• Published : 2009.10.31


In this paper, we consider the existence of solutions to some generalized vector quasi-equilibrium-like problem under a c-diagonal quasi-convexity assumptions, but not monotone concepts. For an example, in the proof of Theorem 1, the c-diagonally quasi-convex concepts of a set-valued mapping was used but monotone condition was not used. Our problem is a new kind of equilibrium problems, which can be compared with those of Hou et al. [4].


  1. Q. H. Ansari and F. F. Bazan, Generalized vector quasi-equilibrium problems with applications, J. Math. Anal. Appl. 277 (2003), 246–256
  2. Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to the system of variational inequalities, Bull. Aust. Math. Soc. 59 (1990), 433–442
  3. F. E. Browder, The fixed point theory of multivalued mappings in topological vector spaces, Math. Annal. 177 (1968), 283–301
  4. S. H. Hou, H. Yu, and G. Y. Chen, On vector quasi-equilibrium problems with set-valued maps, J. Optim. Th. Appl. 119 (2003), 485–498
  5. L.-J. Lin, S. Park, and Z.-T. Yu, Remarks on fixed points, maximal elements, and equilibria of general games, J. Math. Anal. Appl. 233 (1999), 581–596
  6. Q. M. Liu, L. Fan, and G. Wang, Generalized vector quasi-equilibrium problems with set-valued mappings, Appl. Math. Lett. 21 (2008), 946–950
  7. M. A. Noor, Mixed quasi-equilibrium-like problems, J. Appl. Math. Stoch. Anal. (2006), Art. ID 70930, 1–8
  8. H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3, Springer, New York, 1971
  9. X. Q. Yang, Generalized convex functions and vector variational inequalities, J. Optim. Th. Appl. 79 (1993), 563–580
  10. G. X.-Z. Yuan, G. Isac, K.-K. Tan, and J. Yu, The study of minimax inequalities, abstract economics and applications to variational inequalities and Nash equilibria, Acta Appl. Math. 54 (1998), 135–166