Existence and uniqueness theorem — Picard-Lindelöf (statement): Foundations

This page is the toolbox. Before you read the Picard–Lindelöf statement, make sure every symbol below feels obvious. We build each one from nothing, in an order where each rests only on the ones before it.

1. The variables $x$ and $y$ , and the function $y (x)$

Why the topic needs it: the ODE asks us to find such a curve $y (x)$ . Everything else is machinery for pinning down which curve.

2. The derivative $y^{'} (x)$ — the slope of the curve

Picture: at each point of the curve, lay a tiny straight ruler tangent to it. The tilt of that ruler is $y^{'}$ there.

Why the topic needs it: an ODE is a rule for the slope. It tells you, at every point, which way the curve must tilt.

3. The slope rule $f (x, y)$

Picture — a slope field. At many points $(x, y)$ of the plane, draw a short dash tilted by the value $f (x, y)$ . A solution curve is any curve that stays tangent to these dashes as it threads through them.

Why the topic needs it: $f$ is the given data of the problem. The whole theorem is a promise about $f$ : "if $f$ is well-behaved, a unique thread exists."

4. The starting point $(x_{0}, y_{0})$ and the IVP

Picture: a pin stuck at $(x_{0}, y_{0})$ ; the solution curve must pass through that pin and follow the dashes from there.

Why the topic needs it: without the pin there is no "the" solution to be unique. Picard–Lindelöf is about the one thread through one pin.

5. Absolute value $∣ \cdot ∣$ — measuring distance

Picture: a ruler laid between two dots; $∣ a - b ∣$ is the gap you read off, and it can't be negative.

Why the topic needs it: the theorem constantly says things like " $y$ stays within $b$ of $y_{0}$ " — written $∣ y - y_{0} ∣ \leq b$ — and " $f$ can't jump too far", written $∣ f (x, y_{1}) - f (x, y_{2}) ∣$ . Absolute value is our tape measure.

6. Continuity of $f$ on $R$ , the rectangle $R$ , and the bound $M$

Picture: a rectangle drawn around the pin; the number $M$ is a speed limit — no dash in the box tilts more than a line of slope $M$ or $- M$ .

7. The safe step-size $h = min (a, b / M)$

Picture: starting at the pin, the curve can climb or fall at rate no more than $M$ (section 6). So after a horizontal run of $∣ x - x_{0} ∣$ it has moved vertically at most $M ∣ x - x_{0} ∣$ . To keep inside the box's height $b$ we need $M ∣ x - x_{0} ∣ \leq b$ , i.e. $∣ x - x_{0} ∣ \leq b / M$ ; and we can't run past the box's width $a$ either. The tighter limit wins, which is why $h$ is the minimum of $a$ and $b / M$ .

Why the topic needs it: this is the interval on which everything lives — the solution $y (x)$ , and the machine $T$ below, all act on $x$ ranging over $[x_{0} - h, x_{0} + h]$ . The theorem is local: it promises an answer only this close to the pin. See Maximal interval of existence for how far you can push beyond it.

8. The Lipschitz condition — the star of the show

Picture: freeze $x$ , walk straight up along the $y$ -direction, and watch $f$ . Lipschitz says the graph of $f$ -versus- $y$ is trapped between two straight lines of slopes $+ L$ and $- L$ — it can wobble, but never steeper than $L$ .

Why the topic needs it: this single tameness controls both that the Picard machine settles down (existence) and that it can't settle to two different answers (uniqueness). It is the difference between the full theorem and the weaker Peano existence theorem.

9. The integral $\int_{x_{0}}^{x}$ — undoing a slope

Picture: shade the region under the $g$ -curve from $x_{0}$ to $x$ ; the integral is that shaded area (below-axis counts as negative).

Why the topic needs it: differentiation is fragile, but adding up slopes is sturdy. Rewriting the ODE by integrating turns it into $y (x) = y_{0} + \int_{x_{0}}^{x} f (t, y (t)) d t,$ which says "start at $y_{0}$ , then accumulate all the little slopes." This integral form is the workbench for Picard iteration (method of successive approximations).

