Picard–Lindelöf theorem

Picard–Lindelöf theorem Differential equations Navier–Stokes differential equations used to simulate airflow around an obstruction Scope show Fields Classification show Types show Relation to processes Solution show Existence and uniqueness show General topics show Solution methods People show List vte 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.

Contents 1 Theorem 2 Proof sketch 3 Example of Picard iteration 4 Example of non-uniqueness 5 Detailed proof 6 Optimization of the solution's interval 7 Other existence theorems 8 See also 9 Notes 10 References 11 External links Theorem Let {displaystyle Dsubseteq mathbb {R} times mathbb {R} ^{n}} be a closed rectangle with {displaystyle (t_{0},y_{0})in D} . Let {displaystyle f:Dto mathbb {R} ^{n}} be a function that is continuous in {displaystyle t} and Lipschitz continuous in {displaystyle y} . Then, there exists some ε > 0 such that the initial value problem {displaystyle y'(t)=f(t,y(t)),qquad y(t_{0})=y_{0}.} has a unique solution {displaystyle y(t)} on the interval {displaystyle [t_{0}-varepsilon ,t_{0}+varepsilon ]} .[1][2] Note that {displaystyle D} is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within {displaystyle D} .

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 {displaystyle 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 {displaystyle varphi _{0}(t)=y_{0}} and {displaystyle 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" 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).

See Newton's method of successive approximation for instruction.

Example of Picard iteration Let {displaystyle y(t)=tan(t),} the solution to the equation {displaystyle y'(t)=1+y(t)^{2}} with initial condition {displaystyle y(t_{0})=y_{0}=0,t_{0}=0.} Starting with {displaystyle varphi _{0}(t)=0,} we iterate {displaystyle varphi _{k+1}(t)=int _{0}^{t}(1+(varphi _{k}(s))^{2}),ds} so that {displaystyle varphi _{n}(t)to y(t)} : {displaystyle varphi _{1}(t)=int _{0}^{t}(1+0^{2}),ds=t} {displaystyle varphi _{2}(t)=int _{0}^{t}(1+s^{2}),ds=t+{frac {t^{3}}{3}}} {displaystyle varphi _{3}(t)=int _{0}^{t}left(1+left(s+{frac {s^{3}}{3}}right)^{2}right),ds=t+{frac {t^{3}}{3}}+{frac {2t^{5}}{15}}+{frac {t^{7}}{63}}} and so on. Evidently, the functions are computing the Taylor series expansion of our known solution {displaystyle y=tan(t).} Since {displaystyle tan } has poles at {displaystyle pm {tfrac {pi }{2}},} this converges toward a local solution only for {displaystyle |t|<{tfrac {pi }{2}},} not on all of {displaystyle mathbb {R} } . Example of non-uniqueness To understand uniqueness of solutions, consider the following examples.[3] A differential equation can possess a stationary point. For example, for the equation dy / dt = ay ( {displaystyle a<0} ), 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 solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. However, for an equation in which the stationary solution is reached after a finite time, the uniqueness fails. This happens for example for the equation dy / dt = ay  2 / 3 , which has at least two solutions corresponding to the initial condition y(0) = 0 such as: y(t) = 0 or {displaystyle y(t)={begin{cases}left({tfrac {at}{3}}right)^{3}&t<0\ 0&tgeq 0,end{cases}}} so the previous state of the system is not uniquely determined by its state after t = 0. The uniqueness theorem does not apply because the function  f (y) = y  2 / 3 has an infinite slope at y = 0 and therefore is not Lipschitz continuous, violating the hypothesis of the theorem. Detailed proof Let {displaystyle C_{a,b}={overline {I_{a}(t_{0})}}times {overline {B_{b}(y_{0})}}} where: {displaystyle {begin{aligned}{overline {I_{a}(t_{0})}}&=[t_{0}-a,t_{0}+a]\{overline {B_{b}(y_{0})}}&=[y_{0}-b,y_{0}+b].end{aligned}}} This is the compact cylinder where  f  is defined. Let {displaystyle M=sup _{C_{a,b}}|f|,} this is, the supremum of (the absolute values of) the slopes of the function. Finally, let L be the Lipschitz constant of  f  with respect to the second variable. We will proceed to apply the Banach fixed-point theorem using the metric on {displaystyle {mathcal {C}}(I_{a}(t_{0}),B_{b}(y_{0}))} induced by the uniform norm {displaystyle |varphi |_{infty }=sup _{tin I_{a}}|varphi (t)|.} We define an operator between two function spaces of continuous functions, Picard's operator, as follows: {displaystyle Gamma :{mathcal {C}}(I_{a}(t_{0}),B_{b}(y_{0}))longrightarrow {mathcal {C}}(I_{a}(t_{0}),B_{b}(y_{0}))} defined by: {displaystyle Gamma varphi (t)=y_{0}+int _{t_{0}}^{t}f(s,varphi (s)),ds.} 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 {displaystyle a} , {displaystyle Gamma } takes {displaystyle {overline {B_{b}(y_{0})}}} into itself in the space of continuous functions with the uniform norm. Here, {displaystyle {overline {B_{b}(y_{0})}}} is a closed ball in the space of continuous (and bounded) functions "centered" at the constant function {displaystyle y_{0}} . Hence we need to show that {displaystyle |varphi -y_{0}|_{infty }leq b} implies {displaystyle left|Gamma varphi (t)-y_{0}right|=left|int _{t_{0}}^{t}f(s,varphi (s)),dsright|leq int _{t_{0}}^{t'}left|f(s,varphi (s))right|dsleq int _{t_{0}}^{t'}M,ds=Mleft|t'-t_{0}right|leq Maleq b} where {displaystyle t'} is some number in {displaystyle [t_{0}-a,t_{0}+a]} where the maximum is achieved. The last inequality in the chain is true if we impose the requirement {displaystyle a<{frac {b}{M}}} . Now let's prove that this operator is a contraction mapping. Given two functions {displaystyle 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 require {displaystyle left|Gamma varphi _{1}-Gamma varphi _{2}right|_{infty }leq qleft|varphi _{1}-varphi _{2}right|_{infty },} for some {displaystyle 0leq q<1} . So let {displaystyle t} be such that {displaystyle |Gamma varphi _{1}-Gamma varphi _{2}|_{infty }=left|left(Gamma varphi _{1}-Gamma varphi _{2}right)(t)right|.} Then using the definition of {displaystyle Gamma } , {displaystyle {begin{aligned}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)dsright|\&leq int _{t_{0}}^{t}left|fleft(s,varphi _{1}(s)right)-fleft(s,varphi _{2}(s)right)right|ds\&leq Lint _{t_{0}}^{t}left|varphi _{1}(s)-varphi _{2}(s)right|ds&&{text{since }}f{text{ is Lipschitz-continuous}}\&leq Lint _{t_{0}}^{t}left|varphi _{1}-varphi _{2}right|_{infty },ds\&leq Laleft|varphi _{1}-varphi _{2}right|_{infty }end{aligned}}} This is a contraction if {displaystyle a<{tfrac {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 {displaystyle varphi in {mathcal {C}}(I_{a}(t_{0}),B_{b}(y_{0}))} 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 {displaystyle a

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