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# General existence theorems

Let \(|\cdot|\) denote the Euclidean norm on \(\R^n\).

### Initial-value problems

Let \((t_0,x_0)\) be a fixed point in an open subset \(I \times U \subset \R \times \R^n\), and \(f \in C(I \times U, \R^n)\) a continuous vector-valued function on this subset. The problem of finding \(x \in C^1(J,U)\) such that
\[
\dot x(t) = f(t,x(t)), \qquad x(t_0) = x_0, \qquad\qquad \mathrm{(IVP)}
\]
for some possibly smaller interval \(J \subset I\) is called an **initial-value problem**. Here, \(\dot x = \frac{d}{dt} x\).

### › Reformulation of real-valued ODEs as first-order systems

Any ordinary differential equation \[ x^{(n)}(t) = g(t,x(t),\dot x(t), \ldots, x^{(n-1)}(t)), \] with initial conditions \[ x(t_0) = x_1, \quad \dot x(t_0) = x_2, \quad \ldots, \quad x^{(n-1)}(t_0) = x_{n}, \] and \(g\) continuous in some open set \(I \times U \subset \R \times \R^n\) containing \((t_0, x_1, \ldots, x_{n})\), can be reformulated in the form (IVP).

**Ex.**

- The second-order ordinary differential equation \[ \ddot x + \sin(x) = 0, \qquad x(0) = 1,\quad \dot x(0) = 2, \] is equivalent to the system \[ \begin{bmatrix} \dot y_0 \\ \dot y_1 \end{bmatrix} = \begin{bmatrix} y_1 \\ -\sin(y_0) \end{bmatrix} \quad\text{ with }\quad \begin{bmatrix} y_0 \\y_1 \end{bmatrix}_{t=0} = \begin{bmatrix}1 \\2\end{bmatrix}. \] In this case \[f \colon \R^2 \to \R^2, \qquad \begin{bmatrix} y_0 \\ y_1 \end{bmatrix} \mapsto \begin{bmatrix} y_1 \\ -\sin(y_0) \end{bmatrix} \] is independent of time.

### › The Peano existence theorem

For any \((t_0,x_0) \in I \times U\) there exists \(\varepsilon > 0\) such that the initial-value problem (IVP) has a solution defined for \(|t-t_0| < \varepsilon\). The solution \(x = x(\cdot;t_0,x_0) \in C^1(B_\varepsilon(t_0),U)\).

**N.b.** The Peano existence theorem guarantees the existence of (local) solutions, but not their uniqueness.

**Ex.**

- The initial-value problem \[ \begin{cases} \dot x = {\textstyle\frac{3}{2}} x^{1/3}, \quad &t \geq 0,\\ \dot x = 0, \quad &t < 0,\end{cases} \qquad x(0) = 0, \] has the trivial solution \(x \equiv 0\), but also the ones given by \[ \begin{cases} x(t) = \pm t^{3/2}, \quad &t \geq 0, \\ x(t) = 0, \quad &t<0. \end{cases}\]

To remedy this lack of uniqueness in Peano's theorem one needs the concept of Lipschitz continuity.

### Lipschitz continuity

A continuous function \(f \in C(I \times U, \R^n)\) is said to be locally **Lipschitz continuous** with respect to its second variable \(x \in U\) if for any \((t_0,x_0) \in I \times U\) there exists \(\varepsilon, L >0\) with
\[
| f(t,x) - f(t,y) | \leq L\, | x-y |, \qquad \text{ for all }\quad (t,x), (t,y) \in B_{\varepsilon}(t_0,x_0).
\]
The set of locally Lipschitz continuous functions on \(I \times U\) form a vector space, \(Lip(I \times U, \R^n)\). *If the Lipschitz constant* \(L\) *does not depend on the point* \((t_0,x_0)\), *then the Lipschitz condition is said to be uniform* (\(f\) is then uniformly Lipschitz continuous).

*A locally Lipschitz continuous function is uniformly Lipschitz continuous on any compact set.*

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**N.b.** Any continuously differentiable function is also locally Lipschitz continuous, and hence unformly Lipschitz on any compact set.

**Ex.**

Consider \(f \colon \R \to \R\) (one spatial variable, no time).

