Abstract: We considered nonempty  boundness of solution sets for the quasiconvex function minimization problems in reflexive Banach spaces, gave the definition of q-asymptotic function of quasiconvex function on the nonempty closed convex set, and obtained the equivalent characterization of the existence of the solution set of the quasiconvex function minimization problems. The nonempty boundness of the solution set of the classical minimization problem of  convex function was extended to quasiconvex function, and a sufficient condition for the nonempty boundness of the solution set of equilibrium problems was given.

Key words: asymptotic cone, asymptotic function, quasiconvex function, quasiconvex minimization problem, equilibrium problem