(Luận văn thạc sĩ) các điều kiện tối ưu cấp hai với hiện tượng envelope like cho các bài toán tối ưu vectơ không trơn trong các không gian vô hạn chiều

i
BỘ GIÁO DỤC VÀ ĐÀO TẠO

TRƯỜNG ĐẠI HỌC KINH TẾ TP.HCM
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ĐỀ TÀI NGHIÊN CỨU KHOA HỌC CẤP TRƯỜNG
CÁC ĐIỀU KIỆN TỐI ƯU CẤP HAI VỚI HIỆN TƯỢNG
ENVELOPE-LIKE CHO CÁC BÀI TỐN TỐI ƯU VECTƠ
KHƠNG TRƠN TRONG CÁC KHƠNG GIAN VƠ HẠN
CHIỀU

Mã số: CS – 2014 - 43
Chủ nhiệm: TS. Nguyễn Đình Tuấn

Tp. Hồ Chí Minh - 2014


ệ ệ
ữỡ
ữỡ ợ t t ❝ù✉ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ô♥❣ ♥❤÷ ♠ët
sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✺
❈❤÷ì♥❣ ✷✿ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✾
❈❤÷ì♥❣ ✸✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✶✸
❈❤÷ì♥❣ ✹✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tố ữ ừ
t ữợ ự ♠ð rë♥❣ ✤➲ t➔✐✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✷
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✸✸

✶

✷


ữỡ
ỵ ồ t
tè✐ ÷✉ ❝➜♣ ❤❛✐ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ ✈➻ ♥â ❧➔♠ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐
÷✉ ❝➜♣ ♠ët ❤♦➔♥ t❤✐➺♥ ❤ì♥ ❜➡♥❣ ♥❤ú♥❣ t❤ỉ♥❣ t✐♥ ❝➜♣ ❤❛✐ ❣✐ó♣ ➼❝❤ r➜t ♥❤✐➲✉ tr♦♥❣ ✈✐➺❝
♥❤➟♥ r❛ ❝→❝ ♥❣❤✐➺♠ tè✐ ÷✉ ❝ơ♥❣ ♥❤÷ ✤÷❛ r❛ ❝→❝ t❤✉➟t t♦→♥ sè ✤➸ t➼♥❤ ❝→❝ ♥❣❤✐➺♠ ♥➔②✳
❇↔♥ ❝❤➜t ❝õ❛ t❤æ♥❣ t✐♥ ❝➜♣ ❤❛✐ ♥➔② ❧➔ ♥❤÷ s❛✉✳ ◆â✐ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥✱ ❝→❝ ✤✐➲✉ ❦✐➺♥
tè✐ ÷✉ ❝➜♣ ♠ët ❦❤➥♥❣ ✤à♥❤ r➡♥❣ t↕✐ ✤✐➸♠ ❝ü❝ trà✱ t ữợ ừ ủ
ử t✐➯✉ ✈➔ ❝→❝ r➔♥❣ ❜✉ë❝✱ ❦❤æ♥❣ t❤✉ë❝ ✈➲ ♣❤➛♥ tr♦♥❣ ❝õ❛ ♥â♥ ✭❤đ♣✮ ➙♠ tr♦♥❣
t➼❝❤ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ↔♥❤✳ ✣↕♦ t ữợ õ t tr ừ ♥â♥ ♥â✐
tr➯♥✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❝✉♥❣ ❝➜♣ t❤æ♥❣ t✐♥ t❤➯♠✿ ♥â✐
❝❤✉♥❣✱ ✤↕♦ t ữợ ừ r ổ ➙♠✳
❚✉② ♥❤✐➯♥✱ ✈➔♦ ♥➠♠ ✶✾✽✽✱ ❑❛✇❛s❛❦✐ ❬✶✹❪ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ✤➣ ♣❤→t ❤✐➺♥ r❛ r➡♥❣ ❦❤✐
t❛ ①➨t ❜❛♦ ✤â♥❣ ❝õ❛ ♥â♥ ➙♠ ♥â✐ tr➯♥✱ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ r õ t
t ữợ ♠ët ❝õ❛ →♥❤ ①↕ ❤ñ♣ ♥â✐ tr➯♥ ♥➡♠ tr➯♥ ♣❤➛♥ ✤➦❝ ❜✐➺t ❝õ❛
❜✐➯♥ ❝õ❛ ♥â♥ ➙♠✳ ➷♥❣ ❣å✐ ❤✐➺♥ t÷đ♥❣ ♥➔② ❧➔ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳ ◆❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥
❝ù✉ ✈➝♥ ❦❤ỉ♥❣ ú ỵ tữủ s ❧➛♠ ✤→♥❣ t✐➳❝✳ ◆❤✐➲✉ t→❝ ❣✐↔
❦❤→❝ ❝❤➾ ①➨t ♥â♥ ➙♠ ♥â✐ tr➯♥✱ ❦❤æ♥❣ ①➨t ❜❛♦ ✤â♥❣ ❝õ❛ ♥â♥ ♥➔②✱ ✈➔ ✈➻ t❤➳ ❦❤ỉ♥❣ ❝â ❤✐➺♥
t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛✳ ✣➣ ❝â ♥❤✐➲✉ ✤â♥❣ ❣â♣ q✉❛♥ trå♥❣ ❝❤♦ ❤✐➺♥ t÷đ♥❣ t❤ó ✈à ♥➔②✳
❈→❝ ❦➳t q✉↔ ❝õ❛ ❑❛✇❛s❛❦✐ ✤÷đ❝ ♠ð rë♥❣ ✈➔ ♣❤→t tr t q ổ
ữợ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✸✱ ✺✱ ✷✹✱ ✷✺❪✱ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ ❦❤↔ ✈✐ ❝➜♣ ❤❛✐ tr♦♥❣
❬✶✵✱ ✶✶❪✱ q✉② ❤♦↕❝❤ ✤❛ ♠ö❝ t✐➯✉ ✭❤ú✉ ❤↕♥ ❝❤✐➲✉✮ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ ♠ët tr♦♥❣ ❬✼❪ ✈➔ ❝❤♦
q✉② ❤♦↕❝❤ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❝❤➦t tr♦♥❣ ❬✷✵✱ ✷✶❪✳ ❈❤ó♥❣ tỉ✐
♥❤➟♥ t❤➜② r➡♥❣ ❝➛♥ ♣❤↔✐ ❣✐↔✐ t❤➼❝❤ rã r➔♥❣ ❤ì♥ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛
✈➔ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ♥➔② ❦❤ỉ♥❣ ①↔② r❛✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ú tổ s ró ỡ ố ợ
ỳ ữợ r❛ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✳
❍ì♥ ♥ú❛✱ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ✈ỵ✐ ♠ù❝ ✤ë ❦❤ỉ♥❣ trì♥ ❝➜♣ ❝❛♦ ❤ì♥ ❧✉ỉ♥ ❧✉ỉ♥ ❧➔ ♠ët
♥❤✉ ❝➛✉ t❤ü❝ t➳✳ ❉♦ ✤â✱ tr♦♥❣ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ❝→❝ ①➜♣ ①➾ ✤➣ ✤÷đ❝

✤➲ ①✉➜t tr♦♥❣ ❬✶✱ ✶✸❪ ❞ị♥❣ ❧➔♠ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐ ❞ị♥❣
❝→❝ ữủ ự tr ợ t❤✐➳t r➡♥❣ t➜t ❝↔ ❝→❝ ①➜♣ ①➾ ✤÷đ❝ sû ❞ư♥❣
❧➔ ❝♦♠♣❛❝t✳ ❈→❝ ①➜♣ ①➾ ❝â t❤➸ ❦❤æ♥❣ ❜à ❝❤➦♥ ✤➣ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ ❦✐➺♥
tè✐ ÷✉ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ tr♦♥❣ ❬✶✺✱ ✶✼✲✶✾❪ ❝❤♦ ♥❤✐➲✉ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤→❝ ♥❤❛✉✳
✣↕♦ ❤➔♠ s✉② rë♥❣ t❤✉ë❝ ❧♦↕✐ ♥➔② t✐➺♥ ❧ñ✐ ð ❝❤ê ❧➔ ♥❣❛② ❝↔ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐
♠ët ✤✐➸♠ ❝â t❤➸ ❝â ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ t↕✐ ✤✐➸♠ ♥➔②✳ ❚✉② ♥❤✐➯♥✱ ✤➸ t➟♣
tr✉♥❣ tr➯♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➜♣ ❤❛✐ ró ữợ ữủ ①➾ ❣➙②
r❛ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡✱ ❝❤ó♥❣ tỉ✐ ❝❤õ ②➳✉ ①➨t ❝→❝ →♥❤ ①↕ ❦❤↔ ✈✐ ❝➜♣ ♠ët✳
❈→❝ q✉❛♥ s→t tr➯♥ ỗ ự ử ự ừ ❝❤ó♥❣ tỉ✐ tr♦♥❣
✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ →♣ ❞ư♥❣ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ✤➸ t❤✐➳t ❧➟♣
tố ữ ợ ợ tữủ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣
trì♥ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ ❈→❝ →♥❤ ①↕ tr♦♥❣ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❦❤↔
✈✐ ❝❤➦t ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ✮ ✈➔
❦❤ỉ♥❣ ❝➛♥ ❦❤↔ ✈✐ ❧✐➯♥ tư❝✳ ❈→❝ ❦➳t q✉↔ ♥➔② ❝↔✐ t❤✐➺♥ ✈➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉
✸


❣➛♥ ✤➙②✳

✷✳ ▼ư❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✳
❈❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ s❛✉ ✤➙②✳
❈❤♦ X, Z, W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ C Y õ ỗ
õ K Z t ỗ f : X Y g : X → Z ✱ ✈➔ h : X W
t ữợ sỹ ①➨t ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔
✭P✮

♠✐♥C f (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K, h(x) = 0✳

❈❤ó♥❣ tỉ✐ ❞ị♥❣ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ t❤❡♦ ♥❣❤➽❛ ①➜♣ ①➾ ✈ỵ✐ ♠ù❝ ✤ë ❦❤ỉ♥❣ trỡ
ữợ tt t tr ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥✮ ❤❛② ❦❤↔ ✈✐ ✭tr♦♥❣ ❝→❝ ✤✐➲✉

❦✐➺♥ tè✐ ÷✉ ✤õ✮✱ tr→♥❤ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❧✐➯♥ tư❝✱ ✤➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➜♣ ❤❛✐
✈ỵ✐ t➼♥❤ ❝❤➜t ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❜➔✐ t♦→♥ ✭P✮✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔♠ rã ❤ì♥ ✈➜♥
✤➲ ❦❤✐ ♥➔♦ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ①↔② r❛ ✈➔ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤
✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✳ ❈ö t❤➸✱ ✤➲ t➔✐ t❤ü❝ ❤✐➺♥ ❝→❝ ♠ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ✤➙②✳
✰ ❑❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐✳
✰ ❑❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛
❝❤ó♥❣✳
✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐ ✈ỵ✐ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ②➳✉
✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳
✰ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ✭P✮✳
❈→❝ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝
✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì ❦❤ỉ♥❣ trì♥ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥
❝❤✐➲✉✳ ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ✤➣ ✤÷đ❝ t→❝ ❣✐↔ ✈➔ ●❙✳❚❙❑❍✳ P❤❛♥ ◗✉è❝ ❑❤→♥❤✱
tr÷í♥❣ ✣↕✐ ❤å❝ ◗✉è❝ t➳✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❚♣✳ ❍❈▼ ❝æ♥❣ ❜è tr♦♥❣ ♠ët ❜➔✐ ❜→♦ tr➯♥
t↕♣ ❝❤➼ ❦❤♦❛ ❤å❝ q✉è❝ t➳ tr♦♥❣ ❤➺ t❤è♥❣ ■❙■ ❬✷✷❪✿
P✳◗✳ ❑❤❛♥❤ ❛♥❞ ◆✳❉✳ ❚✉❛♥✱ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✇✐t❤ ❡♥✈❡❧♦♣❡✲❧✐❦❡
❡❢❢❡❝t ❢♦r ♥♦♥s♠♦♦t❤ ✈❡❝t♦r ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳
✭✷✵✶✸✮ ✶✸✵✲✶✹✽✳

✼✼

✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳

✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❞ị♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ✈➔ ❦ÿ t❤✉➟t tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ trì♥✱ ❣✐↔✐ t➼❝❤
✤❛ trà ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✳

✹✳ ❑➳t ❝➜✉ ❝õ❛ ✤➲ t➔✐✳

✣➲ t ỗ ữỡ

ã ữỡ ỵ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✱ ♠ö❝ t✐➯✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ự ừ t
ã ữỡ ợ t t ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣
♥❤÷ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ởt
ã ữỡ s rở ởt
ã ữỡ tố ữ
ã ữỡ ✹✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ✤õ ❝➜♣ ❤❛✐✳

✹


ữỡ ợ t t ự ởt sè
❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝ơ♥❣ ♥❤÷ ♠ët sè ❦❤→✐ ♥✐➯♠
✈➲ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐
❈❤♦ X, Z, W ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ổ C Y õ
ỗ õ K Z t ỗ f : X → Y ✱ g : X → Z ✱ ✈➔ h : X → W ❧➔ ❝→❝ →♥❤
①↕✳ ❈❤ó♥❣ tỉ✐ ①➨t ❜➔✐ t♦→♥ tè✐ ÷✉ ✈❡❝tì s❛✉ ✤➙②✿
✭P✮

