Picard-Lindelöf and First-Order Ordinary Differential Equations

Let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open, let $F:D\to {\mathbb{R}}^n$, and let $n\ge 1$. The equation

$x^{\mathrm{\prime }}(t)=F(t,x(t))$

is a first-order system. An initial value problem consists of this equation and data $(t_0,x_0)\in D$, written $x(t_0)=x_0$.

A solution on an interval $I$ is a function $x:I\to {\mathbb{R}}^n$ such that $t_0\in I$, $(t,x(t))\in D$ for every $t\in I$, $x$ is differentiable on $I$ in the domain-relative sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, and

$x(t_0)=x_0,x^{\mathrm{\prime }}(t)=F(t,x(t))(t\in I)\mathrm{.}$

Endpoints of $I$ use the corresponding one-sided, domain-relative derivative. A local solution is one whose interval contains a nondegenerate neighborhood of $t_0$ relative to the time projection of $D$.

Let $n\in \mathbb{N}$ with $n\ge 1$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open, and let $F:D\to {\mathbb{R}}^n$. The map $F$ is locally Lipschitz in the state variable, locally uniformly in time, if every compact time-state cylinder $C\subseteq D$ has a finite $L\ge 0$ such that

$\mathrm{\parallel }F(t,x)-F(t,y){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$

whenever $(t,x),(t,y)\in C$ have the same time coordinate.

On every compact time-state cylinder the state-variable inequality holds with one finite constant $L$. More generally, let $p\ge 1$, let $\mathrm{\Lambda }\subseteq {\mathbb{R}}^p$, let $D_{\mathrm{\Lambda }}\subseteq \mathbb{R}\times {\mathbb{R}}^n\times \mathrm{\Lambda }$ be open in its relative Euclidean topology, and let $F:D_{\mathrm{\Lambda }}\to {\mathbb{R}}^n$. The parameter family is locally Lipschitz in the state variable, locally uniform in time and parameters when every compact time-state-parameter cylinder $C\subseteq D_{\mathrm{\Lambda }}$ has one finite $L\ge 0$ such that

$\mathrm{\parallel }F(t,x,\lambda )-F(t,y,\lambda ){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$

whenever $(t,x,\lambda ),(t,y,\lambda )\in C$ have the same time and parameter coordinates.

Let $n\ge 1$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open, let $F:D\to {\mathbb{R}}^n$ be continuous, let $(t_0,x_0)\in D$, let $I\subseteq \mathbb{R}$ be order-convex with at least two elements, and let $x:I\to {\mathbb{R}}^n$ be continuous with $t_0\in I$ and $(t,x(t))\in D$. A curve solves the IVP if and only if it satisfies the associated Volterra integral equation. Explicitly, the equation is

$x(t)=x_0+{\int }_{t_0}^{t}F(s,x(s))ds(t\in I)\mathrm{.}$

The integral is oriented, so the assertion applies on either side of $t_0$.

Fix $n\in \mathbb{N}$ with $n\ge 1$, $h\ge 0$, $J=[t_0-h,t_0+h]$, an initial state $x_0\in {\mathbb{R}}^n$, and a radius $r>0$. Let

${\mathcal{B}}_r:=\{x\in C(J,{\mathbb{R}}^n):{\sup }_{t\in J}\mathrm{\parallel }x(t)-x_0{\mathrm{\parallel }}_2\le r\}\mathrm{.}$

Let $D_F\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open and let $F:D_F\to {\mathbb{R}}^n$ be continuous. The domain of the Picard operator on ${\mathcal{B}}_r$ consists of those $x\in {\mathcal{B}}_r$ whose whole graph $\{(s,x(s)):s\in J\}$ lies in $D_F$. For such a curve, continuity makes $s\mapsto F(s,x(s))$ integrable on every interval with endpoints $t_0,t\in J$, and its Picard image is

$(\mathcal{T}x)(t):=x_0+{\int }_{t_0}^{t}F(s,x(s))ds\mathrm{.}$

The operator maps into $C(J,{\mathbb{R}}^n)$; it is a self-map of ${\mathcal{B}}_r$ only when its image is known to remain in that ball. Starting from $x^{(0)}(t):=x_0$, define $x^{(m+1)}:=\mathcal{T}{x}^{(m)}$ inductively whenever $x^{(m)}$ is defined and lies in the operator domain; if that condition first fails, no later iterate is defined. When a separate invariant-ball result makes $\mathcal{T}:{\mathcal{B}}_r\to {\mathcal{B}}_r$ a total self-map, The recursion theorem supplies the entire sequence. Every fixed point satisfies the corresponding Volterra equation. When $h>0$, it is exactly a solution of the IVP by A first-order initial value problem is equivalent to its Volterra integral equation; for $h=0$ the fixed-point equation remains defined, but no derivative on the isolated one-point domain is asserted.

