The Peano Existence Theorem

While the Picard-Lindelof theorem requires a Lipschitz condition, the Peano theorem shows that mere continuity suffices for existence (though not uniqueness).

Statement

Theorem8.5Peano existence theorem

Let $\mathbf{F}: \Omega \to \mathbb{R}^n$ be continuous on an open set $\Omega \subset \mathbb{R} \times \mathbb{R}^n$ , and let $(t_0, \mathbf{x}_0) \in \Omega$ . Then the initial value problem

$\mathbf{x}' = \mathbf{F}(t, \mathbf{x}), \quad \mathbf{x}(t_0) = \mathbf{x}_0$

has at least one solution defined on some interval $[t_0 - \delta, t_0 + \delta]$ .

More precisely, if $R = [t_0 - a, t_0 + a] \times \overline{B_b(\mathbf{x}_0)} \subset \Omega$ and $M = \max_R \|\mathbf{F}\|$ , then a solution exists on $[t_0 - \delta, t_0 + \delta]$ with $\delta = \min(a, b/M)$ .

Proof via Euler Approximations

Proof

Step 1: Euler polygonal approximations. For each $N \geq 1$ , define the Euler approximation $\mathbf{x}_N(t)$ on $[t_0, t_0 + \delta]$ with step size $h = \delta/N$ :

$\mathbf{x}_N(t_0) = \mathbf{x}_0, \quad \mathbf{x}_N(t_{k+1}) = \mathbf{x}_N(t_k) + h\mathbf{F}(t_k, \mathbf{x}_N(t_k)),$

where $t_k = t_0 + kh$ , with linear interpolation between grid points. By construction, $\|\mathbf{x}_N(t) - \mathbf{x}_0\| \leq M\delta \leq b$ , so $(t, \mathbf{x}_N(t)) \in R$ for all $t$ .

Step 2: Equicontinuity. For any $t, t' \in [t_0, t_0 + \delta]$ :

$\|\mathbf{x}_N(t) - \mathbf{x}_N(t')\| \leq M|t - t'|,$

so the family $\{\mathbf{x}_N\}$ is uniformly bounded and equicontinuous.

Step 3: Arzela-Ascoli theorem. By the Arzela-Ascoli theorem, there exists a subsequence $\{\mathbf{x}_{N_k}\}$ converging uniformly to a continuous function $\mathbf{x}: [t_0, t_0 + \delta] \to \overline{B_b(\mathbf{x}_0)}$ .

Step 4: Passing to the limit. Each Euler approximation satisfies:

$\mathbf{x}_N(t) = \mathbf{x}_0 + \int_{t_0}^{t} \mathbf{F}(s, \mathbf{x}_N(s))\,ds + \mathbf{e}_N(t),$

where the error $\|\mathbf{e}_N(t)\| \to 0$ uniformly as $N \to \infty$ (using uniform continuity of $\mathbf{F}$ on $R$ ). Since $\mathbf{x}_{N_k} \to \mathbf{x}$ uniformly and $\mathbf{F}$ is continuous:

$\mathbf{x}(t) = \mathbf{x}_0 + \int_{t_0}^{t} \mathbf{F}(s, \mathbf{x}(s))\,ds.$

Differentiating: $\mathbf{x}'(t) = \mathbf{F}(t, \mathbf{x}(t))$ and $\mathbf{x}(t_0) = \mathbf{x}_0$ . $\blacksquare$

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Non-Uniqueness Examples

ExampleNon-uniqueness without Lipschitz

The IVP $x' = 3x^{2/3}$ , $x(0) = 0$ has the trivial solution $x(t) = 0$ and the family of solutions:

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

for any $c \geq 0$ . The right-hand side $F(x) = 3x^{2/3}$ is continuous but not Lipschitz at $x = 0$ (since $F'(x) = 2x^{-1/3} \to \infty$ ). Peano guarantees existence but cannot guarantee uniqueness.

RemarkComparison: Peano vs. Picard-Lindelof

| Property | Peano | Picard-Lindelof | |---|---|---| | Hypothesis on $\mathbf{F}$ | Continuous | Continuous + Lipschitz | | Existence | Yes | Yes | | Uniqueness | No | Yes | | Proof method | Arzela-Ascoli (compactness) | Contraction mapping (completeness) | | Constructive | No (subsequence extraction) | Yes (Picard iterates converge) |

The Lipschitz condition is the precise dividing line between uniqueness and potential non-uniqueness.