# 数学代写|常微分方程代写ordinary differential equation代考|DIFFERENTIABILITY WITH RESPECT TO PARAMETERS

#### Doug I. Jones

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## 数学代写|常微分方程代写ordinary differential equation代考|DIFFERENTIABILITY WITH RESPECT TO PARAMETERS

In the present section we consider systems of equations (E) and initial value problems (I). Given $f \in C(D)$ with $f$ differentiable with respect to $x$, we definine the Jacobian matrix $f_x=\partial f / \partial x$ as the $n \times n$ matrix whose $(i, j)$ th element is $\partial f_i / \partial x_j$, i.e.,
$$f_x=\partial f / \partial x=\left[\partial f_i / \partial x_j\right] .$$
In this section, and throughout the remainder of this book, $E$ will denote the identity matrix. When the dimension of $E$ is to be emphasized, we shall write $E_n$ to denote an $n \times n$ identity matrix.

In the present section we show that when $f_x$ exists and is continuous, then the solution $\phi$ of (I) depends smoothly on the parameters of the problem.

Theorem 7.1. Let $f \in C(D)$, let $f_x$ exist and let $f_x \in C(D)$. If $\phi(t, \tau, \xi)$ is the solution of (E) such that $\phi(\tau, \tau, \xi)=\xi$, then $\phi$ is of class $C^1$ in $(t, \tau, \xi)$. Each vector valued function $\partial \phi / \partial \xi_i$ or $\partial \phi / \partial \tau$ will solve
$$y^{\prime}=f_x(t, \phi(t, \tau, \xi)) y$$
as a function of $t$ while
$$\frac{\partial \phi}{\partial \tau}(\tau, \tau, \xi)=-f(\tau, \xi) \quad \text { and } \quad \frac{\partial \phi}{\partial \xi}(\tau, \tau, \xi)=E_n .$$
Proof. In any small spherical neighborhood of any point $(\tau, \xi) \in D$, the function $f$ is Lipschitz continuous in $x$. Hence $\phi(t, \tau, \xi)$ exists locally, is unique, is continuable while it remains in $D$, and is continuous in $(t, \tau, \xi)$. Note also that $(7.1)$ is a linear equation with continuous coefficient matrix. Thus by Theorem 6.1 solutions of $(7.1)$ exist for as long as $\phi(t, \tau, \xi)$ is defined.

Fix a point $(t, \tau, \xi)$ and define $\xi(h)=\left(\xi_1+h, \xi_2, \ldots, \xi_n\right)^{\mathrm{T}}$ for all $h$ with $|h|$ so small that $(\tau, \xi(h)) \in D$. Define
$$z(t, \tau, \xi, h)=(\phi(t, \tau, \xi(h))-\phi(t, \tau, \xi)) / h, \quad h \neq 0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|COMPARISON THEORY

This is the only section of the present chapter where it is crucial in our treatment of some results that the differential equation in question be a scalar equation. We point out that the results below on maximal solutions could be generalized to vector systems, however, only under the strong assumption that the system of equations is quasimonotone (see the problems at the end of the chapter).

Consider the scalar initial value problem (I’) where $f \in C(D)$ and $D$ is a domain in the $(t, x)$ space. Any solution of $\left(I^{\prime}\right)$ can be bracketed between the two special solutions called the maximal solution and the minimal solution. More precisely, we define the maximal solution $\phi_{\mathrm{M}}$ of $\left(I^{\prime}\right)$ to be that noncontinuable solution of $\left(I^{\prime}\right)$ such that if $\phi$ is any other solution of $\left(I^{\prime}\right)$, then $\phi_M(t) \geq \phi(t)$ for as long as both solutions are defined. The minimal solution $\phi_m$ of $\left(\mathrm{I}^{\prime}\right)$ is defined to be that noncontinuable solution of $\left(\mathrm{I}^{\prime}\right)$ such that if $\phi$ is any other solution of $\left(l^{\prime}\right)$, then $\phi_m(t) \leq \phi(t)$ for as long as both solutions are defined. Clearly, when $\phi_M$ and $\phi_m$ exist, they are unique. Their existence will be proved below.
Given $\varepsilon \geq 0$, consider the family of initial value problems
$$X^{\prime}=f(t, X)+\varepsilon, \quad X(\tau)=\xi+\varepsilon .$$
Let $X(t, \varepsilon)$ be any fixed solution of $(8 . \varepsilon)$ which is noncontinuable to the right. We are now in a position to prove the following result.
Theorem 8.1. Let $f \in C(D)$ and let $\varepsilon \geq 0$.
(i) If $\varepsilon_1>\varepsilon_2$, then $X\left(t, \varepsilon_1\right)>X\left(t, \varepsilon_2\right)$ for as long as both solutions exist and $t \geq \tau$.
(ii) There exist $\beta$ as well as a solution $X^$ of $\left(l^{\prime}\right)$ defined on $[\tau, \beta)$ and noncontinuable to the right such that $$\lim _{\varepsilon \rightarrow 0^{+}} X(t, \varepsilon)=X^(t)$$
with convergence uniform for $t$ on compact subsets of $[\tau, \beta)$.
(iii) $X^$ is the maximal solution of $\left(I^{\prime}\right)$, i.e., $X^=\phi_M$.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|DIFFERENTIABILITY WITH RESPECT TO PARAMETERS

$$f_x=\partial f / \partial x=\left[\partial f_i / \partial x_j\right] .$$

$$y^{\prime}=f_x(t, \phi(t, \tau, \xi)) y$$

$$\frac{\partial \phi}{\partial \tau}(\tau, \tau, \xi)=-f(\tau, \xi) \quad \text { and } \quad \frac{\partial \phi}{\partial \xi}(\tau, \tau, \xi)=E_n .$$

$$z(t, \tau, \xi, h)=(\phi(t, \tau, \xi(h))-\phi(t, \tau, \xi)) / h, \quad h \neq 0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|COMPARISON THEORY

$$X^{\prime}=f(t, X)+\varepsilon, \quad X(\tau)=\xi+\varepsilon .$$

(i)如果$\varepsilon_1>\varepsilon_2$，则$X\left(t, \varepsilon_1\right)>X\left(t, \varepsilon_2\right)$，只要两个解都存在，$t \geq \tau$。
(ii)存在$\beta$以及$\left(l^{\prime}\right)$的解决方案$X^$，该解决方案定义在$[\tau, \beta)$上，并且不可持续到$$\lim _{\varepsilon \rightarrow 0^{+}} X(t, \varepsilon)=X^(t)$$

(iii) $X^$为$\left(I^{\prime}\right)$的极大解，即$X^=\phi_M$。

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