Giải tích đa trị P6

Chia sẻ: Trần Bá Trung1 | Ngày: | Loại File: PDF | Số trang:24

Thêm vào BST

Báo xấu

186
lượt xem 46
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Giải tích đa trị P6 Giải tích (tiếng Anh: mathematical analysis) là ngành toán học nghiên cứu về các khái niệm giới hạn, đạo hàm, tích phân... Nó có vai trò chủ đạo trong giáo dục đại học hiện nay. Phép toán cơ bản của giải tích là "phép lấy giới hạn". Để nghiên cứu giới hạn của một dãy số, hàm số,... ta phải "đo" được "độ xa gần" giữa các đối tượng cần xét giới hạn đó. Do vậy, những khái niệm như là mêtric, tôpô được tạo ra để mô tả một cách chính xác, đầy đủ việc...

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Giải tích đa trị P6

5.8. §èi ®¹o hµm Mordukhovich vµ Jacobian xÊp xØ 195 V× thÕ, kh«ng thÓ so s¸nh kh¸i niÖm ®èi ®¹o hµm víi kh¸i niÖm Jacobian xÊp xØ. §Ó v−ît qua khã kh¨n ®ã, chóng ta cÇn ®Õn ®Þnh nghÜa sau. §Þnh nghÜa 5.8.1. Mét tËp ®ãng kh¸c rçng ∆ ⊂ L(Rn , Rm ) c¸c to¸n tö tuyÕn tÝnh ®−îc gäi lµ mét ®¹i diÖn 20 cña ¸nh x¹ ®èi ®¹o hµm D∗ f (¯)(·) nÕu x (8.2) sup x∗ , u = sup A∗ y ∗ , u ∀u ∈ Rn , ∀y ∗ ∈ Rm . x∗ ∈D ∗ f (¯)(y ∗ ) x A∈∆ Do ®Þnh lý t¸ch c¸c tËp låi, (8.2) t−¬ng ®−¬ng víi ®iÒu kiÖn sau (8.3) coD∗ f (¯)(y ∗ ) = co{A∗ y ∗ : A ∈ ∆} x ∀y ∗ ∈ Rm . NÕu f lµ kh¶ vi chÆt t¹i x, th× ∆ := {f (¯)} lµ mét ®¹i diÖn cña ¸nh x¹ ®èi ¯ x ®¹o hµm D∗ f (¯)(·). x NÕu f : Rn → Rm lµ Lipschitz t¹i x, nghÜa lµ tån t¹i > 0 sao cho ¯ f (x ) − f (x) x − x víi mäi x, x ®−îc lÊy tïy ý trong mét l©n cËn cña x, khi ®ã tËp ¯ JB f (¯) = { lim f (xk ) : {xk } ⊂ Ωf , xk → x}, x ¯ k→∞ ®−îc gäi lµ B-®¹o hµm, lµ mét Jacobian xÊp xØ cña f t¹i x. ë ®©y ¯ Ωf = {x ∈ Rn : ∃ ®¹o hµm FrÐchet f (x) cña f t¹i x}. NhËn xÐt r»ng tËp lín h¬n J Cl f (¯) := co{ lim f (xk ) : {xk } ⊂ Ωf , xk → x} x ¯ k→∞ (Jacobian suy réng Clarke) cña cña f t¹i x, còng lµ Jacobian xÊp xØ cña f t¹i ¯ x. Trong tr−êng hîp m = 1, J ¯ Cl f (¯) = ∂ Cl f (¯) (xem Môc 5.2). x x MÖnh ®Ò 5.8.1. NÕu hµm f : Rn → Rm lµ Lipschitz ®Þa ph−¬ng t¹i x, th× tËp ¯ hîp ∆ := JB f (¯) lµ mét ®¹i diÖn cña ¸nh x¹ ®èi ®¹o hµm D∗ f (¯)(·). x x Chøng minh. Theo c«ng thøc (2.23) trong Mordukhovich (1994b), ta cã A∗ y ∗ : A ∈ J Cl f (¯) = coD∗ f (¯)(y ∗ ) x x ∀y ∗ ∈ Rm . V× J Cl f (¯) = coJB f (¯), tõ ®ã suy ra r»ng x x coD∗ f (¯)(y ∗ ) = co{A∗ y ∗ : A ∈ JB f (¯)}. x x 20 TNTA: representative.
