On the Anti-Monotonicity of Differential Mappings Connected with General Equilibrium problem

Let X be a subset of a Hilbert space H and Ψ: X x X →R, Ψ(x, x) = 0 for all x ϵ X. Let G(x) =∂yΨ(x, y) [y=x denote generalized differential with respect to the second argument at tire point (x, x). We shall be concerned with the properties of the function F sufficient to ensure the anti-monotonicity of the map G(x). It will be shown that for the anti-monotonicity of the map G(x) it is sufficient to assume convexity-concavity of the function V. In the case of the weakly convex-concave function F the map) G(x) is antimonotone under some conditions on the remainder terms. In the case of the quasieon vex- concave function F, the condition similar to the anti-monotonicity condition holds. Furthermore, some properties of weakly convex functions will be proved.