❈❤➢➡♥❣ ✶ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ✭❤÷✉ ❤➵♥ ❝❤✐Ị✉✮ ✶✳✶ ❳➞② ❞ù♥❣ ❜❐❝ ❝đ❛ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ n ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥ tr♦♥❣ R , n ✭❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝đ❛ ❜✐➟♥ ∂Ω✮ tr♦♥❣ R ➤è✐ ✈í✐ t❐♣ Ω ❝ã t❤Ĩ ❤×♥❤ ❞✉♥❣ ❇❐❝ ❝đ❛ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ t➵✐ ♠ét ➤✐Ĩ♠ y f : Ω → Rn , tr♦♥❣ ➤ã Ω f (x) = y tr♦♥❣ Ω ◆Õ✉ ❜❐❝ ❜➺♥❣ t❤× t❛ ❝❤➢❛ ❦Õt ❧✉❐♥ ➤➢ỵ❝ ♥❤✐Ị✉✱ ♥❤➢♥❣ ♥Õ✉ ❜❐❝ ❧➭ ♠ét sè ❦❤➳❝ ✭❝❤➻♥❣ ❤➵♥ ❧➭ sè ❧❰✮ t❤× t❛ ❝❤➽❝ ❝❤➽♥ ♣❤➢➡♥❣ tr×♥❤ f (x) = y ❝ã ♥❣❤✐Ư♠ tr♦♥❣ Ω ♥❤➢ ❧➭ sè ✭➤➵✐ sè✮ ❝➳❝ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ❈ã ♥❤✐Ị✉ ❝➳❝❤ ➤Ĩ ①➞② ❞ù♥❣ ❧ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r✱ ❝❤➻♥❣ ❤➵♥ t➠♣➠ ➤➵✐ sè ❤♦➷❝ ❣✐➯✐ tÝ❝❤✳ ë ➤➞②✱ ❝❤ó♥❣ t➠✐ sÏ ❞ï♥❣ ❝➠♥❣ ❝ơ ❣✐➯✐ tÝ❝❤✱ ♠➭ ❝ơ tể ị ý ợ ị ý ➮♥✱ ➜Þ♥❤ ❧ý ❙❛r❞✱ ➜Þ♥❤ ❧ý ❙❝❤✇❛r③ ✭✈Ị ✈✐Ư❝ ➤ỉ✐ t❤ø tù ❧✃② ➤➵♦ ❤➭♠ r✐➟♥❣✮✳ ➜Ó ①➞② ❞ù♥❣ n ❧ý t❤✉②Õt ❜❐❝ ❝❤♦ ➳♥❤ ①➵ ❧✐➟♥ tô❝ f : Ω → R , t➵✐ ♠ét ➤✐Ó♠ y ✭❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝ñ❛ n ❜✐➟♥ ∂Ω✮ tr♦♥❣ R t❛ ❝❤✐❛ ❧➭♠ ❜❛ ❜➢í❝ ♥❤➢ s❛✉✿ • ¯ Rn ) t➵✐ ➤✐Ó♠ y C (Ω; ♥ã ❦❤➠♥❣ ❝❤ø❛ ➤✐Ó♠ ♥➭♦ ó t ó 0, t ị ĩ ❜❐❝ ❝❤♦ t❛ ➤Þ♥❤ ♥❣❤Ü❛ ❜❐❝ ❝❤♦ ➯♥❤ ❝đ❛ ❜✐➟♥ • ✶✳✶✳✶ ❧➭ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ♠➭ ♥❣❤Þ❝❤ ➯♥❤ ❝đ❛ f ❧➭ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ¯ Rn ) C (Ω; t➵✐ ➤✐Ó♠ y ❦❤➠♥❣ ♥➺♠ tr♦♥❣ f ❧➭ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ¯ Rn ) C(Ω; t➵✐ ➤✐Ĩ♠ y ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ∂Ω, t❛ ➤Þ♥❤ ♥❣❤Ü❛ ❜❐❝ ❝❤♦ ➯♥❤ ❝ñ❛ ❜✐➟♥ f ∂Ω ❳➞② ❞ù♥❣ ❜❐❝ ❝ñ❛ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ¯ Rn ) C (Ω; ¯ Rn )✳ ❚❛ ❦ý ❤✐Ö✉ Ω ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥ tr♦♥❣ Rn ✈➭ f ❧➭ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ C (Ω; S = {x ∈ Ω Jf (x) = 0}✱ t❐♣ ♥Õ♣ ✭❝r❡❛s❡✮ ❝ñ❛ ➳♥❤ ①➵ f ✱ t❐♣ ❣å♠ ♥❤÷♥❣ ➤✐Ĩ♠ x ♥➺♠ tr♦♥❣ Ω ♠➭ ❏❛❝♦❜✐❡♥ ❝ñ❛ ➳♥❤ ①➵ f t➵✐ ➤ã ❜➺♥❣ 0✳ ó ỗ ể y tr f (S) ❝ò♥❣ ♥❤➢ f (∂Ω) t❤× t❐♣ f −1 (y) ỉ ữ tử t ì ❣✐➯ sö ❦❤➠♥❣ −1 ♣❤➯✐ ✈❐②✱ ❞♦ f (y) ⊂ Ω ❧➭ t❐♣ ➤ã♥❣✱ ❜Þ ❝❤➷♥ ✭❝♦♠♣❛❝t✮ ♥➟♥ ♥ã ❝ã ♠ét ❞➲② {xn }∞ n=1 ✭❣å♠ ❈❤♦ ✷ ❝➳❝ ➤✐Ó♠ ♣❤➞♥ ❜✐Ưt✮✱ ❤é✐ tơ ➤Õ♥ x0 ∈ f −1 (y) ❈ã f (x0 ) = y = f (xn ), ✈➭ Jf (x0 ) = ❤❛② ∃||(f (x0 ))−1 ||−1 ❉♦ ➤ã✱ ||f (xn ) − f (x0 ) − f (x0 )(xn − x0 )|| =0 n→∞ ||xn − x0 || ||(f (x0 ))−1 ||−1 ≤ lim ✭✈➠ ❧ý✮ ¯ Rn ) t➵✐ ➤✐Ó♠ y ❦❤➠♥❣ ♥➺♠ tr♦♥❣ C (Ω; ➯♥❤ ❝đ❛ ❜✐➟♥ ∂Ω, ✈➭ ♥❣❤Þ❝❤ ➯♥❤ ❝đ❛ ♥ã ❦❤➠♥❣ ❝❤ø❛ ➤✐Ĩ♠ ♥➭♦ ♠➭ ❏❛❝♦❜✐❡♥ t➵✐ ➤ã ❜➺♥❣ ❚õ ➤ã t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ ❜❐❝ ❝đ❛ f ❧➭ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ Ω ⊂ Rn ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥✱ ✈➭ y ∈ Rn \ (f (S) ∪ f (∂Ω)), f ∈ ¯ Rn ) ❇❐❝ ❝đ❛ ➳♥❤ ①➵ f ➤è✐ ✈í✐ ♠✐Ị♥ Ω t ể y ợ ị s C (Ω; ➜Þ♥❤ ♥❣❤Ü❛ ✶✳ ❈❤♦ x∈f −1 (y) deg(f, Ω, y) := sgn Jf (x), 0, ♥Õ✉f −1 f −1 ♥Õ✉ (y) = ∅, (y) = ∅ ❇➺♥❣ ➤Þ♥❤ ♥❣❤Ü❛ t❛ ❝ã t❤Ĩ tÝ♥❤ ❜❐❝ ❝đ❛ ♠ét sè ➳♥❤ ①➵ ➤➷❝ ❜✐Ưt ✭t✉②Õ♥ tÝ♥❤✮ s❛✉✳ ❱Ý ❞ơ ✶✳ ❇❐❝ ❝đ❛ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t I : Ω → Rn , Ix = x, ➤è✐ ✈í✐ t❐♣ Ω t➵✐ ➤✐Ĩ♠ y ∈ Rn deg(I, Ω, y) := 1, 0, y ∈ Ω, ♥Õ✉ y ∈ Ω ♥Õ✉ ✷✳ ❇❐❝ ❝ñ❛ ➳♥❤ ①➵ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ s✉② ❜✐Õ♥ n R ❧➭ ❱Ý ❞ô y∈ deg(T, Ω, y) := ❱Ý ❞ô