196 5. HÖ bÊt ®¼ng thøc suy réng VËy (8.3) nghiÖm ®óng nÕu ta chän ∆ = JB f (¯). §iÒu ®ã chøng tá r»ng x ∆ = JB f (¯) lµ mét ®¹i diÖn cña ¸nh x¹ ®èi ®¹o hµm D∗ f (¯)(·). 2 x x MÖnh ®Ò 5.8.2. NÕu f lµ Lipschitz t¹i x vµ nÕu ∆ lµ mét ®¹i diÖn cña ¸nh x¹ ¯ ®èi ®¹o hµm D∗ f (¯)(·), th× Jf (¯) := ∆ lµ Jacobian xÊp xØ cña f t¹i x. x x ¯ Chøng minh. Gi¶ sö y ∗ ∈ Rm ®−îc cho tïy ý. Theo MÖnh ®Ò 2.11 trong Mordukhovich (1994b), ta cã (8.4) D∗ f (¯)(y ∗ ) = ∂(y ∗ ◦ f )(¯). x x V× y ∗ ◦ f lµ Lipschitz t¹i x, ¯ (y ∗ ◦ f )o (¯; u) = sup{ x∗ , u : x∗ ∈ ∂ Cl (y ∗ ◦ f )(¯)} ∀u ∈ Rn . x x KÕt hîp ®iÒu ®ã víi (7.4) vµ (8.4), ta thu ®−îc (y ∗ ◦ f )o (x; u) = sup{ x∗ , u : x∗ ∈ D∗ f (¯)(y ∗ )} x = sup{ A∗ y ∗ , u : A ∈ ∆}. Do ®ã, (y ∗ ◦ f )+ (¯; u) x (y ∗ ◦ f )o (x; u) = sup{ y ∗ , Au : A ∈ ∆}. V× tÝnh chÊt ®ã ®óng víi mäi y∗ ∈ Rm vµ u ∈ Rn , ta kÕt luËn r»ng Jf (¯) := ∆ x lµ Jacobian xÊp xØ cña f t¹i x. 2 ¯ Trong mèi liªn hÖ víi MÖnh ®Ò 5.8.2, chóng ta cã c©u hái tù nhiªn sau ®©y. C©u hái 2: Ph¶i ch¨ng nÕu f : Rn → Rm lµ hµm vÐct¬ liªn tôc vµ ∆ lµ mét ®¹i diÖn cña ¸nh x¹ ®èi ®¹o hµm D∗ f (¯)(·) : Rm ⇒ Rn , th× Jf (¯) := ∆ lµ x x Jacobian xÊp xØ cña f t¹i x? ¯ KÕt hîp mÖnh ®Ò sau víi mÖnh ®Ò 5.8.2 ta cã c©u tr¶ lêi kh¼ng ®Þnh cho C©u hái 2. MÖnh ®Ò 5.8.3. NÕu ¸nh x¹ ®èi ®¹o hµm D∗ f (¯)(·) : Rm ⇒ Rn cña hµm sè x liªn tôc f : Rn → Rm cã mét ®¹i diÖn Jf (¯) ⊂ L(Rn , Rm ), th× f lµ Lipschitz x ®Þa ph−¬ng t¹i x. ¯ Chøng minh. Tõ (8.3) suy ra r»ng coD∗ f (¯)(0) = {0}. V× vËy, D ∗ f (¯)(0) = x x {0}. Theo MÖnh ®Ò 2.8 trong Mordukhovich (1988), ®iÒu ®ã kÐo theo x → {f (x)} lµ ¸nh x¹ ®a trÞ gi¶-Lipschitz t¹i (¯, f (¯)). V× f lµ ¸nh x¹ ®¬n trÞ, ta cã f lµ x x Lipschitz ®Þa ph−¬ng t¹i x. 2 ¯ Chóng ta xÐt thªm vµi vÝ dô ë ®ã ta sÏ tÝnh d−íi vi ph©n Mordukhovich vµ ®èi ®¹o hµm cña c¸c hµm sè vµ ¸nh x¹ kh«ng tr¬n.
5.8. §èi ®¹o hµm Mordukhovich vµ Jacobian xÊp xØ 197 VÝ dô 5.8.1. Gi¶ sö hµm vÐct¬ f : R → R2 ®−îc x¸c ®Þnh bëi c«ng thøc f (x) = (|x|1/2 , −|x|) víi mäi x ∈ I . Khi ®ã f lµ hµm sè liªn tôc, kh«ng R Lipschitz t¹i 0, vµ gph f = {(x, |x|1/2 , −|x|) : x ∈ R}. Sö dông (7.3) vµ c«ng thøc tÝnh nãn ph¸p tuyÕn FrÐchet NΩ (x) ®· ®−îc nh¾c l¹i ë Môc 5.7, ta cã thÓ chøng tá r»ng Ngph f ((0, 0, 0)) = Ngph f ((0, 0, 0)) = R × (−∞, 0] × R. V× vËy, víi mçi y ∗ = (y1 , y2 ) ∈ R2 , ∗ ∗ ∗ R nÕu y1 0, D∗ f (0)(y ∗ ) = ∗ ∅ nÕu y1 < 0. V× f kh«ng lµ Lipschitz ®Þa ph−¬ng t¹i x = 0, MÖnh ®Ò 5.8.3 kh¼ng ®Þnh ¸nh x¹ ¯ ®èi ®¹o hµm D ∗ f (0)(·) kh«ng cã ®¹i diÖn d−íi d¹ng mét tËp to¸n tö tuyÕn tÝnh. Mét tÝnh to¸n trùc tiÕp cho thÊy r»ng, víi mçi y∗ = (y1 , y2 ) ∈ R2 vµ u ∈ I , ta ∗ ∗ R cã ⎧ ∗ ⎪ +∞ ⎪ nÕu y1 > 0, u = 0 ⎨ ∗ ∗ −|u|y2 nÕu y1 = 0 (y ∗ ◦ f )+ (0; u) = ∗ < 0, u = 0 ⎪ −∞ ⎪ nÕu y1 ⎩ ∗ 0 nÕu y1 < 0, u = 0. NÕu ta chän Jf (0) = (−∞, 0] × I , x = 0, vµ ®Æt Au = (αu, βu) víi R ¯ mäi A = (α, β) ∈ Jf (0), u ∈ I , th× (7.8) kh«ng ®−îc tháa m·n v× r»ng R sup y ∗ , Au = 0 nÕu y1 > 0, u > 0, y2 = 0, trong khi (y∗ ◦ f )+ (0; u) = ∗ ∗ A∈Jf (0) +∞. T−¬ng tù, nÕu ta chän Jf (0) = [0, +∞) × I vµ x = 0, th× (7.8) R ¯ ∗ ∗ > 0, u < 0, y ∗ = 0, kh«ng ®−îc tháa m·n v× sup y , Au = 0 nÕu y1 2 A∈Jf (0) trong khi (y∗ ◦ f )+ (0; u) = +∞. V× thÕ, c¸c tËp Jf (0) ®· chän ®Òu kh«ng ph¶i lµ Jacobian xÊp xØ cña f t¹i 0. MÆc dï vËy, tËp hîp kiÓu Jf (0) := {(−∞, −1] ∪ [2, +∞)} × I lµ mét Jacobian xÊp xØ cña f t¹i 0. R VÝ dô 5.8.2. XÐt hµm sè f : R → R2 cho bëi c«ng thøc f (x) = (−|x|1/3 , x1/3 ) víi mäi x ∈ I . Ta cã f lµ hµm sè liªn tôc, kh«ng Lipschitz ®Þa ph−¬ng t¹i 0, R vµ gph f = {(x, −|x|1/3 , x1/3 ) : x ∈ R}. ¸p dông c«ng thøc (7.3) vµ c«ng thøc ®Þnh nghÜa nãn ph¸p tuyÕn FrÐchet NΩ (x) ®−îc ®−a ra ngay tr−íc ®ã, ta cã thÓ chøng tá r»ng Ngph f ((0, 0, 0)) = Ngph f ((0, 0, 0)) = R × W, ë ®ã W = {y∗ = (y1 , y2 ) ∈ R2 : −y1 ∗ ∗ ∗ ∗ y2 ∗ y1 }. V× vËy, víi mçi y ∗ = (y1 , y2 ) ∈ R2 ta cã ∗ ∗ R ∗ ∗ nÕu y1 y2 −y1 ∗ D∗ f (0)(y ∗ ) = ∅ trong tr−êng hîp cßn l¹i.
