Peano's Existence TheoremProving Peano's Existence Theorem by approximating with $C^{infty}$ functions using...
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Peano's Existence Theorem
Proving Peano's Existence Theorem by approximating with $C^{infty}$ functions using Weierstrass' Theorem.Mean Value Theorem and Intermediate Value TheoremWeierstrass Approximation Theorem for continuous functions on open intervalDoes the implicit function theorem imply Peano existence theoremSome questions about Weierstrass approximation theoremLecture Notes in Real AnalysisPeano existence theorem & Uniqueness of solutions to IVPsProof without using Stone-Weierstrass theorem.help Extreme Value Theorem proof stepHow to study real analysis?Looking for a “soft” book on second semester real analysis?
$begingroup$
This is an exercise problem given by my TA in my Real Analysis class. However, I am having quite a bit of problem with just understanding the problem. In particular, I am not sure how to show (1) and the motivation for defining such $phi_n(t)$
For (3), I suppose I need to express $phi_n(t)$ as a sequence of real-valued continuous function in order to get the result.
For (4), I thought the function is defined as $phi_n(t)=y_0+int_0^t F(phi_n(s-frac{1}{n}),s)$ already. The question makes me a bit confused.
In summary, I would like to hear anything that can help me understand the question better as well as proceeding with the answers. The textbook I am using is Analysis II by Tao
I have also found a comparable proof where I was only able to understand part of it. Proving Peano's Existence Theorem by approximating with $C^{infty}$ functions using Weierstrass' Theorem.
real-analysis
asked Mar 13 at 6:44
RicoRico
908
$endgroup$
add a comment |
$begingroup$
This is an exercise problem given by my TA in my Real Analysis class. However, I am having quite a bit of problem with just understanding the problem. In particular, I am not sure how to show (1) and the motivation for defining such $phi_n(t)$
For (3), I suppose I need to express $phi_n(t)$ as a sequence of real-valued continuous function in order to get the result.
For (4), I thought the function is defined as $phi_n(t)=y_0+int_0^t F(phi_n(s-frac{1}{n}),s)$ already. The question makes me a bit confused.
In summary, I would like to hear anything that can help me understand the question better as well as proceeding with the answers. The textbook I am using is Analysis II by Tao
I have also found a comparable proof where I was only able to understand part of it. Proving Peano's Existence Theorem by approximating with $C^{infty}$ functions using Weierstrass' Theorem.
real-analysis
asked Mar 13 at 6:44
RicoRico
908
$endgroup$
$begingroup$
Why is part of (3) blurred?
$endgroup$
– Saucy O'Path
Mar 13 at 6:47
$begingroup$
It basically says that the Arzela-Ascoli Theorem is the Corollary 3.7 in the book. There are some website information in there that I would not like to share (such as my school).
$endgroup$
– Rico
Mar 13 at 6:49
$begingroup$
In a nutshell, the idea of the proof is to approximate a solution of $y'(t)=F(y(t),t)$, $y(0)=y_0$, by functions satisfying $y'(t)=F(y(t-tfrac1n),t)$ (plus the IC). Such functions are just $phi_n$ given by the formula in (1). Now, by the Arzela-Ascoli theorem, the sequence $(phi_n)_{n=1}^{infty}$ has a uniformly convergent subsequence, and the limit is just a solution looked for.
$endgroup$
– user539887
Mar 13 at 7:02
$begingroup$
@user539887 Then how may I define a function as (1)? I am not quite clear with the motivation of how to define such function. (or how to define such function as the hint)
$endgroup$
– Rico
Mar 13 at 7:05
$begingroup$
Observe that in the integrand in $(1)$ for $tin[0,tfrac1n]$ you need only to know the values of $phi_n$ on $[-tfrac1n,0]$ (which is $y_0$), for $tin[tfrac1n,tfrac2n]$ you need only to know the values of $phi_n$ on $[0,tfrac1n]$ (which you know already, by the first step), and so on.
$endgroup$
– user539887
Mar 13 at 13:37
add a comment |
$begingroup$
This is an exercise problem given by my TA in my Real Analysis class. However, I am having quite a bit of problem with just understanding the problem. In particular, I am not sure how to show (1) and the motivation for defining such $phi_n(t)$
For (3), I suppose I need to express $phi_n(t)$ as a sequence of real-valued continuous function in order to get the result.
For (4), I thought the function is defined as $phi_n(t)=y_0+int_0^t F(phi_n(s-frac{1}{n}),s)$ already. The question makes me a bit confused.
