Existence Theorems for Generalized Noncoercive Equilibrium Problems: The Quasi-Convex Case

We provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems (an extension of generalized quasi-variational inequalities) defined in reflexive Banach spaces in the quasi-convex case. In addition, several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given. Our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems. In fact, we find some particular cones as estimates for the recession cone of the solution set. These cones (for the ones containing the latter set) are proved to be sharp enough to encompass several special situations found in the literature.