Implicit functions and sensitivity of stationary points

Abstract

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ℝ n ,D being an open bounded subset of ℝ n . LetF belong toL(D) and suppose that $$\bar x$$ solves the equationF(x) = 0. In case that the generalized Jacobian ofF at $$\bar x$$ is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of $$\bar x$$ Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.