A unique solution for $y'=cosleft(yright)$
$begingroup$
I've considered the non-linear problem
$$ displaystyle left(starright) begin{cases}
displaystyle y'(x)=cosleft(y(x)right)\
y(0)=alpha, alpha in mathbb{R}
end{cases}$$
I want to prove that it admits a unique solution on $mathbb{R}$. I've used the following lemma :
Grönwall Lemma : If $y$ satisfies for all $t in left[0,+inftyright[$ the inequality
$$
y'left(tright) leq betaleft(tright)yleft(tright)
$$
then for all $t in left[0,+inftyright[$
$$
yleft(tright) leq yleft(0right) text{exp}left(int_{0}^{t}betaleft(sright)text{d}sright)
$$
Then let $y_1$ and $y_2$ be two solutions of $(star)$ then we have
$$
(y_1-y_2)'=cosleft(y_1left(tright)right)-cosleft(y_2left(tright)right)=-2sinleft(frac{y_1+y_2}{2}right)sinleft(frac{y_1-y_2}{2}right)
$$
So letting $Y=(y_1-y_2)/2$ we have
$$
Y' leq sinleft(frac{y_1left(tright)+y_2left(tright)}{2}right)sinleft(-Yright)
$$
Using that $Yleft(0right)=0$, we have
$$
y_1(t)-y_2(t) leq 0
$$
I can then use the same argument with $Z=y_2-y_1$ to obtain $y_2(t)-y_1(t) leq 0$ to have
$$
y_1=y_2
$$
Can somebody tell me if it's true ( or correct me to fill the proof )? My problem is the majoration of $Y'$ because I dont know the sign of the sin ( can't be maxed by $1$ then ), and it depends on $y_1$ and $y_2$.
ordinary-differential-equations
$endgroup$
add a comment |
$begingroup$
I've considered the non-linear problem
$$ displaystyle left(starright) begin{cases}
displaystyle y'(x)=cosleft(y(x)right)\
y(0)=alpha, alpha in mathbb{R}
end{cases}$$
I want to prove that it admits a unique solution on $mathbb{R}$. I've used the following lemma :
Grönwall Lemma : If $y$ satisfies for all $t in left[0,+inftyright[$ the inequality
$$
y'left(tright) leq betaleft(tright)yleft(tright)
$$
then for all $t in left[0,+inftyright[$
$$
yleft(tright) leq yleft(0right) text{exp}left(int_{0}^{t}betaleft(sright)text{d}sright)
$$
Then let $y_1$ and $y_2$ be two solutions of $(star)$ then we have
$$
(y_1-y_2)'=cosleft(y_1left(tright)right)-cosleft(y_2left(tright)right)=-2sinleft(frac{y_1+y_2}{2}right)sinleft(frac{y_1-y_2}{2}right)
$$
So letting $Y=(y_1-y_2)/2$ we have
$$
Y' leq sinleft(frac{y_1left(tright)+y_2left(tright)}{2}right)sinleft(-Yright)
$$
Using that $Yleft(0right)=0$, we have
$$
y_1(t)-y_2(t) leq 0
$$
I can then use the same argument with $Z=y_2-y_1$ to obtain $y_2(t)-y_1(t) leq 0$ to have
$$
y_1=y_2
$$
Can somebody tell me if it's true ( or correct me to fill the proof )? My problem is the majoration of $Y'$ because I dont know the sign of the sin ( can't be maxed by $1$ then ), and it depends on $y_1$ and $y_2$.
ordinary-differential-equations
$endgroup$
$begingroup$
How do you know that there exists such a solution on $mathbb{R}$?
$endgroup$
– Eclipse Sun
Jan 27 at 22:39
$begingroup$
Because I've found it
$endgroup$
– Atmos
Jan 27 at 22:45
$begingroup$
If you have found a global solution, then the conclusion follows from Picard existence theorem, or known as Cauchy-Lipschitz theorem. However you don't need to know the solution because a stronger version of the theorem (mentioned in lonza's answer) applies as well.