♠✐♥C f (x)✱ s❛♦ ❝❤♦ g(x) ∈ −K, h(x) = 0✳

❈❤ó♥❣ tỉ✐ ũ ỵ ỡ N = {1, 2, ..., n, ...} ✈➔ R ❧➔ t➟♣ ❤ñ♣ ❝→❝ sè
t❤ü❝✳ ❱ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X ✱ X ∗ ❧➔ ✤è✐ ♥❣➝✉ t♦♣♦ ❝õ❛ ♦❢ X ❀ ., . ❧➔ t➼❝❤ ✤è✐
♥❣➝✉✳ . ❧➔ ❝❤✉➞♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❜➜t ❦ý ✈➔ d(y, S) ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø
✤✐➸♠ y ✤➳♥ t➟♣ S ✳ Bn (x, r) = {y ∈ Rn : x − y < r}❀ Sn = {y ∈ Rn : y = 1}❀
BX (x, r) = {y ∈ X : x − y < r}✱ SX = {y ∈ X : y = 1} ✈➔ ✤è✐ ✈ỵ✐ BX (0, 1) t❛
✈✐➳t ✤ì♥ ❣✐↔♥ ❧➔ BX ✳ L(X, Y ) ỵ ổ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø
X ✈➔♦ Y ✈➔ B(X, X, Y ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕ s♦♥❣ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø X × X
✈➔♦ Y ✱ tr♦♥❣ ✤â X ✈➔ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❱ỵ✐ Pn ✱ P tr♦♥❣ L(X, Y )✱ t❛ ✈✐➳t
♣
Pn −

→ P ❤❛② P = ♣✲lim Pn ♥➳✉ Pn ❤ë✐ tử P ỵ tữỡ tỹ ữủ ❞ị♥❣ ❝❤♦
Mn , M ∈ B(X, X, Y )✳ ❱ỵ✐ õ C X ỵ C = {c∗ ∈ X ∗ : c∗ , c ≥ 0, ∀c ∈ C} ❧➔
♥â♥ ✤è✐ ❝ü❝ ❞÷ì♥❣ ❝õ❛ C ✳ ợ A X ỵ rA tA ❝❧A✱ ❜❞A✱ ❝♦♥❡A✱ ❝♦A ✈➔
A(x) ❧➛♥ ❧÷đt ❧➔ ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ✤è✐✱ ♣❤➛♥ tr♦♥❣✱ ❜❛♦ ✤â♥❣✱ ❜✐➯♥✱ ❜❛♦ ♥â♥✱ ❜❛♦ ỗ ừ
A õ ừ A + x✳ ❱ỵ✐ t > 0 ✈➔ r ∈ N✱ o(tr ) ỵ ừ ởt
ử tở t s❛♦ ❝❤♦ o(tr )/tr → 0 ❦❤✐ t → 0+ ✳ C 1,1 ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ →♥❤ ①↕
❦❤↔ ✈✐ ❋r➨❝❤❡t s❛♦ ❝❤♦ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✳
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ ①➨t X ✱ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ h : X → Y ❧➔ →♥❤ ①↕✳
❚❛ ♥â✐ h ❧➔ ê♥ ✤à♥❤ t↕✐ x0 tỗ t ởt U ừ x0 κ > 0 s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐
x ∈ U✱

h(x) − h(x0 ) ≤ κ x − x0 ✳
h ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ∈ X ♥➳✉ ♥â ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t h (x0 ) t↕✐ x0 ✈➔
limy→x0 ,y →x0

h(y) − h(y ) − h (x0 )(y − y )
= 0✳
y−y

❍✐➸♥ ♥❤✐➯♥ r➡♥❣ ♥➳✉ h ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ t❤➻ h ❧➔ ▲✐♣s❝❤✐t③ ❣➛♥ x0 ✳
❑➳t q✉↔ s❛✉ ✤➙② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü ♥❤÷ ❇ê ✤➲ ✸ ❝õ❛ ❬✼❪✳

▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ h ❧➔ →♥❤ ①↕ ❦❤↔ ✈✐ ❋r➨❝❤❡t q✉❛♥❤ x

∈ X ✈ỵ✐ h ❧➔ ê♥ ✤à♥❤ t↕✐ x0 ✱
✈➔ u, w ∈ X ✳ ◆➳✉ (tn , rn ) → (0 , 0 )✱ tn /rn → 0 ✱ ✈➔ wn := (xn − x0 − tn u)/ 12 tn rn → w✱
t❤➻
h(xn ) − h(x0 ) − tn h (x0 )u
yn :=
→ h (x0 )w✳

tn rn /2
+

+

0

+

❚❛ ♥❤ỵ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ♥â♥ t✐➳♣ ①ó❝ ✈➔ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ s❛✉ ✤➙②✳
✺


✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❈❤♦ x , u ∈ X ✈➔ S ⊂ X ✳
0

✭❛✮ ◆â♥ ❝♦♥t✐♥❣❡♥t ✭❤❛② ❇♦✉❧✐❣❛♥❞✮ ❝õ❛ S t↕✐ x0 ❧➔

T (S, x0 ) = {v ∈ X | ∃tn → 0+ , ∃vn → v, ∀n ∈ N, x0 + tn vn ∈ S}✳
✭❜✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ✭♥â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❈❧❛r❦❡✱ t÷ì♥❣ ù♥❣✮ ❝õ❛ S t↕✐ x0 ❧➔

IT (S, x0 ) = {v ∈ X | ∀tn → 0+ , ∀vn → v, ∀n ✤õ ❧ỵ♥, x0 + tn vn ∈ S}
✭ITC (S, x0 ) = {v ∈ X | ∀xn →S x0 , ∀tn → 0+ , ∀vn → v, ∀n ✤õ ❧ỵ♥, xn + tn vn ∈ S}✮✳
✭❝✮ ❚➟♣ ❝♦♥t✐♥❣❡♥t ✭t➟♣ ❦➲✱ t÷ì♥❣ ù♥❣✮ ❝➜♣ ❤❛✐ ❝õ❛ S t↕✐ (x0 , u) ❧➔

T 2 (S, x0 , u) = {w ∈ X | ∃tn → 0+ , ∃wn → w, ∀n ∈ N, x0 + tn u + 21 t2n wn ∈ S}
✭A2 (S, x0 , u) = {w ∈ X | ∀tn → 0+ , ∃wn → w, ∀n ∈ N, x0 + tn u + 21 t2n wn ∈ S}✮✳
✭❞✮ ◆â♥ t✐➳♣ ①ó❝ ✭♥â♥ ❦➲✱ t÷ì♥❣ ù♥❣✮ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ S t↕✐ (x0 , u) ❧➔

T (S, x0 , u) = {w ∈ X | ∃(tn , rn ) → (0+ , 0+ ) :

tn
rn

→ 0, ∃wn → w✱

∀n ∈ N, x0 + tn u + 12 tn rn wn ∈ S}
✭A (S, x0 , u) = {w ∈ X | ∀(tn , rn ) → (0+ , 0+ ) :

tn
rn

→ 0, ∃wn → w,

∀n ∈ N, x0 + tn u + 12 tn rn wn ∈ S}✮✳
✭❡✮ ❚➟♣ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ ❝õ❛ S t↕✐ (x0 , u) ❧➔

IT 2 (S, x0 , u) = {w ∈ X | ∀tn → 0+ , ∀wn → w, ∀n ✤õ ❧ỵ♥,
x0 + tn u + 21 t2n wn ∈ S}✳
✭❢✮ ◆â♥ t✐➳♣ ①ó❝ tr♦♥❣ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ❝õ❛ S t↕✐ (x0 , u) ❧➔
IT (S, x0 , u) = {w ∈ X | ∀(tn , rn ) → (0+ , 0+ ) :

tn
rn

→ 0, ∀wn → w,

∀n ✤õ ❧ỵ♥, x0 + tn u + 12 tn rn wn ∈ S}✳
❈→❝ ♥â♥ T (S, x0 )✱ IT (S, x0 ) ✈➔ ITC (S, x0 ) ✈➔ ❝→❝ t➟♣ T 2 (S, x0 , u)✱ A2 (S, x0 , u) ✈➔ IT 2 (S, x0 , u)
✤÷đ❝ ❜✐➳t rã✳ ❈→❝ ♥â♥ A (S, x0 , u) ✈➔ T (S, x0 , u) ✤÷đ❝ P❡♥♦t ❬✷✺✱ ✷✻❪ sû ❞ư♥❣✳ ❈❤ó♥❣

tỉ✐ ✤à♥❤ ♥❣❤➽❛ ♥â♥ IT (S, x0 , u) ♠ët ❝→❝❤ tü ữ ỵ r x0 clS t t➜t ❝↔
❝→❝ t➟♣ t✐➳♣ ①ó❝ ð tr➯♥ ❧➔ ré♥❣✳ ❱➻ t❤➳✱ ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ①➨t ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝❤➾ t↕✐ ♥❤ú♥❣
✤✐➸♠ t❤✉ë❝ ❜❛♦ ✤â♥❣ ❝õ❛ t➟♣ ✤❛♥❣ ①➨t✳
❈❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ t➟♣ t✐➳♣ ①ó❝ ❝➜♣ ♠ët ✈➔ ❝➜♣ ❤❛✐ ð
tr➯♥ tr♦♥❣ ❜❛ ♠➺♥❤ ✤➲ s❛✉ ✤➙②✳

▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ S ⊂ X ✈➔ x , u ∈ X ✳ ❑❤✐ ✤â✱ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤÷đ❝ ❜✐➳t rã
0

✭✐✮ IT (S, x0 , u) ⊂ A (S, x0 , u) ⊂ T 2 (S, x0 , u) ⊂ clcone[cone(S − x0 ) − u]❀
2

2

✭✐✐✮ IT 2 (S, x0 , u) = IT 2 (intS, x0 , u) ✈➔ ♥➳✉ u ∈ bd[cone(S −x0 )]✱ t❤➻ 0 ∈ IT 2 (S, x0 , u)❀
✭✐✐✐✮ ♥➳✉ u ∈ T (S, x0 )✱ t❤➻ T 2 (S, x0 , u) = ∅✳

●✐↔ sû✱ t❤➯♠ ♥ú❛✱ S ỗ tS = u T (S, x0 )✳ ❚❛ ❝â ✤✐➲✉ s❛✉ ✭❬✶✶✱ ✷✸✱ ✷✾❪✮✿
✭✐✈✮ ✐♥t❝♦♥❡(S − x0 ) = IT (intS, x0 ) = ITC (intS, x0 ✮ ✈➔ ❞♦ ✤â 0 ∈ intcone(S − x0 ) ✈ỵ✐
x0 ∈ intS ❀
✭✈✮ ♥➳✉ A2 (S, x0 , u) = ∅✱ t❤➻
✻


IT 2 (S, x0 , u) = intA2 (S, x0 , u), clIT 2 (S, x0 , u) = A2 (S, x0 , u);
✭✈✐✮ ♥➳✉ u ∈ cone(S − x0 )✱ t❤➻
✭❛✮ IT 2 (S, x0 , u) = intcone[cone(S − x0 ) − u]❀
✭❜✮ A2 (S, x0 , u) = clcone[cone(S − x0 ) − u]✳

▼➺♥❤ ✤➲ ✶✳✹✳ ❈❤♦ S ⊂ X ✈➔ x , u ∈ X ✳

0

✭✐✮ IT (S, x0 , u) ⊂ A (S, x0 , u) ⊂ T (S, x0 , u) ⊂ clcone[cone(S − x0 ) − u]✳
✭✐✐✮ IT (S, x0 , u) = IT (intS, x0 , u) ✈➔ ♥➳✉ u ∈ bd[cone(S−x0 )]✱ t❤➻ 0 ∈ IT (S, x0 , u)✳

✭✐✐✐✮ ◆➳✉ u ∈ T (S, x0 )✱ t❤➻ T (S, x0 , u) = ∅✳
✭✐✈✮ A (S, x0 , u) + ITC (S, x0 ) ⊂ IT (S, x0 , u)✱

✈➔ ❞♦ ✤â✱ ♥➳✉ ITC (S, x0 ) = ∅ ✈➔ A (S, x0 , u) = ∅✱ t❤➻

IT (S, x0 , u) = intA (S, x0 , u), clIT (S, x0 , u) = A (S, x0 , u).
✭✈✮ ◆➳✉ S ỗ x0 S t

A (S, x0 , u) + T (T (S, x0 ), u) ⊂ A (S, x0 , u) ⊂ T (T (S, x0 ), u)
✈➔ ❞♦ ✤â✱ ♥➳✉ A (S, x0 , u) = ∅✱ t❤➻ A (S, x0 , u) = T (T (S, x0 ), u)✳

