Questions about the Picard–Lindelöf theorem for an ODE

Question

1. In the sketch proof of Picard–Lindelöf existence theorem for an ODE:

   It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" $φ_k$ is convergent and that the limit is a solution to the problem. Exploiting the fact that the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function, one can assure global existence of the solution, i.e. the solution exists and is unique until it leaves the domain of definition of the ODE. An application of Grönwall's lemma to $|φ(t) − ψ(t)|$, where $φ$ and $ ψ$ are two solutions, shows that $φ(t) = ψ(t)$, thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point).

   In the same article, it latter says

   The importance of this result is that the interval of definition of the solution does eventually not depend on the Lipschitz constant of the field, but essentially depends on the interval of definition of the field and its maximum absolute value of it.

2. Wikipedia also mentions

   ... which (both Peano existence theorem and Picard–Lindelöf theorem) are both local results. ...

   When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:[20] For each initial condition $(x_0, y_0)$ there exists a unique maximum (possibly infinite) open interval $$ I_{max} = (x_-,x_+), x_\pm \in \mathbb{R}, x_0 \in I_{max} $$ such that any solution which satisfies this initial condition is a restriction of the solution which satisfies this initial condition with domain $I_{max}$. ... Note that the maximum domain of the solution ... may depend on the specific choice of $(x_0, y_0)$.

My questions:

1. Global or local existence and uniqueness: I was wondering what "local" and "global" mean when referring to existence and/or uniqueness of the solution to an ODE?

   In the second quote, the Picard–Lindelöf theorem is mentioned first as a local result, and the maximum domain $I_{max}$ of the solution both contains and depends on $(x_0, y_0)$, which is how I guess what "local" is. So why is the Picard–Lindelöf theorem said to be " a global result" in the second quote, and "global existence" and "global uniqueness" in the first quote ?

2. Dependence on the Lipschitz constant: In the first article, it first says "the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function", but latter says "the interval of definition of the solution does eventually not depend on the Lipschitz constant of the field". I was wondering if it is self-contradicting?

3. Proof of Uniqueness: In the first quote, it says "Exploiting the fact that the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function, one can assure global existence of the solution, i.e. the solution exists and is unique until it leaves the domain of definition of the ODE". Is "unique" assured here, or by an application of Grönwall's lemma mentioned in the next sentence in the quote?

Thanks and regards!

Answer 1

There are also global variants of the Picard-Lindelöf theorem. If $f$ is defined on $I\times \Bbb R^n$ and satisfies a global Lipschitz condition in the second argument there, that is one constant $L$ for the full domain, then unique solutions $I\to\Bbb R^n$ to any IVP exist. This can be directly proven using the Banach fixed point theorem on the integral formulation of the IVP in the norm $$\|x\|_L=\sup_{t\in I}e^{-2L|t-t_0|}\|x(t)\|,$$ as in that norm the Picard iteration has contraction factor $\frac12$.

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See also

• a more detailed proof or here,
• a different method for the same global result,
• or a little more general for a time-dependent Lipschitz condition.

Answer 2

I am not sure if the answer to this question is still sought, but i hope the following would be helpful, nevertheless.

Consider $\ f(t,x) \in \mathcal{C}(U, \mathbb{R}^n)\ $ where $ \ U \subset \mathbb{R}^{n+1} $, moreover let $ U = I \times J \,$ where $t \in I \subset \mathbb{R}$ and $x \in J \subset \mathbb{R}^n$, then for the initial value problem

\begin{equation} \dot{x} = f(t, x), \quad x(t_0) = x_0 \quad \tag{1}\label{1} \end{equation}

the statement of Picard-Lindelöf reads as: If $f$ is locally Lipschitz continuous in $x$, then there is an open subset $I_0$ around $t_0$ where a unique solution $\ \bar{x}(t) \in \mathcal{C}^1(I_0) \ $ exists.

1. Global or local existence and uniqueness: I was wondering what "local" and "global" mean when referring to existence and/or uniqueness of the solution to an ODE? In the second quote, the Picard–Lindelöf theorem is mentioned first as a local result, and the maximum domain $I_{max}$ of the solution both contains and depends on $(x_0,y_0)$, which is how I guess what "local" is. So why is the Picard–Lindelöf theorem said to be " a global result" in the second quote, and "global existence" and "global uniqueness" in the first quote ?

The key point is that if the vector field $\ f$ is locally Lipschitz continuous, a unique solution exists in a neighbourhood of $t_0$. Whereas, if $ \ f$ is Lipschitz continuous for all $x \in J$, then it is possible to extend this solution upto the boundary of $U$, making it a "global" unique solution. This is usually a separate theorem called the Extensibility theorem:

Say the hypothesis is the same as for Picard-Lindelöf theorem above. If $\phi(t)$ is the "local" solution of $\eqref{1}$, then there is a continuation of $\phi$ to a maximal interval of existence. Furthermore if the maximal interval of existence of a solution $\bar{x}$ of $\eqref{1}$ is $a < t < b$, then $(t,\bar{x}(t))$ tends to the boundary of $U$ as $t \to a$ and $t \to b$.

Some remarks on the theorem: If $b < \infty \ $, then $ dist((t, \bar{x}(t)), \partial U) \to 0$ as $t \to t_{max}$, that is the solution gets arbitrarily close to the boundary of $U$ . If $t_{max} =\infty$ the solution $\bar{x}(t)$ exists for all $t \geq t_0$ in which case the solution can be extended upto the boundary of $U$ in the $t-$direction. $ \ \bar{x}(t) \ $ is said to blow up in finite time if there is some finite $T$ such that $\lim_{t \to T} \bar{x}(t) \ \to \infty$. If the solution blows up, then it can be extended upto the boundary of $U$ in the $x-$direction. A similar argument can be made about the case $t \leq t_0$.

The blow up is perhaps more illustrative via example. Consider the initial value problem: $$ \dot{x} = x^2, \quad x(0) = x_0.$$ The solution of this equation is $\ x(t) = \dfrac{x_0}{1-x_0t} $. The solution exhibits blow up in finite time, namely $T = \frac{1}{x_0}$. So if $x \geq 0$, the solution exists on the interval $(- \infty,T)$.

2. Dependence on the Lipschitz constant: In the first article, it first says "the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function", but latter says "the interval of definition of the solution does eventually not depend on the Lipschitz constant of the field". I was wondering if it is self-contradicting?

The unique solution given by Picard-Lindelöf is dependent on the Lipschitz constant and the initial condition for the problem. From the remarks following the extensibility theorem, it is easy to see that unless the solution blows up in finite time, the solution $\bar{x}(t)$ exists on the entire interval $I$.

3. Proof of Uniqueness: In the first quote, it says "Exploiting the fact that the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function, one can assure global existence of the solution, i.e. the solution exists and is unique until it leaves the domain of definition of the ODE". Is "unique" assured here, or by an application of Grönwall's lemma mentioned in the next sentence in the quote?

A small lemma is apropos here:

Let the hypotheses outlined for the initial value problem $\eqref{1}$ be satisfied. Moreover let the Lipschitz constant of $f$ w.r.t. $x$ be $L$. If $x_1$ and $x_2$ are solutions of $\eqref{1}$ on the interval $[t_0, T]$ then,

\begin{equation} |y_1(t) - y_2(t)|^2 \leq |y_1(t_0) - y_2(t_0)|^2 e^{2 L (t - t_0)} \end{equation}

The proof of this lemma utilises Gronwall's inequality. Uniqueness of solutions on the interval can be proved from this lemma by setting $y_1(t_0) = y_2(t_0)$.