$endgroup$
– Eclipse Sun
Jan 28 at 3:05
$begingroup$
I dont want to use this theorem but to complete the proof this way :/
$endgroup$
– Atmos
Jan 28 at 11:33
add a comment |
$begingroup$
I've considered the non-linear problem
$$ displaystyle left(starright) begin{cases}
displaystyle y'(x)=cosleft(y(x)right)\
y(0)=alpha, alpha in mathbb{R}
end{cases}$$
I want to prove that it admits a unique solution on $mathbb{R}$. I've used the following lemma :
Grönwall Lemma : If $y$ satisfies for all $t in left[0,+inftyright[$ the inequality
$$
y'left(tright) leq betaleft(tright)yleft(tright)
$$
then for all $t in left[0,+inftyright[$
$$
yleft(tright) leq yleft(0right) text{exp}left(int_{0}^{t}betaleft(sright)text{d}sright)
$$
Then let $y_1$ and $y_2$ be two solutions of $(star)$ then we have
$$
(y_1-y_2)'=cosleft(y_1left(tright)right)-cosleft(y_2left(tright)right)=-2sinleft(frac{y_1+y_2}{2}right)sinleft(frac{y_1-y_2}{2}right)
$$
So letting $Y=(y_1-y_2)/2$ we have
$$
Y' leq sinleft(frac{y_1left(tright)+y_2left(tright)}{2}right)sinleft(-Yright)
$$
Using that $Yleft(0right)=0$, we have
$$
y_1(t)-y_2(t) leq 0
$$
I can then use the same argument with $Z=y_2-y_1$ to obtain $y_2(t)-y_1(t) leq 0$ to have
$$
y_1=y_2
$$
Can somebody tell me if it's true ( or correct me to fill the proof )? My problem is the majoration of $Y'$ because I dont know the sign of the sin ( can't be maxed by $1$ then ), and it depends on $y_1$ and $y_2$.
ordinary-differential-equations
$endgroup$
I've considered the non-linear problem
$$ displaystyle left(starright) begin{cases}
displaystyle y'(x)=cosleft(y(x)right)\
y(0)=alpha, alpha in mathbb{R}
end{cases}$$
I want to prove that it admits a unique solution on $mathbb{R}$. I've used the following lemma :
Grönwall Lemma : If $y$ satisfies for all $t in left[0,+inftyright[$ the inequality
$$
y'left(tright) leq betaleft(tright)yleft(tright)
$$
then for all $t in left[0,+inftyright[$
$$
yleft(tright) leq yleft(0right) text{exp}left(int_{0}^{t}betaleft(sright)text{d}sright)
$$
Then let $y_1$ and $y_2$ be two solutions of $(star)$ then we have
$$
(y_1-y_2)'=cosleft(y_1left(tright)right)-cosleft(y_2left(tright)right)=-2sinleft(frac{y_1+y_2}{2}right)sinleft(frac{y_1-y_2}{2}right)
$$
So letting $Y=(y_1-y_2)/2$ we have
$$
Y' leq sinleft(frac{y_1left(tright)+y_2left(tright)}{2}right)sinleft(-Yright)
$$
Using that $Yleft(0right)=0$, we have
$$
y_1(t)-y_2(t) leq 0
$$
I can then use the same argument with $Z=y_2-y_1$ to obtain $y_2(t)-y_1(t) leq 0$ to have
$$
y_1=y_2
$$
Can somebody tell me if it's true ( or correct me to fill the proof )? My problem is the majoration of $Y'$ because I dont know the sign of the sin ( can't be maxed by $1$ then ), and it depends on $y_1$ and $y_2$.
ordinary-differential-equations
ordinary-differential-equations
asked Jan 27 at 22:33
AtmosAtmos
4,830420
4,830420
$begingroup$
How do you know that there exists such a solution on $mathbb{R}$?
$endgroup$
– Eclipse Sun
Jan 27 at 22:39
$begingroup$
Because I've found it
$endgroup$
– Atmos
Jan 27 at 22:45
$begingroup$
If you have found a global solution, then the conclusion follows from Picard existence theorem, or known as Cauchy-Lipschitz theorem. However you don't need to know the solution because a stronger version of the theorem (mentioned in lonza's answer) applies as well.
$endgroup$
– Eclipse Sun
Jan 28 at 3:05
$begingroup$
I dont want to use this theorem but to complete the proof this way :/
$endgroup$
– Atmos
Jan 28 at 11:33
add a comment |
$begingroup$
How do you know that there exists such a solution on $mathbb{R}$?