198 5. HÖ bÊt ®¼ng thøc suy réng ¸nh x¹ ®èi ®¹o hµm D∗ f (0)(·) kh«ng cã ®¹i diÖn d−íi d¹ng mét tËp hîp to¸n tö tuyÕn tÝnh. Cã thÓ chøng tá r»ng, víi mäi y∗ = (y1 , y2 ) ∈ R2 vµ u ∈ I , ∗ ∗ R ⎧ ⎪0 ⎪ nÕu u=0 ⎪0 ⎪ ∗ ∗ ⎪ ⎪ nÕu y2 = y1 = 0, u=0 ⎪0 ⎪ ∗ ∗ y2 − y1 = 0, ⎪ ⎪ nÕu u>0 ⎨ ∗ − y ∗ > 0, ∗ +∞ nÕu y2 u>0 (y ◦ f ) (0; u) = + ∗ 1 ∗ ⎪ −∞ ⎪ nÕu y2 − y1 < 0, u>0 ⎪ ⎪0 ∗ + y ∗ = 0, ⎪ ⎪ nÕu y2 u 0, u 0 vµ x2 = 0, th× z ∈ Γ1 ∩ Γ2 . V× TΓ1 (z) = {(v1 , v2 , α) ∈ R3 : v2 0, 0 = v1 − v2 − α}, sö dông Bæ ®Ò Farkas (xem Rockafellar (1970), tr. 200) ta cã NΓ1 (z) = {(η1 , η2 , θ) = −λ(0, 1, 0) − µ(1, −1, −1) : λ 0, µ ∈ R}.
5.8. §èi ®¹o hµm Mordukhovich vµ Jacobian xÊp xØ 199 T−¬ng tù, NΓ2 (z) = {(η1 , η2 , θ) = −λ (0, −1, 0) − µ (1, 1, −1) : λ 0, µ ∈ R}. Do Ngph f (z) = NΓ1 (z) ∩ NΓ2 (z), ta suy ra r»ng Ngph f (z) = {(−µ, µ − λ, µ) : 2µ λ 0}. Râ rµng r»ng nãn ph¸p tuyÕn FrÐchet nµy kh«ng phô thuéc vµo vÞ trÝ cña z = 0 trªn nöa ®−êng th¼ng Γ1 ∩ Γ2 . NÕu x1 < 0 vµ x2 = 0, th× z ∈ Γ3 ∩ Γ4 . LËp luËn t−¬ng tù nh− trªn, ta thu ®−îc Ngph f (z) = {(µ, λ − µ, µ) : 2µ λ 0}. NÕu x1 = 0 vµ x2 > 0, th× z ∈ Γ1 ∩ Γ4 vµ Ngph f (z) = {(−λ − µ, µ, µ) : −2µ λ 0}. NÕu x1 = 0 vµ x2 < 0, th× z ∈ Γ2 ∩ Γ3 vµ Ngph f (z) = {(−λ − µ, −µ, µ) : −2µ λ 0}. NÕu x1 = 0 vµ x2 = 0, th× z = (¯, 0) ∈ Γ1 ∩ Γ2 ∩ Γ3 ∩ Γ4 . V× x TΓ1 (¯, 0) = {(v1 , v2 , α) : v1 x 0, v2 0, 0 = v1 − v2 − α}, do Bæ ®Ò Farkas ta cã NΓ1 ((¯, 0)) = {−λ1 (1, 0, 0)−λ2 (0, 1, 0)−µ(1, −1, −1) : λ1 x 0, λ2 0, µ ∈ I R}. LËp luËn t−¬ng tù, ta tÝnh ®−îc c¸c nãn ph¸p tuyÕn NΓi ((¯, 0)) (i = 2, 3, 4). x Khi ®ã, sö dông c«ng thøc 4 Ngph f (¯, 0) = x NΓi ((¯, 0)) x i=1 ta cã thÓ chøng tá r»ng Ngph f (¯, 0) = {(0, 0, 0)}. x KÕt hîp c¸c kÕt qu¶ ®· thu ®−îc víi c«ng thøc (2.3), ta cã Ngph f ((¯, 0)) = lim sup Ngph f (z) x z→(¯,0) x = cone{(1, −1, −1), (1, 1, −1), (−1, 1, −1), (−1, −1, −1)} ∪{(−µ, µ − λ, µ) : 2µ λ 0} ∪{(µ, λ − µ, µ) : 2µ λ 0} ∪{(−λ − µ, µ, µ) : −2µ λ 0} ∪{(−λ − µ, −µ, µ) : −2µ λ 0}.
