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Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces

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Abstract

The purpose of this paper is to investigate nonemptiness and boundedness of the solution set for a vector equilibrium problem with strict feasibility in reflexive Banach spaces. We introduce the concept of strict feasibility for a vector equilibrium problem, which recovers the existing concepts of strict feasibility introduced for variational inequalities. We prove that a pseudomonotone vector equilibrium problem has a nonempty bounded solution provided that it is strictly feasible in the strong sense.

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References

  1. Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 1–23 (1993)

    MathSciNet  Google Scholar 

  2. Bianchi M., Schaible S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bianchi M., Pini R.: A note on equilibrium problems with properly quasimonotone bifunctions. J. Global Optim. 20(1), 67–76 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bianchi M., Pini R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124(1), 79–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chadli O., Schaible S., Yao J.C.: Regularized equilibrium problems with application to noncoercive hemivariational inequalities. J. Optim. Theory Appl. 121(3), 571–596 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bianchi M., Hadjisavvas N., Schaible S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92(3), 527–542 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hadjisavvas N., Schaible S.: From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96, 297–309 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Oettli W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Mathematica Vietnam. 22, 213–221 (1997)

    MATH  MathSciNet  Google Scholar 

  9. Fakhar M., Zafarani J.: Equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 126(1), 125–136 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ansari Q.H., Konnov I.V., Yao J.C.: Characterizations of solutions for vector equilibrium problems. J. Optim. Theory Appl. 113(3), 435–447 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giannessi F.: Theorems of alterative, quadratic progams 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 

  12. Chen G.Y., Huang X.X., Yang X.Q.: Vector Optimization: Set-valued and Variational Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  13. Chen G.Y., Yang X.Q.: The vector complementary problem and its equivalence with weak minimal elements in ordered spaces. J. Math. Anal. Appl. 153, 136–158 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen G.Y.: Existence of solutions for a vector variational inequality: an extension of the Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yang X.Q.: Vector complementarity and minimal element problems. J. Optim. Theory Appl. 77, 483–495 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Yang X.Q.: Vector variational inequality and its duality. Nonlinear Anal. 95, 729–734 (1993)

    Google Scholar 

  17. Konnov I.V., Yao J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lalitha C.S., Mehta M.: Vector variational inequalities with cone-pseudomonotone bifunctions. Optimization 54(3), 327–338 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fang Y.P., Huang N.J.: Feasibility and solvability for vector complementarity problems. J. Optim. Theory Appl. 129(3), 373–390 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Harker P.T., Pang J.S.: Finite-dimensional variational and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Prog. 48, 161–220 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Daniilidis A., Hadjisavvas N.: Coercivity conditions and variational inequalities. Math. Prog. Ser. A 86(2), 433–438 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  23. Crouzeix J.P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Prog. Ser. A 78(3), 305–314 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. He Y.R., Ng K.F.: Strict feasibility of generalized complementarity problems. J. Austral. Math. Soc. Ser. A 81(1), 15–20 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. He Y.R., Mao X.Z., Zhou M.: Strict feasibility of variational inequalities in reflexive Banach spaces. Acta Math. Sin. English Ser. 23, 563–570 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hu R., Fang Y.P: Feasibility-solvability theorem for a generalized system. J. Optim. Theory Appl. 142(3), 493–499 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Auslender A., Teboulle M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)

    MATH  Google Scholar 

  28. Fang Y.P, Huang N.J.: Vector equilibrium problems, minimal element problems and least element problems. Positivity 11(2), 251–268 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Adly S., Théra M., Ernst E.: Stability of the solution set of non-coercive variational inequalities. Commun. Contemp. Math. 4(1), 145–160 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Adly S., Théra M., Ernst E.: On the closedness of the algebraic difference of closed convex sets. J. Math. Pures Appl. 82(9), 1219–1249 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Adly S., Théra M., Ernst E.: Well-positioned closed convex sets and well-positioned closed convexfunctions. J. Global Optim. 29, 337–351 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan, People’s Republic of China

    Rong Hu

  2. Department of Mathematics, Sichuan University, Chengdu, Sichuan, People’s Republic of China

    Ya Ping Fang

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  1. Rong Hu
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  2. Ya Ping Fang
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Correspondence to Ya Ping Fang.

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Hu, R., Fang, Y.P. Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces. Optim Lett 5, 505–514 (2011). https://doi.org/10.1007/s11590-010-0215-9

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  • DOI: https://doi.org/10.1007/s11590-010-0215-9

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