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

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

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

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|MATH350 The Implicit Function Theorem

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

隐函数定理是分析中最有用的定理之一。我们将证明它是一致收缩定理的推论。
定理1.182 (隐函数定理) 。假设 $X, Y$ ,和 $Z$ 是 Banach 空间, $U \subseteq X, V \subseteq Y$ 是
开集, $F: U \times V \rightarrow Z$ 是一个 $C^{1}$ 功能,和 $\left(x_{0}, y_{0}\right) \in U \times V$ 和 $F\left(x_{0}, y_{0}\right)=0$.
如果 $F_{x}\left(x_{0}, y_{0}\right): X \rightarrow Z$ 有界逆,则有积邻域 $U_{0} \times V_{0} \subseteq U \times V$ 和
$\left(x_{0}, y_{0}\right) \in U_{0} \times V_{0}$ 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 $x=\beta(y)$.
证明。定义 $L: Z \rightarrow X$ 经过 $L z=\left[F_{x}\left(x_{0}, y_{0}\right)\right]^{-1} z$ 和 $G: U \times V \rightarrow X$ 经过
$G(x, y)=x-L F(x, y)$. 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
$G_{x}\left(x_{0}, y_{0}\right)=I-L F_{x}\left(x_{0}, y_{0}\right)=0$.
自从 $G$ 是 $C^{1}$ ,有一个产品邻域 $U_{0} \times V_{1}$ 其因数是两个公制球, $U_{0} \subseteq U$ 以 $x_{0}$ 和 $V_{1} \subseteq V$ 以 $y_{0}$ ,这样
$$
\left|G_{x}(x, y)\right|<\frac{1}{2}
$$
每当 $(x, y) \in U_{0} \times V_{1}$.


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


在本节中,我们将证明微分方程的基本存在性和唯一性定理。我们还将证明一个关于解
的可扩展性的定理。虽然存在性、唯一性和可扩展性定理是常微分方程理论研究的基
础,但研究它们的证明还有另一个原因。事实上,本节中使用的技术在我们学科的现代
发展中非常重要。特别是隐函数定理在微扰理论中得到了广泛的应用,收缩原理的各种
扩展是用来证明不变流形的存在性和光滑性的基本技术。我们将通过以两种不同的方式
证明微分方程的基本存在定理来演示这些工具。
假设 $J \subseteq \mathbb{R}, \Omega \subseteq \mathbb{R}^{n}$ ,和 $\Lambda \subseteq \mathbb{R}^{m}$ 都是开集,并且
$$
f: J \times \Omega \times \Lambda \rightarrow \mathbb{R}^{n}
$$
由 $(t, x, \lambda) \mapsto f(t, x, \lambda)$ 是一个连续函数。回想一下,如果 $\lambda \in \Lambda$ ,然后是常微分方 程的解
$$
\dot{x}=f(t, x, \lambda)
$$
是可微函数 $\sigma: J_{0} \rightarrow \Omega$ 在某个开放子区间上定义 $J_{0} \subseteq J$ 这样
$$
\frac{d \sigma}{d t}(t)=f(t, \sigma(t), \lambda)
$$
对所有人 $t \in J_{0}$. 为了 $t_{0} \in J, x_{0} \in \Omega$ ,和 $\lambda_{0} \in \Lambda$ ,与微分方程 (1.55) 相关的初始 值问题由微分方程和解的初始值一起给出,如下所示:
$$
\dot{x}=f\left(t, x, \lambda_{0}\right), \quad x\left(t_{0}\right)=x_{0}
$$

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