200 5. HÖ bÊt ®¼ng thøc suy réng Tõ ®ã suy ra ⎧ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎪ {(y , −y ), (y , y ), (−y , y ), (−y , −y )} ⎪ ⎪ ⎪ ∪{(−λ∗ + y ∗ , −y ∗ ) : 2y ∗ λ∗ 0} ⎪ ⎪ ⎪ ∪{(−λ∗ + y ∗ , y ∗ ) : 2y ∗ λ∗ 0} ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ nÕu y∗ > 0, D ∗ f (¯)(y ∗ ) = {(y ∗ , −y ∗ ), (y ∗ , y ∗ ), (−y ∗ , y ∗ ), (−y ∗ , −y ∗ )} x ⎪ ⎪ ∪{(y ∗ , −y ∗ − λ∗ ) : −2y ∗ λ∗ 0} ⎪ ⎪ ⎪ ⎪ ∪{(−y ∗ , y ∗ + λ∗ ) : −2y ∗ λ∗ 0} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ nÕu y∗ < 0, ⎩ {(0, 0)} nÕu y∗ = 0. Nh− vËy, víi mçi y∗ , D ∗ f (0)(y ∗ ) lµ mét tËp comp¾c kh¸c rçng (th−êng lµ kh«ng låi). Còng b»ng ph−¬ng ph¸p trªn, ta thu ®−îc Nepi f ((¯, 0, ) = lim sup Nepi f (z) x z→(¯,0) x = cone{(1, −1, −1), (1, 1, −1), (−1, 1, −1), (−1, −1, −1)} ∪{(−λ − µ, µ, µ) : −2µ λ 0} ∪{(−λ − µ, −µ, µ) : −2µ λ 0}. Do ®ã, ∂f (¯) = {x∗ : (x∗ , −1) ∈ Nepi f ((¯, 0))} x x = {(1, −1), (1, 1), (−1, 1), (−1, −1)} ∪{(−λ∗ + 1, −1) : 2 λ∗ 0} ∪ {(−λ∗ + 1, 1) : 2 λ∗ 0} = {(λ∗ , 1) : −1 λ∗ 1} ∪ {(λ∗ , −1) : −1 λ∗ 1}. VËy ∂f (¯) lµ tËp comp¾c, kh«ng låi. TËp hîp nµy lµ d−íi vi ph©n J-L cña f t¹i x x. Tuy vËy, ®ã kh«ng ph¶i d−íi vi ph©n J-L tèi thiÓu, v× r»ng tËp hîp ¯ ∂ JL f (¯) := {(1, −1), (−1, 1)} x còng lµ mét d−íi vi ph©n J-L cña f t¹i x (xem Jeyakumar vµ Luc (1999)). ¯
Phô lôc A 201 Phô lôc A §Ò thi hÕt m«n gi¶i tÝch ®a trÞ ë ViÖn To¸n häc (Ngµy thi: 26/8/2002. Líp Cao häc kho¸ 8) Bµi 1 (3 ®iÓm). (a) Nªu ®Þnh nghÜa ¸nh x¹ ®a trÞ, ®å thÞ cña ¸nh x¹ ®a trÞ, miÒn h÷u hiÖu vµ tËp ¶nh cña ¸nh x¹ ®a trÞ. (b) X¸c ®Þnh c¸c tËp gph F, dom F , rge F víi F : R ⇒ I ®−îc cho bëi R c«ng thøc F (x) = co{sin x, cos x} ∀x ∈ R. (c) XÐt ph−¬ng tr×nh ®¹i sè xn + a1 xn−1 + . . . + an−1 x + an = 0, ë ®ã n 2 lµ sè nguyªn cho tr−íc vµ a = (a1 , . . . , an ) lµ vÐct¬ thùc. Ký hiÖu F (a) lµ tËp hîp c¸c nghiÖm phøc cña ph−¬ng tr×nh ®· cho. ¸nh x¹ F : Rn ⇒ C, a → F (a), cã ph¶i lµ ¸nh x¹ ®a trÞ - cã gi¸ trÞ kh¸c rçng? - cã gi¸ trÞ comp¾c? - cã gi¸ trÞ låi? - cã gi¸ trÞ ®ãng? - trµn (tøc lµ rge F = C)? (Gîi ý: LÇn l−ît chøng tá r»ng: (i) Víi n = 2 th× F lµ trµn, (ii) Víi n > 2 th× F lµ trµn.) Bµi 2 (2 ®iÓm). (a) Ph¸t biÓu kh¸i niÖm ¸nh x¹ ®a trÞ nöa liªn tôc trªn vµ ¸nh x¹ ®a trÞ nöa liªn tôc d−íi. Cho hai vÝ dô ®Ó chøng tá r»ng ®ã lµ hai kh¸i niÖm cã néi dung hoµn toµn kh¸c nhau. (b) Ph¸t biÓu vµ chøng minh ®Þnh lý vÒ sù b¶o tån tÝnh liªn th«ng t«p« qua ¸nh x¹ ®a trÞ nöa liªn tôc d−íi. Bµi 3 (2 ®iÓm). (a) Ph¸t biÓu ®Þnh lý ®iÓm bÊt ®éng Kakutani. (b) Cho c¸c vÝ dô thÝch hîp ®Ó chøng tá r»ng nÕu trong ph¸t biÓu cña ®Þnh lý ta bá ®i mét trong 4 ®iÒu kiÖn sau (nh−ng vÉn gi÷ nguyªn 3 ®iÒu kiÖn kia) th× kÕt luËn cña ®Þnh lý cã thÓ kh«ng cßn ®óng n÷a: (i) G lµ ¸nh x¹ ®a trÞ nöa liªn tôc trªn, (ii) G cã gi¸ trÞ låi, (iii) G cã gi¸ trÞ ®ãng, (iv) G cã gi¸ trÞ kh¸c rçng, ë ®ã G lµ ¸nh x¹ ®a trÞ ®−îc xÐt.
