On the domain of the implicit function and applications
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The implicit function theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign-preserving condition, we prove that an implicit function exists in the case of a set of inequalities. Also in this case, we state an estimate for the size of the domain. An application to the local Lipschitz behavior of solution maps is discussed. 1. Introduction The implicit function theorem is one of the fundamental results in multivariable analysis [1, 8, 11]. It asserts that if Fi (x, y), i = 1,...,n, x ∈ Rm , y ∈ Rn , are countinuously differentiable functions in a neighborhood of a point (x0 , y0 ), where Fi (x0 , y0 ) = 0, for i = 1,...,n, and the Jacobian
D y F x0 , y 0 =
∂Fi x0 , y 0 ∂y j
1≤i, j ≤n
(1.1)
is invertible, then there exist a positive number r > 0 and continuous functions g1 (x),..., gn (x), defined in the domain B = {x ∈ Rm : |x − x0 | < r }, such that gi (x0 ) = y0i and Fi (x,g1 (x),...,gn (x)) = 0, for i = 1,...,n, in B. This theorem has been extended to Lipschitz functions by Clarke [4, 5]. In this case F = (F1 ,...,Fn ) is a locally Lipschitz function in a neighborhood of (x0 , y0 ) and the invertibility assumption is required for all the matrices of the generalized Jacobian of F at (x0 , y0 ). Despite the central role played by this result in analysis, multidimensional nonlinear optimization algorithms [2, 7, 16, 17], and in developing Newton-type methods for solving nonsmooth equations [12, 13, 18], a lower bound for the size of the domain B has not been sufficiently investigated in the literature. The first nontrivial estimate has been reported in [3] for the case of complex analytic functions. The authors base their result on the Roche theorem to derive a lower bound in the case n = 1, then they recursively extend this estimate to the general case. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 221–234 DOI: 10.1155/JIA.2005.221
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The domain of the implicit function
Some important problems, like those which appear in sensitivity and stability analysis of systems of equations and inequalities [14, 15], do not show strong regularity properties. Therefore, over the years, a great deal of attention has been focused on developing new tools for maps not necessarily differentiable. Given the high relevance played by the Lipschitz continuity, one of the purposes of this paper is to establish an estimate for the size of the domain B and consequently of the set of values of the implicit function, in the context of Lipschitz continuous maps. These estimates can be applied for proving the upper-Lipschitz continuity [19] of some set-valued maps. This is a recently introduced concept of regularity which turns out to be quite natural in nonlinear optimization. For illustration, we consider the question of the local Lipschitz b
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