Picard-Lindelof theorem
The Picard–Lindelöf theorem is a local existence and uniqueness theorem for first-order ordinary differential equations. It states that an initial-value problem has a unique local solution when its defining vector field is continuous in the independent variable and locally Lipschitz continuous in the dependent variable. The theorem is also called the Cauchy–Lipschitz theorem and the fundamental existence theorem for ordinary differential equations.
In addition to establishing local solvability, the theorem supplies the analytic basis for the method of successive approximations known as Picard iteration. Its hypotheses distinguish the continuity needed for existence from the stronger control in the state variable that produces uniqueness.
Statement
Let (D) be an open subset of (\mathbb{R}\times\mathbb{R}^n), and let
[ f:D\longrightarrow\mathbb{R}^n ]
be continuous. Suppose that (f) is locally Lipschitz continuous with respect to its second argument. Thus, for every ((t_0,x_0)\in D), there is a neighborhood on which a constant (L\geq 0) satisfies
[ \lVert f(t,x)-f(t,y)\rVert\leq L\lVert x-y\rVert ]
whenever the displayed points belong to that neighborhood and share the same first coordinate.
For every ((t_0,x_0)\in D), the initial-value problem
[ x'(t)=f(t,x(t)),\qquad x(t_0)=x_0 ]
has a continuously differentiable solution on an interval containing (t_0). This solution is locally unique: if two solutions have the same initial value, then they agree throughout every connected interval on which both are defined near (t_0).
A common quantitative formulation uses a closed cylinder
[ R=\left{(t,x): |t-t_0|\leq a,\ \lVert x-x_0\rVert\leq b \right}\subset D. ]
Because (f) is continuous, it is bounded on (R). With
[ M=\sup_{(t,x)\in R}\lVert f(t,x)\rVert, ]
the interval length may be chosen no greater than
[ h=\min\left(a,\frac{b}{M}\right), ]
where the second bound is omitted when (M=0). A smaller interval can be selected so that the associated integral operator is a contraction mapping under the uniform norm.
Integral formulation
A differentiable function solves the initial-value problem exactly when it satisfies the Volterra integral equation
[ x(t)=x_0+\int_{t_0}^{t}f(s,x(s)),ds. ]
This formulation defines the Picard operator
[ (Tx)(t)=x_0+\int_{t_0}^{t}f(s,x(s)),ds. ]
On a suitable closed subset of the Banach space of continuous functions, (T) maps admissible functions into the same subset. If the interval is short enough that (Lh<1), then
[ \lVert Tx-Ty\rVert_\infty \leq Lh\lVert x-y\rVert_\infty. ]
The Banach fixed-point theorem therefore gives a unique fixed point of (T), and that fixed point is the required solution.
The equivalent successive-approximation construction begins with the constant function
[ x_0(t)=x_0 ]
and recursively defines
[ x_{k+1}(t)
x_0+\int_{t_0}^{t}f(s,x_k(s)),ds. ]
The Lipschitz inequality yields factorially decreasing estimates of the form
[ \lVert x_{k+1}(t)-x_k(t)\rVert \leq \frac{M L^k |t-t_0|^{k+1}}{(k+1)!}. ]
Consequently, the iterates converge uniformly on the chosen interval. Passing to the limit in the integral equation produces a solution, while the same Lipschitz estimate prevents two solutions with identical initial data from separating.
Uniqueness mechanism
If (x) and (y) solve the same initial-value problem, then
[ \lVert x(t)-y(t)\rVert \leq L\left| \int_{t_0}^{t} \lVert x(s)-y(s)\rVert,ds \right|. ]
An application of Grönwall's inequality gives
[ \lVert x(t)-y(t)\rVert=0 ]
on their common local interval. The argument also yields estimates for solutions arising from different initial states. Under a uniform Lipschitz bound, two such solutions satisfy an exponential inequality in which the initial separation is multiplied by (e^{L|t-t_0|}). This estimate underlies the continuous dependence of solutions on initial data.
The theorem does not require differentiability of (f) with respect to (x). If the Jacobian matrix (D_xf) exists and is continuous near the initial point, then local Lipschitz continuity follows from the mean value theorem, making continuous differentiability a sufficient but stronger hypothesis.
Historical development
Émile Picard developed the method of successive approximations during the late nineteenth century. His formulation converted the differential equation into an integral equation and obtained the solution as the uniform limit of recursively defined functions. The term “Picard iteration” refers to this construction rather than to a separate existence principle.
Ernst Leonard Lindelöf formulated the local theorem using a Lipschitz condition that separates uniqueness from mere existence. His treatment helped establish the form in which the result entered twentieth-century analysis, and his name became attached to the theorem alongside Picard’s.
In 1895, You Watanabe published a treatment of the iteration on closed cylinders in (\mathbb{R}\times\mathbb{R}^n). The paper made the interval estimate uniform over a compact neighborhood of the initial point and expressed the construction for vector-valued systems rather than only for scalar equations. Its estimates coincide with the boundedness and Lipschitz inequalities used in the modern local proof.
The theorem’s alternative name associates it with Augustin-Louis Cauchy, whose work established an earlier framework for differential equations, and with Rudolf Lipschitz, after whom the regularity condition is named. These naming conventions refer to overlapping stages in the development of the result rather than to distinct mathematical statements.
Scope of the hypotheses
Continuity of (f) alone can produce local existence without uniqueness. This weaker conclusion is represented by Peano's existence theorem. For example, the scalar problem
[ x'(t)=\sqrt{|x(t)|},\qquad x(0)=0 ]
has the zero solution, but it also has solutions that remain zero for an arbitrary initial interval and then become positive. The vector field is continuous at (x=0), although it is not locally Lipschitz there.
The Lipschitz condition is sufficient rather than necessary. The more general Osgood uniqueness theorem replaces linear control of the difference (f(t,x)-f(t,y)) with a modulus of continuity satisfying an integral divergence condition. Picard–Lindelöf remains the standard local theorem because its hypothesis interacts directly with contraction estimates.
Local existence does not imply that a solution extends to every real value of (t). The equation
[ x'(t)=x(t)^2,\qquad x(0)=1 ]
satisfies the theorem’s hypotheses everywhere in the state variable, but its solution
[ x(t)=\frac{1}{1-t} ]
becomes unbounded as (t) approaches (1). A local solution extends to a maximal interval of existence, whose finite endpoint can occur when the solution leaves every compact subset of the equation’s domain.
A global Lipschitz condition in the state variable, combined with suitable growth control, excludes finite-time escape and produces a solution on the full time interval under consideration. This global conclusion is an extension of the local theorem rather than part of its basic statement.
Autonomous systems and flows
For an autonomous system
[ x'(t)=f(x(t)), ]
local uniqueness gives a well-defined trajectory through each initial state. Where the solutions remain defined, these trajectories determine a local flow
[ \Phi_t(x_0)=x(t;x_0). ]
Uniqueness implies the composition law
[ \Phi_{t+s}(x_0)
\Phi_t\bigl(\Phi_s(x_0)\bigr) ]
whenever both sides are defined. Additional smoothness of (f) leads to corresponding differentiability of the flow with respect to its initial state, connecting the theorem with the theory of dynamical systems.