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

4.6.8 · D1 · Maths › Ordinary Differential Equations › Existence and uniqueness theorem — Picard-Lindelöf (statemen

Yeh page ek toolbox hai. Picard–Lindelöf statement padhne se pehle, ensure karo ki neeche diya har symbol obvious lagta ho. Hum har ek ko kuch se bhi nahi banate, ek aisi order mein jahan har ek sirf unhi par tikta hai jo pehle aaye hain.

1. Variables $x$ aur $y$ , aur function $y (x)$

Yeh topic ko kyun chahiye: ODE humse aisi curve $y (x)$ dhundhwata hai. Baaki sab machinery hai yeh pin karne ke liye ki kaun si curve.

2. Derivative $y^{'} (x)$ — curve ka slope

Picture: curve ke har point par ek tiny seedha ruler tangent rakho. Us ruler ka tilt hi $y^{'}$ hai wahan.

Yeh topic ko kyun chahiye: ek ODE ek slope ka rule hai. Yeh tumhe batata hai, har point par, curve ko kis taraf tilt karna chahiye.

3. Slope rule $f (x, y)$

Picture — slope field. Plane ke bahut saare points $(x, y)$ par, $f (x, y)$ ki value se tilted ek chhoti dash kheencho. Solution curve koi bhi aisi curve hai jo in dashes ke saath tangent rehti hai jab woh unse guzarti hai.

Yeh topic ko kyun chahiye: $f$ problem ka given data hai. Saara theorem $f$ ke baare mein ek promise hai: "agar $f$ well-behaved hai, toh ek unique thread exist karta hai."

4. Starting point $(x_{0}, y_{0})$ aur IVP

Picture: $(x_{0}, y_{0})$ par gada ek pin; solution curve us pin se guzarni chahiye aur wahan se dashes follow karni chahiye.

Yeh topic ko kyun chahiye: pin ke bina "the" solution hota hi nahi jो unique ho. Picard–Lindelöf ek pin se guzarne wale ek thread ke baare mein hai.

5. Absolute value $∣ \cdot ∣$ — distance measure karna

Picture: do dots ke beech rakha ek ruler; $∣ a - b ∣$ woh gap hai jo tum padhte ho, aur yeh negative nahi ho sakta.

Yeh topic ko kyun chahiye: theorem hamesha aisi cheezein kehta hai jaise " $y$ , $y_{0}$ se $b$ ke andar rehta hai" — likha jaata hai $∣ y - y_{0} ∣ \leq b$ — aur " $f$ bahut zyada jump nahi kar sakta", likha jaata hai $∣ f (x, y_{1}) - f (x, y_{2}) ∣$ . Absolute value hamara tape measure hai.

6. $f$ ki continuity on $R$ , rectangle $R$ , aur bound $M$

Picture: pin ke around kheencha ek rectangle; number $M$ ek speed limit hai — box mein koi bhi dash slope $M$ ya $- M$ se zyada nahi tilt karta.

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

Picture: pin se shuru karke, curve at most $M$ ki rate se climb ya fall kar sakti hai (section 6). Isliye $∣ x - x_{0} ∣$ ki horizontal run ke baad woh vertically at most $M ∣ x - x_{0} ∣$ move hui hogi. Box ki height $b$ ke andar rehne ke liye humein $M ∣ x - x_{0} ∣ \leq b$ chahiye, yaani $∣ x - x_{0} ∣ \leq b / M$ ; aur hum box ki width $a$ se bhi zyada nahi ja sakte. Tighter limit jeet jaati hai, isliye $h$ , $a$ aur $b / M$ ka minimum hai.

Yeh topic ko kyun chahiye: yeh woh interval hai jis par sab kuch rehta hai — solution $y (x)$ , aur neeche wali machine $T$ , sab $x$ par act karte hain jo $[x_{0} - h, x_{0} + h]$ par range karta hai. Theorem local hai: yeh sirf pin ke is qadar paas answer ka promise karta hai. Isse aage kitna push kar sakte ho, uske liye Maximal interval of existence dekho.

8. Lipschitz condition — show ka star

Picture: $x$ freeze karo, $y$ -direction mein seedha upar chalo, aur $f$ dekho. Lipschitz kehta hai $f$ -versus- $y$ ka graph slopes $+ L$ aur $- L$ ki do seedhi lines ke beech trapped hai — woh wobble kar sakta hai, lekin kabhi $L$ se zyada steep nahi.

Yeh topic ko kyun chahiye: yeh single tameness dono control karti hai — ki Picard machine settle hoti hai (existence) aur ki woh do alag answers par settle nahi ho sakti (uniqueness). Yeh full theorem aur weaker Peano existence theorem ke beech ka fark hai.

9. Integral $\int_{x_{0}}^{x}$ — slope ko undo karna

Picture: $g$ -curve ke neeche $x_{0}$ se $x$ tak region shade karo; integral woh shaded area hai (axis ke neeche negative count hota hai).

