My Blog: Anh-Tourguide: The Implicit Function Theorem: How To Prove A Continuously Dependence On Parameters For Solutions Of ODEs

It is clear that the implicit function theorem plays an important role in analysis. From now on, I am going to demonstrate this significant matter from the theory of differential equations, both ODE and PDE, point of view.

Let us start with the following ODE

$-u''-\alpha^2 u^{-q-1}+\beta^2u^{q-1}=0$

on some domain $\Omega \subset \mathbb R^n$ with $\alpha \not\equiv 0$ and $\beta \not\equiv 0$ . We assume the existence result on $W_+^{2,p}$ is proved for some $p>1$ . We prove the following

Theorem. The solution $u \in W_+^{2,p}$ depends continuously on $(\alpha, \beta) \in L^\infty \times L^\infty$ .

Proof. Consider the map

$\mathcal N : W_+^{2,p} \times (L^\infty \times L^\infty) \to L^p$

taking

$(u,\alpha,\beta) \mapsto -u''-\alpha^2 u^{-q-1}+\beta^2u^{q-1}$ .

This map is evidently continuous (since $W_+^{2,p}$ is an algebra). One readily shows that its Fréchet derivative at $(u, \alpha, \beta)$ with respect to $u$ in the direction $h$ is

$\mathcal N'[u,\alpha ,\beta ]h = - h'' + \left[ {(q + 1){\alpha ^2}{u^{ - q - 2}} + (q - 1){\beta ^2}{u^{q - 2}}} \right]h$ .

The continuity of the map

$(u,\alpha,\beta) \mapsto \mathcal N'[u,\alpha ,\beta ]$

follows from the fact that $W_+^{2,p}$ is an algebra continuously embedded in $C^0(\Omega)$ .

Since $\alpha \not\equiv 0$ and $\beta \not\equiv 0$ , the potential

$V={(q + 1){\alpha ^2}{u^{ - q - 2}} + (q - 1){\beta ^2}{u^{q - 2}}}$

is not identically zero. Thus it is well-known that the map

$-\Delta +V : W^{2,p} \to L^p$

is an isomorphism.

The implicit function theorem then implies that if $u_0$ is a solution for data $(\alpha_0, \beta_0)$ , there is a continuous map defined near $(\alpha_0, \beta_0)$ taking $(\alpha, \beta)$ to the corresponding solution of the ODE. This establishes the conclusion.

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The Implicit Function Theorem: How To Prove A Continuously Dependence On Parameters For Solutions Of ODEs

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