Abstract and Applied Analysis

Connectedness of Solution Sets for Weak Vector Variational Inequalities on Unbounded Closed Convex Sets

Ren-you Zhong, Yun-liang Wang, and Jiang-hua Fan

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Abstract

We study the connectedness of solution set for set-valued weak vector variational inequality in unbounded closed convex subsets of finite dimensional spaces, when the mapping involved is scalar C-pseudomonotone. Moreover, the path connectedness of solution set for set-valued weak vector variational inequality is established, when the mapping involved is strictly scalar C-pseudomonotone. The results presented in this paper generalize some known results by Cheng (2001), Lee et al. (1998), and Lee and Bu (2005).

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 431717, 5 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511851

Digital Object Identifier
doi:10.1155/2013/431717

Mathematical Reviews number (MathSciNet)
MR3045042

Zentralblatt MATH identifier
1272.49023

Citation

Zhong, Ren-you; Wang, Yun-liang; Fan, Jiang-hua. Connectedness of Solution Sets for Weak Vector Variational Inequalities on Unbounded Closed Convex Sets. Abstr. Appl. Anal. 2013 (2013), Article ID 431717, 5 pages. doi:10.1155/2013/431717. https://projecteuclid.org/euclid.aaa/1393511851


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References

  • F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in Variational Inequalities and Complementarity Problems, pp. 151–186, Wiley, Chichester, UK, 1980.
  • G. Y. Chen, “Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem,” Journal of Optimization Theory and Applications, vol. 74, no. 3, pp. 445–456, 1992.
  • G. Y. Chen and S. J. Li, “Existence of solutions for a generalized vector quasivariational inequality,” Journal of Optimization Theory and Applications, vol. 90, no. 2, pp. 321–334, 1996.
  • A. Daniilidis and N. Hadjisavvas, “Existence theorems for vector variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 54, no. 3, pp. 473–481, 1996.
  • F. Giannessi, G. Mastronei, and L. Pellegrini, “On the theory of vector optimization and variational inequalities,” in Image Space Analysis and Separation, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, F. F. Giannessi, Ed., Kluwer Academic, Dordrecht, The Netherlands, 1999.
  • Y. Cheng, “On the connectedness of the solution set for the weak vector variational inequality,” Journal of Mathematical Analysis and Applications, vol. 260, no. 1, pp. 1–5, 2001.
  • X. H. Gong, “Connectedness of the solution sets and scalarization for vector equilibrium problems,” Journal of Optimization Theory and Applications, vol. 133, no. 2, pp. 151–161, 2007.
  • X. H. Gong and J. C. Yao, “Connectedness of the set of efficient solutions for generalized systems,” Journal of Optimization Theory and Applications, vol. 138, no. 2, pp. 189–196, 2008.
  • B. Chen, Q.-y. Liu, Z.-b. Liu, and N.-j. Huang, “Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces,” Fixed Point Theory and Applications, vol. 2011, article 36, 2011.
  • G. M. Lee, D. S. Kim, B. S. Lee, and N. D. Yen, “Vector variational inequality as a tool for studying vector optimization problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 5, pp. 745–765, 1998.
  • G. M. Lee and I. J. Bu, “On solution sets for a$\pm $ne vector variational inequality,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, pp. 1847–1855, 2005.
  • G. Jameson, Ordered Linear Spaces, vol. 141 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1970.
  • R. Hu and Y.-P. Fang, “On the nonemptiness and compactness of the solution sets for vector variational inequalities,” Optimization, vol. 59, no. 7, pp. 1107–1116, 2010.
  • X. X. Huang, Y. P. Fang, and X. Q. Yang, “Characterizing the nonemptiness and compactness of the solution set of a vector variational inequality by scalarization,” Journal of Optimization Theory and Applications, 2012.
  • J.-P. Crouzeix, “Pseudomonotone variational inequality problems: existence of solutions,” Mathematical Programming, vol. 78, no. 3, pp. 305–314, 1997.
  • J.-P. Aubin, Mathematical Methods of Game and Economic Theory, vol. 7 of Studies in Mathematics and its Applications, North-Holland, Amsterdam, The Netherlands, 1979.
  • A. R. Warburton, “Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives,” Journal of Optimization Theory and Applications, vol. 40, no. 4, pp. 537–557, 1983.