  In this paper, we extend the definition of qx-asymptotic function, for
extended real-valued function that define on an infinite dimensional
topological normed space without lower semicontinuity or quasi-convexity
condition. As the main result, by using some asymptotic conditions, we obtain
sufficient optimality conditions for existence of solutions to equilibrium
problems, under weaker assumptions of continuity and convexity, when the
feasible set is an unbounded subset of an infinite dimensional space. Also, as
a corollary, we obtain a necessary and sufficient optimality conditions for
existence of solutions to equilibrium problems with unbounded feasible set.
Finally, as an application, we establish a result for existence of solutions to
minimization problems.
