How Does the Lemma Support the Uniqueness Theorem in ODEs?

[yifli]

I'm reading the differential equations chapter of Advanced Calculus by Loomis, and have some questions.

First it proved the following theorem:
Let A be and open subset of a Banach space W, let I be an open interval in R, and let F be a continuous mapping from I X A to W which has a continuous second partial differential. Then for any point [itex]<t_0, \alpha_0>[/itex] in I X A, from some neighborhood U of [itex]\alpha_0[/itex] and for any sufficiently small interval J containing [itex]t_0[/itex], there is a unique function f from J to U which is a solution of the differential equation passing through the point [itex]<t_0, \alpha_0>[/itex]

Then it states the following Lemma:
Let [itex]g_1[/itex] and [itex]g_2[/itex] be any two solutions of [itex]d\alpha/dt=F(t,\alpha)[/itex] through [itex]<t_0, \alpha_0>[/itex]. Then [itex]g_1(t)=g_2(t)[/itex] for all t in the intersection [itex]J=J_1\cap J_2[/itex] of their domains.

What is the above lemma useful for? The theorem says there is only one solution through [itex]<t_0, \alpha_0>[/itex], why does the lemma say "g1 and g2 be any two solutions"?

 

[rmehta]

I think the difference is that the theorem says the solution exists and is unique in some "sufficiently small interval". The lemma asserts that the uniqueness (but not the existence) is more robust.

The lemma could be rephrased as: "if a solution exists on any interval, then it is the unique solution on that interval." And this is true even for large intervals, possibly even for all of R (if a solution exists for all t).

 

[yifli]

I think the difference is that the theorem says the solution exists and is unique in some "sufficiently small interval". The lemma asserts that the uniqueness (but not the existence) is more robust.

The lemma could be rephrased as: "if a solution exists on any interval, then it is the unique solution on that interval." And this is true even for large intervals, possibly even for all of R (if a solution exists for all t).

Thanks.

I just noticed the book says that "the lemma allows us to remove restriction on the range of f", so the theorem can be stated as follows:
Let A, I, and F be as in the above theorem. Then for any point [itex]<t_0,\alpha_0>[/itex] in I X A and any sufficiently small interval neighborhood J of [itex]t_0[/itex], there is a unique solution from J to A passing through [itex]<t_0,\alpha_0>[/itex]

Why does the lemma allow us to remove the restriction on the range of f?

 

[csco]

I don't understand your question. The lemma allow us to remove the restriction on the range of f because that what the lemma is about. Uniqueness on arbitrary intervals follows from local uniqueness that's what the lemma says.

There's a short proof of this fact. Let there be two solutions defined on interescting intervals. Let J be their intersection. Let A be the subset of J such that the solutions are equal there and B the subset of J such that the solutions are not equal. Observe that A and B are disjoint sets and that their union is J. We have to prove B is the empty set. Take a in A. Both solutions take the same value on a so by local existence and uniqueness the solutions are equal on a small interval around a. Then A is open. But B is open too because the difference of the solutions is a continuous function and takes a value different from zero for any b in B so there is a small interval around b where the solutions are different. Now A is not empty because the initial time is in A, if B where non empty too then this would imply J is not connected and this absurd so B is empty and this completes the proof.

A more important result is the existence of a maximal solution and that also follows from local uniqueness and existence easily.

 

[blue_raver22]

The fundamental theorems of ODE (ordinary differential equations) are essential tools for understanding and solving differential equations, which are mathematical equations that describe how a system changes over time. These theorems provide rigorous mathematical proofs for the existence and uniqueness of solutions to differential equations, which are crucial for making accurate predictions and understanding the behavior of a system.

The first theorem you mentioned states that for a given differential equation with continuous and well-behaved functions, there exists a unique solution passing through a given point. This is useful because it guarantees that there is a well-defined solution to the equation, which can be used to make predictions or analyze the system.

The lemma, on the other hand, is useful for proving the uniqueness of the solution. It states that if two different solutions of the same differential equation pass through the same point, then they must be equal on the intersection of their domains. This is important because it shows that the solution is not affected by the choice of initial conditions, and it is the same for any two solutions passing through the same point.

The lemma also allows us to prove the uniqueness of the solution by contradiction. If we assume that there are two different solutions that do not agree on the intersection of their domains, then this would contradict the lemma and thus the uniqueness of the solution.

In summary, the fundamental theorems of ODE, including the theorem and the lemma, are essential for proving the existence and uniqueness of solutions to differential equations. They provide a solid mathematical foundation for understanding and solving these equations, which are fundamental in many areas of science and engineering.