In summary, I would like to hear anything that can help me understand the question better as well as proceeding with the answers. The textbook I am using is Analysis II by Tao
I have also found a comparable proof where I was only able to understand part of it. Proving Peano's Existence Theorem by approximating with $C^{infty}$ functions using Weierstrass' Theorem.
real-analysis
asked Mar 13 at 6:44
RicoRico
908
$endgroup$
This is an exercise problem given by my TA in my Real Analysis class. However, I am having quite a bit of problem with just understanding the problem. In particular, I am not sure how to show (1) and the motivation for defining such $phi_n(t)$
For (3), I suppose I need to express $phi_n(t)$ as a sequence of real-valued continuous function in order to get the result.
For (4), I thought the function is defined as $phi_n(t)=y_0+int_0^t F(phi_n(s-frac{1}{n}),s)$ already. The question makes me a bit confused.
In summary, I would like to hear anything that can help me understand the question better as well as proceeding with the answers. The textbook I am using is Analysis II by Tao
I have also found a comparable proof where I was only able to understand part of it. Proving Peano's Existence Theorem by approximating with $C^{infty}$ functions using Weierstrass' Theorem.
real-analysis
real-analysis
asked Mar 13 at 6:44
RicoRico
908
asked Mar 13 at 6:44
RicoRico
908
asked Mar 13 at 6:44
RicoRico
908
asked Mar 13 at 6:44
RicoRico
908
asked Mar 13 at 6:44
RicoRico
908
908
$begingroup$
Why is part of (3) blurred?
$endgroup$
– Saucy O'Path
Mar 13 at 6:47
$begingroup$
It basically says that the Arzela-Ascoli Theorem is the Corollary 3.7 in the book. There are some website information in there that I would not like to share (such as my school).
$endgroup$
– Rico
Mar 13 at 6:49
$begingroup$
In a nutshell, the idea of the proof is to approximate a solution of $y'(t)=F(y(t),t)$, $y(0)=y_0$, by functions satisfying $y'(t)=F(y(t-tfrac1n),t)$ (plus the IC). Such functions are just $phi_n$ given by the formula in (1). Now, by the Arzela-Ascoli theorem, the sequence $(phi_n)_{n=1}^{infty}$ has a uniformly convergent subsequence, and the limit is just a solution looked for.
$endgroup$
– user539887
Mar 13 at 7:02
$begingroup$
@user539887 Then how may I define a function as (1)? I am not quite clear with the motivation of how to define such function. (or how to define such function as the hint)
$endgroup$
– Rico
Mar 13 at 7:05
$begingroup$
Observe that in the integrand in $(1)$ for $tin[0,tfrac1n]$ you need only to know the values of $phi_n$ on $[-tfrac1n,0]$ (which is $y_0$), for $tin[tfrac1n,tfrac2n]$ you need only to know the values of $phi_n$ on $[0,tfrac1n]$ (which you know already, by the first step), and so on.
$endgroup$
– user539887
Mar 13 at 13:37
add a comment |
$begingroup$
Why is part of (3) blurred?
$endgroup$
– Saucy O'Path
Mar 13 at 6:47
$begingroup$
It basically says that the Arzela-Ascoli Theorem is the Corollary 3.7 in the book. There are some website information in there that I would not like to share (such as my school).
$endgroup$
– Rico
Mar 13 at 6:49
$begingroup$
In a nutshell, the idea of the proof is to approximate a solution of $y'(t)=F(y(t),t)$, $y(0)=y_0$, by functions satisfying $y'(t)=F(y(t-tfrac1n),t)$ (plus the IC). Such functions are just $phi_n$ given by the formula in (1). Now, by the Arzela-Ascoli theorem, the sequence $(phi_n)_{n=1}^{infty}$ has a uniformly convergent subsequence, and the limit is just a solution looked for.
$endgroup$
– user539887
Mar 13 at 7:02
$begingroup$
@user539887 Then how may I define a function as (1)? I am not quite clear with the motivation of how to define such function. (or how to define such function as the hint)
$endgroup$
– Rico
Mar 13 at 7:05
$begingroup$
Observe that in the integrand in $(1)$ for $tin[0,tfrac1n]$ you need only to know the values of $phi_n$ on $[-tfrac1n,0]$ (which is $y_0$), for $tin[tfrac1n,tfrac2n]$ you need only to know the values of $phi_n$ on $[0,tfrac1n]$ (which you know already, by the first step), and so on.
$endgroup$
– user539887
Mar 13 at 13:37
$begingroup$
Why is part of (3) blurred?