❈❤ù♥❣ ♠✐♥❤✳ ❈→❝ ♣❤➛♥ ✭✐✮✲✭✐✐✐✮ ữủ s r tứ ợ ①❡♠

❇ê ✤➲ ✹✳✶ ❝õ❛ ❬✷✽❪✳ ●✐í ✤➙②✱ t❛ ①➨t ♣❤➛♥ ✭✐✈✮✳ ❈❤♦ w ∈ A (S, x0 , u)✱ v ∈ ITC (S, x0 ) ✈➔
z := w + v ✳ ❈❤♦ (tn , rn ) → (0+ , 0+ )✿ tn /rn → 0✱ ✈➔ zn → z ✳ õ tỗ t wn w s
xn := x0 + tn u + 21 tn rn wn ∈ S ✳ ❱➻ vn := zn − wn → v ✱ t❛ ❝â z ∈ IT (S, x0 , u) ✈➻✱ ✈ỵ✐
n ❧ỵ♥✱

x0 + tn u + 21 tn rn zn = xn + 12 tn rn vn ∈ S ✳

▼➺♥❤ ✤➲ ✶✳✺✳ ●✐↔ sû r➡♥❣ X = R

✈➔ x0 ∈ S ⊂ X ✳ ◆➳✉ xn ∈ S \ {x0 } ở tử x0
t tỗ t u ∈ T (S, x0 ) \ {0} ❝â ❝❤✉➞♥ ❜➡♥❣ ởt ởt ỵ xn ✱ s❛♦
❝❤♦

m

✭✐✮ ✭❝ê ✤✐➸♥✮ (xn − x0 )/tn → u✱ tr♦♥❣ ✤â tn = xn − x0 ❀
✭✐✐✮ ✭❬✶✶❪✮ ❤♦➦❝ z ∈ T 2 (S, x0 , u) ∩ u⊥ tỗ t s (xn x0 tn u)/ 12 t2n → z ❤♦➦❝ z ∈
T (S, x0 , u)u \{0} rn 0+ tỗ t s rtnn → 0+ ✈➔ (xn −x0 −tn u)/ 12 tn rn → z ✱
tr♦♥❣ ✤â u⊥ ❧➔ ♣❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ u ∈ Rm ✳

✼


✽


❈❤÷ì♥❣ ✷✿ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦✐➸✉ ①➜♣ ①➾ ❝➜♣ ♠ët ✈➔
❝➜♣ ❤❛✐
✣à♥❤ ♥❣❤➽❛ ✷✳✶ ✭❬✶✱ ✶✸❪✮✳ ❳➨t h : X → Y

❧➔ →♥❤ ①↕✳

✭✐✮ ❚➟♣ Ah (x0 ) ⊂ L(X, Y ) ✤÷đ❝ ❣å✐ ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ♥➳✉✱ ✈ỵ✐ x tr♦♥❣
♠ët ❧➙♥ ❝➟♥ ừ x0 tỗ t r 0+ s r x − x0 −1 → 0 ❦❤✐ x → x0 ✈➔✱

h(x) − h(x0 ) ∈ Ah (x0 )(x − x0 ) + rBY ✳
✭✐✐✮ ❈➦♣ (Ah (x0 ), Bh (x0 ))✱ ✈ỵ✐ Ah (x0 ) ⊂ L(X, Y ) ✈➔ Bh (x0 ) ⊂ B(X, X, Y )✱ ✤÷đ❝ ❣å✐
❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ♥➳✉ Ah (x0 ) ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ✱ ✈➔ ✈ỵ✐ x tr♦♥❣
♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 tỗ t r 0+ s r x − x0 −1 → 0 ❦❤✐ x → x0 ✈➔

h(x) − h(x0 ) ∈ Ah (x0 )(x − x0 ) + Bh (x0 )(x − x0 , x − x0 ) + r2 BY .

◆❤➟♥ ①➨t ✷✳✷✳ ✭✐✮ ◆➳✉ h : X → Y ❝â ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ h (x )✱ t❤➻ (h (x ),

0

❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ✳

0

1
h
2

(x0 ))

✭✐✐✮ ✭❬✶✱ ✶✸❪✮ ◆➳✉ h : Rn → Rm ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ x0 ✱ t❤➻ ❏❛❝♦❜✐❛♥ ❈❧❛r❦❡
❬✹❪ ∂C h(x0 ) ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ✳ ◆➳✉✱ t❤➯♠ ♥ú❛✱ h t❤✉ë❝ ❧ỵ♣ C 1,1 t↕✐ x0 ✱ t❤➻
(h (x0 ), 12 ∂C2 g(x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ✱ tr♦♥❣ ✤â ∂C2 h(x0 ) ❧➔ ❍❡ss✐❛♥ ❈❧❛r❦❡ ❬✽❪
❝õ❛ h t↕✐ x0 ✳
✭✐✐✐✮ ✭❬✶✺❪✮ ◆➳✉ h : Rn → Rm ❧➔ ❧✐➯♥ tö❝ ✈➔ ❝â →♥❤ ①↕ tü❛ ❏❛❝♦❜✐❛♥ ❬✾❪ ∂h(.) ❧➔ ♥ú❛
❧✐➯♥ tö❝ tr➯♥ t↕✐ x0 ✱ t❤➻ ❝♦∂h(x0 ) ❧➔ ①➜♣ ①➾ ❝➜♣ ♠ët ❝õ❛ h t↕✐ x0 ✳ ◆➳✉ h ❧➔ ❦❤↔ ✈✐ ❧✐➯♥
tö❝ ❋r➨❝❤❡t tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 ✈➔ ❝â →♥❤ ①↕ tü❛ ❍❡ss✐❛♥ ❬✾❪ ∂ 2 h(.) ❧➔ ♥ú❛ ❧✐➯♥ tö❝
tr➯♥ t↕✐ x0 ✱ t❤➻ (h (x0 ), 12 ❝♦∂ 2 h(x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ h t↕✐ x0 ✳
❉♦ ✤â✱ ❝→❝ ①➜♣ ①➾ ❧➔ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ r➜t tê♥❣ q✉→t✳ ❍ì♥ ♥ú❛✱ ♠é✐ →♥❤ ①↕ h ✤➲✉
❝â ♠ët ①➜♣ ①➾ t➛♠ t❤÷í♥❣ t↕✐ ❜➜t ❝ù ✤✐➸♠ ♥➔♦✱ ❧➔ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥ L(X, Y )✳ ❈→❝ ✤↕♦
❤➔♠ ❦✐➸✉ ①➜♣ ①➾ t✐➺♥ ❧đ✐ ❦❤✐ ❞ị♥❣ ❤ì♥ s♦ ✈ỵ✐ ❝→❝ ✤↕♦ ❤➔♠ s rở õ
t tỗ t ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ♥❣❛② ❝↔ ❝❤♦ →♥❤ ①↕ ❦❤ỉ♥❣ ❧✐➯♥ tư❝✳ ❱➼ ❞ư ❝❤♦ h : R → R
✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
 √
 x ♥➳✉ x > 0,
h(x) =
0
♥➳✉ x = 0,

 −1
x
♥➳✉ x < 0.
❑❤✐ ✤â h ❧➔ ❦❤æ♥❣ ❧✐➯♥ tö❝ t↕✐ ✵ ✈➔ t❛ ❝â t❤➸ ❧➜② Ah (0) = (α, +∞) ✈ỵ✐ ❜➜t ❦ý α > 0 ✈➔
Bh (0) = {0}✱ →♥❤ ①↕ ❦❤æ♥❣ tø R ✈➔♦ R✳
❚✉② ♥❤✐➯♥✱ t❛ ❦❤æ♥❣ ❝â t➼♥❤ ❞✉② ♥❤➜t ❝❤♦ ❝→❝ ①➜♣ ①➾✳ ✣➦❝ ❜✐➺t✱ ❜➜t ❦ý t➟♣ ♥➔♦ ❝❤ù❛
♠ët ①➜♣ ①➾ t❤➻ ♥â ❝ơ♥❣ ❧➔ ♠ët ①➜♣ ①➾✳
❈→❝ ✈➼ ❞ư ữợ ự tọ r s rở ❝➜♣ ♠ët ð tr➯♥ ❝â t❤➸ ❜➡♥❣
♥❤❛✉ ❤❛② ❦❤→❝ ♥❤❛✉ ❬✶✺❪✳

❱➼ ❞ö ✷✳✶✳ ❈❤♦ h : R

→ R ①→❝ ✤à♥❤ ❜ð✐
x2 sin(1/x) + |y| ♥➳✉ x = 0,
h(x, y) =
|y|
♥➳✉ x = 0.

2

❑❤✐ ✤â✱ h ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ ✭✵✱ ✵✮ ✈➔ t❛ ❝â tü❛ ❏❛❝♦❜✐❛♥
✾


∂h(0, 0) = Ah (0, 0) = {(0, β) : β ∈ {−1, 1}}✱
❝ô♥❣ ❧➔ tü❛ ❏❛❝♦❜✐❛♥ ❋r➨❝❤❡t ❬✾❪✳ ❚✉② ♥❤✐➯♥✱

❱➼ ❞ö ✷✳✷✳ ❈❤♦ h : R

∂hC (0, 0) = {(α, β) : α, β ∈ [−1, 1]}✳

→ R2 ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ h(x, y) = (|x| − |y|, |y| − |x|)✳ ❑❤✐ ✤â✱
h ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ (0, 0) ✈➔
2

1 −1
,
−1 1

∂h(0, 0) ❂

−1 1
1 −1

❧➔ tü❛ ❏❛❝♦❜✐❛♥ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ tü❛ ❏❛❝♦❜✐❛♥ ❋r➨❝❤❡t✳ ❍ì♥ ♥ú❛✱ ①➜♣ ①➾ ❝➜♣ ♠ët ❧➔
1
1
−1 −1
Ah (0, 0) = ∂h(0, 0)
,
−1 −1
1
1
❝ô♥❣ ❧➔ tü❛ ❏❛❝♦❜✐❛♥ ❋r➨❝❤❡t✳ ❚❛ ❝ô♥❣ ❝â ∂C h(0, 0) = ❝♦Ah (0, 0)✳

❱➼ ❞ö ✷✳✸✳ ❈❤♦ h : R

→ R2 ♥❤÷ s❛✉ h(x, y) = (|x|1/2 s✐❣♥(x), y 1/3 + |x|)✳ ❑❤✐ ✤â✱ h ❧➔
❧✐➯♥ tư❝ ♥❤÷♥❣ ❦❤ỉ♥❣ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣ t↕✐ (0, 0) ✈➔ ①➜♣ ①➾ ❝➜♣ ♠ët
2

α 0
β γ

Ah (0, 0) ❂

: α > 0, β = ±1, γ > 0

t❤➻ ❦❤→❝ tü❛ ❏❛❝♦❜✐❛♥ ❋r➨❝❤❡t

∂F h(0, 0) ❂

α 0
β γ

: α ≥ 0, β ∈ [−1, 1], γ ∈ R ✳

❍❛✐ ✈➼ ❞ö s❛✉ ✤➙② ❝❤ù♥❣ tä ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ ❤❛✐ ð tr➯♥ ❝ô♥❣ ①↔② r❛ t➻♥❤
❤✉è♥❣ t÷ì♥❣ tü ❬✶✺❪✳

❱➼ ❞ư ✷✳✹✳ ❈❤♦ h : R

h ∈ C 1,1

→ R ✤à♥❤ ♥❣❤➽❛ ❜ð✐ h(x, y) = 21 x2 s✐❣♥(x) + 12 y 2 s✐❣♥(y)✳ ❑❤✐ ✤â✱
t↕✐ (0, 0)✳ ❚❛ ❝â h (x, y) = (|x|, |y|) ✈➔ ❜❛ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❦❤→❝ ♥❤❛✉✿
2

∂C2 h(0, 0) ❂
1 0

,
0 1

∂ 2 h(0, 0) ❂
Bh (0, 0) ❂

1/2 0
,
0 1/2

❱➼ ❞ö ✷✳✺✳ ⑩♥❤ ①↕ h : R

α 0
0 β

: α, β ∈ [−1, 1] ✱

1 0
,
0 −1

1/2
0
,
0 −1/2

−1 0
,
0 1

−1 0
0 −1

−1/2 0
,
0
1/2

✱

−1/2
0
0
−1/2

✳

→ R ❝❤♦ ❜ð✐ h(x, y) = 32 |x|3/2 + 12 y 2 t❤✉ë❝ ợ C 1 ữ
ổ tở ợ C 1,1 õ C2 h ổ tỗ t ❝➜♣ ❤❛✐ ①→❝ ✤à♥❤ ❜ð✐
2

∂ 2 h(0, 0) ❂
Bh (0, 0) ❂

α 0
0 1
α 0
0 1/2

:α≥0 ✱

:α>0 ✳

✣à♥❤ ♥❣❤➽❛ ✷✳✸ ✭❬✶✺✱ ✶✼❪✮✳ ❚➟♣ A ⊂ L(X, Y ) (B ⊂ B(X, X, Y )) ✤÷đ❝ ❣å✐ ❧➔ ❝♦♠♣❛❝t

✤✐➸♠ t✐➺♠ ❝➟♥ ✭t❤❡♦ ❞➣②✮ ✭✈✐➳t t➢t ♣✲❝♦♠♣❛❝t✮ ♥➳✉ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② t❤ä❛✿
✭✐✮ ♠é✐ ❞➣② ❜à ❝❤➦♥ t❤❡♦ ❝❤✉➞♥ (Mn ) ⊂ A ✭⊂ B ✱ t÷ì♥❣ ù♥❣✮ ✤➲✉ ❝â ❞➣② ❝♦♥ ❤ë✐ tö
✤✐➸♠❀
✶✵


