Osgood’s uniqueness theorem for differential equations

October 1, 2014

Osgood’s uniqueness theorem for differential equations

Filed under: advanced mathematics, analysis, calculus, differential equations — Tags: existence and uniqueness, Osgood's uniqueness theorem — collegemathteaching @ 1:59 am

I am teaching a numerical analysis class this semester and we just started the section on differential equations. I want them to understand when we can expect to have a solution and when a solution satisfying a given initial condition is going to be unique.

We had gone over the “existence” theorem, which basically says: given $y' = f(x,y)$ and initial condition $y(x_0) = y_0$ where $(x_0,y_0) \in int(R)$ where $R$ is some rectangle in the $x,y$ plane, if $f(x,y)$ is a continuous function over $R$ , then we are guaranteed to have at least one solution to the differential equation which is guaranteed to be valid so long as $(x, y(x)$ stays in $R$ .

I might post a proof of this theorem later; however an outline of how a proof goes will be useful here. With no loss of generality, assume that $x_0 = 0$ and the rectangle has the lines $x = -a, x = a$ as vertical boundaries. Let $\phi_0 = f(0, y_0)x$ , the line of slope $f(0, y_0)$ . Now partition the interval $[-a, a]$ into $-a, -\frac{a}{2}, 0, \frac{a}{2}, a$ and create a polygonal path as follows: use slope $f(0, y_0)$ at $(0, y_0)$ , slope $f(\frac{a}{2}, y_0 + \frac{a}{2}f(0, y_0))$ at $(\frac{a}{2}, y_0 + \frac{a}{2}f(0, y_0))$ and so on to the right; reverse this process going left. The idea: we are using Euler’s differential equation approximation method to obtain an initial piecewise approximation. Then do this again for step size $\frac{a}{4},$

In this way, we obtain an infinite family of continuous approximation curves. Because $f(x,y)$ is continuous over $R$ , it is also bounded, hence the curves have slopes whose magnitude are bounded by some $M$ . Hence this family is equicontinuous (for any given $\epsilon$ one can use $\delta = \frac{\epsilon}{M}$ in continuity arguments, no matter which curve in the family we are talking about. Of course, these curves are uniformly bounded, hence by the Arzela-Ascoli Theorem (not difficult) we can extract a subsequence of these curves which converges to a limit function.

Seeing that this limit function satisfies the differential equation isn’t that hard; if one chooses $t, s \in (-a.a)$ close enough, one shows that $| \frac{\phi_k(t) - \phi_k(s)}{(t-s)} - f(t, \phi(t))| 0$ where $|f(x,y_1)-f(x,y_2)| \le K|y_1-y_2|$ then the differential equation $y'=f(x,y)$ has exactly one solution where $\phi(0) = y_0$ which is valid so long as the graph $(x, \phi(x) )$ remains in $R$ .

Here is the proof: $K > 0$ where $|f(x,y_1)-f(x,y_2)| \le K|y_1-y_2| < 2K|y_1-y_2|$ . This is clear but perhaps a strange step.
But now suppose that there are two solutions, say $y_1(x)$ and $y_2(x)$ where $y_1(0) = y_2(0)$ . So set $z(x) = y_1(x) -y_2(x)$ and note the following: $z'(x) = y_1(x) - y_2(x) = f(x,y_1)-f(x,y_2)$ and $|z'(x)| = |f(x,y_1)-f(x,y_2)| 0$ . A Mean Value Theorem argument applied to $z$ means that we can assume that we can select our $x_1$ so that $z' > 0$ on that interval (since $z(0) = 0$ ).

So, on this selected interval about $x_1$ we have $z'(x) < 2Kz$ (we can remove the absolute value signs.).

Now we set up the differential equation: $Y' = 2KY, Y(x_1) = z(x_1)$ which has a unique solution $Y=z(x_1)e^{2K(x-x_1)}$ whose graph is always positive; $Y(0) = z(x_1)e^{-2Kx_1}$ . Note that the graphs of $z(x), Y(x)$ meet at $(x_1, z(x_1))$ . But $z'(x) 0$ where $z(x_1 - \delta) > Y(x_1 - \delta)$ .

