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Semicontinuity of Solution Sets to Parametric Quasivariational Inclusions with Applications to Traffic Networks II: Lower Semicontinuities Applications

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Abstract

We propose some notions related to semicontinuity of a multivalued mapping and provide a clear insight for various semicontinuity-related definitions. We establish verifiable sufficient conditions for solution sets of general quasivariational inclusion problems to have these semicontinuity-related properties. Our results are proved to include and improve recent ones in the literature by corollaries and examples. Part I is devoted to upper semicontinuity properties of solution sets. Part II discusses lower semicontinuities of these sets and applications, where we discuss in details a traffic network problem as a sample for employing the main results in practical situations

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References

  1. Ait Mansour, M., Scrimali, L.: Hölder continuity of solutions to elastic traffic network models. J. Global Optim. 40, 175–184 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Anh, L.Q., Khanh, P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anh, L.Q., Khanh, P.Q.: Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces. J. Optim. Theory Appl. (2008, accepted)

  5. Anh, L.Q., Khanh, P.Q.: Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions. J. Global Optim. (2007, online first)

  6. Anh, L.Q., Khanh, P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks I: upper semicontinuities. Set-Valued Anal. (2008, online first)

  7. Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. De Luca, M.: Generalized quasivariational inequalities and traffic equilibrium problems. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 45–54. Plenum Press, New York, NY (1995)

    Google Scholar 

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

    Google Scholar 

  12. Hai, N.X., Khanh, P.Q.: The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328, 1268–1277 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hai, N.X., Khanh, P.Q.: Systems of set-valued quasivariational inclusion problems. J. Optim. Theory Appl. 135, 55–67 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hai, N.X., Khanh, P.Q., Quan, N.H.: On the existence of solutions to quasivariational inclusion problems (submitted for publication)

  15. Huang, N.J., Li, J., Thompson, H.B.: Stability for parametric implicit vector equilibrium problems. Math. Comput. Modelling 43, 1267–1274 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Khanh, P.Q., Luu, L.M.: On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J. Optim. Theory Appl. 123, 533–548 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Khanh, P.Q., Luu, L.M.: Some existence results for vector quasivariational inequalities involving multifunctions and applications to traffic equilibrium problems. J. Global Optim. 32, 551–568 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Khanh, P.Q., Luu, L.M.: Upper semicontinuity of the solution set of parametric multivalued vector quasivariational inequalities and applications. J. Global Optim. 32, 569–580 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Khanh, P.Q., Luu, L.M.: Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities. J. Optim. Theory Appl. 133, 329–339 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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  MATH  MathSciNet  Google Scholar 

  21. Maugeri, A.: Variational and quasivariational inequalities in network flow models: recent developments in theory and algorithms. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 195–211. Plenum Press, New York, NY (1995)

    Google Scholar 

  22. Smith, M.J.: The existence, uniqueness and stability of traffic equilibrium. Trans. Res. 138, 295–304 (1979)

    Google Scholar 

  23. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proceedings of the Institute of Civil Engineers, part II, pp. 325–378 (1952)

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Authors and Affiliations

  1. Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam

    Lam Quoc Anh

  2. Department of Mathematics, International University of Hochiminh City, Khu pho 6, Linh Trung, Thu Duc, Hochiminh City, Vietnam

    Phan Quoc Khanh

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  1. Lam Quoc Anh
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  2. Phan Quoc Khanh
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Correspondence to Phan Quoc Khanh.

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Dedicated to Professor Boris S. Mordukhovich.

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Anh, L.Q., Khanh, P.Q. Semicontinuity of Solution Sets to Parametric Quasivariational Inclusions with Applications to Traffic Networks II: Lower Semicontinuities Applications. Set-Valued Anal 16, 943–960 (2008). https://doi.org/10.1007/s11228-008-0082-z

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  • DOI: https://doi.org/10.1007/s11228-008-0082-z

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