Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications
نویسندگان
چکیده
By using a Danes̆’ drop theorem in locally convex spaces we obtain a vectorial form of Ekelandtype variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk’s fixed-point theorem, Takahashi’s nonconvex minimization theorem, and Oettli-Théra’s theorem. Furthermore, we show that these results are equivalent to each other. Also, the existence of solution of vector equilibrium problem is given.
منابع مشابه
$(varphi_1, varphi_2)$-variational principle
In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm, such that $f + g $ attains its strong minimum on $X. $ This result extends some of the well-known varitional principles as that of Ekeland [On the variational principle, J. Ma...
متن کاملCritical Point Theorems and Ekeland Type Variational Principle with Applications
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 con...
متن کاملA Minimization Theorem in Quasi-metric Spaces and Its Applications
We prove a new minimization theorem in quasi-metric spaces, which improves the results of Takahashi (1993). Further, this theorem is used to generalize Caristi's fixed point theorem and Ekeland's ε-variational principle. 1. Introduction. Caristi [1] proved a fixed point theorem on complete metric spaces which generalizes the Banach contraction principle. Ekeland [3] also obtained a non-convex m...
متن کاملPhelps ' Lemma , Dane s ' Drop Theorem and Ekeland ' sPrinciple in Locally Convex
We present a generalization of the Phelps lemma to locally convex topological vector spaces and show the equivalence of this theorem, Ekeland's principle and Dane s' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto eeciency theorem due to Isac. Concerning the drop theorem this solves a problem proposed by G. Isac in 1997. We show that another formulation ...
متن کاملTopological number for locally convex topological spaces with continuous semi-norms
In this paper we introduce the concept of topological number for locally convex topological spaces and prove some of its properties. It gives some criterions to study locally convex topological spaces in a discrete approach.
متن کامل