✭✐✐✮ ♥➳✉ (Mn ) ⊂ A ✭⊂ B ✱ t÷ì♥❣ ù♥❣✮ ✈ỵ✐ lim Mn = ∞✱ t❤➻ (Mn / Mn ) ❝â ❞➣② ❝♦♥
❤ë✐ tư ✤✐➸♠ ✤➳♥ ♠ët ❣✐ỵ✐ ❤↕♥ ❦❤→❝ ❦❤ỉ♥❣✳
◆➳✉ ✏❤ë✐ tư ✤✐➸♠✑ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ✤÷đ❝ t❤❛② ❜ð✐ ✏❤ë✐ tö✑✱ t❤➻ t❛ ♥â✐ r➡♥❣ A
✭❤❛② B t t t ữ ỵ r ♥➳✉ Y = R✱ t❤➻ ❤ë✐ tư ✤✐➸♠ trị♥❣
✈ỵ✐ ❤ë✐ tö s❛♦✲②➳✉✳ ❑❤→✐ ♥✐➺♠ ❝♦♠♣❛❝t t❤❡♦ ❞➣② ♥â✐ tr➯♥ ❦❤→❝ ❦❤→✐ ♥✐➺♠ ♣✲❝♦♠♣❛❝t✳
❚✉② ♥❤✐➯♥✱ tr♦♥❣ ✤➲ t➔✐ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤➾ sû ❞ö♥❣ ❦❤→✐ ♥✐➺♠ ♣✲❝♦♠♣❛❝t t❤❡♦ ❞➣② ✈➔ ❜ä
✤✐ tt ỳ t ữ ỵ r X Y ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ t❤➻ ❜➜t ❦ý t➟♣ A ❤❛②
B ♥â✐ tr➯♥ ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✳
❱ỵ✐ A ⊂ L(X, Y ) ✈➔ B ⊂ B(X, X, Y ) t❛ ũ ỵ
A = {P L(X, Y ) | ∃(Pn ) ⊂ A, P = ♣✲lim Pn }✱
♣✲❝❧B = {M ∈ B(X, X, Y ) | ∃(Mn ) ⊂ B, M = ♣✲lim Mn }✱

A∞ = {P ∈ L(X, Y ) | ∃(Pn ) ⊂ A, ∃tn → 0+ , P = lim tn Pn }✱
♣✲A∞ = {P ∈ L(X, Y ) | ∃(Pn ) ⊂ A, ∃tn → 0+ , P = ♣✲lim tn Pn }✱
♣✲B∞ = {M ∈ B(X, X, Y ) | ∃(Mn ) ⊂ B, ∃tn → 0+ , M = ♣✲lim tn Mn }✳

✶✶


✶✷

❈❤÷ì♥❣ ✸✿ ❈→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐
❈❤ó♥❣ t❛ ❤➣② ♥❤ỵ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ♥❣❤✐➺♠ tè✐ ữ ừ t P ỵ
G = g 1 (−K) ✈➔ H = h−1 (0)✳ ❑❤✐ ✤â✱ t➟♣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ✭P✮ ❧➔

S = G ∩ H = {x ∈ X | g(x) ∈ −K, h(x) = 0}.
✣✐➸♠ x0 ∈ S ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣ ữỡ tữỡ ự ừ
P tỗ t ởt ❧➙♥ ❝➟♥ U ❝õ❛ x0 s❛♦ ❝❤♦✱ ∀x ∈ U ∩ S ✱

f (x) − f (x0 ) ∈ −✐♥t C
✭f (x) − f (x0 ) ∈ (−C) \ C ✱ t÷ì♥❣ ù♥❣✮✳
❚➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ữỡ ữỡ tữỡ ự ừ P
ữủ ỵ f, S f, S tữỡ ự ợ m ∈ N✱ x0 ∈ S ✤÷đ❝ ❣å✐ ❧➔
♥❣❤✐➺♠ ❝❤➢❝ ữỡ m ữủ ỵ x0 m, f, S tỗ t
> 0 ✈➔ ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ x0 s❛♦ ❝❤♦✱ ∀x ∈ U ∩ S \ {x0 }✱

(f (x) + C) ∩ BY (f (x0 ), γ x − x0

m

) = ∅✱

❤❛②✱ t÷ì♥❣ ✤÷ì♥❣✱

d(f (x) − f (x0 ), −C) ≥ x x0

m

ữ ỵ r ợ p m
m, f, S) ⊂ ▲❋❊✭p, f, S) ⊂ ▲❊✭f, S) ⊂ ▲❲❊✭f, S ✮✳
❉♦ ✤â✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ♥❣❤✐➺♠ ②➳✉ ❝ô♥❣ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝á♥ ❧↕✐✱ ✈➔
✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ♥❣❤✐➺♠ ❝❤➢❝ ❝❤➢♥ ❝ô♥❣ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ ❝→❝ ♥❣❤✐➺♠ ❝á♥ ❧↕✐✳
✣➸ t❤✐➳t ❧➟♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐ ❝❤♦ ❜➔✐ t♦→♥ P t ũ
ữợ q tr s

♥❣❤➽❛ ✸✳✶ ✭❬✷✺❪✮✳ ❈❤♦ x , u ∈ X ✿ u = 0✱ T ⊂ Y

✈➔ h : X → Y õ r h
ữợ q tr t ữợ t (x0 , u) ố ợ T tỗ t à > 0 > 0 s
ợ ♠å✐ t ∈ (0, ρ) ✈➔ v ∈ BX (u, ρ)✱ t❛ ❝â
0

✭❉▼❙❘u ✮

d(x0 + tv, h−1 (T )) ≤ àd(h(x0 + tv), T )

õ r h ữợ q tr t x0 ố ợ T tỗ t à > 0 > 0 s
ợ ồ x ∈ BX (x0 , ρ)✱ t❛ ❝â
✭▼❙❘✮

d(x, h−1 (T )) àd(h(x), T )

ữ ỵ r ữợ q tr ữủ ự
sỷ ử ữợ tt ỳ h(x0 ) ∈ T ✱ t❤➻ ✤✐➲✉ ❦✐➺♥ ✭▼❙❘✮ trị♥❣
✈ỵ✐ ữợ q tr ừ trà x → h(x) − T t↕✐ (x0 , 0) ✤÷đ❝
✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❬✻❪✳ ❱➻ t❤➳ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ tt ỳ ữợ q ú t
q st r ✈ỵ✐ ❜➜t ❦ý u = 0✱ ✤✐➲✉ ❦✐➺♥ ✭❉▼❙❘u ✮ ❧➔ ❤➺ q✉↔ ❝õ❛ ✭▼❙❘✮✳ ❚r♦♥❣

♣❤➛♥ s❛✉✱ ❝❤ó♥❣ tỉ✐ s➩ ũ ỵ u u = 0 ữ ổ õ ữợ
q tr t ữợ 0 ổ ởt ữợ õ 0 tữỡ ữỡ
ợ ỡ ỳ T ỗ õ h ❧➔ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ t↕✐ x0 ✱ ✤✐➲✉ ❦✐➺♥ ✭▼❙❘✮
✶✸


❧➔ ❤➺ q✉↔ ❝õ❛ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ▼❛♥❣❛s❛r✐❛♥✲❋r♦♠♦✈✐t③✿
✭▼❋✮

h (x0 )X (T h(x0 )) = Y

rữợ t ú tỉ✐ t❤✐➳t ❧➟♣ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉ ❝➛♥ ❝➜♣ ❤❛✐ P tr ổ
ố

ỵ ♣❤➛♥ tr♦♥❣ ❝õ❛ C ✈➔ K ❧➔ ❦❤→❝ trè♥❣ ✈➔ x
✤â✱ ✈ỵ✐ ♠å✐ u ∈ X ✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳

0

∈ ▲❲❊✭f, S ✮✳ ❑❤✐

✭✐✮ ❈❤♦ (f, g, h) ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ ✈➔ h ữợ q tr t ữợ t
(x0 , u) ✤è✐ ✈ỵ✐ T = {0} ❦❤✐ u = 0✳ ❑❤✐ ✤â✱ (f, g, h) (x0 )u ∈ −int[C × K(g(x0 ))] × {0}✳
✭✐✐✮ ❈❤♦ (f, g, h) ❧➔ ❦❤↔ ✈✐ t t x0 h ữợ q tr t ữợ t (x0 , u)
ố ợ T = {0} ✈➔ ((f, g, h) (x0 ), B(f,g,h) (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g, h) t↕✐ x0 ✈ỵ✐
B(f,g,h) (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✳ ◆➳✉ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C ì
K(g(x0 )))] ì {0} t
tỗ t (M, N, P ) ∈ ♣✲❝❧B(f,g,h) (x0 ) s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ w ∈ X ✱

(f, g, h) (x0 )w + 2(M, N, P )(u, u) ∈ −intcone[C + f (x0 )u] × IT 2 (−K, g(x0 ), g (x0 )u) × {0}

tỗ t (M, N, P ) B(f,g,h) (x0 )∞ \ {0} s❛♦ ❝❤♦

(M, N, P )(u, u) ∈ −intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) × {0}✳
✭✐✐✐✮ ❈❤♦ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ (f (x0 ), Bf (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0 ✈ỵ✐
Bf (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✱ t❤➻
✈ỵ✐ ồ w T (S, x0 , u) tỗ t↕✐ M ∈ ♣✲Bf (x0 )∞ s❛♦ ❝❤♦

f (x0 )w + M (u, u) ∈ −intcone[C + f (x0 )u]✱
❤♦➦❝ tỗ t M Bf (x0 ) \ {0} s ❝❤♦

M (u, u) ∈ −intcone[C + f (x0 )u]✳

❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ◆➳✉ u = 0✱ t❤➻ ❦➳t q✉↔ ❧➔ rã r➔♥❣✳ ●✐↔ sû ♣❤↔♥ ❝❤ù♥❣ r➡♥❣✱ ✈ỵ✐ u ∈ X
❦❤→❝ ❦❤ỉ♥❣✱

(f, g, h) (x0 )u ∈ −✐♥t[C × K(g(x0 ))] × {0}✳
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ tn → 0+ ✱

h(x0 + tn u)
h (x0 )u = 0
tn
tt ữợ q tr ừ h tỗ t à > 0 ρ > 0 s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ t ∈ (0, ρ)
✈➔ v ∈ BX (u, ρ)✱ t❛ ❝â d(x0 + tv, H) ≤ µ h(x0 + tv) ✳ ❉♦ ✤â✱ ợ n ợ tỗ t yn H
ợ (x0 + tn u − yn )/tn → 0✳ ❚❛ ❝â un := (yn − x0 )/tn → u ✈➔ x0 + tn un ∈ H ✳
❱➻
f (x0 + tn un ) − f (x0 )
g(x0 + tn un ) − g(x0 )
→ f (x0 )u ∈ −intC,
→ g (x0 )u ∈ −intK(g(x0 ))✱
tn

tn
✈ỵ✐ n ✤õ ❧ỵ♥✱ t❛ ❝â

f (x0 + tn un ) − f (x0 ) ∈ −intC ✱
g(x0 + tn un ) ∈ −intK ⊂ −K ✱
tù❝ ❧➔✱ t❛ ✤÷đ❝ ✤✐➲✉ ♠➙✉ t❤✉➝♥✳
✶✹


✭✐✐✮ ▲➜② u ∈ X s❛♦ ❝❤♦

(f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C × K(g(x0 )))] ì {0}
ợ tn 0+ tỗ t (Mn , Nn , Pn ) ∈ B(f,g,h) (x0 ) s❛♦ ❝❤♦✱ ✈ỵ✐ n ❧ỵ♥✱

(f, g, h)(x0 + tn u) − (f, g, h)(x0 ) = tn (f, g, h) (x0 )u + t2n (Mn , Nn , Pn )(u, u) + o(t2n )✳
♣

✭❛✮ ◆➳✉ {(Mn , Nn , Pn )} ❜à ❝❤➦♥✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ (Mn , Nn , Pn ) −
→ (M, N, P ) ∈
♣✲❝❧B(f,g,h) (x0 )✳ ❉♦ ✤â✱

(f, g, h)(x0 + tn u) − (f, g, h)(x0 ) − tn (f, g, h) (x0 )u
→ 2(M, N, P )(u, u)✳
t2n /2
❱ỵ✐ ❜➜t ❦ý w ∈ X ✱ ❜ð✐ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❝❤➦t ❝õ❛ f ✱ t❛ ❝â

f (x0 + tn u + 21 t2n w) − f (x0 + tn u)
f (x0 + tn u + 21 t2n w) − f (x0 ) − tn f (x0 )u
=
t2n /2

t2n /2
+

f (x0 + tn u) − f (x0 ) − tn f (x0 )u
→ f (x0 )w + 2M (u, u)✳
t2n /2

❚÷ì♥❣ tü✱ t❛ ✤↕t ✤÷đ❝

g(x0 + tn u + 21 t2n w) − g(x0 ) − tn g (x0 )u
→ g (x0 )w + 2N (u, u)✱
t2n /2
h(x0 + tn u + 21 t2n w) − h(x0 ) − tn h (x0 )u
→ h (x0 )w + 2P (u, u)✳
t2n /2
●✐↔ sû

(f, g, h) (x0 )w + 2(M, N, P )(u, u)
∈ −intcone[C + f (x0 )u] × IT 2 (−K, g(x0 ), g (x0 )u) × {0}✳

✭✶✮

❱➻ h(x0 ) = 0 ✈➔ h (x0 )u = 0✱ ✤✐➲✉ ♥➔② s✉② r❛ r➡♥❣ h(x0 + tn u + 12 t2n w)/ 21 t2n 0 t
ữợ q tr ừ h ợ n ợ tỗ t yn H s (x0 +tn u+ 12 t2n w−yn )/ 12 t2n →
0✳ ❉♦ ✤â✱ wn := (yn − x0 − tn u)/ 12 t2n → w ✈➔ x0 + tn u + 12 t2n wn ∈ H ✳
▼➦t ❦❤→❝✱ t➼♥❤ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ❝õ❛ f ❣➛♥ x0 ✈➔ ✭✶✮ s✉② r❛ r➡♥❣

f (x0 + tn u + 21 t2n wn ) − f (x0 ) − tn f (x0 )u
f (x0 + tn u + 21 t2n wn ) − f (x0 + tn u + 12 t2n w)
=

t2n /2
t2n /2
f (x0 + tn u + 21 t2n w) − f (x0 ) − tn f (x0 )u
+
→ f (x0 )w + 2M (u, u)
t2n /2
∈ −intcone[C+f (x0 )u].

