Skip to main content
SpringerLink
Log in

Existence Criteria for the Solutions of Two Types of Variational Relation Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The variational relation problems have been introduced in 2008, by Dinh The Luc, as general models for a large class of problems in nonlinear analysis and applied mathematics. Since this manner of approach proved to be a powerful tool for studying a wide class of problems in nonlinear analysis and applied mathematics, several types of variational relation problems or systems of variational relation problems have been investigated in many recent papers. The present paper fits into this interesting group of works, establishing existence criteria for the solutions of two very general types of variational relation problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Luc, D.T.: An abstract problem in variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lin, L.J., Wang, S.Y.: Simultaneous variational relation problems and related applications. Comput. Math. Appl. 58, 1711–1721 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lin, L.J., Ansari, Q.H.: Systems of quasi-variational relations with applications. Nonlinear Anal. 72, 1210–1220 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Luc, D.T., Sarabi, E., Soubeyran, A.: Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364, 544–555 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balaj, M., Lin, L.J.: Equivalent forms of a generalized KKM theorem and their applications. Nonlinear Anal. 73, 673–682 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balaj, M., Lin, L.J.: Generalized variational relation problems with applications. J. Optim. Theory Appl. 148, 1–13 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balaj, M., Luc, D.T.: On mixed variational relation problems. Comput. Math. Appl. 60, 2712–2722 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Balaj, M., Coroianu, L.: Matching theorems and simultaneous relation problems. Bull. Korean Math. Soc. 48, 939–949 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khanh, P.Q., Luc, D.T.: Stability of solutions in parametric variational relation problems. Set-Valued Anal. 16, 1015–1035 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pu, Y.J., Yang, Z.: Stability of solutions in parametric variational relation problems with applications. Nonlinear Anal. 75, 516–525 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang, Z., Pu, Y.J.: Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications. Nonlinear Anal. 75, 516–525 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Park, S., Kim, H.: Foundations of the KKM theory on generalized convex spaces. J. Math. Anal. Appl. 209, 551–571 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fan, K.: Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38, 121–126 (1952)

    Article  MATH  Google Scholar 

  14. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)

    Article  MathSciNet  Google Scholar 

  15. Yu, S.J., Yao, J.C.: On the generalized nonlinear variational inequalities and quasi-variational inequalities. J. Math. Anal. Appl. 198, 178–193 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cubiotti, P.: Generalized quasi-variational inequalities without continuities. J. Optim. Theory Appl. 92, 477–495 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, L.J., Park, S.: On some generalized quasi-equilibrium problems. J. Math. Anal. Appl. 224, 167–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fu, J.Y.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 52, 57–64 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chiang, Y., Chadli, O., Yao, J.C.: Generalized vector equilibrium problems with trifunctions. J. Glob. Optim. 30, 135–154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tan, N.X.: On the existence of solutions of quasivariational inclusion problems. J. Optim. Theory Appl. 123, 619–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hou, S.H., Gong, X.H., Yang, X.M.: Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions. J. Optim. Theory Appl. 146, 387–398 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hou, S.H., Yu, H., Chen, G.Y.: On vector quasi-equilibrium problems with set-valued maps. J. Optim. Theory Appl. 119, 485–498 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Peng, J.W., Zhu, D.L.: Generalized vector quasi-equilibrium problems with set-valued mappings. J. Inequal. Appl. 2006, 6925 (2006)

    Google Scholar 

  24. Li, S.J., Teo, K.L., Yang, X.Q., Wu, S.Y.: Gap functions and existence of solutions to generalized vector quasi-equilibrium problems. J. Glob. Optim. 34, 427–440 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sach, P.H., Tuan, L.A.: Existence results for set-valued vector quasiequilibrium problems. J. Optim. Theory Appl. 133, 229–240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, X.B., Li, S.J.: Existence of solutions for generalized vector quasi-equilibrium problems. Optim. Lett. 4, 17–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Plubtieng, S., Sitthithakerngkiet, K.: Existence result of generalized vector quasiequilibrium problems in locally G-convex spaces. Fixed Point Theory Appl. 2011, 967515 (2011)

    MathSciNet  Google Scholar 

  28. Tuan, L.A., Lee, G.M., Sach, P.H.: Upper semicontinuity in a parametric general variational problem and application. Nonlinear Anal. 72, 1500–1513 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sach, P.H., Tuan, L.A., Lee, G.M.: Sensitivity results for a general class of generalized vector quasi-equilibrium problems with set-valued maps. Nonlinear Anal. 71, 571–586 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lee, G.M., Kum, S.: On implicit vector variational inequalities. J. Optim. Theory Appl. 104, 409–425 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Fu, J.Y., Wan, A.H.: Generalized vector equilibrium problems with set-valued mappings. Math. Methods Oper. Res. 56, 259–268 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. Lin, L.J., Ansari, Q.H., Huang, Y.J.: Some existence results for solutions of generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 65, 85–98 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hai, N.X., Khanh, P.Q.: Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133, 317–327 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sach, P.H., Tuan, L.A.: Generalizations of vector quasivariational inclusion problems with set-valued maps. J. Glob. Optim. 43, 23–45 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Park, S., Singh, S.P., Watson, B.: Some fixed point theorems for composites of acyclic maps. Proc. Am. Math. Soc. 121, 1151–1158 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their suggestions which have improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oradea, 410087, Oradea, Romania

    M. Balaj

  2. Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan

    L. J. Lin

Authors
  1. M. Balaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. J. Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. J. Lin.

Additional information

Communicated by Dinh The Luc.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Balaj, M., Lin, L.J. Existence Criteria for the Solutions of Two Types of Variational Relation Problems. J Optim Theory Appl 156, 232–246 (2013). https://doi.org/10.1007/s10957-012-0136-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0136-0

Keywords

Search

Navigation