Isaac Scientific Publishing

Journal of Advances in Applied Mathematics

Quasi-equilibrium Problems and Fixed Point Theorems of the Product Mapping of Lower and Upper Semicontinuous Mappings

Download PDF (562.6 KB) PP. 89 - 100 Pub. Date: March 23, 2017

DOI: 10.22606/jaam.2017.22002

Author(s)

  • Nguyen Xuan Tan*
    Institute of Mathematics, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam
  • Nguyen Quynh Hoa
    University of economics and business administration of Thai Nguyen

Abstract

In this paper, we introduce generalized quasi-equilibrium problems. These contain several problems in the optimization theory as special cases. We give sufficient conditions on the existence of their solutions. In particular, we establish several results on the existence of fixed p oints for product mappings of lower and upper semicontinuous mappings. These results generalize some well-known fixed point theorems obtained by previous authors as F. E. Browder and Ky Fan, X. Wu, L. J. Lin, and Z. T. Yu etc.

Keywords

Generalized quasi-equilibrium problems, upper and lower semi-continuous mappings, fixed point theorems

References

[1] L. Brouwer, “Ueber abbildingen von mannigfaltigkeiten,” Math. Ann., vol. 71, pp. 97 – 115, 1911.

[2] Schauder, “Der fixpunktsatz in funktionalraeumen,” Studia Math., vol. 2, pp. 171 – 180, 1934.

[3] S. Kakutani, “A generalization of brouwer’ fixed point theorem,” Duke Math. Journal, vol. 8, pp. 457 – 459, 1941.

[4] K. Fan, “A generalization of tychonoff’s fixed point theorem,” Math. ann., vol. 142, pp. 305 – 310, 1961.

[5] F. E. Browder, “The fixed point theory of multi-valued mappings in topological vector spaces,” Math. Ann., vol. 177, pp. 283–301, 1968.

[6] N. C. Yannelis and N. D. Prabhakar, “Existence of maximal elements and equilibria in linear topological spaces,” J. Math. Economics, vol. 233, pp. 233 – 245, 1983.

[7] H. Ben-El-Mechaiekh, “Fixed points for compact set-valued maps, questions answers gen,” Topology, vol. 10, pp. 153 – 156, 1992.

[8] W. K. K. X. P. Ding and K. K. Tan, “A selection theorem and its applications,” Bull. Austra. Math. Soc., vol. 46, pp. 205 – 212, 1992.

[9] C. D. Horvath, “Existension and selection theorems in topological vector spaces with a generalized convexity structure,” Ann. Fac. Sci., Toulouse, vol. 2, pp. 253 – 269, 1993.

[10] X. Wu, “A new fixed point theorem and its applications,” Proc. Amer. Math. Soc., vol. 125, pp. 1779 – 1783, 1997.

[11] S. Park, “Continuous selection theorems in generalized convex spaces,” Numer. Funct. Anal. Optim.l, vol. 25, pp. 567 – 583, 1999.

[12] Z. T. Yu and L. J. Lin, “Continuous selection and fixed point theorems,” Nonlinear Anal., vol. 52, pp. 445 – 455, 2003.

[13] T. T. T. Duong and N. X. Tan, “On the existence of solutions to generalized quasi-equilibrium problems of type i and related problems,” Ad. in Nonlinear Variational Inequalities, vol. 13, pp. 29 – 47, 2010.

[14] ——, “On the existence of solutions to generalized quasi-equilibrium problems of type ii and related problems,” Acta Math. Vietnamica, vol. 36, pp. 29 – 47, 2011.

[15] N. X. Tan, “On the existence of solutions of quasi-variational inclusion problems,” Jour. of Opt. Theory and Appl., vol. 123, pp. 619–638, 2004.

[16] W. Rudin, Principles of Mathematical Analysis. McGraw-hill, 1987.

[17] M.Sion, “On general minimax theorems,” Pacific J. Math., vol. 8, p. 171 aAS176, 1958.