✭✷✮

❚÷ì♥❣ tü✱ t❛ ❝â

g(x0 + tn u + 21 t2n wn ) − g(x0 ) − tn g (x0 )u
→ g (x0 )w + 2N (u, u)
t2n /2
∈ IT 2 (−K, g(x0 ), g (x0 )u).
✶✺

✭✸✮


❱➻ IT (−✐♥tC, f (x0 )u) = −✐♥t❝♦♥❡(C + f (x0 )u)✱ ✭✷✮ s✉② r❛ r➡♥❣✱ ✈ỵ✐ n ❧ỵ♥✱

1 f (x0 + tn u + 12 t2n wn ) − f (x0 ) − tn f (x0 )u
f (x0 )u + tn
∈ −✐♥tC ✱
2
t2n /2
✈➔ ✈➻ t❤➳

f (x0 +tn u+ 21 t2n wn )−f (x0 ) ∈ −intC ✳
❚÷ì♥❣ tü✱ ❜ð✐ ✭✸✮ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ IT 2 ✱ t❛ ❝â✱ ✈ỵ✐ n ✤õ ❧ỵ♥✱

✭✹✮

1 g(x0 + tn u + 21 t2n wn ) − g(x0 ) − tn g (x0 )u
g(x0 ) + tn g (x0 )u + t2n
∈ −K ✱
2
t2n /2
✈➔ ✈➻ ✈➟②

g(x0 +tn un + 12 t2n wn ) ∈ −K ✳

✭✺✮

❈→❝ ❝æ♥❣ t❤ù❝ ✭✹✮ ✈➔ ✭✺✮ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t x0 ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣✳
✭❜✮ ◆➳✉ {(Mn , Nn , Pn )} ❦❤ỉ♥❣ ❜à ❝❤➦♥✱ t❛ ❣✐↔ sû r➡♥❣ αn := (Mn , Nn , Pn ) → ∞ ✈➔
1
♣
(Mn , Nn , Pn ) −
→ (M, N, P ) ∈ ♣✲B(f,g,h) (x0 )∞ \ {0}✳ ❉♦ ✤â✱
αn
(f, g, h)(x0 + tn u) − (f, g, h)(x0 ) − tn (f, g, h) (x0 )u
→ (M, N, P )(u, u)✳
αn t2n
●✐↔ sû

(M, N, P )(u, u) ∈ −intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) × {0}✳

✭✻✮

❱➻ (f, g) (x0 )u ∈ −[C×❝❧K(g(x0 ))\ ✐♥t(C ×K(g(x0 )))]✱ t❛ ❝â ❤♦➦❝ f (x0 )u ∈ −❜❞C ❤♦➦❝
g (x0 )u ∈ −❜❞K(g(x0 ))✳ ❱➻ t❤➳✱ ❜ð✐ ▼➺♥❤ ✤➲ ✶✳✸ ✭✐✈✮ ✈➔ ✶✳✹ ✭✐✐✮✱ ❤♦➦❝ M (u, u) = 0 ❤♦➦❝
N (u, u) = 0✱ ✈➔ ❞♦ ✤â αn tn → 0+ ✳
❱➻ h(x0 ) = 0 ✈➔ h (x0 )u = 0✱ t❛ ❝â h(x0 + tn u)/αn t2n → 0✳ tt ữợ
q tr ừ h ợ n ợ tỗ t yn H s (x0 + tn u − yn )/αn t2n → 0✳
✣➦t un := (yn − x0 )/tn ✱ t❛ ❝â (un − u)/αn tn → 0 ✈➔ x0 + tn un ∈ H ✳
▼➦t ❦❤→❝✱ ✈➻ f ❧➔ ▲✐♣s❝❤✐t③ ❣➛♥ x0 ✱ ✭✻✮ ❞➝♥ ✤➳♥

f (x0 + tn un ) − f (x0 ) − tn f (x0 )u
f (x0 + tn un ) − f (x0 + tn u)
=
2
αn tn
αn t2n
+

f (x0 + tn u) − f (x0 ) − tn f (x0 )u
→ M (u, u) ∈ −intcone[C+f (x0 )u]✳
αn t2n

✭✼✮

❚÷ì♥❣ tü✱ t❛ ❝â

g(x0 + tn un ) − g(x0 ) − tn g (x0 )u
→ N (u, u) ∈ IT (−K, g(x0 ), g (x0 )u)✳
αn t2n

✭✽✮

❱➻ IT (−✐♥tC, f (x0 )u) = −✐♥t❝♦♥❡(C + f (x0 )u)✱ tø ✭✼✮ t ữủ ợ n ợ

f (x0 )u + n tn

f (x0 + tn un ) − f (x0 ) − tn f (x0 )u
∈ −✐♥tC ✱
αn t2n

✈➔ ✈➻ t❤➳

f (x0 +tn un )−f (x0 ) ∈ −intC ✳
✶✻

✭✾✮


❚÷ì♥❣ tü✱ ❜ð✐ ✭✽✮ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ IT ✱ t❛ ❝â✱ ✈ỵ✐ n ❧ỵ♥✱

g(x0 + tn un ) − g(x0 ) − tn g (x0 )u
1
g(x0 ) + tn g (x0 )u + tn (2αn tn )
∈ −K ✱
2
αn t2n
✈➔ ❞♦ ✤â

g(x0 +tn un ) ∈ −K ✳

✭✶✵✮

❈→❝ ❝æ♥❣ t❤ù❝ ✭✾✮ ✈➔ ✭✶✵✮ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t ✈➲ x0
t q ữủ s r tứ ỵ ✹✳✶ ✭❜✮ ❝õ❛ ❬✶✼❪✳
✣➸ ❝â ❞↕♥❣ ✤è✐ ♥❣➝✉ ❝õ❛ ✣à♥❤ ỵ t tỷ r t t q✉↔ s❛✉
✤➙② ✈➲ t→❝❤✳

❇ê ✤➲ ✸✳✸✳ ❈❤♦ E

✈➔ G ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ F ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱
(y0 , z0 ) ∈ F × G✱ ✈➔ B ❧➔ t ỗ ừ F ợ tB = ϕ : E → F ✈➔ ψ : E → G
❧➔ ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝✳ ●✐↔ sû r➡♥❣✱ ✈ỵ✐ ♠å✐ x ∈ E ✱

(ϕ, ψ)(x) + (y0 , z0 ) ∈ −intB × {0}✳
◆➳✉ ψ(E) = G✱ t tỗ t (y , z ) F ì G ợ y = 0 s ❝❤♦✱ ✈ỵ✐ ♠å✐ b ∈ B ✱

y ∗ ◦ ϕ + z ∗ ◦ ψ = 0✱
y ∗ , y0 + z ∗ , z0 + y ∗ , b ≥ 0.
❚❤➯♠ ♥ú❛ ♥➳✉ B ❧➔ ♥â♥✱ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð t❤➔♥❤ y ∗ ∈ B ∗ ✈➔ y ∗ , y0 + z ∗ , z0 ≥ 0✳

❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣ t➟♣ ❤ñ♣

A := {(y, z) ∈ F × G | ∃x ∈ E : y − ϕ(x) ∈ y0 + ✐♥tB, z − ψ(x) = z0 }
ỗ ổ ự (0, 0) ồ b0 ✐♥tB ✳ ❚❛ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ (b0 + y0 , z0 ) tA
t tỗ t ởt U ❝õ❛ ❦❤æ♥❣ tr♦♥❣ F s❛♦ ❝❤♦ b0 + U + U tB
tỗ t r > 0 s ❝❤♦ −ϕ(x) ∈ U ✈ỵ✐ ♠å✐ x ∈ BE (0, r) ởt tỗ
t ởt ❧➙♥ ❝➟♥ V ❝õ❛ ❦❤æ♥❣ tr♦♥❣ G s❛♦ ❝❤♦ V ⊂ ψ(BE (0, r))✳ ❚❛ ❝❤ù♥❣ tä r➡♥❣
(b0 + y0 + U ) × (z0 + V ) ⊂ A✳ ▲➜② y ∈ U ✈➔ z ∈ V ✳ ❑❤✐ õ tỗ t x BE (0, r) s
z = ψ(x)✳ ❍ì♥ ♥ú❛✱ t❛ ❝â b0 + y0 + y − ϕ(x) ∈ b0 + y0 + U + U ⊆ y0 + ✐♥tB ✳ ❑❤✐ ✤â✱
(b0 + y0 + y, z0 + z) ∈ A ✈➔ ❞♦ ✤â (b0 + y0 , z0 ) tA

ỵ t tổ tữớ tỗ t (y , z ) ∈ F ∗ × G∗ \ {(0, 0)} s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐
(y, z) ∈ A✱

y ∗ , y + z ∗ , z ≥ 0✳

✭✶✶✮

❱ỵ✐ ♠♦✐ x ∈ E ✈➔ b ∈ ✐♥tB ✱ t❛ ❝â (y, z) := (ϕ(x) + b + y0 , ψ(x) + z0 ) ∈ A✳ ❚ø ✭✶✶✮ t❛
s✉② r❛

y ∗ , ϕ(x) + z ∗ , ψ(x) + y ∗ , y0 + z ∗ , z0 + y ∗ , b ≥ 0✳
❉♦ ✤â✱ ✈➻ ϕ ✈➔ ψ ❧➔ t✉②➳♥ t➼♥❤✱

y ∗ ◦ ϕ + z ∗ ◦ ψ = 0✱
y ∗ , y0 + z ∗ , z0 + y ∗ , b ≥ 0, ∀b ∈ B ✳
◆➳✉ y ∗ = 0✱ t❤➻ ✤➥♥❣ t❤ù❝ tr➯♥ ❞➝♥ ✤➳♥ z ∗ = 0✱ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳

0✳

◆➳✉ B ❧➔ ♥â♥✱ tø ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ s✉② r❛ y ∗ ∈ B ∗ ✱ ✈➔ ❞♦ ✤â y ∗ , y0 + z ∗ , z0 ≥
✶✼


ũ ỵ s t tû ❋r✐t③ ❏♦❤♥

Λ(x0 ) := {(c∗ , k ∗ , h∗ ) ∈ X ∗ × Y ∗ × Z ∗ : (c∗ , k ∗ , h∗ ) = (0, 0, 0), c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 )
+ h∗ ◦ h (x0 ) = 0, c∗ ∈ C ∗ , k ∗ ∈ N (K, g(x0 ))}

ỵ ợ t P ❝❤♦ ✐♥tC ✈➔ ✐♥tK ❧➔ ❦❤→❝ ré♥❣ ✈➔ x

✤â✱ ✈ỵ✐ ♠å✐ u ∈ X ✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳

0

∈ ▲❲❊✭f, S ✮✳ ❑❤✐

✭✐✮ ❈❤♦ f, g, h ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ ✈➔ h (x0 )(X) = W õ tỗ t (c , k ∗ , h∗ ) ∈
Λ(x0 ) ✈ỵ✐ (c∗ , k ∗ ) = (0, 0)❀
✭✐✐✮ ❈❤♦ f, g, h ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ h (x0 )(X) = W ✱ ((f, g, h) (x0 ), B(f,g,h) (x0 )) ❧➔ ①➜♣
①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g, h) t↕✐ x0 ✈ỵ✐ B(f,g,h) (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ A (−K, g(x0 ), g (x0 )u)
❦❤→❝ ré♥❣✳ ◆➳✉ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C ì K(g(x0 )))] ì {0} t
tỗ t (M, N, P ) ∈ ♣✲❝❧B(f,g,h) (x0 ) ✈➔ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) ✈ỵ✐ (c∗ , k ∗ ) = (0, 0)
s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 21 supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ,
✈➔ c∗ = 0 ♥➳✉✱ t❤➯♠ ♥ú❛✱ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❝➜♣ ❤❛✐ s❛✉ ✤➙② t❤ä❛
✭❚❘u ✮