202 Phô lôc A Bµi 4 (2 ®iÓm). x b x (a) Ph¸t biÓu ®Þnh nghÜa c¸c nãn tiÕp tuyÕn TM (¯), TM (¯), CM (¯). Nªu x mèi quan hÖ gi÷a c¸c h×nh nãn ®ã vµ h×nh nãn cone(M − x). Nªu 3 vÝ dô (kh«ng ¯ b x b x cÇn tr×nh bµy c¸c tÝnh to¸n) ®Ó chøng tá r»ng CM (¯) = TM (¯), TM (¯) = x TM (¯), TM (¯) = cone(M − x). x x ¯ (b) Cho ¸nh x¹ ®a trÞ F : R ⇒ I , R F (x) = {y ∈ R : x2 + y 2 1, x − y + 1 0} ∀x ∈ R. - Hái F cã ph¶i lµ ¸nh x¹ ®a trÞ låi hay kh«ng? - TÝnh c¸c tËp Tgph F (¯) vµ Tgph F (z), ë ®ã z = (−1, 0) vµ z = (0, 1). z ¯ - ViÕt c«ng thøc cña c¸c ®¹o hµm DFz , DFz , CFz , vµ CFz . Hái nh÷ng ¯ ¯ ®¹o hµm ®ã cã ph¶i c¸c qu¸ tr×nh låi ®ãng hay kh«ng? cã ph¶i lµ c¸c ¸nh x¹ trµn hay kh«ng? Bµi 5 (1 ®iÓm). Chän gi¶i mét trong hai bµi tËp sau: 1. Cho X, Y lµ c¸c kh«ng gian t«p«, F : X ⇒ Y lµ ¸nh x¹ ®a trÞ nöa liªn tôc trªn ë trong X. Chøng minh r»ng nÕu dom F lµ tËp comp¾c vµ F lµ ¸nh x¹ cã gi¸ trÞ comp¾c, th× rge F lµ tËp comp¾c. 2. Cho X, Y lµ c¸c kh«ng gian t«p«, F : X ⇒ Y lµ ¸nh x¹ ®a trÞ cã ®å thÞ ®ãng. Chøng minh r»ng F (x) lµ tËp ®ãng víi mäi x ∈ X.
Phô lôc B 203 Phô lôc B §Ò thi hÕt m«n gi¶i tÝch ®a trÞ ë §¹i häc S− ph¹m Tp. Hå ChÝ Minh (Ngµy thi: 28/8/2003. Líp Sinh viªn chän, §HSP Tp. Hå ChÝ Minh) Bµi 1 (2 ®iÓm). Cho ¸nh x¹ ®a trÞ F : R ⇒ R, F (x) = {y ∈ R : y x3 }. (a) X¸c ®Þnh c¸c tËp dom F vµ rge F . (b) F cã ph¶i lµ ¸nh x¹ ®a trÞ låi hay kh«ng? (c) F cã ph¶i lµ ¸nh x¹ ®a trÞ ®ãng (tøc lµ ¸nh x¹ cã ®å thÞ ®ãng) hay kh«ng? (d) ViÕt c«ng thøc tÝnh tËp F −1 (y) víi y ∈ I . R (e) X¸c ®Þnh tËp hîp gph (F −1 ◦ F ). TÝnh tËp (F −1 ◦ F )(x) víi x ∈ I . R Bµi 2 (2 ®iÓm). Cho M = {x = (x1 , x2 ) ∈ R2 : x1 + x2 2, x2 x3 }, 1 x = (1, 1). ¯ TÝnh h×nh nãn Bouligand TM (¯). Gäi G : R ⇒ I lµ ¸nh x¹ ®a trÞ cã ®å thÞ x R trïng víi h×nh nãn TM (¯) ®ã. X¸c ®Þnh c¸c tËp dom G vµ rge G. x Bµi 3 (2 ®iÓm). Cho X, Y lµ c¸c kh«ng gian t«p«, F : X ⇒ Y lµ ¸nh x¹ ®a trÞ. Chøng minh r»ng nÕu (i) dom F lµ tËp liªn th«ng, (ii) F (x) lµ tËp liªn th«ng víi mäi x ∈ dom F , vµ (iii) F lµ nöa liªn tôc d−íi ë trong X, th× rge F lµ tËp liªn th«ng. Bµi 4 (1 ®iÓm). Cho X, Y lµ c¸c kh«ng gian tuyÕn tÝnh, A : X → Y lµ ¸nh x¹ tuyÕn tÝnh, K ⊂ Y lµ h×nh nãn låi. Chøng minh r»ng F : X ⇒ Y cho bëi c«ng thøc F (x) = Ax + K (x ∈ X) lµ ¸nh x¹ ®a trÞ låi. Chøng minh r»ng F lµ ¸nh x¹ ®a trÞ thuÇn nhÊt d−¬ng, tøc lµ F (λx) = λF (x) (∀x ∈ X, ∀λ 0). Bµi 5 (1 ®iÓm). Cho X, Y lµ c¸c kh«ng gian t«p«, F : X ⇒ Y lµ ¸nh x¹ ®a trÞ cã ®å thÞ ®ãng. Chøng minh r»ng F (x) lµ ®ãng víi mäi x ∈ X. Bµi 6 (1 ®iÓm). Cho X, Y lµ c¸c kh«ng gian t«p«, F : X ⇒ Y lµ ¸nh x¹ ®a trÞ nöa liªn tôc trªn ë trong X. Chøng minh r»ng nÕu dom F lµ tËp comp¾c vµ F lµ ¸nh x¹ ®a trÞ cã gi¸ trÞ comp¾c th× rge F lµ tËp comp¾c. Bµi 7 (1 ®iÓm). Cho X, Y , Z lµ c¸c kh«ng gian ®Þnh chuÈn, F : X ⇒ Y vµ F : Y ⇒ Z lµ c¸c ¸nh x¹ ®a trÞ låi. Chøng minh r»ng G ◦ F : X ⇒ Z lµ ¸nh x¹ ®a trÞ låi. L−u ý: NÕu sè ng−êi gi¶i ®−îc c¸c c©u 5-7 kh«ng nhiÒu, th× ®iÓm cho c¸c c©u nµy sÏ ®−îc nh©n ®«i.
