Open Access

Critical Point Theorems and Ekeland Type Variational Principle with Applications

Fixed Point Theory and Applications20102011:914624

https://doi.org/10.1155/2011/914624

Received: 28 September 2010

Accepted: 9 December 2010

Published: 15 December 2010

Abstract

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.

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Authors’ Affiliations

(1)
Department of Mathematics, National Changhua University of Education
(2)
Department of Mathematics, Aligarh Muslim University

Copyright

© Lai-Jiu Lin et al. 2011

This article is published under license to BioMed Central Ltd. 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.