By invoking the Implicit Function Theorem
, there exists a function f such that the identity [y'.sub.1] - E'Qy = f(t,Qy) holds.
To illustrate that [GAMMA] (V, [[PHI].sup.*.sub.0], [[PHI].sup.*.sub.1]) satisfies the implicit function theorem
, the following theorem is given.
and the implicit function theorem
, there exists a subfamily of variations satisfying restriction equation (43).
(2) In order to handle the nonaffine coupling terms, the implicit function theorem
and the mean value theorem are invoked, respectively.
and using the analytic implicit function theorem
, one solves w in terms of z, [bar.z], and [bar.w] getting a representation of M as
By Propositions 3.1, 4.1 we can apply the implicit function theorem
(see, e.g., [4, Theorem 1.20]) to conclude that F maps a sufficiently small neighborhood of a positive, constant function c in [C.sup.[alpha]][(0, 1).sub.0] homeomorphically onto a neighborhood of [square root of (2c')] in [C.sup.[alpha]+[1/2]] [(0, 1).sub.0].
Advanced topics included vector-valued functions, the implicit function theorem
, extremal problems, matrix-valued holomorphic functions, matrix equations, realization theory, eigenvalue location and zero location problems, convexity, and some special results relating to matrices with nonnegative entries.
The implicit function theorem
is applied in a manner that requires further qualification because the optimization problem is of an unconstrained kind without any redundant variables.
By implicit function theorem
, there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (10) has a unique solution ([lambda], [M.sub.2], [M.sub.3]) satisfying [lambda], [M.sub.2] > 0 and [M.sub.3] > 0.
The continuity of the constant equilibria follows from the Implicit Function Theorem
and the hypothesis of normal hyperbolicity.
We prove that the first-order condition is an implicit function [R.sub.i] = [R.sub.i]([R.sub.j]), except possibly at a point of measure zero using the Implicit Function Theorem
. We first establish two lemmas where we define