204 Phô lôc B
Tµi liÖu tham kh¶o 205 Tµi liÖu tham kh¶o 1. J.-P. Aubin (1981), Contingent derivatives of set-valued maps and ex- istence of solutions to nonlinear inclusions and differential inclusions, Advances in Mathematics, Supplementary studies (L. Nachbin, Ed.), 160– 232. 2. J.-P. Aubin (1984), Lipschitz behavior of solutions to convex minimization problems, Mathematics of Operations Research Vol. 9, 87–111. 3. J.-P. Aubin and A. Cellina (1984), Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin-Heidelberg. 4. J.-P. Aubin and I. Ekeland (1984), Applied Nonlinear Analysis, John Wiley & Sons, Wiley-Interscience. 5. J.-P. Aubin and H. Frankowska (1987), On inverse function theorem for set-valued maps, Journal de Math´ matiques Pures et Appliqu´ es Vol. 66, e e 71–89. 6. J.-P. Aubin and H. Frankowska (1990), Set-Valued Analysis, Birkhauser, Berlin. 7. A. Auslender (1979), Differential stability in nonconvex and nondifferen- tiable programming, Mathematical Programming Study Vol. 10, 29–41. 8. A. Auslender and M. Teboulle (2003), Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer, New York. 9. C. Berge (1959), Espaces topologiques: Fonctions multivoques, Dunod, Paris. 10. J. F. Bonnans and A. Shapiro (2000), Perturbation Analysis of Optimization Problems, Springer, New York. 11. J. M. Borwein (1986), Stability and regular points of inequality systems, Journal of Optimization Theory and Applications Vol. 48, 9–52. 12. J. M. Borwein and Q. J. Zhu (2005), Techniques of Variational Analysis, Springer, New York. 13. J. M. Borwein and D. M. Zhuang (1988), Verifiable necessary and suffi- cient conditions for regularity of set-valued and single-valued maps, Jour- nal of Mathematical Analysis and Applications Vol. 134, 441–459.
206 Tµi liÖu tham kh¶o 14. G. Bouligand (1930), Sur les surfaces dÐpourvues de points hyperlimits, Ann. Soc. Polon. Math. Vol. 9, 32–41. 15. C. Castaing and M. Valadier (1977), Convex Analysis and Measurable Functions, Springer-Verlag. 16. NguyÔn Huy Chiªu (2004), Sù tån t¹i l¸t c¾t ®Æc biÖt cña ¸nh x¹ ®a trÞ vµ kh¸i niÖm tÝch ph©n Aumann, LuËn v¨n Th¹c sÜ to¸n häc, §¹i häc Vinh, 2004. 17. N. H. Chieu (2006a), A Newton-Leibniz formula for the integration of the Clarke subdifferential mapping (b¶n th¶o ®· göi ®¨ng). 18. N. H. Chieu (2006b), The contingent cone of the product of two sequential sets in the real line (b¶n th¶o ®· göi ®¨ng). 19. N. H. Chieu (2006c), Integral of subdifferential mappings and subdiffer- ential of integral functionals (b¶n th¶o ®· göi ®¨ng). 20. F. H. Clarke (1983), Optimization and Nonsmooth Analysis, Wiley, New York. 21. B. D. Craven (1978), Mathematical Programming and Control Theory, Chapman and Hall, London. 22. P. H. Dien (1982), Locally Lipschitzian set-valued maps and generalized extremal problems, Acta Mathematica Vietnamica Vol. 8, 109–122. 23. P. H. Dien (1985), On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints, Applied Mathematics and Optimization Vol. 13, 151–161. 24. P. H. Dien and P. H. Sach (1989), Further properties of the regularity of inclusion systems, Nonlinear Analysis Vol. 13, 1251–1267. 25. P. H. Dien and N. D. Yen (1991), On implicit function theorems for set- valued maps and their applications to mathematical programming under inclusion constraints, Applied Mathematics and Optimization Vol. 24, 35–54. 26. A. L. Donchev and R. T. Rockafellar (1996), Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM Journal on Optimization Vol. 6, 1087–1105. 27. I. Ekeland (1974), On the variational principle, Journal of Mathematical Analysis and Applications Vol. 47, 324–353.
Tµi liÖu tham kh¶o 207 28. J. Gauvin (1979), The generalized gradient of a marginal function in math- ematical programming, Mathematics of Operations Research Vol. 4, 458– 463. 29. J. Gauvin and F. Dubeau (1982), Differential properties of the marginal function in mathematical programming, Mathematical Programming Study Vol. 19, 101–119. 30. J. Gauvin and F. Dubeau (1984), Some examples and counterexamples for the stability analysis of nonlinear programming problems, Mathematical Programming Study Vol. 21, 69–78. 31. J. Gauvin and J. W. Tolle (1977), Differential stability in nonlinear pro- gramming, SIAM Journal on Control and Optimization Vol. 15, 294–311. 32. B. Gollan (1984), On the marginal function in nonlinear programming, Mathematics of Operations Research Vol. 9, 208–221. 33. V. V. Gorokhovik and P. P. Zabreiko (2005), On Frechet differentiability ´ of multifunctions, Optimization Vol. 54, 391–409. 34. T. X. D. Ha (2005), Lagrange multipliers for set-valued problems asso- ciated with coderivatives, Journal of Mathematical Analysis and Applica- tions Vol. 311, 647–663. 35. R. B. Holmes (1974), Geometric Functional Analysis and Its Applications, Springer. 36. A. D. Ioffe (2000), Codirectional compactness, metric regularity and sub- differential calculus, Canadian Mathematical Society Conference Proceed- ings Vol. 27, 123–163. 37. A. D. Ioffe and V. M. Tihomirov (1979), Theory of Extremal Problems, North-Holland Publishing Company. 38. V. Jeyakumar and D. T. Luc (1998), Approximate Jacobian matrices for nonsmooth continuous maps and C1 -optimization, SIAM Journal on Con- trol and Optimization Vol. 36, 1815–1832. 39. V. Jeyakumar and D. T. Luc (1999), Nonsmooth calculus, minimality, and monotonicity of convexificators, Journal of Optimization Theory and Applications Vol. 101, 599–621. 40. V. Jeyakumar and D. T. Luc (2002a), An open mapping theorem using unbounded generalized Jacobians, Nonlinear Analysis Vol. 50, 647–663.
