Talk:Picard–Lindelöf theorem

Global existence

The section on global existence and uniqueness is confusing. It seems to imply that we can always get a global solution which is not true. 90.229.231.115 (talk) 16:35, 18 March 2009 (UTC)

Is this the "right" theorem?

I thought the Picard-Lindelöf theorem was the global exist&uniq theorem, while the version show here is the local exist&uniq theorem I know just as Picard theorem, which in this article is later (and rather obscurely) extended to global uniqueness through the use of Grönwall's lemma. The version of the P-L theorem I know is this:

Let I×D be the definition domain of f(t,y), where I=[t₁,t₂] is a real interval and D is a real domain. Suppose f is bounded and Lipschitz-continuous in I×ℝ. Then, for every pair (t₀,y₀) ∈ I×D there exists an unique solution to the IVP for all t ∈ I.

Of course, my version might have some additional restriction or culprit I'm not aware of and I'm not sure of having expressed it right, as I'm taking ODEs formally for the first time in an univ course. Last time I entered a math discussion here I ended up pretty bad, so I've learned to leave math things to math people :D Habbit (talk) 13:30, 7 January 2009 (UTC)

A somewhat related question: I don't see any mention here that the hypotheses for (local) existence are weaker than those for uniqueness (no Lipschitz condition necessary). Is that in some other article? Or should it be mentioned here? -- Spireguy (talk) 18:22, 10 February 2009 (UTC)

That's the Cauchy-Peano theorem, aka the existence theorem of ODEs. Its article, in turn, does mention this theorem. Habbit (talk) 19:02, 10 February 2009 (UTC)

It's mentioned in the 'see also' section as the Peano existence theorem. Algebraist 19:15, 10 February 2009 (UTC)

I think it should be mentioned here, so I added a section. -- Jitse Niesen (talk) 17:10, 11 February 2009 (UTC)

I think that a discussion about the name of the theorem is not important at all. For instance, the statement of the theorem in this article for me is the Picard's existence and uniqueness of solutions of the IVP theorem and what Habbit says above is the Picard's Corollary applying the theorem to each point of the interval of t. So, I mean that everyone might have learned these results with different "equivalent" names, so let's talk about important things. What an engineer or mathematician must have in mind is all the concepts, results, properties of DE's, the name is the least of the problem. Now I would like to end up my intervention saying that a formal proof of this theorem is not hard at all, we could build up the proof using the Picard iterates and applying the Banach fixed point theorem. By the way, I read someone on the article saying that the domaim of the solution depends entirely on the Lipschitz constant, that's actually not true, one can proof that the optimal interval of t where there exists a solution depends on the supremum of f in $t\in [a,b]$, where this is the interval where t is defined on f, because a corollary of Banach fixed point theorem says "if $f^{n}$ is contractive for some $n$ then $f$ has a unique fixed point", so for a large enough Picard iterate we will always be able to asure that it is contractive without requiring additional conditions over the Lipschitz constant L. Fouri87 (talk)

I agree about the names, but I think that's missing the point -- this comment wasn't about the name, but rather the standard result that is meant under these names is often a result for global existence. Looking at 3 different textbooks, their version of this ODE existence result (the one they all use the Banach fixed point theorem to prove) has global existence on the entire interval, but also require that $f$ is (jointly) continuous (i.e., in both arguments). This global result is the standard result seen in an undergraduate ODE textbook, but almost none of them call it "Picard" -- they simply don't give it any name at all! That version of the theorem is absent from wikipedia currently. Lavaka (talk) 16:38, 1 November 2020 (UTC)

Proof

I have added the proof of the theorem which I think it is not hard at all for students going through DE's courses (taking for granted that basic notions on Functional Analysis are gained), as well as I have made a brief sketch of how to optimize the interval of definition of the solution. If you find any language or grammas mistake, or formal mathematical errors or have another proof of the theorem, please say it. I think that's all for now :) Nice job everyone! Fouri87 (talk) —Preceding undated comment added 20:59, 4 October 2010 (UTC).

hyphen v. dash

hello. names should be hyphenated, not dashed. --emerson7 18:00, 13 January 2009 (UTC)

That contradicts both the hyphen article and the guideline WP:ENDASH. Hyphenated names are only appropriate when both names refer to the same person. JackSchmidt (talk) 18:08, 13 January 2009 (UTC)

i'm not so sure it's all that cut-and-dried. the two sections seem, contradictory, showing McCain-Feingold with a dash and a hyphen as both correct. --emerson7 18:31, 13 January 2009 (UTC)

What do you mean? I find no mention of McCain Feingold in either hyphen nor WP:ENDASH. The wiki article Dash notes as a caveat that the "Chicago manual of style" would use a hyphen in McCain Feingold. However, wikipedia does not follow the chicago manual of style, it follows the wikipedia MOS at Wikipedia:Manual of Style, and on this page is the germane paragraph (linked to above as WP:ENDASH). This page also contains WP:HYPHEN that discusses the use of hyphens on wikipedia. RobHar (talk) 19:24, 13 January 2009 (UTC)

well, i guess it all hinges on the definition of 'simple compound'. wp:mos is clearly conflicted on the issue. --emerson7 23:22, 13 January 2009 (UTC)

No, it isn't. The MOS is uncharacteristically clear on this. Where are you seeing conflict? Algebraist 23:24, 13 January 2009 (UTC)

emerson7, why do you keep quoting things from the wiki article on dashes? we are talking about the wiki manual of style. WP:ENDASH, part 1, second bullet discusses this use. Furthermore, in WP:HYPHEN, the subsection "Image filenames and redirects" mentions how the Michelson–Morley experiment is correctly punctuated with an endash as opposed to a hyphen, as in the experiment of Michelson and Morley. To me, the MOS seems quite clear, and the people here seem to agree. Please move it back, or bring this to MOS people if you want. RobHar (talk) 23:51, 13 January 2009 (UTC)