Let $J$ be a nonempty compact interval and $n\ge 1$. Continuous ${\mathbb{R}}^n$-valued curves on $J$ are complete in the supremum metric

$d_{\mathrm{\infty }}(x,y):={\sup }_{t\in J}\mathrm{\parallel }x(t)-y(t){\mathrm{\parallel }}_2\mathrm{.}$

Let $C=[t_0-h,t_0+h]\times \bar{B}(x_0,r)$ be contained in the domain of a continuous vector field $F$, with $h\ge 0$ and $r>0$. If $\mathrm{\parallel }F(t,x){\mathrm{\parallel }}_2\le M$ on $C$ and $hM\le r$, then the Picard operator maps the closed curve ball ${\mathcal{B}}_r$ into itself.

On the invariant curve ball of A bounded vector field makes the Picard operator preserve a sufficiently short closed curve ball, suppose $F$ has state-Lipschitz constant $L\ge 0$. Then

$d_{\mathrm{\infty }}(\mathcal{T}x,\mathcal{T}y)\le Lh{d}_{\mathrm{\infty }}(x,y)\mathrm{.}$

In particular, if $Lh<1$, the Picard operator is a contraction.

Let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open, let $F:D\to {\mathbb{R}}^n$ be continuous and locally Lipschitz in the state variable, and let $(t_0,x_0)\in D$. Then a unique local solution of the IVP exists on some interval around the initial time. More quantitatively, if $h,r>0$ make

$[t_0-h,t_0+h]\times \bar{B}(x_0,r)\subseteq D,$

if $\mathrm{\parallel }F{\mathrm{\parallel }}_2\le M$ and $F$ has state-Lipschitz constant $L$ on this cylinder, and if $hM\le r$ and $Lh<1$, then there is exactly one solution on $[t_0-h,t_0+h]$ through $(t_0,x_0)$ whose graph lies in the cylinder. Any two solutions through the same initial data agree on every common subinterval containing $t_0$.

In particular, a unique local solution exists on an interval around the initial time.

Under the hypotheses of Picard-Lindelof at $(t_0,x_0)$, there are $h,r,\eta >0$ such that every initial value $(s,y)$ with $\mathrm{\mid }s-t_0\mathrm{\mid }<\eta$ and $\mathrm{\parallel }y-x_0{\mathrm{\parallel }}_2<\eta$ has a unique solution on $[s-h,s+h]$, and all these solution graphs lie in one compact time-state cylinder.

Let $(X,d)$ be a nonempty complete metric space and let $T:X\to X$. Suppose nonnegative reals $(a_m{)}_{m\ge 1}$ satisfy ${\sum }_{m\ge 1}{a}_m<\mathrm{\infty }$ and

$d(T^{m}x,T^{m}y)\le a_{m}d(x,y)(x,y\in X,m\ge 1)\mathrm{.}$

Summably contracting iterates have a unique fixed point and the iteration tail bounds its error. Explicitly, for $x_{k+1}=T{x}_k$ and the fixed point $x^{\ast }$,

$d(x_n,x^{\ast })\le d(x_1,x_0){\sum }_{m\ge n}{a}_m(n\ge 1)\mathrm{.}$

Let $\mathcal{T}$ be a Picard operator on an invariant curve ball over a time interval of half-length $h$, and let $L$ be a state-Lipschitz constant. Starting at $x^{(0)}(t)=x_0$, the iterates converge uniformly to the unique fixed point. If $q=Lh<1$, the Banach a priori and a posteriori bounds hold. Without $Lh<1$,

$d_{\mathrm{\infty }}({\mathcal{T}}^{m}x,{\mathcal{T}}^{m}y)\le \frac{(Lh{)}^m}{m!}{d}_{\mathrm{\infty }}(x,y),$

and the corresponding factorial-series tail bounds the error.

Let $t_0\le T$, let $u,a,b:[t_0,T]\to \mathbb{R}$ be continuous, with $u,b\ge 0$, and suppose

$u(t)\le a(t)+{\int }_{t_0}^{t}b(s)u(s)ds\mathrm{.}$

Then

$u(t)\le a(t)+{\int }_{t_0}^{t}a(s)b(s)\exp \text{\hspace{0.17em}⁣}({\int }_s^{t}b(r)dr)ds\mathrm{.}$

If $a$ is nondecreasing, this gives $u(t)\le a(t)\exp ({\int }_{t_0}^{t}b)$. The time-reflected form assumes $u(t)\le a(t)+{\int }_t^{t_0}b(s)u(s)ds$ for $t\le t_0$. In particular, when $a=A\ge 0$ and $b=B\ge 0$ are constant, the two orientations give $u(t)\le A{e}^{B\mathrm{\mid }t-t_0\mathrm{\mid }}$.

