tailieunhanh - Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 914624,

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 914624, 21 pages doi: Research Article Critical Point Theorems and Ekeland Type Variational Principle with Applications Lai-Jiu Lin,1 Sung-Yu Wang,1 and Qamrul Hasan Ansari2 1 2 Department of Mathematics, National Changhua University of Education, Changhua 50058, China Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, Taiwan Correspondence should be addressed to Lai-Jiu Lin, maljlin@ Received 28 September 2010; Accepted 9 December 2010 Academic Editor: S. Al-Homidan Copyright q 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. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 914624 21 pages doi 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 1 Department of Mathematics National Changhua University of Education Changhua 50058 China 2 Department of Mathematics Aligarh Muslim University Aligarh 202 002 Taiwan Correspondence should be addressed to Lai-Jiu Lin maljlin@ 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. We introduce the notion of l-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 l-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 l-spaces and cone metric spaces. As applications of our results we characterize the completeness of l-space cone metric spaces and quasimetric spaces are special cases of l-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. 1. Introduction In the last three decades the famous Ekeland s variational principle in short EVP 1 see also 2 3 emerged as one of the most important tools and results in nonlinear analysis due to its wide applications in optimization optimal control theory game theory nonlinear equations dynamical systems and so forth see for example 2-8 and references therein. It has been extended and generalized in different .