Sensitivity of Solutions to a Parametric Generalized Equation

Abstract

In the present paper we prove that, under some suitable conditions on multifunctions $C\!: {I\!\!R}^l\rightrightarrows {I\!\!R}^n$, $F\!:{I\!\!R}^d\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$, and $K\!:{I\!\!R}^l\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$, the generalized perturbation multifunction $G\!: {I\!\!R}^d\times{I\!\!R}^l\times {I\!\!R}^m\rightrightarrows {I\!\!R}^n$, of the form

$$G(\mu,\lambda,\nu)=\{x\in C(\lambda)\ |\ \nu\in F(\mu,x)+K(\lambda,x)\},$$

is proto-differentiable at (μ, λ, ν) relative to x ∈ G(μ, λ, ν). Moreover, in a special case, where K(λ, x) is a normal cone to C(λ) at x, we also provide sufficient conditions for G(·) to be single-valued on a neighborhood of (μ, λ, ν) and semi-differentiable at (μ, λ, ν).