Abstract and Applied Analysis

Well-Posedness for Generalized Set Equilibrium Problems

Yen-Cherng Lin

Full-text: Open access

Abstract

We study the well-posedness for generalized set equilibrium problems (GSEP) and propose two types of the well-posed concepts for these problems in topological vector space settings. These kinds of well-posedness arise from some well-posedness in the vector settings. We also study the relationship between these well-posedness concepts and present several criteria for the well-posedness of GSEP. Our results are new or include as special cases recent existing results.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 419053, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449684

Digital Object Identifier
doi:10.1155/2013/419053

Mathematical Reviews number (MathSciNet)
MR3121406

Zentralblatt MATH identifier
1291.90229

Citation

Lin, Yen-Cherng. Well-Posedness for Generalized Set Equilibrium Problems. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 419053, 7 pages. doi:10.1155/2013/419053. https://projecteuclid.org/euclid.aaa/1393449684


Export citation

References

  • A. N. Tihonov, “Stability of a problem of optimization of functionals,” Akademija Nauk SSSR, vol. 6, pp. 631–634, 1966.
  • L. C. Ceng and Y. C. Lin, “Metric characterizations of $\alpha $-well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 264721, 22 pages, 2012.
  • A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Springer, Berlin, Germany, 1993.
  • E. Bednarczuk and J.-P. Penot, “Metrically well-set minimization problems,” Applied Mathematics and Optimization, vol. 26, no. 3, pp. 273–285, 1992.
  • G. P. Crespi, A. Guerraggio, and M. Rocca, “Well posedness in vector optimization problems and vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 132, no. 1, pp. 213–226, 2007.
  • M. Furi and A. Vignoli, “About well-posed minimization problems for functionals in metric spaces,” Journal of Optimization Theory and Applications , vol. 5, no. 3, pp. 225–229, 1970.
  • T. Zolezzi, “Well-posedness criteria in optimization with application to the calculus of variations,” Nonlinear Analysis:Theory, Methods & Applications, vol. 25, no. 5, pp. 437–453, 1995.
  • E. Miglierina and E. Molho, “Well-posedness and convexity in vector optimization,” Mathematical Methods of Operations Research, vol. 58, no. 3, pp. 375–385, 2003.
  • E. Bednarczuk, “An approach to well-posedness in vector optimization: consequences to stability,” Control and Cybernetics, vol. 23, no. 1-2, pp. 107–122, 1994.
  • M. Bianchi, G. Kassay, and R. Pini, “Well-posedness for vector equilibrium problems,” Mathematical Methods of Operations Research, vol. 70, no. 1, pp. 171–182, 2009.
  • L. C. Ceng, N. Hadjisavvas, S. Schaible, and J. C. Yao, “Well-posedness for mixed quasivariational-like inequalities,” Journal of Optimization Theory and Applications, vol. 139, no. 1, pp. 109–125, 2008.
  • L. C. Ceng and J. C. Yao, “Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4585–4603, 2008.
  • Y.-P. Fang, N.-J. Huang, and J.-C. Yao, “Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems,” Journal of Global Optimization, vol. 41, no. 1, pp. 117–133, 2008.
  • K. Kimura, Y.-C. Liou, S.-Y. Wu, and J.-C. Yao, “Well-posedness for parametric vector equilibrium problems with applications,” Journal of Industrial and Management Optimization, vol. 4, no. 2, pp. 313–327, 2008.
  • L. Q. Anh, P. Q. Khanh, D. T. M. van, and J.-C. Yao, “Well-posed-ness for vector quasiequilibria,” Taiwanese Journal of Mathematics, vol. 13, no. 2B, pp. 713–737, 2009.
  • C. Berge, Topological Spaces, Macmillan, New York, NY, USA, 1963.
  • J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, Germany, 1984.
  • F. Ferro, “Optimization and stability results through cone lower semicontinuity,” Set-Valued Analysis, vol. 5, no. 4, pp. 365–375, 1997.
  • Y.-C. Lin, Q. H. Ansari, and H.-C. Lai, “Minimax theorems for set-valued mappings under cone-convexities,” Abstract and Applied Analysis, vol. 2012, Article ID 310818, 26 pages, 2012.