VECTORIAL EKELAND＇S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

By using the properties of w-distances and Gerstewitz’s functions,we first give a vectorial Takahashi’s nonconvex minimization theorem with a w-distance.From this,we deduce a general vectorial Ekeland’s variational principle,where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value.From the general vectorial variational principle,we deduce a vectorial Caristi’s fixed point theorem with a w-distance.Finally we show that the above three theorems are equivalent to each other.The related known results are generalized and improved.In particular,some conditions in the theorems of [Y.Araya,Ekeland’s variational principle and its equivalent theorems in vector optimization,J.Math.Anal.Appl.346(2008),9-16] are weakened or even completely relieved.