Characterization of Monotone Operators by Using a Special Class of Preinvex Functions

Abstract

For every operator F: R ⁿ → R ⁿ the scalar increment ◊F(x):R ⁿ →R will be introduced at every point x ∈ R ⁿ, by the relation

◊F(x)(u) = <F(x + u) - F(x), u>

where <. , .> denotes the canonical scalar product of R ⁿ. If F is hemicontinuous, it will be proved that F is monotone (with respect to duality) if and only if its scalar increment is preinvex in every point of R ⁿ. A special class of preinvex functions, called starvex functions, will be introduced. In the differentiable case this class of preinvex functions coincide with the corresponding class of invex functions. In the noncontinuoUS case it will be shown that the monotonicity of F coincides with the starvexity of its scalar increment at every point of R ⁿ. If F is differentiable the monotonicity of F will be proved to be equivalent with the invexities of all its scalar increments. Finally, a characterization for strictly monotone operators will be given.