208 Tµi liÖu tham kh¶o 41. V. Jeyakumar and D. T. Luc (2002b), Convex interior mapping theorems for continuous nonsmooth functions and optimization, Journal of Nonlinear and Convex Analysis Vol. 3, 251–266. 42. V. Jeyakumar and X. Wang (1999), Approximate Hessian matrices and second-order optimality conditions for nonlinear programming problems with C 1 -data, Journal of the Australian Mathematical Society Series B Vol. 40, 403–420. 43. V. Jeyakumar and N. D. Yen (2004), Solution stability of nonsmooth con- tinuous systems with applications to cone-constrained optimization, SIAM Journal on Optimization Vol. 14, 1106–1127. 44. A. Jourani (2000), Hoffman’s error bound, local controllability, and sen- sitivity analysis, SIAM Journal on Control and Optimization Vol. 38, 947–970. 45. J. L. Kelley (1957), General Topology, D. Van Nostrand Company, New York. 46. P. K. Khanh (1986), An induction theorem and general open mapping theorems, Journal of Mathematical Analysis and Applications Vol. 118, 519–534. 47. P. K. Khanh (1988), An open mapping theorem for families of multi- functions, Journal of Mathematical Analysis and Applications Vol. 132, 491–498. 48. P. K. Khanh (1989), On general open mapping theorems, Journal of Math- ematical Analysis and Applications Vol. 144, 305–312. 49. B. T. Kien, J.-C. Yao and N. D. Yen (2007), On the solution existence of pseudomonotone variational inequalities, Journal of Global Optimization (®· ®−îc nhËn ®¨ng). 50. A. Ja. Kruger and B. Mordukhovich (1980), Extremal points and the Euler equation in nonsmooth optimization problems (in Russian), Dokl. Akad. Nauk BSSR Vol. 24, 684–687 (tiÕng Nga). 51. G. M. Lee, N. N. Tam and N. D. Yen (2005), Quadratic Programming and Affine Variational Inequalities: A Qualitative Study, Series: “Nonconvex Optimization and its Applications”, Vol. 78, Springer, New York. 52. D. T. Luc (1989), Theory of Vector Optimization, Lecture Notes in Eco- nomics and Mathematical Systems Vol. 319, Springer, Berlin-Heidelberg.
Tµi liÖu tham kh¶o 209 53. D. T. Luc (2003), A Multiplier rule for multiobjective programming prob- lems with continuous data, SIAM Journal on Optimization Vol. 13, 168– 178. 54. D. T. Luc and C. Malivert (1992), Invex optimisation problems, Bulletin of the Australian Mathematical Society Vol. 46, 47–66. 55. Y. Lucet and J. J. Ye (2001, 2002), Sensitivity analysis of the value function for optimization problems with variational inequality constraints, SIAM Journal on Control and Optimization Vol. 40, 699–723; Erratum. SIAM Journal on Control and Optimization Vol. 41, 1315–1319. 56. Z. Q. Luo, J.-S. Pang and D. Ralph (1996), Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK. 57. O. L. Mangasarian and T. H. Shiau (1987), Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems, SIAM Journal on Control and Optimization Vol. 25, 583–595. 58. H. Maurer and J. Zowe (1979), First and second-order necessary and suf- ficient optimality conditions for infinite-dimensional programming prob- lems, Mathematical Programming Vol. 16, 98–110. 59. B. S. Mordukhovich (1976), Maximum principle in the problem of time response with nonsmooth constraints, Journal of Applied Mathematics and Mechanics Vol. 40, 960–969. 60. B. S. Mordukhovich (1988), Approximation Methods in Problems of Op- timization and Control (in Russian), Nauka, Moscow. 61. B. S. Mordukhovich (1992), Sensitivity analysis in nonsmooth optimization, in “Theoretical Aspects of Industrial Design” (D. A. Field and V. Komkov, Eds.), pp. 32–46, SIAM Publications. 62. B. S. Mordukhovich (1993), Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Transactions of the American Mathematical Society Vol. 340, 1–36. 63. B. S. Mordukhovich (1994a), Lipschitzian stability of constraint systems and generalized equations, Nonlinear Analysis, Theory, Methods & Ap- plications Vol. 22, 173–206. 64. B. S. Mordukhovich (1994b), Generalized differential calculus for non- smooth and set-valued mappings, Journal of Mathematical Analysis and Applications Vol. 183, 250–288.
210 Tµi liÖu tham kh¶o 65. B. S. Mordukhovich (1994c), Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Transac- tions of the American Mathematical Society Vol. 343, 609–658. 66. B. S. Mordukhovich (1994d), Sensitivity analysis for constraint and vari- ational systems by using set-valued differentiation, Optimization Vol. 31, 13–46. 67. B. S. Mordukhovich (2006a), Variational Analysis and Generalized Dif- ferentiation, I: Basic Theory, Springer. 68. B. S. Mordukhovich (2006b), Variational Analysis and Generalized Dif- ferentiation, II: Applications, Springer. 69. B. S. Mordukhovich and N. M. Nam (2005a), Variational stability and marginal functions via generalized differentiation, Mathematics of Oper- ations Research Vol. 30, 800–816. 70. B. S. Mordukhovich and N. M. Nam (2005b), Subgradient of distance functions with some applications to Lipschitzian stability, Mathematical Progrgamming Vol. 104, 635–668. 71. B. S. Mordukhovich and N. M. Nam (2006), Subgradients of distance functions at out-of-state points, Taiwanese Journal of Mathematics Vol. 10, 299–326. 72. B. S. Mordukhovich, N. M. Nam and N. D. Yen (2006), Frechet subdif- ´ ferential calculus and optimality conditions in nondifferentiable program- ming, Optimization Vol. 55, 685–708. 73. B. S. Mordukhovich, N. M. Nam and N. D. Yen (2007), Subgradients of marginal functions in parametric mathematical programming, Mathemat- ical Programming (®· ®−îc nhËn ®¨ng). 74. B. S. Mordukhovich and Y. Shao (1995), Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces, Nonlinear Analysis Vol. 25, 1401–1424. 75. B. S. Mordukhovich and Y. Shao (1996a), Nonsmooth analysis in Asplund spaces, Transactions of the American Mathematical Society Vol. 348, 1230–1280. 76. B. S. Mordukhovich and Y. Shao (1996b), Nonconvex differential calculus for infinite-dimensional multifunctions, Set-Valued Analysis Vol. 4, 205– 236.