(g, h) (x0 )X − T (T (−K, g(x0 )), g (x0 )u) × {0} = Z × W

tỗ t (M, N, P ) ♣✲B(f,g,h) (x0 )∞ \ {0} ❛♥❞ (c∗ , k ∗ , h∗ ) ∈ C ∗ × K(g(x0 ))∗ ×
W ∗ \ {(0, 0, 0)} ✈ỵ✐ c∗ , f (x0 )u = k ∗ , g (x0 )u = 0 s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 0✱
✈➔ (c∗ , k ∗ ) = (0, 0) ♥➳✉ h = 0✳
✭✐✐✐✮ ❈❤♦ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ (f (x0 ), Bf (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0
✈ỵ✐ Bf (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✳
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ w T (S, x0 , u) tỗ t M ∈ ♣✲Bf (x0 )∞ ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐
c∗ , f (x0 )u = 0 s❛♦ ❝❤♦

c∗ , f (x0 )w + M (u, u) ≥ 0✱
❤♦➦❝ tỗ t M Bf (x0 ) \ {0} c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0 s❛♦ ❝❤♦

c∗ , M (u, u) 0

ự ỵ ❞ư♥❣ ❇ê ✤➲ ✸✳✸ ✈ỵ✐ E = X, G = W, F = Y × Z ✱

ϕ = (f (x0 ), g (x0 )), ψ = h (x0 )✱ (y0 , z0 ) = (0, 0)✱ ✈➔ B = C × K(g(x0 ))✱ t❛ ❝â t❤➸ t➻♠
✤÷đ❝ (c∗ , k ∗ ) ∈ [C × K(g(x0 ))]∗ = C ∗ × N (−K, g(x0 )) ✈ỵ✐ (c∗ , k ∗ ) = (0, 0) ✈➔ h∗ ∈ W ∗
s❛♦ ❝❤♦ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = 0✳
✭✐✐✮ ✭❛✮ ●✐↔ sû A2 (−K, g(x0 ), g (x0 )u) = ∅ ✭♥➳✉ ❦❤ỉ♥❣✱ ❦❤➥♥❣ ✤à♥❤ ❧➔ t➛♠ t❤÷í♥❣✮✳
❇ð✐ ✣à♥❤ ỵ tỗ t (M, N, P ) ∈ ♣✲❝❧B(f,g,h) (x0 ) s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ w ∈ X ✱

(f, g, h) (x0 )w + 2(M, N, P )(u, u) ∈ −intcone[C + f (x0 )u] × IT 2 (−K, g(x0 ), g (x0 )u) × {0}✳
⑩♣ ❞ư♥❣ ❇ê ✤➲ ✸✳✸ ✈ỵ✐ E = X, G = W, F = Y × Z ✱ ϕ = (f (x0 ), g (x0 )), ψ = h (x0 )✱
y0 = 2(M, N )(u, u), z0 = 2P (u, u)✱ B = ❝♦♥❡[C + f (x0 )u] × [−IT 2 (−K, g(x0 ), g (x0 )u)]
❝❤♦ (c∗ , k ∗ , h ) X ìY ìZ ợ c∗ ◦f (x0 )+k ∗ ◦g (x0 )+h∗ ◦h (x0 ) = 0 ✈➔ (c∗ , k ∗ ) = (0, 0)
s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ c ∈ ❝♦♥❡[C + f (x0 )u] ✈➔ k ∈ −IT 2 (−K, g(x0 ), g (x0 )u)]✱

c∗ , 2M (u, u) + k ∗ , 2N (u, u) + h∗ , 2P (u, u) + c∗ , c + k ∗ , k ≥ 0✳

✭✶✷✮

❱➻ ❝♦♥❡[C + f (x0 )u] ❧➔ ♥â♥✱ ✭✶✷✮ ❞➝♥ ✤➳♥ c∗ , c ≥ 0✱ ✈ỵ✐ ♠å✐ c ∈ ❝♦♥❡[C + f (x0 )u] ✈➔ ✈➻
✶✽


t❤➳ c∗ ∈ C ∗ ✈➔ c∗ , f (x0 )u = 0✳ ✣➦t α := c∗ , 2M (u, u) + k ∗ , 2N (u, u) + h∗ , 2P (u, u) ✱
tø ✭✶✷✮ t❛ ❝â k ∗ , k ≤ α ✈ỵ✐ ♠å✐ k ∈ IT 2 (−K, g(x0 ), g (x0 )u)✱ tù❝ ❧➔✱ ✈ỵ✐ ♠å✐ k ∈
A2 (−K, g(x0 ), g (x0 )u) ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✸ ✭✈✮✮✳ ✣✐➲✉ ♥➔② ❝ị♥❣ ✈ỵ✐ ✭✶✷✮ s✉② r❛ r➡♥❣

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 21 supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✳
✣➸ t❤➜② r➡♥❣ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 )✱ t❛ q✉❛♥ s→t ▼➺♥❤ ✤➲ ✸✳✶ tr♦♥❣ ❬✺❪

A2 (−K, g(x0 ), g (x0 )u) + T (T (−K, g(x0 )), g (x0 )u) ⊂ A2 (−K, g(x0 ), g (x0 )u)✳
❉♦ ✤â✱ ✈ỵ✐ ♠å✐ k ∈ A2 (−K, g(x0 ), g (x0 )u) ✈➔ k1 ∈ T (T (−K, g(x0 )), g (x0 )u)✱

k ∗ , k + k1 ≤ α ✳
❱➻ T (T (−K, g(x0 )), g (x0 )u) ❧➔ ♥â♥✱ s✉② r❛ r➡♥❣

k ∗ ∈ −[T (T (−K, g(x0 )), g (x0 )u)]∗ = {k ∗ ∈ N (−K, g(x0 )) | k ∗ , g (x0 )u = 0}✳
❇➙② ❣✐í✱ ❣✐↔ sû ✭❚❘u ✮✳ ◆➳✉ c∗ = 0✱ t❤➻✱ ✈ỵ✐ ♠å✐ (y, z) ∈ Y ì Z tỗ t x X
k T (T (−K, g(x0 )), g (x0 )u) s❛♦ ❝❤♦ (g, h) (x0 )x − (k, 0) = (y, z)✳ ❉♦ ✤â✱

(k ∗ , h∗ ), (y, z) = k ∗ , g (x0 )x + h∗ , h (x0 )x − k ∗ , k = − k ∗ , k ≥ 0✱
✈➻ (c∗ , k ∗ , h∗ ) (x0 ) (y, z) tũ ỵ (k ∗ , h∗ ) = (0, 0)✱ ♠ët ✤✐➲✉ t
ỵ tỗ t (M, N, P ) ∈ ♣✲B(f,g,h) (x0 )∞ \ {0} s❛♦ ❝❤♦✱

(M, N, P )(u, u) ∈ −intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) ì {0}
õ trữớ ủ s

ã P (u, u) = 0 t ỵ t tổ tữớ tỗ t (c , k ) Y ×Z ∗ \{0, 0}
s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ w ∈ ✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ k ∈ −IT (−K, g(x0 ), g (x0 )u)✱
c∗ , M (u, u) + k ∗ , N (u, u) + c∗ , w + k ∗ , k ≥ 0✳

✭✶✸✮

❱➻ ✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ −IT (−K, g(x0 ), g (x0 )u) ❧➔ ❝→❝ ♥â♥✱ ✭✶✸✮ ❞➝♥ ✤➳♥ c∗ ∈ C ∗ ✱
c∗ , f (x0 )u = 0✱ ✈➔ k ∗ , k ≤ 0 ✈ỵ✐ ♠å✐ k ∈ IT (−K, g(x0 ), g (x0 )u)✱ tù❝ ❧➔✱ ✈ỵ✐ ♠å✐
k ∈ T (T (−K, g(x0 )), g (x0 )u) ✭①❡♠ ▼➺♥❤ ✤➲ ✶✳✹ ✭✐✈✮ ✈➔ ✭✈✮✮✳ ❱➻ t❤➳✱ k ∗ ∈ K(g(x0 ))∗ ✈➔

k ∗ , g (x0 )u = 0✱ ✈➔ ❜ð✐ ✭✶✸✮ t❛ ❝â

c∗ , M (u, u) + k ∗ , N (u, u) ≥ 0✳
❈❤å♥ h∗ ∈ W tũ ỵ t ữủ t q

ã P (u, u) = 0 t ró r r tỗ t h ∈ W ∗ ❦❤→❝ ❦❤æ♥❣ s❛♦ ❝❤♦ h∗ , P (u, u) ≥
0✳ ❱ỵ✐ (c∗ , k ∗ ) = (0, 0)✱ t❛ ❝â ❦➳t ❧✉➟♥✳
✭✐✐✐✮ ❑➳t q✉↔ ✤÷đ❝ s✉② r tứ ỵ ỵ t t❤ỉ♥❣ t❤÷í♥❣✳
❑➳t q✉↔ s❛✉ ✤➙② ❧➔ ♠ët ❤➺ q✉↔ trü❝ t ừ ỵ

q ợ

t P ❝❤♦ f, g ✈➔ h ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐ x0 ✈➔
h (x0 )(X) = W ✳ ◆➳✉ ✐♥tC ✈➔ ✐♥tK ❧➔ ❦❤→❝ ré♥❣ ✈➔ x0 ∈ LWE(f, S) t
tỗ t (c , k , h ) ∈ Λ(x0 ) s❛♦ ❝❤♦ (c∗ , k ∗ ) = (0, 0)❀
✭✐✐✮ ✈ỵ✐ ♠å✐ u ∈ X ✈ỵ✐ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C ì K(g(x0 )))] ì {0}
tỗ t (c , k ∗ , h∗ ) ∈ Λ(x0 ) ✈ỵ✐ (c∗ , k ∗ ) = (0, 0) s❛♦ ❝❤♦

c∗ , f (u, u) + k ∗ , g (u, u) + h∗ , h (u, u) ≥ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✱
✈➔ ♥➳✉✱ t❤➯♠ ♥ú❛✱ ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ t❤ä❛✱ t❤➻ c∗ = 0❀
✭✐✐✐✮ ♥➳✉ f (x0 )u ∈ −bdC ✈➔ w ∈ T (S, x0 , u) ✭✈➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ intK = ∅ ✈➔
✶✾


h (x0 )(X) = W ❝â t❤➸ ❜ä ✤✐✮✱ t❤➻ tỗ t c C \ {0} ợ c , f (x0 )u = 0 s❛♦ ❝❤♦
c∗ , f (x0 )w 0
ữ ỵ r q✉② ❙❧❛t❡r ❬✶✹❪ t❤ä❛✱ t❤➻ ❜ð✐ ◆❤➟♥ ①➨t ✻✳✷ ❝õ❛ ❬✶✷❪✱
✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ ❝ô♥❣ t❤ä❛✳ ❱➻ t❤➳✱ ❍➺ q✉↔ rở ỵ ừ tr
õ Y = R✱ K ❧➔ ♥â♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❙❧❛t❡r ✤÷đ❝ ❞ị♥❣✳

◆❤➟♥ ①➨t ✸✳✻✳ ✭✐✮ ❇➡♥❣ ❝→❝❤ t❤➯♠ ỵ tt r (g, h) ❧➔ ❦❤↔ ✈✐

❋r➨❝❤❡t q✉❛♥❤ x0 ✈➔ (f, g) ❧➔ ê♥ t x0 ữợ q tr t ữợ t
(x0 , u) ố ợ K ì {0} t ❝õ❛ ✭✐✐✐✮ trð ♥➯♥ ♠↕♥❤ ❤ì♥ ♥❤÷ s❛✉✿ ♥➳✉ g (x0 )w ∈
T (−K, g(x0 ), g (x0 )u) ✈➔ h (x0 )w = 0 t tỗ t c C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0
s❛♦ ❝❤♦ c∗ , f (x0 )w ≥ 0✳ t ử ữợ ợ (g, h) t❤❛② ❜ð✐
g ✱ ✈➔ −K × {0} t❤❛② ❜ð✐ −K ✱ t❛ ❝â✱ ✈ỵ✐ S = (g, h)−1 (−K × {0}) = G ∩ H ✱

T (S, x0 , u) = {w ∈ X | (g, h) (x0 )w ∈ T (−K × {0}, (g, h)(x0 ), (g, h) (x0 )u)}
= {w ∈ X | g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u), h (x0 )w = 0}✳
✭✐✐✮ ▼➦❝ ❞ò ❜✐➸✉ t❤ù❝ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ❧➔ ❦❤ỉ♥❣ ❞÷ì♥❣✱ t❛ ✤÷❛ r❛ ❧í✐ ❝➢t
♥❣❤➽❛ ✤ì♥ ❣✐↔♥ ✈➻ t➛♠ q✉❛♥ trå♥❣ ❝õ❛ ♥â✳ ❇ð✐ ▼➺♥❤ ✤➲ ✶✳✸ ✭✐✮✱ t❛ ❝â

