The Local Existence Theorem for Solutions to Initial Value Problems
The Local Existence Theorem for Solutions to Initial Value Problems of First Order ODEs
| Theorem (Local Existence Theorem): Let $x' = f(t, x)$ with $x(\tau) = \xi$ be an IVP with $f \in C(D, \mathbb{R})$. Then there exists a $c > 0$ such that this IVP has a solution on $[\tau - c, \tau + c]$. |
- Proof: As before, since $f$ is continuous on $D$ (nonempty, open, connected subset of $\mathbb{R}^2$) and $(\xi, \tau) \in D$, let $S \subseteq D$ be a closed rectangle given by:
\begin{align} \quad S = \{ (t, x) : | t - \tau | \leq a, | x - \xi | \leq b \} \end{align}
- And since $f$ is continuous on the closed and bounded rectangle $S$ we have that $f$ is bounded on $S$ and so there exists an $M > 0$ such that $| f(t, x) | \leq M$ for all $(t, x) \in S$. Furthermore, we let:
\begin{align} \quad c = \min \left \{ a, \frac{b}{M} \right \} \end{align}
- Now let $(\epsilon_m)_{m=1}^{\infty}$ be a decreasing sequence of positive real numbers such that $\displaystyle{\lim_{m \to \infty} \epsilon_m = 0}$. For each $\epsilon_m > 0$, by the theorem presented on the ϵ-Approximate Solutions to Initial Values Problems of First Order ODEs page, there exists an $\epsilon_m$-approximate solution $\phi_m$.
- By the construction of each of the $\phi_m$s we have that:
\begin{align} \quad | \phi_m(t) - \phi_m(s) | \leq M | t - s | \quad \forall t, s \in [\tau - c, \tau + c], \: \forall m \in \mathbb{N} \end{align}
- Part 1: Proving that the set of functions $\mathcal F = (\phi_m)_{m=1}^{\infty}$ is equicontinuous.
- Let $\epsilon > 0$ and let $\displaystyle{\delta = \frac{\epsilon}{M} > 0}$. Then if $|t - s| < \delta$ we have by the construction of the $\epsilon_m$-approximate solutions that:
\begin{align} \quad | \phi_m(t) - \phi_m(s) | < M \delta = M \frac{\epsilon}{M} = \epsilon \end{align}
- So for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $|t - s| < \delta$ then $| \phi_m(t) - \phi_m(s) | < \delta$ for all $t, s \in [\tau - c, \tau + c]$ and for all $m \in \mathbb{N}$, so the set $\mathcal F = (\phi_m)_{m=1}^{\infty}$ is equicontinuous.
- Part 2: Proving that the set of functions $\mathcal F = (\phi_m)_{m=1}^{\infty}$ is uniformly bounded.
- Let $M^* = Mc + |\xi |$. Then for each $m \in \mathbb{N}$ we have by the triangle inequality that:
\begin{align} \quad | \phi_m(t) | = | \phi_m(t) - \phi_m(\tau) + \phi_m(\tau) | \leq | \phi_m(t) - \phi_m(\tau) | + | \phi_m(\tau) | = M|t - \tau| + | \xi | \leq Mc + | \xi | = M^* \end{align}
- Note that $M^*$ is fixed, so $\mathcal F = (\phi_m)_{m=1}^{\infty}$ is uniformly bounded.
- Part 3: Applying the Arzela-Ascoli theorem.
- By the Arzela-Ascoli theorem, since $[\tau - c, \tau + c]$ is a compact subset of $\mathbb{R}$ and $\mathcal F = (\phi_m)_{m=1}^{\infty}$ is a collection of continuous functions on $S$ that is uniformly equicontinuous and uniformly bounded there exists a subsequence $(\phi_{m_k})_{k=1}^{\infty}$ of $\mathcal F$ and a function $\phi \in C([\tau - c, \tau + c], \mathbb{R})$ such that this $(\phi_{m_k})_{k=1}^{\infty}$ converges uniformly to $\phi$ on $[\tau - c, \tau + c]$.
- Part 4: Showing that $\phi$ is a solution to the IVP $x' = f(t, x)$ with $x(\tau) = \xi$.
