On the anti-monotonicity of differential mappings connected with general equilibrium problem
Stan Uryasev · Optimization · 1988
Let X be a subset of a Hilbert space H and for all denote generalized differential with respect to the second argument at the point (x, x).We shall be concerned with the properties of the function IP 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 Ψ.In the case of the weakly convex-concave function Ψ the map G(x) is anti-monotone under some conditions on the remainder terms.In the case of the quasiconvex-concave function Ψ, the condition similar to the anti-monotonicity condition holds.Furthermore, some properties of weakly convex functions will be proved.