- \(x \mapsto \sin(x)\) is continuously differentiable. It is also uniformly Lipschitz, since \[ |\sin(x) - \sin(y)| \leq \max_{\xi \in \R} |\cos(\xi)| |x-y|.\]
- \( x \mapsto x^2\) is continuously differentiable. It is locally Lipschitz, since \[ |x^2-y^2| = |x+y| |x-y|.\]
- \(x \mapsto |x|\) is not continuously differentiable. It is however (uniformly) Lipschitz, since \[ ||x|-|y|| \leq |x-y|.\]
- \(x \mapsto \sqrt{|x|}\) is continuous but not locally Lipschitz, since it cannot have a finite Lipschitz constant at \(x_0 = 0\): \[\frac{\sqrt{|x|}}{|x|} \to \infty \text{ as } x \to 0.\]

In particular, this shows that \(C^1(\R) \subsetneq Lip(\R) \subsetneq C^0(\R)\).^{2)}

## The Banach fixed-point theorem and its applications

To solve the initial-value problem (IVP) we shall reformulate it as
\[
x(t) = x_0 + \int_{t_0}^t f(s,x(s))\,ds, \qquad x \in BC(I,U),
\]
where the right-hand side defines a (not necessarily linear) mapping
\[
T \colon BC(J,U) \to BC(J,U), \quad x \mapsto x_0 + \int_{t_0}^t f(s,x(s))\,ds,
\]
for some smaller interval \(J = [t_0-\varepsilon,t_0 + \varepsilon] \subset I\). This is because, if \(x\) and \(f\) are continuous, so is \(s \mapsto f(s,x(s))\), so the integral \(\int_{t_0}^t f(s,x(s))\,ds\) is continuous (even \(C^1\)) and bounded on compact intervals. The idea then is that, if \(f\) is also Lipschitz, then \(T\) **contracts** points for small \(|t-t_0| \leq \varepsilon\):
\[
\begin{align*}
|Tx(t) - Ty(t)| = \Big| \int_{t_0}^t \big( f(s,x(s)) - f(s,y(s)) \big)\,ds \Big| &\leq \int_{t_0}^t \big| f(s,x(s)) - f(s,y(s)) \big|\,ds\\ &\leq \int_{t_0}^t L |x(s) - y(s)|\,ds \leq L |t-t_0| \|x - y\|_{BC(J,U)}
\end{align*}
\]
Thus, if \(\varepsilon L < 1\), taking the maximum over \(t \in J\) yields
\[
\|Tx-Ty\|_{BC(J,U)} \leq \lambda \|x-y\|_{BC(J,U)}, \quad\text{ for } \lambda = \varepsilon L < 1,
\]
so that \(Tx\) and \(Ty\) are closer to each other than \(x\) and \(y\). As we shall now see, that gives us a local and unique solution of our problem.

### Contractions

Let \((X,d)\) be metric space. A mapping \(T: X \to X\) is called a **contraction** if there exists \(\lambda < 1\) such that
\[
d(T(x),T(y)) \leq \lambda\, d(x,y), \qquad\text{ for all }\quad x,y \in X.
\]
In particular, contractions are continuous.

**N.b.** The uniformity of the constant \(\lambda < 1\) is important; it is not enough that \(d(T(x),T(y)) < d(x,y)\) for each pair \((x,y) \in X \times X\).

### › The Banach fixed-point theorem

Let \(T\) be a contraction on a complete metric space \((X,d)\) with \(X \neq \emptyset\). Then there exists a unique \(x \in X\) such that \(T(x) = x\).

### Well-posedness for the initial-value problem (IVP)

### › The Picard–Lindelöf theorem

Let \(f: I \times U \to \R^n\) be locally Lipschitz continuous with respect to its second variable and \((t_{0},x_{0})\) a point in \(I \times U\) determining the initial data. Then, for each \(\eta > 0\), there exists \(\varepsilon > 0\) such that the initial-value problem (IVP) has a unique solution \(x \in C^1(\overline{B_\varepsilon(t_0)},\overline{B_\eta(x_0)})\).

### › Picard iteration

Under the assumptions of the Picard-Lindelöf theorem, the sequence given by
\[
x_0 = x(t_0), \qquad x_n = T x_{n-1}, \quad n \in \N; \qquad (Tx)(t) = x_0 + \int_{t_0}^t f(s,x(s))\,ds,
\]
converges uniformly and exponentially fast to the unique solution \(x\) on \(J = [t_0-\varepsilon, t_0+ \varepsilon]\):
\[
\|x_n - x\|_{BC(J,\R^n)} \leq \frac{\lambda^n}{1-\lambda}\| x_1 - x_0 \|_{BC(J,\R^n)},
\]
where \(\lambda = \varepsilon L\) is the contraction constant used in the proof of the Picard–Lindelöf theorem.^{3)}

**Ex.**

The first Picard iteration for the initial-value problem \[ \dot x = \sqrt{x} + x^3, \qquad x(1) = 2,\] is given by \[ x_1(t) = 2 + \int_1^t \big( \sqrt{2} + 2^3 \big)\,ds = 2 + (\sqrt{2} + 8)(t-1). \] The second is \[ x_2(t) = 2 + \int_1^t \big( \sqrt{x_1(s)} + (x_1(s))^3\big)\,ds. \] (This indicates that Picard iteration, in spite of its simplicity and fast convergence, is better suited as a theoretical and computer-aided tool, than as a way to solve ODE's by hand.)

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