But since $z(0) = 0 z'(x)$ on that interval.

So, no such point $x_1$ can exist.

Note that we used the fact that the solution to $Y' = 2KY, Y(x_1) > 0$ is always positive. Though this is an easy differential equation to solve, note the key fact that if we tried to separate the variables, we’d calculate $\int_0^y \frac{1}{Kt} dt$ and find that this is an improper integral which diverges to positive $\infty$ hence its primitive cannot change sign nor reach zero. So, if we had $Y' =2g(Y)$ where $\int_0^y \frac{1}{g(t)} dt$ is an infinite improper integral and $g(t) > 0$ , we would get exactly the same result for exactly the same reason.

Hence we can recover Osgood’s Uniqueness Theorem which states:

If $f(x,y)$ is continuous on $R$ and for all $(x, y_1), (x, y_2) \in R$ we have a $K > 0$ where $|f(x,y_1)-f(x,y_2)| \le g(|y_1-y_2|)$ where $g$ is a positive function and $\int_0^y \frac{1}{g(t)} dt$ diverges to $\infty$ at $y=0$ then the differential equation $y'=f(x,y)$ has exactly one solution where $\phi(0) = y_0$ which is valid so long as the graph $(x, \phi(x) )$ remains in $R$ .

Comments (6)

6 Comments »

sure we see as say prof dr mircea orasanu and prof horia orasanu as followed
OSGOOD THEOREM AND LAGRANGIAN
Author Horia Orasanu

ABSTRACT
It has often been repeated that a Hamiltonian whose kinetic term depends on the position variable has in principle many different operator versions. For an arbitrary model Hamiltonian, this is of course, true
Printer rezultatele noi obtinute si relatiile dintre noile teoreme cerute, avem pe acelea care
Se refera la functiile mentionate mai sus cu o raza data in planul in care un patrat se afla
In acest plan ,si este perpendicular pe planul cercului si care se afla in miscare supuse la
Legaturi neolonome

Aici mentionam ca se poate construi un boiler inafara unui cilindru marginit de doua emis
Fere si cu pereti de o anumita grosime

Comment by perringu — February 6, 2018 @ 4:09 pm

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also the Osgood theorem can be used to other as say prof dr mircea orasanu as for followed
LAGRANGIAN AND PARTIAL DERIVATIVES
Author Horia Orasanu
ABSTRACT

Okay, we’ve worked two of the four cases that would need to be solved in order to completely solve (1). As we’ve seen each case was very similar and yet also had some differences. We saw the use of both separation constants and that sometimes we need to use a “shifted” solution in order to deal with one of the boundary conditions.

Before moving on let’s note that we used prescribed temperature boundary conditions here, but we could just have easily used prescribed flux boundary conditions or a mix of the two. No matter what kind of boundary conditions we have they will work the same.

Comment by perringu — February 6, 2018 @ 7:28 pm

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,

Comment by gasagu — December 25, 2018 @ 6:56 pm

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first is mentioned that Osgood Theorem is essential in many situations as observed prof dr mircea orasanu and prof drd horia orasanu specially in cases of Riemann Hilbert problem that lead to differential equations ,or differential equations that approach to LAGRANGIAN OF CONSTRAINTS OPTIMIZATION ,and these must attached of other and also must development for Hermite ,Legendre ,and Laguerre differential equations ,indeed

Comment by codogiu — December 25, 2018 @ 7:09 pm

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- as to observed the Osgood Theorem uniqueness appear as an instrument to study of non holonomic question and equations that are equations of LAGRANGIAN ,and these have been considered by prof dr Const . Udriste in many situations
  
  Comment by tusangu — January 1, 2019 @ 6:48 pm
  
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an important result of Osgood theorem as observed prof dr mircea orasanu appear exceptional case of situations when followed problem of LEGENDRE ordinary differential Equation that has been treated using method of GALOIS theory and using with successfully from prof dr GIGEL MILITARU

Comment by ciunaciu — January 19, 2019 @ 3:56 pm

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College Math Teaching

October 1, 2014