Picard–Lindelöf theorem

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In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem, Picard's existence theorem, Cauchy–Lipschitz theorem, or existence and uniqueness theorem gives a set of conditions under which an initial value problem has a unique solution.

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

Consider the initial value problem

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 .[1]

Proof sketch[edit]

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

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



It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" φk 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).


Let and , so and

and so on.

This is calculating the series expansion of , an alternative method is to use Bernoulli's numbers.

Intuitive understanding of the theorem[edit]

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) = y0 ≠ 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 = ay2/3, the solution corresponding to the initial condition y(0) = 0 can be either y(t) = 0 or

One can note that the function f (y) = y2/3 has an infinite slope at y = 0 and therefore is not Lipschitz continuous. The Lipschitz continuity condition rules out this type of differential equation.

Detailed proof[edit]



This is the compact cylinder where f is defined. Let

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 induced by the uniform norm

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

defined by:

We must show that this operator maps a complete non-empty metric space X into itself and also is a contraction mapping.

We first show that, given certain restrictions on , takes into itself in the space of continuous functions with uniform norm. Here, is a closed ball in the space of continuous (and bounded) functions "centered" at the constant function . Hence we need to show that


where is some number in where the maximum is achieved. The last step is true if we impose the requirement a < b/M.

Now let's try to prove that this operator is a contraction.

Given two functions , in order to apply the Banach fixed point theorem we want

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

then using the definition of Γ

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

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

Optimization of the solution's interval[edit]

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.


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:

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

and hence Γm 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[edit]

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 = y1/3 with initial condition y(0) = 0 is continuous but not Lipschitz continuous. Indeed, rather than being unique, this equation has three families of solutions:[3]

, where is any nonnegative real number.

Even more general is Carathéodory's existence theorem, which proves existence (in a more general sense) under weaker conditions on f. 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]

See also[edit]


  1. ^ Coddington & Levinson (1955), Theorem I.3.1
  2. ^ Arnold, V. I. (1978). Ordinary Differential Equations. The MIT Press. ISBN 0-262-51018-9.
  3. ^ Coddington & Levinson (1955), p. 7
  4. ^ Agarwal, Ravi P.; Lakshmikantham, V. (1993). Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific. p. 159. ISBN 981-02-1357-3.


External links[edit]