On conditions warranting Φ-subdifferentiability

Abstract

Let X, Y be two normed spaces. A multifunction Γ:Y→2^X is called γ-paraconvex (1<γ≤2) if there is a constant C>0 such that for all y ₁, y ₂∃dom γ and all numbers α, β≥0, α+β=1, the following inclusion holds

$$\alpha \Gamma y_1 + \beta \Gamma y_2 \subset \Gamma (\alpha y_1 + \beta y_2 ) + C\left\| {y_1 - y_2 } \right\|^\gamma B.$$

In the paper properties of γ-paraconvex multifunctions are described. It is proved that if Y is a Hilbert space, Γy is a closed valued 2-paraconvex multifunction, and y ₀∃Int dom Γy, x ₀∃Γy ₀, then Γy is locally Lipschitzian at (x ₀, y ₀) (i.e. there are a neighbourhood Q of x ₀, a neighbourhood W of y ₀ and a constant K>0 such that for y∃W

$$\bar Q \cap \Gamma y \subset \bar Q \cap \Gamma y_0 + K\left\| {y - y_0 } \right\|. B_X .$$

The investigations of γ-paraconvexity were stimulated by investigations of Hölder and Lipschitz differentiability of Lagrangians considered by Dolecki and Kurcyusz in [8].