Yeh topic ko kyun chahiye: differentiation fragile hai, lekin slopes add karna sturdy hai. ODE ko integrate karke rewrite karna ise badal deta hai $y (x) = y_{0} + \int_{x_{0}}^{x} f (t, y (t)) d t,$ jo kehta hai " $y_{0}$ se shuru karo, phir saare chhote slopes accumulate karo." Yeh integral form Picard iteration (method of successive approximations) ka workbench hai.

10. Operator $T$ aur ek fixed point

Picture: ek machine jisme upar se ek curve jaati hai aur neeche se (usually alag) curve nikalaती hai. Jab input aur output identical hoon, tum solution dhundh chuke ho.

Yeh topic ko kyun chahiye: IVP solve karna $=$ $T$ ka fixed point dhundhna. Poora proof yeh hai: $T$ baar baar apply karo (Picard iteration) aur dikhao ki outputs single unchanged curve par home in karte hain, Banach fixed-point theorem (contraction mapping) ke zariye.

11. Curves ki sequences aur woh kaise settle hoti hain

Picture: curves ka ek stack, har ek agle ke karib, sab ek final curve par press ho rahe hain jaise sheets flat settle ho rahi hoon — aur har jagah ek saath settle ho rahi hoon, sirf yahan ya wahan nahi.

Intuition Woh do tools jo settling force karte hain — named aur sketched

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

Factorial bound. Ek dikhaya ja sakta hai ki har step ka change tiny hota hai: $∣ y_{n + 1} (x) - y_{n} (x) ∣ \leq M \frac{L ^{n} ∣ x - x _{0} ∣ ^{n + 1}}{( n + 1 )!} .$ Denominator mein $(n + 1)!$ terms ko itni tezi se crush karta hai ki unka total finite hota hai — wahi finiteness exactly woh hai jo curves ko settle karvati hai.
Bielecki weighted norm. Plain supremum norm ki jagah, ise ek decaying exponential se weight karo: $∥ u ∥_{B} = max_{∣ x - x_{0} ∣ \leq h} e^{- 2 L ∣ x - x_{0} ∣} ∣ u (x) ∣.$ Is norm se gaps measure karna $T$ ko poore interval $[x_{0} - h, x_{0} + h]$ par ek genuine contraction bana deta hai, chahe $L h$ kitna bhi bada ho.

Dono ek hi cheez prove karne ke tarike hain — iterates converge karte hain. Twin tool jo uniqueness nail karta hai uske liye Grönwall's inequality dekho.

Prerequisite map

Equipment checklist

Right side cover karo aur zor se jawab do.

y^{'} (x)

ek point par kya measure karta hai?

Curve

y (x)

ki steepness (slope) wahan — ek infinitely small run ke liye uski rise-over-run.

ODE

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

tumhe kya batata hai?

Har point

(x, y)

par curve ka slope rule

f

jo value return karta hai uske barabar hona chahiye.

Kaun sa extra piece ek ODE ko IVP mein badalta hai?

Ek initial condition

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

— ek pin jo curve ko ek point par fix karta hai.

∣ y - y_{0} ∣ \leq b

geometrically kya matlab hai?

Height

y

y_{0}

b

distance ke andar rehti hai — box

R

ke andar.

R

par

f

continuous hone ka kya matlab hai?

(x, y)

ko kisi bhi direction mein thoda nudge karna

f (x, y)

ko sirf thoda move karta hai — box mein koi bhi jumps nahi.

Number

M

kahan se aata hai?

f

closed box par continuous hai, isliye Extreme Value Theorem se woh maximum size attain karta hai;

M

bound karta hai

∣ f ∣

ko, jo steepest allowed slope hai.

h

kya hai aur woh minimum kyun hai?

Safe interval

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

ki half-width;

h = min (a, b / M)

kyunki box ki width

a

aur slope-limit

b / M

dono hold karni chahiye, isliye tighter wali jeet jaati hai.

Lipschitz condition ko words mein state karo.

f

-values mein change at most

L

times

y

mein change hai:

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

Kya Lipschitz

x

mein required hai ya

y

mein?

y

mein — comparison ek fixed

x

par hota hai.

ODE ko

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

ke roop mein kyun rewrite karte hain?

Integration robust hai; yeh slope rule ko curves par act karne wale operator

T

ke liye ek fixed-point problem mein badal deta hai

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

par.

T

ka fixed point kya hai?

Ek curve jo

T

unchanged return karta hai,

T [y] = y

— jo exactly IVP ka solution hai.

∥ u - v ∥_{\infty}

kya measure karta hai?

Interval par kahin bhi curves

u

aur

v

ke beech sabse bada vertical gap.

"Picard iterates uniformly converge karte hain" ka kya matlab hai?

Worst gap

∥ y_{n} - y ∥_{\infty}

0

ki taraf shrink hota hai — poori curve ek saath settle hoti hai, sirf scattered points par nahi.