530: Sufficient Conditions for Existence of Unique Global Solution on Interval for Euclidean-Normed Euclidean Vectors Space ODE

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description/proof of sufficient conditions for existence of unique global solution on interval for Euclidean-normed Euclidean vectors space ODE

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Topics

About: normed vectors space
About: differential equation

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description 1
• 2: Natural Language Description 1
• 3: Proof 1
• 4: Note 1

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Starting Context

• The reader knows a definition of Euclidean-normed Euclidean vectors space.
• The reader admits the local unique solution existence for Euclidean-normed Euclidean vectors space ordinary differential equation.

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Target Context

• The reader will have a description and a proof of some sufficient conditions with which there is the unique solution on the whole interval with any initial condition for any Euclidean-normed Euclidean vectors space ordinary differential equation.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description 1

Here is the rules of Structured Description.

Entities:
${\mathbb{R}}^d$: $=\text{ the Euclidean-normed Euclidean vectors space }$
$\mathbb{R}$: $=\text{ the Euclidean-normed Euclidean vectors space }$
$J$: $\subseteq \mathbb{R}$, $=[t_1,t_e]$
$f$: $:{\mathbb{R}}^d\times J\to {\mathbb{R}}^d$, $\in \{\text{ the }{C}^0\text{ maps }\}$
$\frac{dx}{dt}=f(x,t)$: = the ordinary differential equation with any initial condition, $x(t_1)=x_{t_1}$
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Statements:
$\mathrm{\forall }{t}_0\in J$
(
$\mathrm{\exists }[t_0-{ϵ}_{t_0,1},t_0+{ϵ}_{t_0,2}]$ that satisfies the conditions for the local solution existence of the equation for the initial condition with any initial value, $x_{t_0}$, for $t_0$, where ${ϵ}_{t_0,j}$ is not zero except ${ϵ}_{t_1,1}$ and ${ϵ}_{t_e,2}$ and does not depend on $x_{t_0}$ although can depend on $t_0$, which means that there is a map, $x_{t_0}\mapsto x(t)$ for each $t\in [t_0-{ϵ}_{t_0,1},t_0+{ϵ}_{t_0,2}]$, and the map is bijective
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$⟹$
The ordinary differential equation has the unique solution on the entire $J$.
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2: Natural Language Description 1

For any Euclidean-normed Euclidean vectors space, ${\mathbb{R}}^d$ and $\mathbb{R}$, any closed interval, $J\subseteq \mathbb{R}=[t_1,t_e]$, and any $C^0$ map, $f:{\mathbb{R}}^d\times J\to {\mathbb{R}}^d$, the ordinary differential equation, $\frac{dx}{dt}=f(x,t)$ with any initial condition $x(t_1)=x_{t_1}$, has the unique solution on the entire $J$ if at each point, $t_0\in J$, there is a closed interval, $[t_0-{ϵ}_{t_0,1},t_0+{ϵ}_{t_0,2}]$, that satisfies the conditions for the local solution existence of the equation for the initial condition with any initial value, $x_{t_0}$, for $t_0$, where ${ϵ}_{t_0,j}$ is not zero except ${ϵ}_{t_1,1}$ and ${ϵ}_{t_e,2}$, and does not depend on $x_{t_0}$ although can depend on $t_0$, which means that there is a map, $x_{t_0}\mapsto x(t)$ for each $t\in [t_0-{ϵ}_{t_0,1},t_0+{ϵ}_{t_0,2}]$, and the map is bijective.

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3: Proof 1

The open intervals, $(t_0-{ϵ}_{t_0,1},t_0+{ϵ}_{t_0,2})$ for $t_0\ne t_1,t_e$, $[t_1,t_1+{ϵ}_{t_1,2})$, and $(t_e-{ϵ}_{t_e,1},t_e]$, cover compact $J$, so, there is a finite subcover, $[t_1,t_1+{ϵ}_{t_1,2}),(t_2-{ϵ}_{t_2,1},t_2+{ϵ}_{t_2,2}),...,(t_e-{ϵ}_{t_e,1},t_e]$.

There is the local unique solution for $[t_1,t_1+{ϵ}_{t_1,2})$ with the initial condition, $\bar{x}(t_1)=x_{t_1}$, $\bar{x}:[t_1,t_1+{ϵ}_{t_1,2})\to {\mathbb{R}}^d$. Let us define the restriction of $x$, $x{|}_{[t_1,t_1+{ϵ}_{t_1,2})}=\bar{x}$.

There is an interval that intersects $[t_1,t_1+{ϵ}_{t_1,2})$, $(t_{j_2}-{ϵ}_{t_{j_2},1},t_{j_2}+{ϵ}_{t_{j_2},2})$, with an intersection point denoted as $t_{1,j_2}$, and there is the local unique solution for it with the initial condition, $\bar{x}(t_{j_2})=x_{t_{j_2}}$, where $x_{t_{j_2}}$ can be chosen such that $\bar{x}(t_{1,j_2})=x(t_{1,j_2})$, which is because the bijective map mentioned in Description is supposed to exist. In fact, $\bar{x}$ coincides with $x$ on the whole intersection area, because there is an interval around $t_{1,j_2}$ and there is the unique solution on it with the initial condition, which has to agree with $x$ and $\bar{x}$ there, and if $x$ and $\bar{x}$ did not agree on the whole intersection area, $(t_{j_2}-{ϵ}_{t_{j_2},1},t_{j_2}+{ϵ}_{t_{j_2},2})$ would have another solution by switching to $x$ after passing $t_{1,j_2}$ or $[t_1,t_1+{ϵ}_{t_1,2})$ would have another solution by switching to $\bar{x}$ after passing $t_{1,j_2}$. Let us extend $x$ to $[t_1,t_{j_2}+{ϵ}_{t_{j_2},2})$ with $\bar{x}$.

Thus, $x$ can be extended to the whole $J$. The extension is unique, because at each point, $t_{j_k}$, $x_{t_{j_k}}$ is determined uniquely.

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4: Note 1

It is crucial that such a bijective map exists for this proposition. Just that there is a local solution around each point does not guaranteed the existence of any global solution, as is described in another article.

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References

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