Nonlinear Analysis 69 (2008) 126–139 www.elsevier.com/locate/na Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory ✩ S. Al-Homidan a , Q.H. Ansari a,b,∗ , J.-C. Yao c a Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 1169, Dhahran 31261, Saudi Arabia b Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India c Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan Received 18 January 2007; accepted 18 May 2007 Abstract In this paper, we introduce the concept of a Q-function defined on a quasi-metric space which generalizes the notion of a τ -function and a w-distance. We establish Ekeland-type variational principles in the setting of quasi-metric spaces with a Q-function. We also present an equilibrium version of the Ekeland-type variational principle in the setting of quasi-metric spaces with a Q-function. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. We also extend the Nadler’s fixed point theorem for multivalued maps to a Q-function and in the setting of complete quasi-metric spaces. As a consequence, we prove the Banach contraction theorem for a Q-function and in the setting of complete quasi-metric spaces. The results of this paper extend and generalize many results appearing recently in the literature. c  2008 Published by Elsevier Ltd Keywords: Ekeland-type variational principle; Equilibrium problems; Q-functions; Fixed point theorems; Caristi–Kirk type fixed point theorem; Nadler’s fixed point theorem; Banach contraction theorem; Quasi-metric spaces 1. Introduction In 1972, Ekeland [15] (see also, [16,17]) discovered a pioneer result, now known as Ekeland’s variational principle (in short, EVP), that provides an approximate minimizer of a bounded below and lower semicontinuous function in a given neighborhood of a point. This localization property is very useful and explains the importance of this result. EVP is one of the most important results obtained in nonlinear analysis and it has appeared as one of the most useful tools to solve problems in optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, etc; See for example [2,6,16,17,21,24,37] and references therein. Many famous results, namely the Krasnosel’skii–Zabrjeko ✩ In this research, first two authors were supported by the Fast Track Research Grants No. 26-3 (2005–2006). They express their thanks to the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. ∗ Corresponding author at: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 1169, Dhahran 31261, Saudi Arabia. Tel.: +966 3 860 2538; fax: +966 3 860 2340. E-mail addresses: homidan@kfupm.edu.sa (S. Al-Homidan), qhansari@kfupm.edu.sa (Q.H. Ansari), yaojc@math.nsysu.edu.tw (J.-C. Yao). 0362-546X/$ - see front matter c  2008 Published by Elsevier Ltd doi:10.1016/j.na.2007.05.004