Let $F:D\to {\mathbb{R}}^n$ be continuous on an open ODE domain, let $I\subseteq \mathbb{R}$ be an order-convex interval with at least two elements, and let $x,y:I\to {\mathbb{R}}^n$ solve $z^{\mathrm{\prime }}=F(t,z)$ with both graphs in $D$. Fix $t_0\in I$, and suppose that for every $t\in I$,

$\mathrm{\parallel }F(t,x(t))-F(t,y(t)){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x(t)-y(t){\mathrm{\parallel }}_2\mathrm{.}$

Then

$\mathrm{\parallel }x(t)-y(t){\mathrm{\parallel }}_2\le e^{L\mathrm{\mid }t-t_0\mathrm{\mid }}\mathrm{\parallel }x(t_0)-y(t_0){\mathrm{\parallel }}_2\mathrm{.}$

Coincident initial values give uniqueness on every common interval.

Let $F(t,x,\lambda )$ be continuous on an open time-state-parameter domain and locally state-Lipschitz with one constant on compact cylinders. Near fixed data $(t_0,x_0,{\lambda }_0)$, the solutions supplied by Picard-Lindelof exist on one common compact time interval and depend jointly and uniformly continuously there on the initial time, initial state, and parameter. Quantitatively, if the common time interval has length at most $H$, $\mathrm{\parallel }F{\mathrm{\parallel }}_2\le M$, and $F$ has state-Lipschitz constant $L$, then solutions through $(t_0,x_0)$ and $(s,y_0)$ satisfy

$\mathrm{\parallel }x(t)-y(t){\mathrm{\parallel }}_2\le e^{LH}(\mathrm{\parallel }{x}_0-y_0{\mathrm{\parallel }}_2+M\mathrm{\mid }s-t_0\mathrm{\mid }+H\omega (\mathrm{\parallel }\lambda -\mu {\mathrm{\parallel }}_2)),$

where $\omega (r)\to 0$ is a uniform modulus for the parameter dependence of $F$ on that cylinder.

Let $x:I\to {\mathbb{R}}^n$ and $y:J\to {\mathbb{R}}^n$ solve the same IVP. The solution $y$ is an extension of $x$ when $I\subseteq J$ and $y{\mathrm{\mid }}_I=x$. It is a proper extension when $I⊊J$.

A solution is maximal when it has no proper extension. Its domain is its maximal interval of existence. A solution is global when its domain is the entire time projection allowed by the ODE domain; for a vector field on $\mathbb{R}\times {\mathbb{R}}^n$, this means domain $\mathbb{R}$.

Locally unique solutions through the same data agree on every overlap. If two such solutions are defined on adjacent intervals, agree at their common endpoint, and solve the same continuous ODE there, their piecewise union is a solution on the union interval.

Every initial value problem satisfying the Picard-Lindelof hypotheses has a unique maximal solution. Its domain is an open interval containing the initial time, and every other solution through the same data is its restriction.

Let $x:(\alpha ,\beta )\to {\mathbb{R}}^n$ solve a Picard-Lindelof ODE and suppose $\beta <\mathrm{\infty }$. If there are $t_j\uparrow \beta$ for which $(t_j,x(t_j))$ lies in one compact subset $K$ of the open ODE domain, then $x$ extends to a solution beyond $\beta$. A solution whose graph has a sequence approaching a compact interior endpoint state extends past that endpoint. The reflected statement holds at a finite left endpoint.

Let $x:(\alpha ,\beta )\to {\mathbb{R}}^n$ be the maximal solution of an IVP whose continuous vector field is locally Lipschitz in the state variable on its open ODE domain. If the positive maximal endpoint is finite, the solution must eventually leave every compact set $K$ contained in that domain: some $s<\beta$ satisfies $(t,x(t))\mathrm{\notin }K$ for every $s<t<\beta$. The analogous assertion holds as $t\downarrow \alpha$ when $\alpha$ is finite.

At a finite maximal endpoint the solution leaves every compact subset of the ODE domain. If the positive maximal endpoint is finite, the solution must eventually leave every compact set.

Let $F:\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ be continuous and suppose one $L\ge 0$ satisfies $\mathrm{\parallel }F(t,x)-F(t,y){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$ for all $t,x,y$. Then every IVP for $x^{\mathrm{\prime }}=F(t,x)$ has a unique global solution.

Let $I\subseteq \mathbb{R}$ be order-convex with at least two elements, let $x_0\in I$, and let $p,q:I\to \mathbb{R}$ be continuous. The IVP $y^{\mathrm{\prime }}+py=q$, $y(x_0)=y_0$, has exactly one solution on $I$, namely

$y(x)=\exp (-P(x))(y_0+{\int }_{x_0}^x\exp (P(t))q(t)dt),P(x)={\int }_{x_0}^{x}p(t)dt\mathrm{.}$