$endgroup$
– Eclipse Sun
Jan 27 at 22:39
$begingroup$
Because I've found it
$endgroup$
– Atmos
Jan 27 at 22:45
$begingroup$
If you have found a global solution, then the conclusion follows from Picard existence theorem, or known as Cauchy-Lipschitz theorem. However you don't need to know the solution because a stronger version of the theorem (mentioned in lonza's answer) applies as well.
$endgroup$
– Eclipse Sun
Jan 28 at 3:05
$begingroup$
I dont want to use this theorem but to complete the proof this way :/
$endgroup$
– Atmos
Jan 28 at 11:33
$begingroup$
How do you know that there exists such a solution on $mathbb{R}$?
$endgroup$
– Eclipse Sun
Jan 27 at 22:39
$begingroup$
How do you know that there exists such a solution on $mathbb{R}$?
$endgroup$
– Eclipse Sun
Jan 27 at 22:39
$begingroup$
Because I've found it
$endgroup$
– Atmos
Jan 27 at 22:45
$begingroup$
Because I've found it
$endgroup$
– Atmos
Jan 27 at 22:45
$begingroup$
If you have found a global solution, then the conclusion follows from Picard existence theorem, or known as Cauchy-Lipschitz theorem. However you don't need to know the solution because a stronger version of the theorem (mentioned in lonza's answer) applies as well.
$endgroup$
– Eclipse Sun
Jan 28 at 3:05
$begingroup$
If you have found a global solution, then the conclusion follows from Picard existence theorem, or known as Cauchy-Lipschitz theorem. However you don't need to know the solution because a stronger version of the theorem (mentioned in lonza's answer) applies as well.
$endgroup$
– Eclipse Sun
Jan 28 at 3:05
$begingroup$
I dont want to use this theorem but to complete the proof this way :/
$endgroup$
– Atmos
Jan 28 at 11:33
$begingroup$
I dont want to use this theorem but to complete the proof this way :/
$endgroup$
– Atmos
Jan 28 at 11:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Since the function $ cos $ satisfies the Lipschitz condition $ mid cosleft(y_2right)-
cosleft(y_2right)mid le mid y_1 - y_2,mid$ for all $ y_1, y_2inmathbb R $, the existence and uniqueness of a solution follows from the global Cauchy-Lipschitz theorem.
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3090206%2fa-unique-solution-for-y-cos-lefty-right%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Since the function $ cos $ satisfies the Lipschitz condition $ mid cosleft(y_2right)-
cosleft(y_2right)mid le mid y_1 - y_2,mid$ for all $ y_1, y_2inmathbb R $, the existence and uniqueness of a solution follows from the global Cauchy-Lipschitz theorem.
$endgroup$
add a comment |
$begingroup$
Since the function $ cos $ satisfies the Lipschitz condition $ mid cosleft(y_2right)-
cosleft(y_2right)mid le mid y_1 - y_2,mid$ for all $ y_1, y_2inmathbb R $, the existence and uniqueness of a solution follows from the global Cauchy-Lipschitz theorem.
$endgroup$
add a comment |
$begingroup$
Since the function $ cos $ satisfies the Lipschitz condition $ mid cosleft(y_2right)-
cosleft(y_2right)mid le mid y_1 - y_2,mid$ for all $ y_1, y_2inmathbb R $, the existence and uniqueness of a solution follows from the global Cauchy-Lipschitz theorem.
$endgroup$
Since the function $ cos $ satisfies the Lipschitz condition $ mid cosleft(y_2right)-
cosleft(y_2right)mid le mid y_1 - y_2,mid$ for all $ y_1, y_2inmathbb R $, the existence and uniqueness of a solution follows from the global Cauchy-Lipschitz theorem.
answered Jan 28 at 0:45
lonza leggieralonza leggiera
1,20828
1,20828
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3090206%2fa-unique-solution-for-y-cos-lefty-right%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
How do you know that there exists such a solution on $mathbb{R}$?
$endgroup$
– Eclipse Sun
Jan 27 at 22:39
$begingroup$
Because I've found it
$endgroup$
– Atmos
Jan 27 at 22:45
$begingroup$
If you have found a global solution, then the conclusion follows from Picard existence theorem, or known as Cauchy-Lipschitz theorem. However you don't need to know the solution because a stronger version of the theorem (mentioned in lonza's answer) applies as well.
$endgroup$
– Eclipse Sun
Jan 28 at 3:05
$begingroup$
I dont want to use this theorem but to complete the proof this way :/
$endgroup$
– Atmos
Jan 28 at 11:33