Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 914624, 21 pages
doi:10.1155/2011/914624
Research Article

Critical Point Theorems and Ekeland Type Variational Principle with Applications

Lai-Jiu Lin,1 Sung-Yu Wang,1 and Qamrul Hasan Ansari2

1Department of Mathematics, National Changhua University of Education, Changhua 50058, China
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, Taiwan

Received 28 September 2010; Accepted 9 December 2010

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.

Abstract

We introduce the notion of 𝜆 -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 𝜆 -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 𝜆 -spaces and cone metric spaces. As applications of our results, we characterize the completeness of 𝜆 -space (cone metric spaces and quasimetric spaces are special cases of 𝜆 -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.