Generalized equilibrium problems with marginally 0-upper semicontinuous bifunctions

Abstract In this paper, we study the existence of solutions for equilibrium problems in which the bifunction involved is defined on the Cartesian product of two distinct sets. Such a problem will be called a generalized equilibrium problem. The proofs we propose for establishing the existence theorems do not rely on classical tools, such as KKM-type theorems or separation theorems. Instead, as we will see, they are based on an intersection theorem established by Bassanezi and Greco [11]. Upper semicontinuity in the first variable of a given bifunction is a standard condition when seeking solutions to a classical equilibrium problem. In this paper, we introduce a concept that refines upper semicontinuity, called marginal r -upper semicontinuity. In our existence results, it will suffice as the bifunction to be marginally 0-upper semicontinuous in the first variable.