Tµi liÖu tham kh¶o 211 77. N. M. Nam and N. D. Yen (2007), Relationships between approximate Jacobians and coderivatives, Journal of Nonlinear and Convex Analysis (®· ®−îc nhËn ®¨ng). 78. H. V. Ngai, D. T. Luc and M. Thera (2000), Approximate convex functions, Journal of Nonlinear and Convex Analysis Vol. 1, 155–176. 79. J. V. Outrata, M. Kocvara and J. Zowe (1998), Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer, Dordrecht, The Netherlands. 80. J.-P. Penot (1989), Metric regularity, openness, and Lipschitzian behavior of multifunctions, Nonlinear Analysis, Theory, Methods & Applications Vol. 13, 629–643. 81. R. R. Phelps (1993), Convex Functions, Monotone Operators and Differ- entiability, 2nd Edition, Springer, Berlin. 82. H. T. Phung and P. H. Dien (1991), Solving nonsmooth inclusions in the convex case, Z. Oper. Res. Vol. 35, 401–424. 83. S. M. Robinson (1976a), Regularity and stability for convex multivalued functions, Mathematics of Operations Research Vol. 1, 130–143. 84. S. M. Robinson (1976b), Stability theory for systems of inequalities, Part 2: Differentiable nonlinear systems, SIAM Journal on Numerical Analysis Vol. 13, 497–513. 85. S. M. Robinson (1979), Generalized equations and their solutions, Part I: Basic theory, Mathematical Programming Study Vol. 10, 128–141. 86. S. M. Robinson (1981), Some continuity properties of polyhedral multi- functions, Mathematical Programming Study Vol. 14, 206–214. 87. R. T. Rockafellar (1970), Convex Analysis, Princeton University Press, Princeton, New Jersey. 88. R. T. Rockafellar (1982), Lagrange multipliers and subderivatives of op- timal value functions in nonlinear programming, Mathematical Program- ming Study Vol. 17, 28–66. 89. R. T. Rockafellar (1985), Extensions of subgradient calculus with appli- cations to optimization, Nonlinear Analysis Vol. 9, 665–698. 90. R. T. Rockafellar and R. J-B. Wets (1998), Variational Analysis, Springer- Verlag, Berlin-Heidelberg.
212 Tµi liÖu tham kh¶o 91. W. Rudin (1976), Principles of Mathematical Analysis, Third Edition, McGraw-Hill. 92. W. Rudin (1987), Real and Complex Analysis, Third Edition, McGraw- Hill. 93. W. Rudin (1991), Functional Analyis, Second Edition, McGraw-Hill. 94. P. H. Sach (1988a), Differentiability of set-valued maps in Banach spaces, Mathematische Nachrichten Vol. 139, 215–235. 95. P. H. Sach (1988b), Regularity, calmness and support principle, Optimiza- tion Vol. 19, 13–27. 96. P. H. Sach (1996), Sufficient conditions for generalized convex set-valued maps, Optimization Vol. 37, 293–304. 97. P. H. Sach and N. D. Yen (1997), Convexity criteria for set-valued maps, Set-Valued Analysis Vol. 5, 37–45. 98. F. Severi (1930), Su ancune questioni di topologia infinitesimale, Ann. Soc. Polon. Math. Vol. 9, 97–108. 99. NguyÔn Xu©n TÊn vµ NguyÔn B¸ Minh (2006), Mét sè vÊn ®Ò trong lý thuyÕt tèi −u vÐct¬ ®a trÞ, Nhµ xuÊt b¶n Gi¸o dôc, Hµ Néi. 100. L. Thibault (1991), On subdifferentials of optimal value functions, SIAM Journal on Control and Optimization Vol. 29, 1019–1036. 101. L. Thibault and D. Zagrodny (1995), Integration of subdifferentials of lower semicontinuous functions on Banach spaces, Journal of Mathemat- ical Analysis and Applications Vol. 189, 33–58. 102. Hoµng Tôy (2003), Hµm thùc vµ Gi¶i tÝch hµm (Gi¶i tÝch hiÖn ®¹i), NXB ®¹i häc Quèc gia Hµ Néi. 103. C. Ursescu (1975), Multifunctions with convex closed graph, Cechoslovak Mathematical Journal Vol. 25, 438–441. 104. D. W. Walkup and R. J.-B. Wets (1969), A Lipschitzian characterization of convex polyhedra, Proceedings of the American Mathematical Society Vol. 23, 167–173. 105. X. Wang (2000), A Generalized Jacobian and Nonsmooth Optimization, Ph. D. Thesis, University of New South Wales, Sydney.
Tµi liÖu tham kh¶o 213 106. X. Wang and V. Jeyakumar (2000), A Sharp Lagrange multiplier rule for nonsmooth mathematical programming problems involving equality constraints, SIAM Journal on Optimization Vol. 10, 1136–1148. 107. A. R. Warburton (1983), Quasiconcave vector maximization: Connected- ness of the sets of Pareto-optimal and weak Pareto-optimal alternatives, Journal of Optimization Theory and Applications Vol. 40, 537–557. 108. Z. Wu and J. J. Ye (2000), Some results on integration of subdifferentials, Nonlinear Analyis Vol. 39, 955–976. 109. J. J. Ye (2001), Multiplier rules under mixed assumptions of differentia- bility and Lipschitz continuity, SIAM Journal on Optimization Vol. 39, 1441–1460. 110. N. D. Yen (1987), Implicit function theorems for set-valued maps, Acta Mathematica Vietnamica Vol. 12, No. 2, 17–28. 111. N. D. Yen (1997), Stability of the solution set of perturbed nonsmooth inequality systems and application, Journal of Optimization Theory and Applications Vol. 93, 199–225. 112. E. Zeidler (1986), Nonlinear Functional Analysis and Its Applications, I. Fixed-Point Theorems, Springer-Verlag, Berlin.
214 Tµi liÖu tham kh¶o