Critical Point Theorems and Ekeland Type Variational Principle with Applications
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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, [email protected] 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 reproduction in any medium, provided the original work is properly cited. 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.
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 directions and in different settings. See, for example, 4, 5, 7–22 and references therein. The vectorial version of EVP in short, VEVP is considered and studied in 5, 22, 23 and references therein. Aubin and Frankowska 4 presented the equilibrium version of EVP in short, EEVP in the setting of complete metric spaces. Such version of EVP is further studied in 9, 11, 21 with applications to an equilibrium problem which is a unified model of several problems, namely, variational inequalities, complementarity problems, fixed point problem, optimization problem, Nash equilibrium problem, saddle point problem, and so forth; see, for example, 24, 25 and references therein. Al-Homidan et al. 9 established EEVP in the setting of quasimetric
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Fixed Point Theory and Applications
spaces with a Q-function which generalizes the notions of τ-function 17 and a w-distance 8. They proved some equivalences of EEVP with a fixed point theorem of Caristi-Kirk type for multivalued maps 12, Takahashi’s minimization theorem 8, and some other related results. As applications, they derived the existence results for solutions of equilibr
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