Comparison of two different types of pseudomonotone mappings

The purpose of this paper is to show connections between two different types of pseudomonotone mappings. The first type of pseudomonotonic-ity notion, called in this paper A-pseudomonotonicity, was introduced by S. Karamardian [8] and has been frequently used in optimization problems [3, 4, 6, 11]. The second type, called in this paper B-pseudomonotonicity, is a generaliza-tion of the pseudomonotonicity notion introduced by H. Brézis [2] and used in the study of integral equations and partial differential equations [7, 9, 10]. We will follow some of the methods elaborated in [5] during the study of functional variational inequalities. Throughout this paper X will be a Banach space, X ∗ its dual space, and K ⊂ X a nonempty, closed, convex set. Definition 1 A mapping T: K → X ∗ is said to be monotone if 〈T (x) − T (y), x − y 〉 ≥ 0, ∀ x, y ∈ K. We notice that if T is monotone mapping, the