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  • Research Article
  • Open Access

Critical Point Theorems and Ekeland Type Variational Principle with Applications

Fixed Point Theory and Applications20102011:914624

  • Received: 28 September 2010
  • Accepted: 9 December 2010
  • Published:


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.


  • Variational Principle
  • Point Theorem
  • Differential Geometry
  • Computational Biology
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Authors’ Affiliations

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


© 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.