Taiwanese Journal of Mathematics

SYSTEMS OF GENERALIZED VECTOR QUASI-VARIATIONAL INCLUSION PROBLEMS AND APPLICATION TO MATHEMATICAL

X. P. Ding, T. C. Lai, and S. J. Yu

Full-text: Open access

Abstract

In this paper, we introduce and study some new systems of generalized vector quasi-variational inclusion problems involving condensing mappings in locally $FC$-uniform spaces. These systems contain many known systems of generalized vector quasi-variational inclusion problems, systems of generalized vector quasi-equilibrium problems and systems of vector quasi-optimization problems as special cases. By applying an existence theorem of maximal elements of a family of set-valued mappings involving condensing mapping due to author, we prove some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems. As applications, some existence results of solutions of the mathematical programs with systems of generalized vector quasi-variational inclusion constraints are established in noncompact locally $FC$-uniform spaces.

Article information

Source
Taiwanese J. Math., Volume 13, Number 5 (2009), 1515-1536.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405557

Digital Object Identifier
doi:10.11650/twjm/1500405557

Mathematical Reviews number (MathSciNet)
MR2554474

Zentralblatt MATH identifier
1188.49008

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 49J53: Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06] 90C48: Programming in abstract spaces

Keywords
systems of generalized vector quasi-variational inclusion problems condensing mappings maximal element $\Psi_i$-$FC$-quasiconvex locally $FC$-uniform spaces

Citation

Ding, X. P.; Lai, T. C.; Yu, S. J. SYSTEMS OF GENERALIZED VECTOR QUASI-VARIATIONAL INCLUSION PROBLEMS AND APPLICATION TO MATHEMATICAL. Taiwanese J. Math. 13 (2009), no. 5, 1515--1536. doi:10.11650/twjm/1500405557. https://projecteuclid.org/euclid.twjm/1500405557


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References

  • \item[1.] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, Springer-Verlag, New York, 1994.
  • \item[2.] Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Aust. Math. Soc., 59 (1999), 433-442.
  • \item[3.] Q. H. Ansari, S. Schaible and J. C. Yao, Systems of vector equilibrium problems and its applications, J. Optim. Theory Appl., 107 (2000), 547-557.
  • \item[4.] Q. H. Ansari, S. Schaible and J. C. Yao, The system of generalized vector equilibrium problems with applications, J. Glob. Optim., 22 (2002), 3-16.
  • \item[5.] H. Ben-El-Mechaiekh, S. Chebbi, M. Flornzano and J. V. Llinares, Abstract convexity and fixed points, J. Math. Anal. Appl., 222 (1998), 138-150.
  • \item[6.] S. Chebbi and M. Florenzano, Maximal elements and equilibria for condensing correspondences, J. Math. Anal. Appl., 38(3) (1999), 995-1002.
  • \item[7.] X. P. Ding, Maximal element theorems in product $FC$-spaces and generalized games, J. Math. Anal. Appl., 305(1) (2005), 29-42.
  • \item[8.] X. P. Ding, Generalized game and system of generalized vector quasi-equilibrium problems in $G$-convex spaces, Acta Math. Scientia, 26(4) (2006), 506-515.
  • \item[9.] X. P. Ding, System of generalized vector quasi-equilibrium problems in locally $FC$-spaces, Acta Math. Sinica, 22(5) (2006), 1529-1538.
  • \item[10.] X. P. Ding, System of generalized vector quasi-equilibrium problems on product $FC$-spaces, Acta Math. Scientia, 27(3) (2007), 522-534.
  • \item[11.] X. P. Ding, Maximal elements and generalized games involving condensing mappings in locally $FC$-uniform spaces and applications (I), Appl. Math. Mech., 28(12) (2007), 1561-1568.
  • \item[12.] X. P. Ding, Maximal elements and generalized games involving condensing mappings in locally $FC$-uniform spaces and applications (II), Appl. Math. Mech., 28(12) (2007), 1569-1580.
  • \item[13.] X. P. Ding, Maximal elements of $G_{KKM}$-majorized mappings in product $FC$-spaces and applications (I), Nonlinear Anal., 67(3) (2007), 963-973.
  • \item[14.] X. P. Ding, Maximal elements of $G_{KKM}$-majorized mappings in product $FC$-spaces and applications (II), Nonlinear Anal., 67(12) (2007), 3411-3423.
  • \item[15.] X. P. Ding, The generalized game and system of generalized vector quasi-equilibrium problems in locally $FC$-uniform spaces, Nonlinear Anal., 68(4) (2008), 1028-1036.
  • \item[16.] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.
  • \item[17.] N. X. Hai and P. Q. Khanh, Systems of set-valued quasivariational inclusion problems, J. Optim. Theory Appl., 135 (2007), 55-67.
  • \item[18.] C. D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl., 156 (1991), 341-357.
  • \item[19.] L. J. Lin and Q. H. Ansari, Collective fixed points and maximal elements with applications to abstract economies, J. Math. Anal. Appl., 296(2) (2004), 455-472.
  • \item[20.] L. J. Lin and Y. H. Liu, Existence theorems for systems of generalized vector quasi-equilibrium problems and optimization problems, J. Optim. Theory Appl., 130 (2006), 461-475.
  • \item[21.] L. J. Lin, System of Generalized Vector Quasi-Equilibrium Problems with Applications to Fixed Point Theorems for a Family of Nonexpansive Multivalued Mappings, J. Glob. Optim., 34 (2006), 15-32.
  • \item[22.] L. J. Lin, Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems, J. Glob. Optim., 38(1) (2007), 21-39.
  • \item[23.] L. J. Lin and C. I. Tu, The studies of systems of variational inclusions problems and applications, Nonlinear Anal., 2007, doi:10.1016/j.na.2007.07.041.
  • \item[24.] D. T. Luc, Theory of Vector Optimization, Lectures Notes in Economics and Mathematical Systems, Vol 319, Springer Verlag, Berlin, 1989.
  • \item[25.] S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl., 209 (1997), 551-571.
  • \item[26.] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.