# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MAP2302 Integration in Banach Spaces

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## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Integration in Banach Spaces

This section is a brief introduction to integration on Banach spaces following the presentation in [106]. As an application, we will give an alternative proof of the mean value theorem and a proof of a version of Taylor’s theorem.

Let $I$ denote a closed interval of real numbers and $X$ a Banach space with norm $|$. A simple function $f: I \rightarrow X$ is a function with the following property: There is a finite cover of $I$ consisting of disjoint subintervals such that $f$ restricted to each subinterval is constant. Here, each subinterval can be open, closed, or half open.

A sequence $\left{f_{n}\right}_{n=1}^{\infty}$ of not necessarily simple functions, each mapping $I$ to $X$, converges uniformly to a function $f: I \rightarrow X$ if for each $\epsilon>0$ there is an integer $N>0$ such that $\left|f_{n}(t)-f_{m}(t)\right|<\epsilon$ whenever $n, m>N$ and $t \in I$.

Definition 1.158. A regulated function is a uniform limit of simple functions.
Lemma 1.159. Every continuous function $f: I \rightarrow X$ is regulated.

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|The Contraction Principle

In this section, let us suppose that $(X, d)$ is a metric space. A point $x_{0} \in X$ is a fixed point of a function $T: X \rightarrow X$ if $T\left(x_{0}\right)=x_{0}$. The fixed point $x_{0}$ is called globally attracting if $\lim {n \rightarrow \infty} T^{n}(x)=x{0}$ for each $x \in X$.

Definition 1.170. Suppose that $T: X \rightarrow X$, and $\lambda$ is a real number such that $0 \leq \lambda<1$. The function $T$ is called a contraction (with contraction constant $\lambda$ ) if
$$d(T(x), T(y)) \leq \lambda d(x, y)$$
whenever $x, y \in X$.
The next theorem is fundamental; it states that a contraction, viewed as a dynamical system, has a globally attracting fixed point.

Theorem $1.171$ (Contraction Mapping Theorem). If the function $T$ is a contraction on the complete metric space $(X, d)$ with contraction constant $\lambda$, then $T$ has a unique fixed point $x_{0} \in X$. Moreover, if $x \in X$, then the sequence $\left{T^{n}(x)\right}_{n=0}^{\infty}$ converges to $x_{0}$ as $n \rightarrow \infty$ and
$$d\left(T^{n}(x), x_{0}\right) \leq \frac{\lambda^{n}}{1-\lambda} d\left(x, x_{0}\right) .$$
Proof. Let us prove first that fixed points of $T$ are unique. Indeed, if $T\left(x_{0}\right)=x_{0}$ and $T\left(x_{1}\right)=x_{1}$, then, by virtue of the fact that $T$ is a contraction, $d\left(T\left(x_{0}\right), T\left(x_{1}\right)\right) \leq \lambda d\left(x_{0}, x_{1}\right)$, and, by virtue of the fact that $x_{0}$ and $x_{1}$ are fixed points, $d\left(T\left(x_{0}\right), T\left(x_{1}\right)\right)=d\left(x_{0}, x_{1}\right)$. Thus, we have that
$$d\left(x_{0}, x_{1}\right) \leq \lambda d\left(x_{0}, x_{1}\right)$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代 考|The Contraction Principle

d(T(x), T(y)) \leq \lambda d(x, y)
$$每当 x, y \in X. 下一个定理是基本的；它指出，收缩，被视为一个动力系统，有一个全球吸引的固定 点。 定理1.171 (收缩映射定理)。如果函数 T 是完整度量空间的收缩 (X, d) 收缩常数 \lambda, 然后 T 有一个唯一的不动点 x_{0} \in X. 此外，如果 x \in X ，那么序列$$
d\left(T^{n}(x), x_{0}\right) \leq \frac{\lambda^{n}}{1-\lambda} d\left(x, x_{0}\right) .
$$证明。让我们首先证明 T 是独一无二的。确实，如果 T\left(x_{0}\right)=x_{0} 和 T\left(x_{1}\right)=x_{1} ， 那么，由于 T 是收缩， d\left(T\left(x_{0}\right), T\left(x_{1}\right)\right) \leq \lambda d\left(x_{0}, x_{1}\right) ，并且，由于 x_{0} 和 x_{1} 是固 定点， d\left(T\left(x_{0}\right), T\left(x_{1}\right)\right)=d\left(x_{0}, x_{1}\right). 因此，我们有$$
d\left(x_{0}, x_{1}\right) \leq \lambda d\left(x_{0}, x_{1}\right)