Picard–Lindelöf theorem
In mathematics, specifically the study of differential equations, the Picard–Lindelöf 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.
[1][2] A standard proof relies on transforming the differential equation into an integral equation, then applying the Banach fixed-point theorem to prove the existence of a solution, and then applying Grönwall's lemma to prove the uniqueness of the solution.
, this integral operator is a contraction and so the Banach fixed-point theorem proves that a solution can be obtained by fixed-point iteration of successive approximations.
Set and It follows from the Banach fixed-point theorem that the sequence of "Picard iterates"
is convergent and that its limit is a solution to the original initial value problem.
Evidently, the functions are computing the Taylor series expansion of our known solution
it is not Lipschitz continuous in the neighborhood of those points, and the iteration converges toward a local solution for
To understand uniqueness of solutions, contrast the following two examples of first order ordinary differential equations for y(t).
[3] Both differential equations will possess a single stationary point y = 0.
By contrast for an equation in which the stationary point can be reached after a finite time, uniqueness of solutions does not hold.
Consider the homogeneous nonlinear equation dy/dt = ay 2/3, which has at least these two solutions corresponding to the initial condition y(0) = 0: y(t) = 0 and so the previous state of the system is not uniquely determined by its state at or after t = 0.
We will proceed to apply the Banach fixed-point theorem using the metric on
induced by the uniform norm We define an operator between two function 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.
The last inequality in the chain is true if we impose the requirement
, in order to apply the Banach fixed-point theorem we require for some
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.
This function is the unique solution of the initial value problem, valid on the interval Ia where a satisfies the condition We wish to remove the dependence of the interval Ia on L. To this end, there is a corollary of the Banach fixed-point theorem: if an operator Tn is a contraction for some n in N, then T has a unique fixed point.
Before applying this theorem to the Picard operator, recall the following: Lemma —
So by the previous corollary Γ will have a unique fixed point.
Finally, we have been able to optimize the interval of the solution by taking α = min{a, b/M}.
In the end, this result shows the interval of definition of the solution does not depend on the Lipschitz constant of the field, but only on the interval of definition of the field and its maximum absolute value.
Indeed, rather than being unique, this equation has at least three solutions:[4] 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.
[5] The Picard–Lindelöf theorem ensures that solutions to initial value problems exist uniquely within a local interval
The behavior of solutions beyond this local interval can vary depending on the properties of f and the domain over which f is defined.
For instance, if f is globally Lipschitz, then the local interval of existence of each solution can be extended to the entire real line and all the solutions are defined over the entire R. If f is only locally Lipschitz, some solutions may not be defined for certain values of t, even if f is smooth.
For instance, the differential equation dy/dt = y 2 with initial condition y(0) = 1 has the solution y(t) = 1/(1-t), which is not defined at t = 1.
Nevertheless, if f is a differentiable function defined over a compact subset of Rn, then the initial value problem has a unique solution defined over the entire R.[6] Similar result exists in differential geometry: if f is a differentiable vector field defined over a domain which is a compact smooth manifold, then all its trajectories (integral curves) exist for all time.
