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In this paper, we introduce the concept of a weak $q$-distance and for this distance we derive a set-valued version of Ekeland's variational principle in the setting of uniform spaces. By using this principle, we prove the existence of solutions to a vector optimization problem with a set-valued map. Moreover, we define the $(p,\varepsilon)$-condition of Takahashi and the $(p,\varepsilon)$-condition of Hamel for a set-valued map. It is shown that these two conditions are equivalent. As an application, we discuss the relationship between an $\varepsilon$-approximate solution and a solution of a vector optimization problem with a set-valued map. Also, a well-posedness result for a vector optimization problem with a set-valued map is given.

EKELAND'S VARIATIONAL PRINCIPLE FOR SET-VALUED MAPS WITH APPLICATIONS TO VECTOR OPTIMIZATION IN UNIFORM SPACES