Kenneth Kuttler

24.1. MOUNTAIN PASS THEOREM IN HILBERT SPACE 815

∥∥∥∥ x− y0

∥x− y0∥R− ∥y− y0∥∥y− y0∥

(y− y0)

∥∥∥∥≤

∥∥∥∥ x− y0

∥x− y0∥R− (y− y0)

∥y− y0∥R∥∥∥∥+

∥∥∥∥ R∥y− y0∥

(y− y0)−∥y− y0∥∥y− y0∥

(y− y0)

∥∥∥∥= A+B

NowB = (R−∥y− y0∥)< ∥x− y0∥−∥y− y0∥ ≤ ∥y− x∥

A≤∥∥∥∥ x− y0

∥x− y0∥R− (y− y0)

∥y− y0∥R∥∥∥∥≤ R

∥(x− y0)∥y− y0∥− (y− y0)∥x− y0∥∥∥x− y0∥∥y− y0∥

≤ R∥x− y0∥∥y− y0∥

(∥(x− y0)∥y− y0∥− (y− y0)∥y− y0∥∥+∥(y− y0)∥y− y0∥− (y− y0)∥x− y0∥∥

)

≤ R∥x− y0∥∥y− y0∥

(∥y− y0∥∥x− y∥+∥y− y0∥∥y− x∥)

≤ R∥x− y0∥

(∥x− y∥+∥y− x∥)< 2∥y− x∥

In case y = y0, you have

∥γ (x)− γ (y)∥=∥∥∥∥ x− y0

∥x− y0∥R∥∥∥∥= ∥∥∥∥ x− y

∥x− y0∥R∥∥∥∥< ∥x− y∥

The only other case is where both x,y are in X \B. In this case, you get

∥γ (x)− γ (y)∥ =

∥∥∥∥y0 +x− y0

∥x− y0∥R−

(y0 +

y− y0

∥y− y0∥R)∥∥∥∥

∥∥∥∥ x− y0

∥x− y0∥R− y− y0

∥y− y0∥R∥∥∥∥≤ 2∥x− y∥

by the same reasoning used above to estimate A.Alternate Proof of Theorem 24.1.8: Let B be a closed ball of radius R centered at

y0 such that f has Lipschitz constant K on B. Let γ be as in Lemma 24.1.9. Considerg(x)≡ f (γ (x)) . Then

∥g(x)−g(y)∥= ∥ f (γ (x))− f (γ (y))∥ ≤ K ∥γ (x)− γ (y)∥ ≤ 3K ∥x− y∥ .

Now consider for y ∈C ([0,T ] ,X)

Fy(t)≡ y0 +∫ t

0g(y(s))ds

Then∥Fy(t)−Fz(t)∥ ≤

∫ t

0K ∥y(s)− z(s)∥ds

24.1. MOUNTAIN PASS THEOREM IN HILBERT SPACE 815X— Yo R— lly — yoll ( ~y |l|x — yol| lly — yol|X— yo (y—yo) RlIx —yoll ~ y= yol|lly — yol|+ (y-yo) — (y—yo)|] =A+B| lly —yoll lly —yollNowB= (R—|ly—yoll) < lx —yoll — lly —yoll < lly — >|as| X— Yo R— (y—yo) a) <rle —yo) ly —yoll — (y—yo) lle — yollllll — yol| lly — voll lx — yoll \ly — yol|< R ( I|(«— yo) lly — yoll — (v— yo) lly — yoll | )~ |lx—yoll ly —yoll \ +1 —yo) lly —yoll — (vy — yo) lla — yoll||“ (|| Il lx — yll + | I ly — >|)7. MILEY VOI! WA Y—Yoll |y — xIe soll soll(llx—yl] + lly — al) < 2 [ly —-l]lx—yoll rlIn case y = yo, you havex—yIx — yol|rr) — 70) =|=ak—yola < liesThe only other case is where both x,y are in X \ B. In this case, you get— Yo ¥— YOwt Een Bont)Dol ly—yolx= p_ YoYoIx—yoll lly —yollIy@)-rO)I| =R| <2»by the same reasoning used above to estimate A. IfAlternate Proof of Theorem 24.1.8: Let B be a closed ball of radius R centered atyo such that f has Lipschitz constant K on B. Let y be as in Lemma 24.1.9. Considerg(x) =f (y())- ThenIIs (~) —s (I= IF) -FVO))I SK ll7@) — YO) I] S 3K |lx—yI].Now consider for y € C((0,7],X)=r | eo(o))aIFv@) Fe@ll < [ Klb() <6} asThen