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Zbl 1142.49005
Al-Homidan, S.; Ansari, Q.H.; Yao, J.-C.
Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. (English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 1, A, 126-139 (2008). ISSN 0362-546X

Summary: We introduce the concept of a $Q$-function defined on a quasi-metric space which generalizes the notion of a $\tau $-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.

MSC 2000:: *49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
91B50 Equilibrium in economics

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

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