ODE-07

Theoretical Foundations

Existence and Uniqueness Theory

Understand the theoretical foundations of ODEs: when solutions exist, when they are unique, how far they can be extended, and how they depend on initial conditions and parameters.

Learning Objectives

State and apply the Picard-Lindelöf theorem
Understand the Lipschitz condition and its role
Distinguish existence from uniqueness
Apply Picard iteration to construct solutions
Use Gronwall's inequality in proofs
Understand continuation and maximal solutions
Analyze continuous dependence on data
Recognize when uniqueness fails

1. The Lipschitz Condition

The Lipschitz condition is the key hypothesis that ensures uniqueness of solutions. It controls how fast the right-hand side of the ODE can change with respect to the dependent variable.

Definition 7.1: Lipschitz Condition

A function $f: D \to \mathbb,{R},^n$ where $D \subset \mathbb,{R}, \times \mathbb,{R},^n$ satisfies aLipschitz condition in $x$ if there exists a constant $L \geq 0$ such that:

|f(t, x_1) - f(t, x_2)| \leq L|x_1 - x_2|

for all $(t, x_1), (t, x_2) \in D$ . The constant $L$ is called the Lipschitz constant.

If this holds only on compact subsets of $D$ , we say $f$ is locally Lipschitz.

Theorem 7.1: Sufficient Condition for Lipschitz

If $f(t, x)$ is continuously differentiable with respect to $x$ on a convex domain $D$ , and $\left|\frac{\partial f}{\partial x}\right| \leq L$ on $D$ , then $f$ is Lipschitz with constant $L$ .

Proof of Theorem 7.1:

By the mean value theorem, for fixed $t$ :

|f(t, x_1) - f(t, x_2)| = \left|\frac{\partial f}{\partial x}(t, \xi)\right| |x_1 - x_2| \leq L|x_1 - x_2|

where $\xi$ is on the line segment between $x_1$ and $x_2$ . ∎

Example 7.1: Checking Lipschitz Condition

Problem: Determine if $f(x) = x^2$ is Lipschitz on $[-1, 1]$ .

Solution:

Since $f'(x) = 2x$ , we have $|f'(x)| \leq 2$ on $[-1, 1]$ .

Therefore, $f$ is Lipschitz with $L = 2$ on this interval.

Alternatively: $|x_1^2 - x_2^2| = |x_1 + x_2||x_1 - x_2| \leq 2|x_1 - x_2|$ for $x_1, x_2 \in [-1, 1]$ .

Remark:

The function $f(x) = \sqrt,{|x|}$ is not Lipschitz at $x = 0$ : $|f(x) - f(0)|/|x - 0| = 1/\sqrt,{|x|}, \to \infty$ as $x \to 0$ . This is why the IVP $x' = \sqrt,{|x|},, x(0) = 0$ has multiple solutions.

2. The Picard-Lindelöf Theorem

Theorem 7.2: Picard-Lindelöf (Existence and Uniqueness)

Consider the initial value problem:

\frac{dx}{dt} = f(t, x), \quad x(t_0) = x_0

Let $f: D o \mathbb{R}^n$ be continuous on $D = {(t,x): |t-t_0| leq a, |x-x_0| leq b}$ and satisfy a Lipschitz condition in $x$ with constant $L$ . Let $M = \sup_D |f|$ .

Then there exists a unique solution $x(t)$ on the interval $|t - t_0| \leq h$ where $h = \min(a, b/M)$ .

Proof of Theorem 7.2:

Step 1: Integral formulation. The IVP is equivalent to:

x(t) = x_0 + \int_{t_0}^{t} f(s, x(s))\,ds

Step 2: Picard iteration. Define the sequence:

\phi_0(t) = x_0, \quad \phi_{n+1}(t) = x_0 + \int_{t_0}^{t} f(s, \phi_n(s))\,ds

Step 3: Show iterates stay in domain.By induction, $|\phi_n(t) - x_0| \leq M|t - t_0| \leq Mh \leq b$ .

Step 4: Convergence. Let $e_n = \sup|_t |\phi_n - \phi_,{n-1},|$ . Then:

e_{n+1} \leq L|t - t_0| e_n \leq Lh \cdot e_n \leq \frac{(Lh)^n}{n!} e_1

Since $\sum (Lh)^n/n!$ converges, $\phi_n$ converges uniformly to some $\phi$ .