I agree—this should be an en-dash. I've undone the change. --Zvika (talk) 06:26, 14 January 2009 (UTC)

Thanks, Zvika. RobHar (talk) 00:04, 17 January 2009 (UTC)

Picard iteration

There should be a section if not an entire article devoted to it.--Gustav Ulsh Iler (talk) 21:16, 23 October 2009 (UTC)

The Picard Maneuver

I recently reverted an edit in which someone added a link to The Picard Maneuver. Since that page doesn't mention mathematics at all, I thought it was someone's idea of a joke. But now an editor is claiming (in the edit summaries) that it is indeed a common name for Picard iteration. Does anyone have a reference or more information about this? Jowa fan (talk) 00:23, 20 September 2011 (UTC)

This seems to be an issue of standard versus colloquial usage. I've certainly heard both "Picard Iteration" and "Picard Maneuver." Although I was taught the process as "Picard Iteration," I have found the relationship to Star Trek to be a very useful teaching aid. Students, at least, remember my lecture more. Plus, as the "Picard Maneuver" itself has an iterative component the term has always seemed appropriate. I'm not certain when application of the term "Picard Maneuver" originated, but I'll ask around my department. SuperDo (talk) 03:22, 20 September 2011 (UTC)

As a math professor, I always refer to it as The Picard Maneuver when I teach it to hundreds of students in a given semester. As SuperDo mentioned above, this is an issue of standard versus colloquial usage, but the colloquial usage is becoming commonplace among mathematicians. Greggo1980

You know, it is a little suspect that another person would happen along to support your claim. No reliable source = No entry. It doe snot matter what you seem to think you know, venerability is key here. The Last Angry Man (talk) 08:50, 20 September 2011 (UTC)

I call it the 'reality vs. Wikipedia' effect. Wikipedia isn't reality. It's a bizarre neutral point of view, on a very opinionated world. — Preceding unsigned comment added by 97.125.208.99 (talk) 08:08, 14 May 2012 (UTC)

I think the statement of the theorem is not correct.

The theorem says "suppose the problem: y'=f=(t,y(t)) y(t0)=y0 t in [t0-e,t0+e] So, if that's the problem given to us, then e=epsilon is fixed. And then later says it exist epsilon. I think it will be correct if we delete in [t0-e,t0+e], in the paragraph: y'=f=(t,y(t)) y(t0)=y0 t in [t0-e,t0+e]

Other existence theorems -example

I think the sentence "Indeed, rather than being unique, this equation has three solutions" is misleading. In fact this y1, y2, y3 are three basic solutions to the problem which we can use to construct infinitely many solutions. y4(x)= y1(x) 0<=x<c, y4(x)= y2(x-c) for x>c, is a different solution to the problem. — Preceding unsigned comment added by 89.77.46.48 (talk) 13:23, 25 September 2015 (UTC)

Names / History

I think it would be nice to have a short "History" section, in order to be able to appreciate the respective attributions to Cauchy/Lipschitz/Picard/Lindelöf: who did what and when? — MFH:Talk 13:56, 3 August 2018 (UTC)

Precision about the domain of $f$

Nothing special is mentioned about the domain $\Omega$ of $f$.

Shouldn't the domain be precised: $\Omega$ is an open of $\mathbb {R} \times E$ where $E$ is a Banach space? Counterexamples (talk) 17:43, 19 November 2019 (UTC)

Optimization of the solution's interval

The proof in this section is incorrect as stated, as the transition between the last two lines is wrong. The claim you actually get by making use of the induction step is:

$L\left|\int _{t_{0}}^{t}\left\|\Gamma ^{m-1}\varphi _{1}(s)-\Gamma ^{m-1}\varphi _{2}(s)\right\|ds\right|\leq {\frac {L^{m}\alpha ^{m}}{(m-1)!}}\left\|\varphi _{1}-\varphi _{2}\right\|$

which is insufficient to finish the induction.

The claim about the optimised interval width is valid, but it requires an alternate proof.

Bornsimi (talk) 14:18, 17 January 2021 (UTC)

Take $\alpha =(t-t_{0})$ to fix the issue. In other words we should be trying to prove

$\left\|\Gamma ^{m}\varphi _{1}(t)-\Gamma ^{m}\varphi _{2}(t)\right\|\leq {\frac {L^{m}(t-t_{0})^{m}}{m!}}\left\|\varphi _{1}(t)-\varphi _{2}(t)\right\|$

by induction and then take sup's over t to prove the lemma.

Radiodont (talk) 21:23, 21 January 2021 (UTC)

Number of solutions to y'=y^(1/3) with y(0)=0.

The article claims there are three solutions to this equation when there are nine. The point is that the solution in x<0 and x>0 need not take the same form, so you can mix-and-match the solutions either side. 2A00:23A8:835:C401:C1DA:A20E:A3DC:DB7B (talk) 16:07, 8 August 2022 (UTC)

In fact there are infinitely many solutions, e.g.:

y(t) = 0 if t < a, (2/3 (t-a))^(3/2) otherwise

for any real parameter a > 0. LockRay (talk) 17:18, 14 November 2023 (UTC)

Closed rectangular.

I think it is not OK to have D a closed rectangular in the statement of the theorem. For instance, if (t_0,x_0) happens to be on the right side, then one cannot solve for t>t_0. So why not require either D to be open, or (t_0,y_0) to be in its interior? AnnZMath (talk) 18:45, 22 November 2022 (UTC)