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By using sequentially lower complete spaces (see [Zhu, J., Wei, L., Zhu, C. C.: Caristi type coincidence point theorem in topological spaces. J. Applied Math., 2013, ID 902692 (2013)]), we give a new version of vectorial Ekeland's variational principle. In the new version, the objective function is defined on a sequentially lower complete space and taking values in a quasi-ordered locally convex space, and the perturbation consists of a weakly countably compact set and a non-negative function p which only needs to satisfy p( x, y) = 0 iff x = y. Here, the function p need not satisfy the subadditivity. From the new Ekeland's principle, we deduce a vectorial Caristi's fixed point theorem and a vectorial Takahashi's non-convex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. By considering some particular cases, we obtain a number of corollaries, which include some interesting versions of fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space (which is a uniform space) has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi's fixed point theorem and a general vectorial Takahashi's nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland's variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi's nonconvex minimization theorem, a generalized Ekeland's variational principle and a generalized Caristi's fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland's variational principle, we deduce a number of particular versions of Ekeland's principle, which include many known versions of the principle and their improvements. [ABSTRACT FROM AUTHOR]