Academic Editor: S. Al-Homidan
Copyright © 2011 Lai-Jiu Lin et al. 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.
We introduce the notion of -spaces which is much weaker than cone
metric spaces defined by Huang and X. Zhang (2007). We establish some critical point
theorems in the setting of -spaces and, in particular, in the setting of complete
cone metric spaces. Our results generalize the critical point theorem proposed by
Dancs et al. (1983) and the results given by Khanh and Quy (2010)
to -spaces and cone metric spaces. As applications of our results, we characterize
the completeness of -space (cone metric spaces and quasimetric spaces are special
cases of -space) and studying the Ekeland type variational principle for single
variable vector-valued functions as well as for multivalued bifunctions in the setting
of cone metric spaces.