- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 267585, 9 pages
Ekeland Variational Principle for Generalized Vector Equilibrium Problems with Equivalences and Applications
1College of Applied Science, Beijing University of Technology, Beijing 100124, China
2College of Mathematics, Jilin Normal University, Siping, Jilin 136000, China
Received 2 April 2013; Accepted 12 July 2013
Academic Editor: Yongsheng S. Han
Copyright © 2013 De-ning Qu and Cao-zong Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- I. Ekeland, “Sur les problèmes variationnels,” Comptes Rendus de l'Académie des Sciences, vol. 275, pp. A1057–A1059, 1972.
- I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp. 324–353, 1974.
- I. Ekeland, “Nonconvex minimization problems,” American Mathematical Society. Bulletin, vol. 1, no. 3, pp. 443–474, 1979.
- J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, NY, USA, 1984.
- G. Y. Chen and X. X. Huang, “Ekeland's -variational principle for set-valued mappings,” Mathematical Methods of Operations Research, vol. 48, no. 2, pp. 181–186, 1998.
- A. Göpfert, H. Riahi, C. Tammer, and C. Zălinescu, Variational Methods in Partially Ordered Spaces, vol. 17 of CMS Books in Mathematics, Springer, New York, NY, USA, 2003.
- T. X. D. Ha, “Some variants of the Ekeland variational principle for a set-valued map,” Journal of Optimization Theory and Applications, vol. 124, no. 1, pp. 187–206, 2005.
- L. J. Lin and W. S. Du, “Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 360–370, 2006.
- C. Gutiérrez, B. Jiménez, and V. Novo, “A set-valued Ekeland's variational principle in vector optimization,” SIAM Journal on Control and Optimization, vol. 47, no. 2, pp. 883–903, 2008.
- P. Q. Khanh and D. N. Quy, “On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings,” Journal of Global Optimization, vol. 49, no. 3, pp. 381–396, 2011.
- P. Q. Khanh and D. N. Quy, “Versions of Ekeland's variational principle involving set perturbations,” Journal of Global Optimization, 2012.
- J. P. Aubin, Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, The Netherlands, 1979.
- I. Ekeland, “Some lemmas about dynamical systems,” Mathematica Scandinavica, vol. 52, no. 2, pp. 262–268, 1983.
- J. T. Xing, “A note about Oden's constitutive variational principle,” Applied Mathematics and Mechanics, vol. 8, no. 10, pp. 975–984, 1987.
- G. Isac, “Nuclear cones in product spaces, Pareto efficiency and Ekeland-type variational principles in locally convex spaces,” Optimization, vol. 53, no. 3, pp. 253–268, 2004.
- E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
- S. Al-Homidan, Q. H. Ansari, and J. C. Yao, “Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 69, no. 1, pp. 126–139, 2008.
- K. R. Kazmi, “A variational principle for vector equilibrium problems,” Indian Academy of Sciences. Proceedings. Mathematical Sciences, vol. 111, no. 4, pp. 465–470, 2001.
- Q. H. Ansari, I. V. Konnov, and J. C. Yao, “Existence of a solution and variational principles for vector equilibrium problems,” Journal of Optimization Theory and Applications, vol. 110, no. 3, pp. 481–492, 2001.
- M. Bianchi, G. Kassay, and R. Pini, “Ekeland's principle for vector equilibrium problems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 66, no. 7, pp. 1454–1464, 2007.
- Q. H. Ansari, “Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 561–575, 2007.
- Y. Araya, K. Kimura, and T. Tanaka, “Existence of vector equilibria via Ekeland's variational principle,” Taiwanese Journal of Mathematics, vol. 12, no. 8, pp. 1991–2000, 2008.
- S. Eshghinezhad and M. Fakhar, “Vectorial form of Ekeland-type variational principle in locally convex spaces and its applications,” Fixed Point Theory and Applications, vol. 2010, Article ID 276294, 15 pages, 2010.
- J. Zeng and S. J. Li, “An Ekeland's variational principle for set-valued mappings with applications,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 477–484, 2009.
- D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989.
- J. P. Aubin and A. Cellina, Differential Inclusion, Springer, Berlin, Germany, 1994.
- O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.
- P. H. Sach and L. A. Tuan, “New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems,” Journal of Optimization Theory and Applications, vol. 157, no. 2, pp. 347–364, 2013.
- L. A. Tuan, G. M. Lee, and P. H. Sach, “Upper semicontinuity in a parametric general variational problem and application,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 3-4, pp. 1500–1513, 2010.
- G. Y. Chen and X. Q. Yang, “Characterizations of variable domination structures via nonlinear scalarization,” Journal of Optimization Theory and Applications, vol. 112, no. 1, pp. 97–110, 2002.
- M. Bianchi and R. Pini, “Coercivity conditions for equilibrium problems,” Journal of Optimization Theory and Applications, vol. 124, no. 1, pp. 79–92, 2005.