Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy.

Consider the initial value problem

y'(t)=f(t,y(t)),\qquad y(t_0)=y_0.

Suppose $f$ is uniformly Lipschitz continuous in $y$ (meaning the Lipschitz constant can be taken independent of $t$ ) and continuous in $t$ . Then, for some value $ε > 0$ , there exists a unique solution $y (t)$ to the initial value problem on the interval $[t_0-\varepsilon, t_0+\varepsilon]$ .^[1]

Proof sketch

The proof relies on transforming the differential equation, and applying fixed-point theory. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation

y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds.

A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration.

Set

\varphi_0(t)=y_0

and

\varphi_{k+1}(t)=y_0+\int_{t_0}^t f(s,\varphi_k(s))\,ds.

It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" $φ k$ is convergent and that the limit is a solution to the problem. An application of Grönwall's lemma to $| φ (t) - ψ (t)|$ , where $φ$ and $ψ$ are two solutions, shows that $φ (t) = ψ (t)$ , thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point).

Intuitive understanding of the theorem

The idea behind the theorem is the following.^[2] A differential equation can possess a stationary point. For example, for the equation $dy / dt = ay$ the stationary solution is $y (t) = 0$ , which is obtained for the initial condition $y (0) = 0$ . Beginning with another initial condition $y (0) = y 0 \neq 0$ , the stationary solution is reached after an infinite time and therefore the uniqueness of solution is guaranteed. However, if the stationary solution is reached after a finite time, the uniqueness is violated. This happens for example for the equation $dy / dt = ay 2 / 3$ , the solution corresponding to the initial condition $y (0) = 0$ can be either $y (t) = 0$ or

y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\ 0 & t \ge 0 \end{cases}

One can note that the function $f (y) = y 2 / 3$ has an infinite slope at $y = 0$ and therefore is not Lipschitz continuous. The Lipschitz continuity condition rules out these type of differential equations.

Detailed proof

Let

C_{a,b}=\overline{I_a(t_0)}\times\overline{B_b(y_0)}

where:

\begin{align} \overline{I_a(t_0)}&=[t_0-a,t_0+a] \\ \overline{B_b(y_0)}&=[y_0-b,y_0+b]. \end{align}

This is the compact cylinder where $f$ is defined. Let

M=\sup_{C_{a,b}}\|f\|,

this is, the maximum slope of the function in modulus. Finally, let L be the Lipschitz constant of $f$ with respect to the second variable.

We will proceed to apply Banach fixed point theorem using the metric on $\mathcal{C}(I_{a}(t_0),B_b(y_0))$ induced by the uniform norm

\| \varphi \|_\infty = \sup_{t \in I_a} | \varphi(t)|.

We define an operator between two functional spaces of continuous functions, Picard's operator, as follows:

\Gamma:\mathcal{C}(I_{a}(t_0),B_b(y_0)) \longrightarrow \mathcal{C}(I_{a}(t_0),B_b(y_0))

defined by:

\Gamma \varphi(t) = y_0 + \int_{t_0}^{t} f(s,\varphi(s)) \, ds.

We impose that it is well-defined, in other words, that its image must be a function taking values on $B_b(y_0)$ , or equivalently, that the norm of

\Gamma\varphi(t)-y_0

is less than b, which can be restated as

\| \varphi_1 \|_\infty \le b.

\left \| \Gamma\varphi(t)-y_0 \right \|=\left \|\int_{t_0}^t f(s,\varphi(s)) \, ds \right \|\leq \int_{t_0}^t \left \|f(s,\varphi(s))\right \| ds \leq M \left |t-t_0 \right|\leq M a\leq b

The last step is the imposition, so we impose the requirement $a < b / M$ .

Now let us impose the Picard's operator to be a contraction under certain hypothesis over a that later on we will be able to omit.

Given two functions $\varphi_1,\varphi_2\in\mathcal{C}(I_{a}(t_0),B_b(y_0))$ , in order to apply the Banach fixed point theorem we want

\left \| \Gamma \varphi_1 - \Gamma \varphi_2 \right\|_\infty \le q \left\| \varphi_1 - \varphi_2 \right\|_\infty,

for some q < 1. So let t be such that

\| \Gamma \varphi_1 - \Gamma \varphi_2 \|_\infty = \left \| \left (\Gamma\varphi_1 - \Gamma\varphi_2 \right )(t) \right \|

then using the definition of Γ

\begin{align} \left \|\left (\Gamma\varphi_1 - \Gamma\varphi_2 \right )(t) \right \| &= \left \|\int_{t_0}^t \left ( f(s,\varphi_1(s))-f(s,\varphi_2(s)) \right )ds \right \|\\ &\leq \int_{t_0}^t \left \|f \left (s,\varphi_1(s)\right )-f\left (s,\varphi_2(s) \right ) \right \| ds \\ &\leq L \int_{t_0}^t \left \|\varphi_1(s)-\varphi_2(s) \right \|ds && f \text{ is Lipschitz-continuous} \\ &\leq L a \left \|\varphi_1-\varphi_2 \right \| \end{align}

This is a contraction if $a < 1 / L$ .

