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An Abstract Problem in Variational Analysis

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Abstract

The purpose of this paper is to present a unifying approach to study various models of equilibrium theory and variational inclusions. A simple condition is established for the existence of solutions of variational relations and is applied to a number of variational problems.

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References

  1. Ansari, A.H., Yao, J.C.: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett. 12, 53–56 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aussel, A., Luc, D.T.: Existence conditions in general quasimonotone variational inequalities. Bull. Aust. Math. Soc. 71, 285–303 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chowdhury, M.S.R., Tan, K.K.: Generalized variational inequalities for quasimonotone operators and applications. Bull. Pol. Acad. Sci. Math. 45, 25–54 (1997)

    MATH  MathSciNet  Google Scholar 

  4. Daniilidis, A., Hadjisavvas, N.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90, 473–481 (1996)

    Article  MathSciNet  Google Scholar 

  5. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, I and II. Springer, New York (2003)

    Google Scholar 

  6. Flores-Bazan, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasiconvex case. SIAM J. Optim. 11, 675–690 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Plenum, New York (1995)

    MATH  Google Scholar 

  8. Hadjisavvas, N., Schaible, S.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90, 95–111 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)

    MATH  Google Scholar 

  11. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  12. Konnov, I.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 99, 165–181 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Luc, D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291–308 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Noor, M.A.: Some recent advances in variational inequalities, Part I, basic concepts. N.Z. J. Math. 26, 53–80 (1997)

    MATH  Google Scholar 

  15. Stampacchia, G.: Formes bilinaires coercives sur les ensembles convexes. C.R. Acad. Sci. Paris 258, 4413–4416 (1964)

    MATH  MathSciNet  Google Scholar 

  16. Tian, G.: Generalizations of the FKKM theorem and the Ky Fan minimax inequality with applications to maximal elements, price equilibrium, and complementarity. J. Math. Anal. Appl. 170, 457–471 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yao, J.C.: Multivalued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhou, J.X., Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J. Math. Anal. Appl. 132, 213–225 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. 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 

  20. Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  21. Guerraggio, A., Tan, N.X.: On general vector quasioptimization problems. Math. Methods Oper. Res. 55, 347–358 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kalmoun, E.M., Riahi, H., Tanaka, T.: On the vector equilibrium problems: remarks on a general existence theorem and applications. Nihonkai Math. J. 12, 149–164 (2001)

    MATH  MathSciNet  Google Scholar 

  23. Lin, L.J., Chen, H.L.: The study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequalities problems. J. Glob. Optim. 32, 135–157 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Luc, D.T., Tan, N.X.: Existence conditions in variational inclusions with constraints. Optimization 53, 505–515 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tuan, L.A., Sach, P.H.: Existence of solutions of generalized quasivariational inequalities with set-valued maps. Acta Math. Vietnam. 29, 309–316 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Tuan, L.A., Sach, P.H.: On some generalized vector equilibrium problems with set-valued maps. Acta Math. Vietnam. 32, 15–32 (2007)

    MathSciNet  Google Scholar 

  27. Aubin, J.P., Celina, A.: Differential Inclusion. Springer, Berlin (1994)

    Google Scholar 

  28. Lin, L.J.: Systems of variational inclusions, disclusions and differential inclusions problems with applications. Preprint (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Browder, F.E.: The fixed point theory of multivalued mappings in topological vector space. Math. Ann. 177, 238–301 (1968)

    Article  MathSciNet  Google Scholar 

  31. Lin, L.J., Chen, L.F., Ansari, Q.H.: Generalized abstract economy and systems of generalized vector quasi-equilibrium problems, J. Comput. Appl. Math. (2008, in press)

  32. Kim, W.K., Yuan, G.X.Z.: Existence of equilibria for generalized games and generalized social systems with coordination. Nonlinear Anal. 45, 169–188 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Michael, E.: Continuous selections. I. Ann. Math. 63, 361–382 (1956)

    Article  MathSciNet  Google Scholar 

  34. Park, S.: Fixed point theorems for better admissible multimaps on almost convex sets. J. Math. Anal. Appl. 329, 690–702 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire ANG, University of Avignon, Avignon, France

    D. T. Luc

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  1. D. T. Luc
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Correspondence to D. T. Luc.

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Communicated by F. Giannessi.

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Luc, D.T. An Abstract Problem in Variational Analysis. J Optim Theory Appl 138, 65–76 (2008). https://doi.org/10.1007/s10957-008-9371-9

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