 137, 121–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, G.Y., Huang, X.X., Yang, X.Q.: In: Vector Optimization: Set-Valued and Variational Analysis. Lecture Notes in Economics and Mathematical System, vol. 541. Springer, Berlin (2005)

    Google Scholar 

  23. Chen, G.Y., Li, S.J.: Existence of solutions for a generalized vector quasivariational inequality. J. Optim. Theory Appl. 90, 321–334 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chiang, Y.: Semicontinuous mappings into T.V.S. with applications to mixed vector variational-like inequalities. J. Optim. Theory Appl. 32, 467–484 (2005)

    MATH  Google Scholar 

  25. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  26. Huang, N.J., Fang, Y.P.: On vector variational inequalities in reflexive Banach spaces. J. Optim. Theory Appl. 32, 495–505 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Yang, X.Q., Yao, J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen, C.R., Li, S.J., Fang, Z.M.: On the solution semicontinuity to a parametric generalized vector quasivariational inequality. Comput. Math. Appl. 60, 2417–2425 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Khanh, P.Q., Luu, L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Glob. Optim. 32, 569–580 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kimura, K., Yao, J.C.: Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. J. Optim. Theory Appl. 138, 429–443 (2008)

    Article  MathSciNet  Google Scholar 

  33. Li, S.J., Chen, C.R.: Stability of weak vector variational inequality. Nonlinear Anal. 70, 1528–1535 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, S.J., Chen, G.Y., Teo, K.L.: On the stability of generalized vector quasivariational inequality problems. J. Optim. Theory Appl. 113, 283–295 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Raghunathan, A.U., Pérez-Correa, J.R., Agosin, E., Biegler, L.T.: Parameter estimation in metabolic flux balance models for batch fermentation–formulation and solution using differential variational inequalities. Ann. Oper. Res. 148, 251–270 (2006)

    Article  MATH  Google Scholar 

  36. Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, New York (1984)

    Book  MATH  Google Scholar 

  37. Fan, J.H., Zhong, R.Y.: Stability analysis for variational inequality in reflexive Banach spaces. Nonlinear Anal. 69, 2566–2574 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fan, K.: A generalization of Tychonoff’ s fixed point theorem. Math. Ann. 142, 305–310 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rochafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  40. Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)

    MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the editor and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171237) and the Key Program of NSFC (Grant No. 70831005).

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P.R. China

    Xing Wang & Nan-Jing Huang

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  1. Xing Wang
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  2. Nan-Jing Huang
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Correspondence to Nan-Jing Huang.

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Communicated by Guang-Ya Chen.

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Wang, X., Huang, NJ. Differential Vector Variational Inequalities in Finite-Dimensional Spaces. J Optim Theory Appl 158, 109–129 (2013). https://doi.org/10.1007/s10957-012-0164-9

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