Study Of Ekeland’s Variational Principle And Related Theorems

At first, we extend the Gerstewiz nonlinear scalarization functionsfrom real topological vector spaces to real vector spaces without anytopological structures. By using our nonlinear scalarization functions,we give several new versions of vectorial Ekeland’s variational princi-ple, where the domain spaces of the objective functions are bornologicalvector spaces, metric spaces and topological vector spaces respectively,and range spaces are extended from real topological vector spaces to realvector spaces. And the perturbations become more general and flexive.Morover, we introduce a real vector cone metric space, where the conemetric is a mapping taking values in a real vector space, and the orderingcone only needs to have a non-empty relatively algebraic interior. Thisconcept is a generalization of the known TVS-cone metric spaces. Wealso establish some new fixed point theorems in real vector cone metricspaces. The details are as follows.1. We establish a vectorial Ekeland’s variational principle where theobjective function is from a bornological vector space into a cone-orderedreal vector space, and the ordering cone needs not be solid. From this, avectorial Caristi’s fixed point theorem and a vectorial Takahashi’s non-convex minimization theorem are deduced and the equivalences betweenthe three theorems are shown.2. Phelps’ lemma is extended to bornological vector spaces. An im-mediate application is to reestablish Wong’s generalization of Ekelandvariational principle to bornological vector spaces. And Ng and Zheng’sresult on existence of efcient points in topological vector spaces is easilydeduced. In bornological linear spaces, a drop theorem is established andMackey drop property is introduced. We give the streaming sequencecharacterizations of the Mackey drop property and the relationship be-tween the Mackey drop property and the problem concerning some classof functionals attaining the extreme values. 3. Prof. Zhong Chengkui gave a version of Ekeland’s variationalprinciple, which is known as Zhong’s variational principle. We obtainedthe vector-valued version of this principle, where the objective functionis from a complete metric space into a cone-ordered real vector space.4. We establish several fixed point theorems in the real vector conemetric spaces. By using nonlinear scalarization functions, we prove thatthese fixed point theorems in cone metric spaces are indeed equivalantto their corresponding counterparts in usual metric spaces. Our resultsimprove essentially some known fixed point theorems in cone metricspaces.5. Inspired by W-distances on metric spaces, we propose W-distanceson topological vector spaces. We deduce a vectorial Ekeland’s variationalprinciple, where the objective function is from a topological vector spacewith a W-distance into a cone-ordered real vector space. From this, avectorial Caristi’s fixed point theorem and a vectorial Takahashi’s non-convex minimization theorem are deduced and the equivalences betweenthe three theorems are shown.