We have established that the Picard's operator is a contraction on the Banach spaces with the metric induced by the uniform norm. This allows us to apply the Banach fixed point theorem to conclude that the operator has a unique fixed point. In particular, there is a unique function

\varphi\in \mathcal{C}(I_a (t_0),B_b(y_0))

such that $Γ φ = φ$ . This function is the unique solution of the initial value problem, valid on the interval I_a where a satisfies the condition

a < \min\{ b/M,1/L\}.

Optimization of the solution's interval

Nevertheless, there is a corollary of the Banach fixed point theorem that states that if an operator Tⁿ is a contraction for some n in N then T has a unique fixed point. We will try to apply this theorem to the Picard's operator. But before doing that, let us recall a lemma that will be very useful to apply the aforementioned corollary.

Lemma: $\left \| \Gamma^m \varphi_1 - \Gamma^m\varphi_2 \right \| \leq \frac{L^m\alpha^m}{m!}\left \|\varphi_1-\varphi_2\right \|$

Proof. We will prove this by induction. For the base of the induction $(m = 1)$ we have already seen this, so suppose the inequality holds for $m - 1$ , then we have:

\begin{align} \left \| \Gamma^m \varphi_1 - \Gamma^m\varphi_2 \right \| &= \left \|\Gamma\Gamma^{m-1} \varphi_1 - \Gamma\Gamma^{m-1}\varphi_2 \right \| \\ &\leq \left| \int_{t_0}^t \left \| f \left (s,\Gamma^{m-1}\varphi_1(s) \right )-f \left (s,\Gamma^{m-1}\varphi_2(s) \right )\right \| ds \right| \\ &\leq L \left| \int_{t_0}^t \left \|\Gamma^{m-1}\varphi_1(s)-\Gamma^{m-1}\varphi_2(s)\right \| ds\right| \\ &\leq \frac{L^m\alpha^m}{m!} \left \|\varphi_1 - \varphi_2 \right \|. \end{align}

Therefore, taking into account this inequality we can assure that for some m large enough,

\frac{L^m\alpha^m}{m!}<1,

and hence Γ^m will be a contraction. So by the previous corollary Γ will have a unique fixed point. So, finally, we have been able to optimize the interval of the solution by taking $α = min{a, b / M}.$

The importance of this result is that the interval of definition of the solution does eventually not depend on the Lipschitz constant of the field, but essentially depends on the interval of definition of the field and its maximum absolute value of it.

Other existence theorems

The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The Peano existence theorem shows only existence, not uniqueness, but it assumes only that $f$ is continuous in $y$ , instead of Lipschitz continuous. For example, the right-hand side of the equation $dy / dt = y 1 / 3$ with initial condition y(0) = 0 is continuous but not Lipschitz continuous. Indeed, rather than being unique, this equation has three solutions:^[3]

y(t) = 0, \qquad y(t) = \pm\left (\tfrac23t\right)^{\frac{3}{2}}.

Even more general is Carathéodory's existence theorem, which proves existence (in a more general sense) under weaker conditions on $f$ . It is also interesting to remark that although these conditions are only sufficient, there also exist necessary and sufficient conditions for the solution of an initial value problem to be unique, such as Okamura's theorem. ^[4]

Notes

↑ Coddington & Levinson (1955), Theorem I.3.1
↑ V. I. Arnold, Ordinary Differential Equations, The MIT Press (1978), ISBN 0-262-51018-9.
↑ Coddington & Levinson (1955), p. 7
↑ Ravi P. Agarwal; V. Lakshmikantham (1993), Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific, ISBN 978-981-02-1357-2 , page 159

References

Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill .
E. Lindelöf, Sur l'application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre; Comptes rendus hebdomadaires des séances de l'Académie des sciences. Vol. 116, 1894, pp. 454–457. Digitized version online via http://gallica.bnf.fr/ark:/12148/bpt6k3074r/f454.table . (In that article Lindelöf discusses a generalization of an earlier approach by Picard.)
Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

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