Variational relation problems were introduced by Luc in [1] as a general model for a large class of problems in nonlinear analysis and applied mathematics. Since this manner of approach provides unified results for several mathematical problems it has been used in many recent papers (see [2] [3] [4] [5] [6] [7] [8] [9] ). In this paper we investigate the existence of solutions for three types of variational relation problems which encompass several generalized equilibrium problems, variational

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... nequalities and variational inclusions studied in a long list of papers in the field. INTRODUCTION Let X, Y and Z be nonempty sets. A nonempty subset R of the product X ×Y ×Z determines a relation R(x, y, z) in a natural manner: we say that R(x, y, z) holds if and only if (x, y, z) ∈ R. When Z is a parameter set, then R is called a variational relation. Variational relation problems were introduced by Luc in [1] as a general model for a large class of problems in nonlinear analysis and applied mathematics, including optimization problems, variational inequalities, variational inclusions, equilibrium problems, etc. Since this manner of approach provides unified results for several mathematical problems it has been used in many recent papers (see [2] [3] [4] [5] [6] [7] [8] [9] [10] ). The present paper fits into this interesting group of works, establishing several existence theorems for the solutions of the following three types of variational relation problems: Assume that X is a convex set in a topological vector space and Y and Z are two sets, endowed for each problem with an adequate topological and/or algebraic structure. Let T : X Y , P : X Z be two set-valued mappings and R(x, y, z) be a relation linking elements x ∈ X, y ∈ Y , z ∈ Z. (VRP 1 a ) Findx ∈ X such that R(x, y, z) holds for all y ∈ T (x) and all z ∈ P (x). (VRP 1 b ) Findx ∈ X such that for each y ∈ T (x) there exists z ∈ P (x) such that R(x, y, z) holds. (VRP 2) Findx ∈ X andz ∈ P (x) such that R(x, y,z) holds for all y ∈ T (x).