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Existence of Solutions and Variational Principles for Generalized Vector Systems

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Abstract

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system.

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References

  1. Chang, S.S., Zhang, Y.: Generalized KKM theorem and variational inequalities. J. Math. Anal. Appl. 159, 208–223 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ansari, Q.H., Konnov, I.V., Yao, J.C.: Existence of a solution and variational principles for vector equilibrium problems. J. Optim. Theory Appl. 110, 481–492 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Tanaka, T.: Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. J. Optim. Theory Appl. 81, 355–377 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Auchmuty, G.: Variational principles for variational inequalities. Numer. Funct. Anal. Optim. 10, 863–874 (1989)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Fan, K.: A Generalization of Tichonoff’s fixed-point theorem. Math. Annal. 142, 305–310 (1961)

    Article  MATH  Google Scholar 

  7. Zeng, L.C., Wu, S.Y., Yao, J.C.: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwan. J. Math. 10, 1497–1514 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Chang, S.S., Lee, B.S., Wu, X., Cho, Y.J., Lee, G.M.: On the generalized quasivariational inequality problems. J. Math. Anal. Appl. 203, 686–711 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, G.Y., Goh, C.J., Yang, X.Q.: On gap functions for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 55–72. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, People’s Republic of China

    L. C. Ceng

  2. Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, Pisa, 56127, Italy

    G. Mastroeni

  3. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    J. C. Yao

Authors
  1. L. C. Ceng
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  2. G. Mastroeni
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

Communicated by F. Giannessi.

Research carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2007.

L.C. Ceng research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118).

J.C. Yao research was partially supported by the National Science Center for Theoretical Sciences at Tainan.

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Ceng, L.C., Mastroeni, G. & Yao, J.C. Existence of Solutions and Variational Principles for Generalized Vector Systems. J Optim Theory Appl 137, 485–495 (2008). https://doi.org/10.1007/s10957-007-9348-0

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  • DOI: https://doi.org/10.1007/s10957-007-9348-0

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