- For each $k \in \mathbb{N}$ let:
\begin{align} \quad E_{m_k}(t) = \phi_{m_k}'(t) - f(t, \phi_{m_k}(t)) \end{align}
- Then each $E_{m_k}$ is piecewise continuous on $[\tau - c, \tau + c]$ and furthermore:
\begin{align} \quad | E_{m_k}(t) | = | \phi_{m_k}'(t) - f(t, \phi_{m_k}(t)) | < \epsilon_{m_k} \end{align}
- Rearranging the equation for $E_{m_k}(t)$ and we get:
\begin{align} \quad \phi_{m_k}'(t) = f(t, \phi_{m_k}(t)) + E_{m_k}(t) \end{align}
- Integrate both sides of the equation above from $\tau$ to $t$ and use the Fundamental theorem of Calculus to get:
\begin{align} \quad \int_{\tau}^{t} \phi_{m_k}'(s) \: ds &= \int_{\tau}^{t} [f(s, \phi_{m_k}(s)) + E_{m_k}(s)] \: ds \\ \quad \phi_{m_k}(t) - \phi_{m_k}(\tau) &= \int_{\tau}^{t} [f(s, \phi_{m_k}(s)) + E_{m_k}(s)] \: ds \\ \quad \phi_{m_k}(t) - \xi &= \int_{\tau}^{t} [f(s, \phi_{m_k}(s)) + E_{m_k}(s)] \: ds \\ \quad \phi_{m_k}(t) &= \xi + \int_{\tau}^{t} [f(s, \phi_{m_k}(s)) + E_{m_k}(s)] \: ds \end{align}
- We now want to show that $\displaystyle{\left ( \int_{\tau}^{t} [f(s), \phi_{m_k}(s)) + E_{m_k}(s)] \: ds \right )_{k=1}^{\infty}}$ converges to $\displaystyle{\int_{\tau}^{t} f(s, \phi(s)) \: ds}$.
- Since $(\phi_m)_{m=1}^{\infty}$ converges to $\phi$ uniformly on $[\tau - c, \tau + c]$ and $f$ is uniformly continuous on $S$ we have that $(f(t, \phi_m(t)))_{m=1}^{\infty}$ converges uniformly to $f(t, \phi(t))$ on $[\tau - c, \tau + c]$. Let:
\begin{align} \quad \alpha_k = \sup_{t \in [\tau - c, \tau + c]} | f(t, \phi_{m_k}(t)) - f(t, \phi(t)) | \end{align}
- Then $\alpha_k \to 0$ as $k \to 0$. Now:
\begin{align} \quad \biggr \lvert \int_{\tau}^{t} [f(s, \phi_{m_k} (s)) + E_{m_k}(s)] - \int_{\tau}^{t} f(s, \phi(s)) \: ds \biggr \rvert & \leq \biggr \lvert \int_{\tau}^{t} [f(s, \phi_{m_k}(s)) - f(s, \phi(s))] \: ds \biggr \lvert + \biggr \lvert \int_{\tau}^{t} E_{m_k} (s) \: ds \biggr \rvert \\ & \leq \int_{\tau}^{t} | f(s, \phi_{m_k}(s)) - f(s, \phi(s)) | \: ds + \int_{\tau}^{t} | E_{m_k}(s) | \: ds \\ & \leq \int_{\tau}^{t} \alpha_k \: ds + \int_{\tau}^{t} \epsilon_{m_k} \\ & \leq [\alpha_k + \epsilon_{m_k}](t - \tau) \\ & \leq [\alpha_k + \epsilon_{m_k}]c \end{align}
- As $k \to \infty$ we have that $[\alpha_k + \epsilon_{m_k}]c \to 0$ since $c$ is constant and $\alpha_k \to 0$ and $\epsilon_k \to 0$ (from earlier). Therefore:
\begin{align} \quad \phi(t) = \lim_{k \to \infty} \phi_{m_k} (t) = \lim_{k \to \infty} \left [ \xi + \int_{\tau}^{t} [f(s, \phi(s)) + E_m(s)] \: ds \right ] = \xi + \int_{\tau}^{t} f(s, \phi(s)) \: ds \end{align}
- Also $\phi(\tau) = \xi$. So $\phi$ is a solution to the IVP $x' = f(t, x)$ with $x(\tau) = \xi$ on $[\tau - c, \tau + c]$. $\blacksquare$