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Variational inclusions problems with applications to Ekeland’s variational principle, fixed point and optimization problems

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Abstract

In this paper, we prove the existence theorems of two types of systems of variational inclusions problem. From these existence results, we establish Ekeland’s variational principle on topological vector space, existence theorems of common fixed point, existence theorems for the semi-infinite problems, mathematical programs with fixed points and equilibrium constraints, and vector mathematical programs with variational inclusions constraints.

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References

  1. Aubin J.P. and Cellina A. (1994). Differential Inclusion. Springer, Berlin, Germany

    Google Scholar 

  2. Birbil S., Bouza G., Frenk J.B.G. and Still G. (2006). Equilibrium constrained optimization problems. Eur. J. Oper. Res. 169: 1108–1127

    Article  Google Scholar 

  3. Deguire P., Tan K.K. and Yuan G.X.Z. (1999). The study of maximal elements, fixed point for L s −majorized mappings and quasi-variational inequalities in product spaces. Nonlinear Anal. 37: 933–951

    Article  Google Scholar 

  4. Ekeland I. (1972). Remarques sur les problems variationals. I. C. R.. Acad. Sci. Paris Ser. A-B 275: 1057–1059

    Google Scholar 

  5. Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324–353

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Gopfer A., Tammer Chr. and Zálinescu C. (2000). On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39: 909–922

    Article  Google Scholar 

  8. Hamel A.H. (2003). Phelp’s lemma, Dane’s drop theorem and Ekeland’s principle in locally convex spaces. Proc. Am. Math. Soc. 131(10): 3025–3038

    Article  Google Scholar 

  9. Himmelberg C.J. (1972). Fixed point of compact multifunctions. J. Math. Anal. Appl. 38: 205–207

    Article  Google Scholar 

  10. Huang N.J. (2001). A new class of generalized set-valued implicit variational inclusions in Banach spaces with applications. Comput. Math. Appl. 41(718): 937–943

    Article  Google Scholar 

  11. Isac G. (2004). Nuclear cones in product spaces, Pareto efficiency and Ekeland’s-type variational principle in locally convex spaces. Optimization 53(3): 253–268

    Article  Google Scholar 

  12. Lin L.J. (2007). Mathematical program with systems of equilibrium constraints. J. Glob. Optim. 37: 275–286

    Article  Google Scholar 

  13. Lin L.J. and Du W.S. (2006). Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 323: 360–370

    Article  Google Scholar 

  14. Lin L.J. and Ansari Q.H. (2004). Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296: 455–472

    Article  Google Scholar 

  15. Lin L.J. and Still G. (2006). Mathematical programs with equilibrium constraints: The existence of feasible points. Optimization 55: 205–216

    Article  Google Scholar 

  16. Lin, L.J.: Systems of generalized quasi-variational inclusions problems with applications to variational analysis and optimization problems. J. Glob. Optim. 38, (2007)

  17. Lin L.J. and Yu Z.T. (2001). On some equilibrium problems for multimaps. J. Comput. Appl. Math. 129: 171–183

    Article  Google Scholar 

  18. Lin L.J. and Hsu H.W. (2007). Existence theorems of vector quasi-equilibrium problems with equilibrium constraints. J. Glob. Optim. 37: 195–213

    Article  Google Scholar 

  19. Lin L.J. and Huang Y.J. (2007). Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems. Nonlinear Anal. Theory Methods Appl. 66: 1275–1289

    Article  Google Scholar 

  20. Lin, L.J., Du, W.S.: Systems of Equilibrium problems with applications to generalized Ekeland’s variational principles and systems of semi-infinite problems. J. Glob. Optim. (2007)

  21. Luc D.T. (1989). Theory of Vector Optimization, Lectures Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin, Germany

    Google Scholar 

  22. Luo Z.Q., Pang J.S. and Ralph D. (1997). Mathematical Program with Equilibrium Constraint. Cambridge University Press, Cambridge

    Google Scholar 

  23. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I,II. Springer, Berlin, Heidelberg, New York (2006)

  24. Robinson S.M. (1979). Generalized equation and their solutions, part I: basic theory. Math. Program. Study 10: 128–141

    Google Scholar 

  25. Tan N.X. (1985). Quasi-variational inequalities in topological linear locally convex Hausdorff spaces. Math. Nachr. 122: 231–245

    Article  Google Scholar 

  26. Wong C-W. (2007). A drop theorem without vector topology. J. Math. Anal. Appl. 329: 452–471

    Article  Google Scholar 

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

    Lai-Jiu Lin

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  1. Lai-Jiu Lin
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Lin, LJ. Variational inclusions problems with applications to Ekeland’s variational principle, fixed point and optimization problems. J Glob Optim 39, 509–527 (2007). https://doi.org/10.1007/s10898-007-9153-1

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  • DOI: https://doi.org/10.1007/s10898-007-9153-1

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