Notes on Existence and Uniqueness of IVPs∗

September 7, 2011

Theorem 1 (Picard’s existence theorem, also known as Picard–Lindelöf theorem) Consider theIVP with n variables

y′(t) = f(y(t), t), y(t0) = y0, t ∈ [t0 − σ, t0 + σ].

Suppose f is Lipschitz continuous in y and continuous in t. Then, for some value ϵ > 0, thereexists a unique solution y(t) to the IVP within the range [t0−ϵ, t0 + ϵ].

We will not go through the detailed proof here. See Appendix A of E&P. The following is a sketchof the proof for first-order DE.

First, we prove the existence by construction. Integrating both sides of

y(t)− y(t0) =

ˆ t

t0

f(y(s), s)ds.

Construct a sequence of successive approximations. Set

x0(t) = y0

xk+1(t) = y0 +

ˆ t

t0

f(xk(s), s)ds

If f is Lipschitz continuous in y, it can be shown that xk(x) converges as k → ∞, and the limit isthe solution to the IVP. See below for more detail of the convergence proof.

To show uniqueness, assume there are two solutions y1 and y2, but it can be shown (usingGronwall’s lemma) that ∥y1(t) − y2(t)∥2 ≤ ∥y1(t0) − y2(t0)∥2e2k(t−a), where k is the constant inLipschitz continuity and a is a lower bound of t. Given the initial condition y1(t0) = y2(t0) = y(t0),∥y1(t)− y2(t)∥ = 0. See Appendix A of E&P for more detail.

Example of successive approximation

dx

dt= Ax, x(0) = b.

Then

x0(t) = b. xn+1(t) = b+

ˆ t

0

Axn(s)ds.

∗Based on Appendix A of textbook “Differential Equations and Boundary Value Problems: Computing andModeling”, 4th edition, by C. Henry Edwards and David E. Penney, Prentice Hall 2008.

1

Thus,

x1(t) = b+

ˆ t

0

Abds = b+Abt = (I +At)b;

x2(t) = b+

ˆ t

0

A (b+Abs) ds = b+Abt+1

2A2bt2 = (I +At+

1

2A2t2)b

x3(t) = b+

ˆ t

0

A

(b+Abs+

1

2A2s2

)ds

= (I +At+1

2A2t2 +

1

6A3t3)b.

It can be shown that

xN (t) =

(N∑

n=0

(At)n

n!

)b,

and

x(t) = limN→∞

xN (t) =

( ∞∑n=0

(At)n

n!

)b = eAtb.

More detail on the convergence of xk(x) Basic idea is telescoping.

x(t) = x0 +∞∑

n=1

(xn(t)− xn−1(t)) .

Let M denote the maximum value of ∥f∥ over R× [0, T ].

∥x1(t)− x0(t)∥ =

∥∥∥∥∥∥tˆ

0

f(x0(s), s)ds

∥∥∥∥∥∥ ≤

∥∥∥∥∥∥tˆ

0

f(b, s)ds

∥∥∥∥∥∥ ≤ Mt

∥x2(t)− x1(t)∥ =

∥∥∥∥∥∥tˆ

0

(f(x1(s), s)− f(x0(s), s)) ds

∥∥∥∥∥∥ ≤ k

tˆ

0

∥x1(s)− x0(s)∥ ds

≤ k

tˆ

0

Msds ≤ Mk

2t2

. . .

∥xn(t)− xn−1(t)∥ ≤ M

k

(kt)n

n!.

Then ∥∥∥∥∥x0 +∞∑

n=1

(xn(t)− xn−1(t))

∥∥∥∥∥ ≤ ∥x0∥+M

k

( ∞∑n=1

(kt)n

n!

)= ∥x0∥+

M

k

(ekt − 1

).

2

Remarks of continuity requirement

• Nonlinearity. The theorem applies to first-order systems of ODEs, either linear or nonlinear.

• The theorems above affirm local existence of ODEs. Global existence is assured between inter-vals of discontinuities. Global existence and uniqueness of nonlinear ODEs is more difficult,

• Higher-order ODEs. A higher-order ODE can be converted into systems of first-order systemsof ODEs. In particularly, in

y(n) = F(x, y(x), y′(x), . . . , y(n−1)(x)

)y(k)(x0) = ak, k = 0, . . . , n− 1,

if F is Lipschitz continuous w.r.t. y(k) over an interval containing ak (or more stronglycontinuously differentiable), then the IVP has a unique solution.

• The theorem is only a sufficient condition.

– Example, y′ = 1/√x is not Lipschitz continuous at x = 0, but the IVP with y(0) = 1

has a unique solution y = 2√x+ 1.

• In fact, existence only requires f(x,y(x)) to be continuous (Peano existence theorem). Pub-lished twice by Italian mathematician Giuseppe Peano.

3