Step 5: Uniqueness. If $\psi$ is another solution, Gronwall's inequality shows $|\phi - \psi| = 0$ . ∎

Algorithm 7.1: Picard Iteration

Input: IVP $x' = f(t, x), x(t_0) = x_0$

Output: Approximate solution (or exact via limit)

1. Set $\phi_0(t) = x_0$

2. For $n = 0, 1, 2, \ldots$ :

Compute $\phi_,{n+1},(t) = x_0 + \int_,{t_0},^t f(s, \phi_n(s))ds$

3. Until convergence or desired accuracy

4. return $\phi_n(t)$

Example 7.2: Picard Iteration

Problem: Apply Picard iteration to $x' = x, x(0) = 1$ .

Solution:

\phi_0(t) = 1

\phi_1(t) = 1 + \int_0^t 1\,ds = 1 + t

\phi_2(t) = 1 + \int_0^t (1+s)\,ds = 1 + t + \frac{t^2}{2}

\phi_3(t) = 1 + \int_0^t \left(1 + s + \frac{s^2}{2}\right)ds = 1 + t + \frac{t^2}{2} + \frac{t^3}{6}

In general: $\phi_n(t) = \sum_,{k=0},^n \frac,{t^k}{k!}$ , which converges to $e^t$ .

3. Gronwall's Inequality

Theorem 7.3: Gronwall's Inequality

Let $u, \beta: [a, b] \to [0, \infty)$ be continuous, and let $\alpha \geq 0$ be a constant. If:

u(t) \leq \alpha + \int_a^t \beta(s) u(s)\,ds \quad \text{for } t \in [a, b]

then:

u(t) \leq \alpha \exp\left(\int_a^t \beta(s)\,ds\right)

Proof of Theorem 7.3:

Let $v(t) = \int_a^t \beta(s) u(s)ds$ . Then $v' = \beta u \leq \beta(\alpha + v)$ .

This gives $v' - \beta v \leq \alpha\beta$ . Multiplying by $e^,{-\int_a^t \beta}$ :

\frac{d}{dt}\left(v \cdot e^{-\int_a^t \beta}\right) \leq \alpha\beta e^{-\int_a^t \beta}

Integrating from $a$ to $t$ and using $v(a) = 0$ :

v(t) \leq \alpha\left(e^{\int_a^t \beta} - 1\right)

Therefore $u(t) \leq \alpha + v(t) \leq \alpha e^,{\int_a^t \beta}$ . ∎

Remark:

Gronwall's inequality is the key tool for proving: (1) uniqueness of solutions, (2) continuous dependence on initial conditions, and (3) continuous dependence on parameters.

4. Peano Existence Theorem

Theorem 7.4: Peano Existence Theorem

If $f(t, x)$ is continuous on $D = {(t,x): |t-t_0| leq a, |x-x_0| leq b}$ , then the IVP $x' = f(t, x), x(t_0) = x_0$ has at least one solution on $|t - t_0| \leq h = \min(a, b/M)$ where $M = \sup_D |f|$ .

Remark:

Peano's theorem guarantees existence but not uniqueness. The proof uses Euler's polygonal approximation and the Arzelà-Ascoli theorem. The Lipschitz condition is essential for uniqueness.

Example 7.3: Non-Unique Solutions

Problem: Show that $x' = 3x^,{2/3},, x(0) = 0$ has multiple solutions.

Solution:

Note that $f(x) = 3x^,{2/3}$ is continuous but not Lipschitz at $x = 0$ .

Solution 1: $x(t) \equiv 0$ (verify: $x' = 0 = 3 \cdot 0^,{2/3}$ ✓)

Solution 2: Try $x(t) = t^3$ :

x'(t) = 3t^2, \quad 3x^{2/3} = 3(t^3)^{2/3} = 3t^2 \quad \checkmark

Both solutions satisfy the IVP, demonstrating non-uniqueness.

In fact, for any $c \geq 0$ , the function:

x(t) = \begin{cases} 0 & t \leq c \\ (t-c)^3 & t > c \end{cases}

is also a solution, showing infinitely many solutions exist.

5. Continuation of Solutions

Definition 7.2: Maximal Solution

A solution $x(t)$ defined on an interval $(\alpha, \omega)$ is called amaximal solution if it cannot be extended to any larger interval while remaining a solution of the ODE.

Theorem 7.5: Blow-Up Theorem

Let $f$ be locally Lipschitz on an open domain $D \subset \mathbb,{R}, \times \mathbb,{R},^n$ . If $x(t)$ is a maximal solution on $(\alpha, \omega)$ with $\omega < \infty$ , then for any compact set $K \subset D$ , there exists $t_K$ such that $(t, x(t)) \notin K$ for $t > t_K$ .

