On Ha.s version of set-valued Ekeland.s variational principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187.206 (2005)] established a new version of Ekeland.s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha.s version of set-valued Ekeland.s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha.s version, we deduce a Caristi-Kirk.s fixed point theorem and a Takahashi.s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.