Vectorial Ekeland variational principle for cyclically antimonotone vector equilibrium problems

Abstract

ABSTRACTIn this paper, we first obtain some results for the vectorial Ekeland variational principle with a W-distance, in which the final space is a real linear space not necessarily endowed with a topology. We introduce a cyclically antimonotone bifunction taking values in a real linear space, show its properties, and then apply the results to deduce a general vector equilibrium version of Ekeland variational principle. Moreover, we apply the properties of the cyclically antimonotone bifunction to study the existence of solutions for vector equilibrium problems defined on compact or noncompact sets. We also prove the existence of solutions for uncountable systems of vector equilibrium problems. Finally, our main results are applied to establish two new general versions of Caristi-Kirk fixed point theorem for set-valued mappings and some new existence theorems for vector variational inequalities under mild conditions.