Generalized Ekeland’s Variational Principle And Its Some Equivalences

In this paper, we present a generalized Ekeland-type variational principle and aCaristi-Kirk type fixed point theorem and a maximal element theorem in the setting ofuniform spaces. We prove some equivalences of our variational principles with Caristi-Kirktype fixed point theorems for multivalued maps, the Takahashi minimization theorem andsome other related results. As applications of our results, we derive existence results forsolutions of equilibrium problems and fixed point theorems for multivalued maps. Thisgeneralizes some existing results for sequentially lower monotone maps in a more generalsetting and our techniques allow us to improve and extend their results in [16,24,25].Furthermore, we present an equilibrium version of vector Ekeland variational principle,and we prove some equivalences among our Ekeland variational principle, existence ofsolutions of Vector equilibrium problem (in short,VEP), Caristi-Kirk type fixed pointtheorem, and Oettli and Th′era type theorem. This generalizes some existing results formultivalued maps in a more general setting and our techniques allow us to improve andextend their results in [3,41,47,49].In Chapter2, First, we collect some known definitions and results which will be usedin the sequel. Second, we present the main results of the paper, namely Theorem2.1andsome corollaries. By using this EVP, we derive the Caristi-Kirk type fixed point theoremand maximal element theorem in the setting of separated and sequentially complete u-niform spaces. Furthermore, we prove some equivalences among our Ekeland variationalprinciple. Last,we provide the equivalence between the equilibrium version of Ekeland-type variational principle, existence of solutions of the equilibrium problem, Caristi Kirktype fixed point theorems and Oettli and Thcera type theorems.In Chapter3, we present a vectorial form of Ekeland variational principle forvector-valued operator whose domain is a complete quasi-metric space and its range is alocally convex space. From this theorem, a Caristi fixed point theorem for vector-valuedmaps is established in a more general setting and we prove some equivalences among our Ekeland variational principle, existence of solutions of Vector equilibrium problem (inshort,VEP), Caristi-Kirk type fixed point theorem, and Oettli and Th′era type theorem.Sitthithakerngkiet and Plubtieng [41] present a vectorial Ekeland Variational Principlefor multivalued bioperator with a w-distance, but there are several errors in the proof ofTheorem3.1. In this paper, we improve and extend some results in [41].