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.