In other words: the solution "escapes to infinity" or approaches the boundary of $D$ .

Example 7.4: Finite-Time Blow-Up

Problem: Solve $x' = x^2, x(0) = 1$ and find the maximal interval.

Solution:

Separating: $\frac,{dx}{x^2}, = dt \implies -\frac,{1}{x}, = t + C$

With $x(0) = 1$ : $C = -1$ , so $x = \frac,{1}{1-t}$

The solution exists on $(-\infty, 1)$ and $x(t) \to \infty$ as $t \to 1^-$ .

This is finite-time blow-up: the solution becomes infinite in finite time.

6. Continuous Dependence

Theorem 7.6: Continuous Dependence on Initial Conditions

Let $f$ be continuous and Lipschitz in $x$ with constant $L$ . Let $x(t)$ and $y(t)$ be solutions of $x' = f(t, x)$ with initial conditions $x(t_0) = x_0$ and $y(t_0) = y_0$ respectively. Then:

|x(t) - y(t)| \leq |x_0 - y_0| e^{L|t - t_0|}

Proof of Theorem 7.6:

We have:

|x(t) - y(t)| \leq |x_0 - y_0| + \int_{t_0}^{t} |f(s, x(s)) - f(s, y(s))|\,ds

\leq |x_0 - y_0| + L\int_{t_0}^{t} |x(s) - y(s)|\,ds

By Gronwall's inequality with $\alpha = |x_0 - y_0|$ and $\beta = L$ :

|x(t) - y(t)| \leq |x_0 - y_0| e^{L|t - t_0|}

∎

Remark:

This theorem shows that solutions depend continuously on initial conditions. The bound grows exponentially in time, which is sharp for some systems (but often pessimistic in practice).

Practice Quiz

Existence and Uniqueness Quiz

Questions

Correct

Accuracy

The Picard-Lindelöf theorem guarantees:

Easy

Not attempted

A function

f(t,x)

is Lipschitz continuous in

x

if:

Easy

Not attempted

The Peano existence theorem requires:

Easy

Not attempted

The IVP

x' = x^{2/3}

x(0) = 0

has:

Medium

Not attempted

The Gronwall inequality states that if

u(t) \leq \alpha + \int_a^t \beta(s)u(s)ds

, then:

Medium

Not attempted

In Picard iteration, starting from

\phi_0(t) = x_0

, the next iterate is:

Medium

Not attempted

A maximal solution is:

Medium

Not attempted

f

C^1

in a neighborhood of

(t_0, x_0)

, then

f

is locally Lipschitz because:

Medium

Not attempted

The blow-up theorem states that if a maximal solution exists on

(\alpha, \omega)

with

\omega < \infty

, then:

Hard

Not attempted

Continuous dependence on initial conditions means that:

Hard

Not attempted

Frequently Asked Questions

What happens when the Lipschitz condition fails?

Without the Lipschitz condition, uniqueness may fail (as in

x' = x^,{2/3}

), but existence is still guaranteed by Peano's theorem if

f

is continuous. Multiple solution curves can pass through the same point.

How do I know if a solution is maximal?

A solution is maximal if attempting to extend it leads to: (1) leaving the domain of

f

, (2) blow-up (solution becomes unbounded), or (3) the solution becomes non-unique. The blow-up theorem characterizes what happens at the endpoints of a maximal solution.

Why is Gronwall's inequality so important?

Gronwall's inequality converts integral inequalities into pointwise bounds. It's used to prove: uniqueness (if two solutions agree at one point, they agree everywhere), continuous dependence (small changes in data produce small changes in solutions), and stability estimates.

Does Picard iteration always converge quickly?

Picard iteration converges, but the rate depends on the Lipschitz constant and interval length. For large

L

or long time intervals, convergence may be slow. In practice, numerical methods are often used instead, but Picard iteration remains important theoretically.

What is the difference between local and global existence?

Local existence (guaranteed by Picard-Lindelöf) means a solution exists on some interval around

t_0

. Global existence means the solution exists for all

t

. Global existence requires additional conditions—for example, linear growth of

f

prevents finite-time blow-up.

How does this theory apply to systems?

The same theorems apply to systems

x' = f(t, x)

where

x \in \mathbb,{R},^n

. The Lipschitz condition becomes

|f(t, x_1) - f(t, x_2)| \leq L|x_1 - x_2|

using vector norms. All results generalize directly.