A2 (−K, g(x0 ), g (x0 )u) ⊂ ❝❧❝♦♥❡❬❝♦♥❡(−K − g(x0 )) − g (x0 )u❪✳
▼➦t ❦❤→❝ ✭①❡♠ ♣❤➛♥ ❝✉è✐ ❝õ❛ ♣❤➨♣ ự ừ ỵ

k −[T (T (−K, g(x0 )), g (x0 )u)]∗ = −[clcone(cone(−K − g(x0 )) − g (x0 )u)]∗ .
✣✐➲✉ ♥➔② s✉② r❛ r➡♥❣ ❜✐➸✉ t❤ù❝ ♥â✐ tr➯♥ ❧➔ ❦❤ỉ♥❣ ❞÷ì♥❣✱ ✈➔ t❤➟♠ ❝❤➼ ➙♠ ✭①❡♠ ❝→❝ ✈➼ ❞ư
s❛✉✮✱ ❦❤ỉ♥❣ ❣✐è♥❣ ♥❤÷ ❦➳t q✉↔ ❝ê ✤✐➸♥✳ ❙ü ❦✐➺♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡
❞♦ ❑❛✇❛s❛❦✐ ❬✶✹❪ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ♣❤→t ❤✐➺♥✳ ❙✉♣r❡♠✉♠ tr♦♥❣ ❜✐➸✉ t❤ù❝ tr➯♥ tr✐➺t t✐➯✉
♥➳✉ 0 ∈ A2 (−K, g(x0 ), g (x0 )u)✳ ✣➦❝ ❜✐➺t✱ ✤✐➲✉ ♥➔② ①↔② r❛ ♥➳✉ t❛ ①➨t ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥
❝➜♣ ữợ u X ợ g (x0 )u ∈ ❝♦♥❡(−K − g(x0 )) = −K(g(x0 )) ✭①❡♠ ▼➺♥❤
✤➲ ✶✳✸ ✭✈✐✮ ✭❜✮✮✱ ♥❤÷ ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ❧➔♠✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡
❦❤ỉ♥❣ ①↔② r❛✳
❚✉② ♥❤✐➯♥✱ ỵ t ữợ u ợ g (x0 )u ∈ −clK(g(x0 ))✳ ◆❣❤➽❛ ❧➔✱ ❤✐➺♥
t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ①↔② r❛ ❝❤♦ ❝→❝ ✤✐➸♠ u tr♦♥❣ ❧é ❤ê♥❣ ❞÷í♥❣ ♥❤÷ ♥❤ä ❝õ❛ ❜❛♦ ✤â♥❣
−clK(g(x0 ))✳ ❚❛ ♥❤➜♥ ♠↕♥❤ r➡♥❣ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣✲❧✐❦❡ ❦❤æ♥❣ ①↔② r❛ ♥➳✉ g (x0 )u ∈
−K(g(x0 ))✱ ♥❣❛② ❝↔ ❦❤✐ ✤✐➸♠ ♥➔② ♥➡♠ tr➯♥ ❜✐➯♥✳ ❱➻ t❤➳✱ ♥➳✉ K ❧➔ ♣♦❧②❤❡❞r❛❧✱ ❤✐➺♥
t÷đ♥❣ ♥➔② ❝ơ♥❣ ❦❤ỉ♥❣ ①↔② r❛ ✭✈➻ K(g(x0 )) õ ữ ỵ r ũ ①➾ t❛
❝â t❤➸ tr→♥❤ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❝➜♣ ♠ët t ữợ

q ✤➳♥ ❤✐➺♥ t÷đ♥❣ ❡♥✈❡❧♦♣❡✲❧✐❦❡ rã ❤ì♥✱ t❛ ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❝➜♣ ♠ët ✭♥❤÷♥❣ ❦❤ỉ♥❣ ❦❤↔
✈✐ ❧✐➯♥ tư❝✮ tr♦♥❣ ✣à♥❤ ỵ
t q s ởt trữ ừ ♥â♥ t✐➳♣ ①ó❝ ❝➜♣ ❤❛✐ t✐➺♠ ❝➟♥ ✈ỵ✐ t➟♣ ❝❤➜♣
♥❤➟♥ ✤÷đ❝ S = g −1 (−K)✳

▼➺♥❤ ✤➲ ✸✳✼✳ ❈❤♦ x , u ∈ X ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t q x

0
0 ợ g ờ t x0
ữợ q tr t ữợ t (x0 , u) ố ✈ỵ✐ −K ✳ ❑❤✐ ✤â✱ ✈ỵ✐ S = g −1 (−K)
t❛ ❝â

T (S, x0 , u) = {w ∈ X | g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)}✳

❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ w ∈ T

(S, x0 , u)✳ õ tỗ t (tn , rn ) (0+ , 0+ ) : tn /rn → 0 ✈➔
wn → w s❛♦ ❝❤♦ xn := x0 + tn u + 21 tn rn wn ∈ S ✳ ❇ð✐ ▼➺♥❤ ✤➲ ✶✳✶✱ t❛ ❝â
✷✵


g(xn ) − g(x0 ) − tn g (x0 )u
→ g (x0 )w✳
tn rn /2
❱➻ g(xn ) ∈ −K ✱ s✉② r❛ r➡♥❣ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u)✳
✣è✐ ✈ỵ✐ ♣❤➛♥ ✤↔♦✱ ❣✐↔ sû r➡♥❣ g (x0 )w ∈ T (−K, g(x0 ), g (x0 )u) õ tỗ t
(tn , rn ) (0+ , 0+ ) : tn /rn → 0✱ ✈➔ zn → g (x0 )w s❛♦ ❝❤♦ g(x0 ) + tn g (x0 )u + 21 tn rn zn ∈
−K ✈ỵ✐ ồ n tt ữợ q tr ợ n ❧ỵ♥✱ t❛ ❝â tn ∈ (0, ρ) ✈➔
un := u + 21 rn w ∈ BX (u, ρ) s❛♦ ❝❤♦

d(x0 + tn un , S) ≤ µd(g(x0 + tn un ), −K)
≤ µ g(x0 + tn un ) − g(x0 ) − tn g (x0 )u − 21 tn rn zn
≤ µ( g(x0 + tn un ) − g(x0 ) − g (x0 )(tn un ) + 12 tn rn g (x0 )w − 21 tn rn zn )
≤ µκ tn un 2 + 12 µtn rn g (x0 )w − zn
1
= µtn rn (2κ(tn /rn ) un 2 + g (x0 )w − zn )
2
✭❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ũ ữủ s r tứ ỵ tr tr ❜➻♥❤ ✈➔ ❣✐↔ t❤✐➳t ê♥ ✤à♥❤
❝õ❛ g ✮✳ ❱➻ zn → g (x0 )w✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ xn ∈ S s❛♦ ❝❤♦ x0 + tn un − xn / 12 tn rn → 0✳
❍ì♥ ♥ú❛✱
xn − x0 − tn u
xn − x0 − tn un
wn :=
=
+ w → w.
tn rn /2
tn rn /2
❉♦ ✤â✱ w ∈ T (S, x0 , u)✳
❑➳t q✉↔ s❛✉ ✤➙② ❧➔ ❤➺ q✉↔ trü❝ t ừ ỵ trữớ ủ h = 0✳ ❚r♦♥❣
tr÷í♥❣ ❤đ♣ ♥➔② t❛ ✤à♥❤ ♥❣❤➽❛

Λ1 (x0 ) := {(c∗ , k ∗ ) ∈ Y ∗ × Z ∗ | (c∗ , k ∗ ) = (0, 0), c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) = 0,
c∗ ∈ C ∗ , k ∗ ∈ N (−K, g(x0 ))}✳

❍➺ q✉↔ ✸✳✽✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ h = 0✳ ◆➳✉ ✐♥tC ✈➔ ✐♥tK ❧➔ ❦❤→❝ ré♥❣ ✈➔ x
▲❲❊(f, S)✱ t❤➻ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳

0

∈

✭✐✮ ❈❤♦ f ✈➔ g ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✳ ❑❤✐ ✤â✱ Λ1 (x0 ) = ∅✳
✭✐✐✮ ❈❤♦ (f, g) ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ ((f, g) (x0 ), B(f,g) (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g)
t↕✐ x0 ✈ỵ✐ B(f,g) (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ u ∈ X ✈ỵ✐ A (−K, g(x0 ), g (x0 )u) = ∅✳
◆➳✉ (f, g) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C × K(g(x0 )))] t
tỗ t (M, N ) ♣✲❝❧B(f,g) (x0 ) ✈➔ (c∗ , k ∗ ) ∈ Λ1 (x0 ) s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) ≥ 21 supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ,
✈➔ c∗ = 0 ♥➳✉✱ t❤➯♠ ♥ú❛✱ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② s❛✉ ✤➙② t❤ä❛
✭❚❘✶u ✮

g (x0 )X − T (T (−K, g(x0 )), g (x0 )u) = Z

tỗ t (M, N ) ∈ ♣✲B(f,g) (x0 )∞ \ {0} ✈➔ (c∗ , k ∗ ) ∈ C ∗ × K(g(x0 ))∗ \ {(0, 0)}
✈ỵ✐ c∗ , f (x0 )u = k ∗ , g (x0 )u = 0 s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) ≥ 0✳
✭✐✐✐✮ ❈❤♦ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ (f (x0 ), Bf (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x0 ✈ỵ✐
Bf (x0 ) ❧➔ ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✱ ✈➔ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤æ♥❣ ❝➛♥ ❦❤→❝
✷✶


ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ w ∈ T (S, x0 , u) tỗ t M Bf (x0 ) ✈➔ c∗ ∈ C ∗ \ {0}
✈ỵ✐ c∗ , f (x0 )u = 0 s❛♦ ❝❤♦

c∗ , f (x0 )w + M (u, u) 0
tỗ t M ♣✲Bf (x0 )∞ \ {0} ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0 s❛♦ ❝❤♦

c∗ , M (u, u) ≥ 0✳
❱➼ ❞ö s❛✉ ởt trữớ ủ tr õ ỵ ✭❤❛② ❍➺ q✉↔ ✸✳✽✮ ❜→❝ ❜ä ✤✐➸♠

♥❣❤✐ ♥❣í ❧➔ ♥❣❤✐➺♠ ②➳✉ ✤à❛ ♣❤÷ì♥❣✱ tr♦♥❣ ❦❤✐ ✤â ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② ❦❤ỉ♥❣ →♣ ❞ư♥❣
✤÷đ❝✳

❱➼ ❞ư ✸✳✶✳

❈❤♦ C = R+ ✱ I = [−1, 1]✱ C(I) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tö❝ tr➯♥
I ✱ C+ (I) := {z ∈ C(I) | z(t) ≥ 0, ∀t ∈ I}✱ ✈➔ (x0 , y0 ) = (0, 0)✳ ❈❤♦ f : R2 → R ✈➔
g : R2 → C(I) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

f (x, y) = x|x| + y, (g(x, y))(t) = y + 3x2 − 2tx + t2 , ∀t ∈ I
✭g ✤÷đ❝ ❧➜② tø ❱➼ ❞ư ✺✳✶ tr♦♥❣ ❬✶✹❪✮✳ ❑❤✐ ✤â✱ f ∈ C 1,1 t↕✐ (0, 0)✱ f (0, 0) = (0, 1)✱
±1 0
Bf (0, 0) =
, g ∈ C 2 t↕✐ (0, 0) ✈➔✱ ✈ỵ✐ ♠å✐ u = (x, y) ∈ R2 ✈➔ t ∈ I ✱
0 0

(g (0, 0)(u))(t) = −2tx + y ✱ (g (0, 0)(u, u))(t) = 6x2 ✳
❱➻ t❤➳✱ ((f, g) (0, 0), Bf (0, 0) × { 21 g (0, 0)}) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ (f, g) ❛t (0, 0) ✈ỵ✐
B(f,g) (0, 0) := Bf (0, 0) × { 12 g (0, 0)} ❜à ❝❤➦♥ ✈➔ ❞♦ ✤â ♣✲❝♦♠♣❛❝t t✐➺♠ ❝➟♥✳
❱➻ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ❙❧❛t❡r t❤ä❛✱ ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ ✭✈➔ ✭❚❘✶u ✮✮ ❝ô♥❣ t❤ä❛✳ ❚➟♣ ♥❤➙♥
tû ❋r✐t③ ❏♦❤♥ ❧➔

Λ1 (0, 0) = {(c∗ , k ∗ ) ∈ R × C(I)∗ | c∗ = α > 0, k ∗ , z = −αz(0), ∀z ∈ C(I)}✳
❈❤å♥ u = (1, 0)✳ ❑❤✐ ✤â✱ f (0, 0)u = 0✳ ❇ð✐ ❇ê ✤➲ ✻✳✶ ❝õ❛ ❬✶✹❪✱ v(·) ∈ −clK(g(0, 0)) =
−clcone(C+ (I) + (·)2 ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ v(t) ≥ 0 ✈ỵ✐ ♠å✐ t ∈ Iξ := {t ∈ I|t2 = 0} = {0}✳
❚❛ ❝â (g (0, 0)u)(t) = −2t ✈➔ ❞♦ ✤â g (0, 0)u ∈ − ❝❧K(g(0, 0))✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❱➼ ❞ö ✺✳✶
tr♦♥❣ ❬✶✹❪✱

