# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH350 The Implicit Function Theorem

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|The Implicit Function Theorem

The implicit function theorem is one of the most useful theorems in analysis. We will prove it as a corollary of the uniform contraction theorem.

Theorem $1.182$ (Implicit Function Theorem). Suppose that $X, Y$, and $Z$ are Banach spaces, $U \subseteq X, V \subseteq Y$ are open sets, $F: U \times V \rightarrow Z$ is a $C^{1}$ function, and $\left(x_{0}, y_{0}\right) \in U \times V$ with $F\left(x_{0}, y_{0}\right)=0$. If $F_{x}\left(x_{0}, y_{0}\right): X \rightarrow Z$ has a bounded inverse, then there is a product neighborhood $U_{0} \times V_{0} \subseteq U \times V$ with $\left(x_{0}, y_{0}\right) \in U_{0} \times V_{0}$ and a $C^{1}$ function $\beta: V_{0} \rightarrow U_{0}$ such that $\beta\left(y_{0}\right)=x_{0}$. Moreover, if $F(x, y)=0$ for $(x, y) \in U_{0} \times V_{0}$, then $x=\beta(y)$.

Proof. Define $L: Z \rightarrow X$ by $L z=\left[F_{x}\left(x_{0}, y_{0}\right)\right]^{-1} z$ and $G: U \times V \rightarrow X$ by $G(x, y)=x-L F(x, y)$. Note that $G$ is $C^{1}$ on $U \times V$ and $F(x, y)=0$ if and only if $G(x, y)=x$. Moreover, we have that $G\left(x_{0}, y_{0}\right)=x_{0}$ and $G_{x}\left(x_{0}, y_{0}\right)=I-L F_{x}\left(x_{0}, y_{0}\right)=0$.

Since $G$ is $C^{1}$, there is a product neighborhood $U_{0} \times V_{1}$ whose factors are two metric balls, $U_{0} \subseteq U$ centered at $x_{0}$ and $V_{1} \subseteq V$ centered at $y_{0}$, such that
$$\left|G_{x}(x, y)\right|<\frac{1}{2}$$
whenever $(x, y) \in U_{0} \times V_{1}$.

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Existence, Uniqueness, and Extensibility

In this section we will prove the basic existence and uniqueness theorems for differential equations. We will also prove a theorem on extensibility of solutions. While the theorems on existence, uniqueness, and extensibility are the foundation for theoretical study of ordinary differential equations, there is another reason to study their proofs. In fact, the techniques used in this section are very important in the modern development of our subject. In particular, the implicit function theorem is used extensively in perturbation theory, and the various extensions of the contraction principle are fundamental techniques used to prove the existence and smoothness of invariant manifolds. We will demonstrate these tools by proving the fundamental existence theorem for differential equations in two different ways.
Suppose that $J \subseteq \mathbb{R}, \Omega \subseteq \mathbb{R}^{n}$, and $\Lambda \subseteq \mathbb{R}^{m}$ are all open sets, and
$$f: J \times \Omega \times \Lambda \rightarrow \mathbb{R}^{n}$$
given by $(t, x, \lambda) \mapsto f(t, x, \lambda)$ is a continuous function. Recall that if $\lambda \in \Lambda$, then a solution of the ordinary differential equation
$$\dot{x}=f(t, x, \lambda)$$
is a differentiable function $\sigma: J_{0} \rightarrow \Omega$ defined on some open subinterval $J_{0} \subseteq J$ such that
$$\frac{d \sigma}{d t}(t)=f(t, \sigma(t), \lambda)$$
for all $t \in J_{0}$. For $t_{0} \in J, x_{0} \in \Omega$, and $\lambda_{0} \in \Lambda$, the initial value problem associated with the differential equation (1.55) is given by the differential equation together with an initial value for the solution as follows:
$$\dot{x}=f\left(t, x, \lambda_{0}\right), \quad x\left(t_{0}\right)=x_{0} .$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|The Implicit Function Theorem

$\left(x_{0}, y_{0}\right) \in U_{0} \times V_{0}$ 和一个 $C^{1}$ 功能 $\beta: V_{0} \rightarrow U_{0}$ 这样 $\beta\left(y_{0}\right)=x_{0}$. 此外，如果 $F(x, y)=0$ 为了 $(x, y) \in U_{0} \times V_{0}$ ，然后 $x=\beta(y)$.

$G(x, y)=x-L F(x, y)$. 注意 $G$ 是 $C^{1}$ 上 $U \times V$ 和 $F(x, y)=0$ 当且仅当
$G(x, y)=x$. 此外，我们有 $G\left(x_{0}, y_{0}\right)=x_{0}$ 和
$G_{x}\left(x_{0}, y_{0}\right)=I-L F_{x}\left(x_{0}, y_{0}\right)=0$.

$$\left|G_{x}(x, y)\right|<\frac{1}{2}$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Existence, Uniqueness, and Extensibility

$$f: J \times \Omega \times \Lambda \rightarrow \mathbb{R}^{n}$$

$$\dot{x}=f(t, x, \lambda)$$

$$\frac{d \sigma}{d t}(t)=f(t, \sigma(t), \lambda)$$

$$\dot{x}=f\left(t, x, \lambda_{0}\right), \quad x\left(t_{0}\right)=x_{0}$$