A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

Let D be a nonempty convex subset of a Banach spaces E. Let F be a bifunction from D ×D to R, where R is the set of all real numbers. The equilibrium problem for F is to find x ∈ D such that F x, y ≥ 0 for all y ∈ D. The set of such solutions is denoted by EP F . The set D is called proximal if for each x ∈ E, there exists an element y ∈ D such that ‖x − y‖ d x,D , where d x,D inf{‖x − z‖ : z ∈ D}. Let CB D , K D , and P D denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of D, respectively. The Hausdorff metric on CB D is defined by