A2 (−K, g(0, 0), g (0, 0)u) = {z ∈ C(I) | z(0) ≥ 2}
❉♦ ✤â✱ (f, g) (0, 0)u ∈ −[C × clK(g(0, 0)) \ int(C ì K(g(0, 0)))] t r ợ ồ

(M, N ) ∈ ♣✲❝❧B(f,g) (0, 0) = B(f,g) (0, 0) ✈➔ (c∗ , k ∗ ) ∈ Λ1 (0, 0)✱

c∗ , M (u, u) + k ∗ , N (u, u) ≤ −2α < −α = 12 supk∈A2 (−K,g(0,0),g (0,0)u) k , k
ỵ q✉↔ ✸✳✽✮✱ (0, 0) ∈ ▲❲❊(f, S)✳ ❱➻ f ❦❤æ♥❣ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣
❤❛✐ t↕✐ (0, 0)✱ ❍➺ q✉↔ ✸✳✺ ỵ ừ ổ ử ữủ
r ♣❤➛♥ ❝á♥ ❧↕✐ t❛ ①➨t tr÷í♥❣ ❤đ♣ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❝õ❛ ✭P✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✤✐➲✉
❦✐➺♥ ❝➛♥ ❦❤→❝ ❞ü❛ tr ỵ t s

ờ ỵ C , C

Rm t ỗ s C1
r õ tỗ t ♠ët s✐➯✉ ♣❤➥♥❣ t→❝❤ C1 ✈➔ C2 r✐➯♥❣ ❜✐➺t ✈➔ ❦❤æ♥❣ ❝❤ù❛ C2
♥➳✉ ✈➔ ❝❤➾ ♥➳✉ C1 ∩ riC2 =
1

2

ỵ ợ t P X, Y, Z ✈➔ W ❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ✐♥tC ✈➔ ✐♥tK ❦❤→❝
ré♥❣ ✈➔ x0 ∈ ▲❲❊✭f, S ✮✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② t❤ä❛✳

✭✐✮ ❈❤♦ f, g, h ❧➔ rt t x0 h ữợ q tr t ữợ



t↕✐ (x0 , u) ✤è✐ ✈ỵ✐ T = {0} ❦❤✐ u = 0 õ tỗ t (c , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦
(c∗ , k ∗ ) = (0, 0)✳
✭✐✐✮ ❈❤♦ f, g, h ❧➔ ❦❤↔ ✈✐ ❝❤➦t t↕✐ x0 ✱ ((f, g, h) (x0 ), B(f,g,h) (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛
(f, g, h) t↕✐ x0 ✱ ✈➔ u ∈ X ✈ỵ✐ A (−K, g(x0 ), g (x0 )u) = ∅✳ h ữợ q tr
t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ T = {0} ✈➔ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C ì
K(g(x0 )))] ì {0} t

tỗ t (M, N, P ) ∈ ❝❧B(f,g,h) (x0 ) ✈➔ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 21 supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ,
tr♦♥❣ ✤â (c∗ , k ∗ ) = (0, 0) ♥➳✉ h = 0✱ ✈➔ c∗ = 0 ♥➳✉ ✤✐➲✉ ❦✐➺♥ ✭❚❘u ✮ tọ
tỗ t (M, N, P ) B(f,g,h) (x0 )∞ \ {0} ✈➔ (c∗ , k ∗ , h∗ ) ∈ C ∗ × K(g(x0 ))∗ ×
W ∗ \ {(0, 0, 0)} ✈ỵ✐ c∗ , f (x0 )u = k ∗ , g (x0 )u = 0 s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ 0
✈➔ ♥➳✉✱ t❤➯♠ ♥ú❛ h = 0✱ t❤➻ (c∗ , k ∗ ) = (0, 0)✳
✭✐✐✐✮ ❈❤♦ f ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x0 ✱ (f (x0 ), Bf (x0 )) ❧➔ ①➜♣ ①➾ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐
x0 ✱ ✈➔ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐
w ∈ T (S, x0 , u) tỗ t M Bf (x0 )∞ ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0 s❛♦
❝❤♦

c∗ , f (x0 )w + M (u, u) 0
tỗ t↕✐ M ∈ Bf (x0 )∞ \ {0} ✈➔ c∗ ∈ C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0 s❛♦ ❝❤♦

c∗ , M (u, u) ≥ 0

ự ỵ ử ❇ê ✤➲ ✸✳✾ ✈ỵ✐ C

= (f, g, h) (x0 )X ✈➔
C2 = −int[C × K(g(x0 ))] × {0}✱ t❛ ❝â ữủ (c , k , h ) X ì Y × Z ∗ ❦❤→❝ ❦❤ỉ♥❣ ✈➔
α ∈ R s❛♦ ❝❤♦✱ ∀(y, z, t) ∈ (f, g, h) (x0 )X ✱ ∀(c, k) ∈ −(C × K(g(x0 )))✱
∗

∗

∗

∗

1
∗

c∗ , y + k ∗ , z + h∗ , t ≥ α✱

✭✶✹✮

c∗ , c + k ∗ , k ≤ α,

✭✶✺✮

✈➔ s✐➯✉ ♣❤➥♥❣

H := {(y, z, t) ∈ X × Y × Z | c∗ , y + k ∗ , z + h∗ , t = α}
❦❤æ♥❣ ❝❤ù❛ C2 ✳ ❱➻ (f, g, h) (x0 )X ✈➔ C × K(g(x0 )) ❧➔ ❝→❝ ♥â♥✱ α = 0✳ ❑❤✐ ✤â✱ ✭✶✹✮ s✉②
r❛ r➡♥❣ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = 0✳ ❈❤♦ k = 0 tr♦♥❣ ✭✶✺✮ t❛ ✤÷đ❝ c∗ ∈ C ∗ ✳
✣➦t c = 0 tr♦♥❣ ✭✶✺✮ t❛ ❝â k ∗ ∈ K(g(x0 ))∗ = N (−K, g(x0 ))✳ ❱➻ s✐➯✉ ♣❤➥♥❣ H ❦❤æ♥❣
❝❤ù❛ C2 ✱ (c∗ , k ∗ ) = (0, 0)✳
✭✐✐✮ ✭❛✮ ❇ð✐ ✣à♥❤ ỵ ử ờ ợ C1 = (f, g, h) (x0 )X +
2(M, N, P )(u, u) ✈➔ C2 = −intcone[C + f (x0 )u] × IT 2 (−K, g(x0 ), g (x0 )u) × {0}✱ t❛ ❝â
✤÷đ❝ (c∗ , k ∗ , h∗ ) ∈ X ∗ × Y ∗ × Z ∗ ❦❤→❝ ❦❤ỉ♥❣ ✈➔ α ∈ R s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ (y, z, t) ∈
(f, g, h) (x0 )X ✱ c ∈ −✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ k ∈ IT 2 (−K, g(x0 ), g (x0 )u)✱

c∗ , y + k ∗ , z + h∗ , t + 2 c∗ , M (u, u) + 2 k ∗ , N (u, u) + 2 h∗ , P (u, u) ≥ α✱ ✭✶✻✮
c∗ , c + k ∗ , k ≤ α ✱

✭✶✼✮

✈➔ H ❦❤ỉ♥❣ ❝❤ó❛ C2 ✳ ❱➻ (f, g, h) (x0 )X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥✱ tø ✭✶✻✮ t❛ ❝â✱ ✈ỵ✐ ♠å✐ (y, z, t) ∈
✷✸


(f, g, h) (x0 )X ✱
c∗ , y + k ∗ , z + h∗ , t = 0✱
✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ c∗ ◦ f (x0 ) + k ∗ ◦ g (x0 ) + h∗ ◦ h (x0 ) = 0 ✈➔
✷ c∗ , M (u, u) +2 k ∗ , N (u, u) +2 h∗ , P (u, u) ≥ α✳

✭✶✽✮

❱➻ −✐♥t❝♦♥❡[C + f (x0 )u] ❧➔ ♥â♥✱ ✭✶✼✮ s✉② r❛ r➡♥❣ c∗ ∈ C ∗ ✈➔ c∗ , f (x0 )u = 0✳ ❈ô♥❣ tø
✭✶✼✮✱ t❛ ❝â k ∗ , k ≤ α✱ ✈ỵ✐ ♠å✐ k ∈ IT 2 (−K, g(x0 ), g (x0 )u)✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ✭✶✽✮✱ ❧➟♣
❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ♣❤➛♥ ❝❤ù♥❣ ỵ t õ t
❣✐í✱ ❣✐↔ sû h = 0✳ ◆➳✉ (c∗ , k ∗ ) = (0, 0)✱ t❤➻ ❜ð✐ ✭✶✻✮ ✈➔ ✭✶✼✮✱ α = 0 ✈➔ ❞♦ ✤â H
❝❤ù❛ C2 ✱ ♠ët t
ỵ ❞ư♥❣ ❇ê ✤➲ ✸✳✾ ✈ỵ✐ C1 = {(M, N, P )(u, u)} ✈➔ C2 =
−intcone[C + f (x0 )u] × IT (−K, g(x0 ), g (x0 )u) × {0}✱ t❛ ♥❤➟♥ ✤÷đ❝ (c∗ , k ∗ , h∗ ) ∈ X ∗ ×
Y ∗ × Z ∗ ❦❤→❝ ❦❤ỉ♥❣ ✈➔ α ∈ R s❛♦ ❝❤♦✱ ✈ỵ✐ ♠å✐ c ∈ −✐♥t❝♦♥❡[C + f (x0 )u] ✈➔ k ∈
IT (−K, g(x0 ), g (x0 )u)✱

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ α✱
c∗ , c + k ∗ , k ≤ α
✈➔ H ❦❤æ♥❣ ❝❤ù❛ C2 ✳ ❚ø ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ♣❤➛♥
❝❤ù♥❣ ♠✐♥❤ ỵ t õ t q
ớ ✈ỵ✐ h = 0✱ ♥➳✉ (c∗ , k ∗ ) = (0, 0)✱ t❤➻ ❜ð✐ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ α = 0 ✈➔ t❛
❝â ✤✐➲✉ ♠➙✉ t❤✉➝♥ t÷ì♥❣ tü ❧➔ H ❝❤ù❛ C2 ✳
✭✐✐✐✮ ❑➳t q✉↔ ✤÷đ❝ s✉② r❛ tứ ỵ ỵ t tổ tữớ
q ữợ ữủ s r trỹ t tứ ỵ ũ ss
s rở r tỹ ss r tữỡ ự

q ợ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ X, Y, Z ✈➔ W

❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ f, g, h t❤✉ë❝ ❧ỵ♣
C 1,1 t↕✐ x0 ∈ X ✱ ✐♥tC ✈➔ ✐♥tK ❦❤→❝ ré♥❣✱ ✈➔ x0 ∈ ▲❲❊(f, S)✳ ❑❤✐ ✤â✱ ♥❤ú♥❣ ❦❤➥♥❣ ✤à♥❤
s❛✉ ✤➙② t❤ä❛✳
✭✐✮ h ữợ q tr t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ T = {0} ❦❤✐ u = 0
õ tỗ t (c , k , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦ (c∗ , k ∗ ) = (0, 0)✳
✭✐✐✮ ❈❤♦ u ∈ X ✳ h ữợ q tr t ữợ t (x0 , u) ✤è✐ ✈ỵ✐ T = {0}
✈➔ (f, g, h) (x0 )u ∈ −[C × clK(g(x0 )) \ int(C ì K(g(x0 )))] ì {0} t tỗ t (M, N, P ) ∈
∂C2 (f, g, h)(x0 ) ✈➔ (c∗ , k ∗ , h∗ ) ∈ Λ(x0 ) s❛♦ ❝❤♦

c∗ , M (u, u) + k ∗ , N (u, u) + h∗ , P (u, u) ≥ supk∈A2 (−K,g(x0 ),g (x0 )u) k ∗ , k ✱

✭✶✾✮

tr♦♥❣ ✤â (c∗ , k ∗ ) = (0, 0) ♥➳✉ h = 0✱ ✈➔ c∗ = 0 ♥➳✉ ✭❚❘u ✮ t❤ä❛✳
✭✐✐✐✮ ❈❤♦ u ∈ X ✈ỵ✐ f (x0 )u ∈ −bdC ✭✈➔ intK ❦❤ỉ♥❣ ❝➛♥ ❦❤→❝ ré♥❣✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐
w ∈ T (S, x0 , u) tỗ t c C ∗ \ {0} ✈ỵ✐ c∗ , f (x0 )u = 0 s❛♦ ❝❤♦

c∗ , f (x0 )w ≥ 0✳
❍➺ q✉↔ ✸✳✶✶ ✭✐✐✮ ❝↔✐ t❤✐➺♥ ❍➺ q✉↔ ✹ ❝õ❛ ❬✼❪✱ tr♦♥❣ ✤â h ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ t↕✐
x0 ✳

❍➺ q✉↔ ✸✳✶✷✳ ❱ỵ✐ ❜➔✐ t♦→♥ ✭P✮✱ ❝❤♦ X, Y, Z ✈➔ W
✷✹

❧➔ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ f, g, h t❤✉ë❝ ❧ỵ♣