$endgroup$
– Saucy O'Path
Mar 13 at 6:47
$begingroup$
Why is part of (3) blurred?
$endgroup$
– Saucy O'Path
Mar 13 at 6:47
$begingroup$
It basically says that the Arzela-Ascoli Theorem is the Corollary 3.7 in the book. There are some website information in there that I would not like to share (such as my school).
$endgroup$
– Rico
Mar 13 at 6:49
$begingroup$
It basically says that the Arzela-Ascoli Theorem is the Corollary 3.7 in the book. There are some website information in there that I would not like to share (such as my school).
$endgroup$
– Rico
Mar 13 at 6:49
$begingroup$
In a nutshell, the idea of the proof is to approximate a solution of $y'(t)=F(y(t),t)$, $y(0)=y_0$, by functions satisfying $y'(t)=F(y(t-tfrac1n),t)$ (plus the IC). Such functions are just $phi_n$ given by the formula in (1). Now, by the Arzela-Ascoli theorem, the sequence $(phi_n)_{n=1}^{infty}$ has a uniformly convergent subsequence, and the limit is just a solution looked for.
$endgroup$
– user539887
Mar 13 at 7:02
$begingroup$
In a nutshell, the idea of the proof is to approximate a solution of $y'(t)=F(y(t),t)$, $y(0)=y_0$, by functions satisfying $y'(t)=F(y(t-tfrac1n),t)$ (plus the IC). Such functions are just $phi_n$ given by the formula in (1). Now, by the Arzela-Ascoli theorem, the sequence $(phi_n)_{n=1}^{infty}$ has a uniformly convergent subsequence, and the limit is just a solution looked for.
$endgroup$
– user539887
Mar 13 at 7:02
$begingroup$
@user539887 Then how may I define a function as (1)? I am not quite clear with the motivation of how to define such function. (or how to define such function as the hint)
$endgroup$
– Rico
Mar 13 at 7:05
$begingroup$
@user539887 Then how may I define a function as (1)? I am not quite clear with the motivation of how to define such function. (or how to define such function as the hint)
$endgroup$
– Rico
Mar 13 at 7:05
$begingroup$
Observe that in the integrand in $(1)$ for $tin[0,tfrac1n]$ you need only to know the values of $phi_n$ on $[-tfrac1n,0]$ (which is $y_0$), for $tin[tfrac1n,tfrac2n]$ you need only to know the values of $phi_n$ on $[0,tfrac1n]$ (which you know already, by the first step), and so on.
$endgroup$
– user539887
Mar 13 at 13:37
$begingroup$
Observe that in the integrand in $(1)$ for $tin[0,tfrac1n]$ you need only to know the values of $phi_n$ on $[-tfrac1n,0]$ (which is $y_0$), for $tin[tfrac1n,tfrac2n]$ you need only to know the values of $phi_n$ on $[0,tfrac1n]$ (which you know already, by the first step), and so on.
$endgroup$
– user539887
Mar 13 at 13:37
add a comment |
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$begingroup$
Why is part of (3) blurred?
$endgroup$
– Saucy O'Path
Mar 13 at 6:47
$begingroup$
It basically says that the Arzela-Ascoli Theorem is the Corollary 3.7 in the book. There are some website information in there that I would not like to share (such as my school).
$endgroup$
– Rico
Mar 13 at 6:49
$begingroup$
In a nutshell, the idea of the proof is to approximate a solution of $y'(t)=F(y(t),t)$, $y(0)=y_0$, by functions satisfying $y'(t)=F(y(t-tfrac1n),t)$ (plus the IC). Such functions are just $phi_n$ given by the formula in (1). Now, by the Arzela-Ascoli theorem, the sequence $(phi_n)_{n=1}^{infty}$ has a uniformly convergent subsequence, and the limit is just a solution looked for.
$endgroup$
– user539887
Mar 13 at 7:02
$begingroup$
@user539887 Then how may I define a function as (1)? I am not quite clear with the motivation of how to define such function. (or how to define such function as the hint)
$endgroup$
– Rico
Mar 13 at 7:05
$begingroup$
Observe that in the integrand in $(1)$ for $tin[0,tfrac1n]$ you need only to know the values of $phi_n$ on $[-tfrac1n,0]$ (which is $y_0$), for $tin[tfrac1n,tfrac2n]$ you need only to know the values of $phi_n$ on $[0,tfrac1n]$ (which you know already, by the first step), and so on.
$endgroup$
– user539887
Mar 13 at 13:37