10. The operator $T$ and a fixed point

Picture: a machine with a curve going in the top and a (usually different) curve coming out the bottom. When input and output are identical, you've found the solution.

Why the topic needs it: solving the IVP $=$ finding a fixed point of $T$ . The whole proof is: repeatedly apply $T$ (Picard iteration) and show the outputs home in on the single unchanged curve, via the Banach fixed-point theorem (contraction mapping).

11. Sequences of curves and how they settle down

Picture: a stack of curves, each closer to the next, all pressing down onto one final curve like sheets settling flat — and settling everywhere at once, not just here and there.

Intuition The two tools that force the settling — named and sketched

Two estimates prove $∥ y_{n} - y ∥_{\infty} \to 0$ :

The factorial bound. One can show each step's change is tiny: $∣ y_{n + 1} (x) - y_{n} (x) ∣ \leq M \frac{L ^{n} ∣ x - x _{0} ∣ ^{n + 1}}{( n + 1 )!} .$ The factorial $(n + 1)!$ in the denominator crushes the terms so fast that their total is finite — that finiteness is exactly what makes the curves settle.
The Bielecki weighted norm. Instead of the plain supremum norm, weight it by a decaying exponential: $∥ u ∥_{B} = max_{∣ x - x_{0} ∣ \leq h} e^{- 2 L ∣ x - x_{0} ∣} ∣ u (x) ∣.$ Measuring gaps with this norm makes $T$ a genuine contraction on the whole interval $[x_{0} - h, x_{0} + h]$ , no matter how big $L h$ is.

Both are ways of proving the same thing — the iterates converge. See Grönwall's inequality for the twin tool that nails down uniqueness.

Prerequisite map

Equipment checklist

Cover the right side and answer aloud.

What does

y^{'} (x)

measure at a point?

The steepness (slope) of the curve

y (x)

there — its rise-over-run for an infinitely small run.

What does the ODE

y^{'} = f (x, y)

tell you?

At every point

(x, y)

the curve's slope must equal the value the rule

f

returns.

What extra piece turns an ODE into an IVP?

An initial condition

y (x_{0}) = y_{0}

— a pin fixing the curve at one point.

What does

∣ y - y_{0} ∣ \leq b

mean geometrically?

The height

y

stays within distance

b

y_{0}

— inside the box

R

What does it mean for

f

to be continuous on

R

Nudging

(x, y)

a tiny amount in either direction moves

f (x, y)

only a tiny amount — no jumps anywhere in the box.

Where does the number

M

come from?

f

is continuous on the closed box, so by the Extreme Value Theorem it attains a maximum size;

M

bounds

∣ f ∣

, the steepest allowed slope.

What is

h

and why is it a minimum?

The half-width of the safe interval

[x_{0} - h, x_{0} + h]

;

h = min (a, b / M)

because both the box's width

a

and the slope-limit

b / M

must hold, so the tighter wins.

State the Lipschitz condition in words.

The change in

f

-values is at most

L

times the change in

y

∣ f (x, y_{1}) - f (x, y_{2}) ∣ \leq L ∣ y_{1} - y_{2} ∣

Is Lipschitz required in

x

or in

y

y

— the comparison happens at a fixed

x

Why rewrite the ODE as

y = y_{0} + \int_{x_{0}}^{x} f (t, y) d t

Integration is robust; it turns the slope rule into a fixed-point problem for an operator

T

acting on curves over

[x_{0} - h, x_{0} + h]

What is a fixed point of

T

A curve that

T

returns unchanged,

T [y] = y

— which is exactly a solution of the IVP.

What does

∥ u - v ∥_{\infty}

measure?

The largest vertical gap between the curves

u

and

v

anywhere on the interval.

What does "the Picard iterates converge uniformly" mean?

The worst gap

∥ y_{n} - y ∥_{\infty}

shrinks to

0

— the whole curve settles at once, not just at scattered points.