Sequentially lower complete spaces and Ekeland.s variational principle

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.