Key concepts:
- Initial value problems Differential equations and integral equations Higher order differential equations
- Lipschitz in the $y$ variable Global Picard Theorem
Outline:
We examine another application of the Banach Contraction Principle, namely, we use it to find solutions of a class of differential equation known as initial value problems. These systems are very important in applied mathematics because many phenomena can be modelled in this way for quantitative descriptions and predictions of how things change over time; for example, the motion of objects under the action of physical forces, the concentrations of a mixture of chemicals as a reaction proceeds, and population growth or collapse of interacting animal species.
We first use the Fundamental Theorem of Calculus to relate solutions of differential equations to integral equations. Thinking about this in the context of discrete dynamical systems and contractions, we recognize that we can define a dynamical system on a space of continuous functions such that the fixed point of this dynamical system is a solution of the initial value problem.
If the map is a contraction, then we can apply BCP: a unique solution to the initial value problem is guaranteed and we can generate a sequence of approximations to the solution by iteratively applying the contraction to any initial guess.
We have an alternative tool in case the map is not a contraction: if the initial value problem satisfies some hypotheses, then we can apply the Global Picard Theorem to determine that a unique solution exists.
(Global Picard Theorem). Let $\Phi:[a, b] \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a continuous function that is Lipschitz in the $y$ variable and let $\Gamma \in \mathbb{R}^{n} .$ Then the initial value problem
$$
F^{\prime}(x)=\Phi(x, F(x)), \quad F(a)=\Gamma
$$
has a unique solution.
Define the map $T: C\left([a, b], \mathbb{R}^{n}\right) \rightarrow C\left([a, b], \mathbb{R}^{n}\right)$ by
$$
T F(x)=\Gamma+\int_{a}^{x} \Phi(t, F(t)) d t
$$
and define a sequence of functions $\left(F_{k}\right){k=1}^{\infty}$ in $C\left([a, b], \mathbb{R}^{n}\right)$ recursively by $$ F{0}(x)=\Gamma, \quad F_{k+1}=T F_{k} \text { for } k \geq 0
$$
Let $L$ denote the Lipschitz constant for $\Phi$ and define $M=\sup {x \in[a, b]}|\Phi(x, \Gamma)|$. $$ \left|F{1}(x)-F_{0}(x)\right|=\left|\int_{a}^{x} \Phi(t, \Gamma) d t\right| \leq M(x-a)=\frac{M(x-a)^{1}}{1 !}
$$
By Lemma $10.3 .3$, we have
$$
\left|F_{2}(x)-F_{1}(x)\right|=\left|T F_{1}(x)-T F_{0}(x)\right| \leq \frac{M L(x-a)^{2}}{2 !}
$$
This also satisfies the condition in the lemma, so we can iterate. We can show by induction that
$$
\left|F_{k+1}(x)-F_{k}(x)\right| \leq \frac{M L^{k}(x-a)^{k+1}}{(k+1) !}
$$
for any integer $k \geq 0$. Following the method in the proof of the Banach Contraction Principle, for any $k, l \geq 0$, we find that
$$
\begin{aligned}
\left|F_{k+l}(x)-F_{k}(x)\right| & \leq \sum_{j=k}^{k+l-1}\left|F_{j+1}(x)-F_{j}(x)\right| \
& \leq \sum_{j=k}^{k+l-1} \frac{M L^{j}(x-a)^{j+1}}{(j+1) !} \
& \leq \frac{M}{L} \sum_{j=k+1}^{\infty} \frac{[L(b-a)]^{j}}{j !}
\end{aligned}
$$
Recall the power series representation of $e^{z}$ is $\sum_{j=0}^{\infty} \frac{z^{j}}{j !}$ so $\sum_{j=0}^{\infty} \frac{[L(b-a)]^{j}}{j !}=e^{L(b-a)}$. Since this series converges, we know that given $\varepsilon>0$, there exists an integer $N$ such that $\frac{M}{L} \sum_{j=N}^{\infty} \frac{[L(b-a)]^{j}}{j !}<\varepsilon$.
Therefore, $\left|F_{p}(x)-F_{q}(x)\right|<\varepsilon$ for all $x \in[a, b]$ and hence $\left|F_{p}-F_{q}\right|_{\infty} \leq \varepsilon$ for all $p, q \geq N$ (cf. proof of Banach Contraction Principle), proving that $\left(F_{k}\right)$ is Cauchy.
But $C\left([a, b], \mathbb{R}^{n}\right)$ is complete with respect to the uniform norm so the sequence must converge to some $\operatorname{limit} F^{}$. By continuity of $T$, we have $$ T F^{}=T\left(\lim {k \rightarrow \infty} F{k}\right)=\lim {k \rightarrow \infty} T F{k}=\lim {k \rightarrow \infty} F{k+1}=F^{} $$ so $F^{}$ is a fixed point of $T$, which is a solution of the IVP.
Proving that the fixed point is unique is left as an exercise.
To conclude, we first handled the case where we could show that the solution of the IVP was the fixed point of a contraction $T$. We then showed that orbits of a map $T$ converge to solutions of the IVP if the function $\Phi$ defining the IVP is Lipschitz in the $y$ variable. This requirement is still quite restrictive. You might encounter functions that are not Lipschitz in the $y$ variable, or at least not over the entire domain. If you are interested, you can read about a few more techniques in the text book (Chapter 12).
