An equilibrium version of vectorial Ekeland variational principle and its applications to equilibrium problems

We present an equilibrium version of vectorial Ekeland variational principle, where the objective bimap F is defined on the product of sequentially lower complete spaces (see Zhu et al., 2013) and taking values in a quasi-ordered locally convex space. Besides, the perturbation consists of a subset of the ordering cone and a non-negative function p which only needs to satisfy p(x,y)=0 iff x=y. Applying the equilibrium version of Ekeland principle to equilibrium problems, we obtain a general existence theorem on solutions of vectorial equilibrium problems in setting of sequentially lower complete spaces, which implies several improvements of known results. Particularly, under the framework of complete metric spaces (Z,d), we obtain an existence result of solutions for equilibrium problems which only requires that the domain X⊂Z is sequentially compact in any Hausdorff topology weaker than that induced by d. By using the theory of angelic spaces, we extend some results in Alleche and Raˇdulescu (2015) from reflexive Banach spaces to the strong duals of weakly compact generated spaces.