Giải tích đa trị P6

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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...

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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
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