s❛✉ ➤➞② sÏ ❝❤♦ t❛ t❤✃② ❧➭ sgn(detT ), 0, T : Ω → Rn , ➤è✐ ✈í✐ t❐♣ Ω t➵✐ ➤✐Ĩ♠ y ∈ Ω, ♥Õ✉ y ∈ Ω ♥Õ✉ deg(f, Ω, y) = ♥❤➢♥❣ ♣❤➢➡♥❣ tr×♥❤ f (x) = y ✈➱♥ ❝ã ✭sè ❝❤➼♥✮ ♥❣❤✐Ö♠✳ f : (−1, 1) → R, f (x) = x2 − , (0 < < 1)✳ ❈ã f (x) = 2x, ∈ f ({−1, 1}) ∪ f ({0}) ✈➭ f −1 (0) = {− , } = ∅, ♥❤➢♥❣ deg(f, (−1, 1), 0) = 0✳ ❱Ý ❞ơ ✸✳ ❈❤♦ ➳♥❤ ①➵ ➜Ĩ ❝❤✉②Ĩ♥ s❛♥❣ ❜➢í❝ t❤ø ❤❛✐ t❛ ❝➬♥ ➤Õ♥ ♠ét ❝➳❝❤ ①➳❝ ➤Þ♥❤ ❦❤➳❝ ❝đ❛ ❜❐❝✳ ▼Ư♥❤ ➤Ị ✶✳ ❈❤♦ Ω ⊂ Rn ¯ Rn ) ❑❤✐ ➤ã✱ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ C (Ω; 0< < 0, y ∈ Rn \ (f (S) ∪ f (∂Ω)), f ∈ s❛♦ ❝❤♦ ✈í✐ ❜✃t ❦ú , ϕ ♥➭♦ ♠➭ ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥✱ ✈➭ ϕ ∈ C0∞ (Rn ; R), s✉♣♣ ϕ ⊂ B(0, ) ϕ (x)dx = ✈➭ Rn t❤× ϕ (f (x) − y)Jf (x)dx deg(f, Ω, y) = Ω ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ✸ −1 ¯ t❤× ❞♦ Ω ¯ ❧➭ t❐♣ ❝♦♠♣❛❝t ✭➤ã♥❣✱ ❜Þ ❝❤➷♥ ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ f (y) = ∅, ❤❛② y ∈ f (Ω), n ¯ ❧➭ t❐♣ ❝♦♠♣❛❝t✱ ♥➟♥ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ < ρ(y, f (Ω)) ¯ ❑❤✐ ➤ã✱ ✈í✐ tr♦♥❣ R ✮ ❝ã f (Ω) 0< < 0, x ∈ Ω ❝ã ϕ (f (x) − y) = ❉♦ ➤ã✱ t❛ ❝ã ϕ (f (x) − y)Jf (x)dx = = deg(f, Ω, y) Ω ◆Õ✉ f −1 (y) = ∅, ♠➭ y ∈ f (S) ∪ f (∂Ω), ♥➟♥ f −1 (y) = {x1 , , xm }, Jf (xi ) = ∀i = 1, , m : Rn ), ỗ i ó Jf (xi ) = 0, ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ❍➭♠ ♥❣➢ỵ❝ tå♥ t➵✐ ❧➞♥ ❝❐♥ f ∈ C (Ω ♠ë Ui ❝ñ❛ xi ✱ ❧➞♥ ❝❐♥ ♠ë Vi ❝ñ❛ y s❛♦ ❝❤♦ ❉♦ f : Ui → Vi ❧➭ ✈✐ ♣❤➠✐ ✈➭ Jf |Ui ❦❤➠♥❣ ➤æ✐ ❞✃✉✳ −1 m (B(y, )) ∩ Ui ❑❤✐ ➤ã✱ ❚å♥ t➵✐ sè ❞➢➡♥❣ s❛♦ ❝❤♦ B(y, ) ⊂ ∩i=1 Vi ❚❛ ➤➷t Wi = f m m ¯ ¯ Ω\(∪i=1 Wi ) ❧➭ t❐♣ ❝♦♠♣❛❝t ♥➟♥ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ < ♠➭ < ρ(y, f (Ω\(∪ i=1 Wi ))) ❑❤✐ ➤ã✱ ✈í✐ ,ϕ ♥➭♦ ♠➭ 0< < 0, ϕ ∈ C0∞ (Rn ; R), s✉♣♣ ϕ ⊂ B(0, ) ϕ (x)dx = ✈➭ Rn x ∈ Wi (∀i = 1, , m) : ρ(y, f (x)) > x ∈ Wi : sgn Jf (x) = sgn Jf (xi ) = 0, ❞♦ ➤ã t❤× ♥Õ✉ > , ❤❛② ϕ (f (x) − y) = 0; ❝ß♥ ♥Õ✉ m ϕ (f (x) − y)Jf (x)dx = Ω ϕ (f (x) − y)Jf (x)dx i=1 m = Wi ϕ (f (x) − y)Jf (x)dx sgn Jf (xi ) f (W i)=B(y, ) i=1 m = sgn Jf (xi ) i=1 ✶✳✶✳✷ ❳➞② ❞ù♥❣ ❜❐❝ ❝đ❛ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ➜Ĩ ①➞② ❞ù♥❣ ❦❤➳✐ ♥✐Ö♠ ❜❐❝ ❝❤♦ ➳♥❤ ①➵ ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝đ❛ ❜✐➟♥ ▼Ư♥❤ ➤Ị ✷✳ ❈❤♦ ρ(y, f (∂Ω)) > 0), ♥Õ✉ f ¯ Rn ) C (Ω; t❤✉é❝ ❧í♣ ¯ Rn ) ➤è✐ ✈í✐ t❐♣ Ω t➵✐ ➤✐Ó♠ y C (Ω; ∂Ω t❛ ❝➬♥ ➤Õ♥ ♠Ư♥❤ ➤Ị s❛✉✳ n n ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥ tr♦♥❣ R ✱ ♠ét ➤✐Ó♠ y ∈ R \ f (∂Ω), (ρ0 ¯ ✈➭ f ∈ C (Ω; Rn ) ❑❤✐ ➤ã✱ ✈í✐ ❜✃t ❦ú y1 , y2 ♥➭♦ t❛ ❝ò♥❣ ❝ã Ω yi ∈ B(y, ρ0 ), yi ∈ f (S), i = 1, 2, t❤× deg(f, Ω, y1 ) = deg(f, Ω, y2 ) = ✹ < δ < ρ0 − |y − yi |, i = 1, ❚❤❡♦ ▼Ư♥❤ ➤Ị ✶✱ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ < δ ✈➭ ♠ét ❤➭♠ ϕ ∈ C0∞ (Rn ; R), s✉♣♣ ϕ ⊂ B(0, ), s❛♦ ❝❤♦ ✈í✐ i = 1, t❛ ➤Ị✉ ❝ã ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ϕ (f (x) − yi )Jf (x)dx deg(f, Ω, yi ) = Ω ❈ã d ϕ (z − y1 + t(y1 − y2 ))dt dt = div(w(z)), ϕ (z − y1 ) − ϕ (z − y2 ) = ✭✶✳✶✮ w(z) = ( dtd ϕ (z − y1 + t(y1 − y2 ))dt)(y1 − y2 ) ❈❤ó ý r➺♥❣✱ ✈í✐ z ∈ f (∂Ω), < t < ❝ã tr♦♥❣ ➤ã✱ ||z − (1 − t)y1 − ty2 || = ||(z − y) + (1 − t)(y − y1 ) + t(y − y2 )|| ≥ δ > ♥➟♥ ✈í✐ x ∈ ∂Ω t❤× wj (f (x)) = ❉♦ ➤ã✱ ♥Õ✉ ➤➷t n j=1 vi (x) = wj (f (x))Aij (x), ♥Õ✉ x ∈ Ω, 0, ♥Õ✉ x ∈ Rn \ Ω, Aij (x) = (−1)i+j det{∂l fk }l=i,k=j ✱ n t❤× vi ∈ C0 (R ; R), s✉♣♣ vi ⊂ Ω, ✈➭ tr♦♥❣ ➤ã✱ n ∂vi (x) ∂wj (f (x)) ∂fk (x) = Aij (x) + ∂xi ∂xk ∂xi j,k=1 ▲➵✐ ❝ã✱ ♥Õ✉ ➤➷t cj,i,k = n wj (f (x)) j=1 ∂Aij (x) ∂xi ✭✶✳✷✮ gj = (f1 , , fj−1 , fj+1 , , fn )t , ✈➭ det(∂1 gj , , ∂i−1 gj , ∂i+1 gj , , ∂k−1 gj , ∂ik gj , ∂k+1 gj , , ∂n gj ), 0, det(∂1 gj , , ∂k−1 gj , ∂ik gj , ∂k+1 gj , , ∂i−1 gj , ∂i+1 gj , , ∂n gj ), ∂Aij (x) = (−1)i+j nk=1 cj,i,k , ♠➭ ∂xi i+k−1 ❤❛② cj,i,k = (−1) cj,k,i ❝ã ¯ Rn ), f ∈ C (Ω; t❤❡♦ ➜Þ♥❤ ❧ý ❙❝❤✇❛r③✱ k > i, ♥Õ✉ k = i, ♥Õ✉ k < i, ♥Õ✉ ∂ik g = ∂ki g ♥➟♥ n i=1 ∂Aij (x) = ∂xi n n (−1) i=1 k=1 n (−1)k+j k=1 (−1)i+k−1 cj,k,i (−1) i=1 k=1 =− i+j cj,i,k = n n n i+j cj,k,i = i=1 ✭✶✳✸✮ ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r n ∂fk (x) i=1 ∂xi Aij (x) ◆❣♦➭✐ r❛✱ n div(v(x)) = i=1 n = ✺ = δjk Jf (x) ♥➟♥ tõ ✭✶✳✷✮✲✭✶✳✸✮ ❝ã n ∂wj (f (x)) ∂fk (x) Aij (x) + ∂x ∂x k i j,k=1 n ∂wj (f (x)) ∂xk j,k=1 i=1 n n wj (f (x)) i=1 j=1 n ∂fk (x) Aij (x) + ∂xi ∂Aij (x) ∂xi n wj (f (x)) j=1 i=1 ∂Aij (x) ∂xi n = ∂wj (f (x)) δjk Jf (x) = div(w(f (x)))Jf (x) ∂xk j,k=1 ♥➟♥ tõ ✭✶✳✶✮ deg(f, Ω, y1 ) − deg(f, Ω, y2 ) = (ϕ (f (x) − y1 ) − ϕ (f (x) − y2 ))Jf (x)dx Ω div(w(f (x)))Jf (x)dx = Ω = div(v(x))dx Ω div(v(x))dx (vi ∈ C0 (Rn ; R), s✉♣♣ vi ⊂ Ω) = Rn m = i=1 ệ ề ỗ ➤✐Ó♠ Rn−1 −∞ ∂i vi (x)) dxi dx = 0, (vi (x) = 0, x ∈ ∂Ω) ∂xi y ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝đ❛ ❜✐➟♥ t❤× tr♦♥❣ ♠ét ❧➞♥ ❝❐♥ n ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ ❊✉❝❧✐❞ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ R ), ❝ñ❛ y trõ r❛ ♠ét B(y, ρ0 ), ρ0 = ρ(y, f (∂Ω))(ρ t❐♣ f (S) ❝ã ➤é ➤♦ ✭➜Þ♥❤ ❧ý f ➤è✐ ✈í✐ t❐♣ Ω ❧➭ ♥❤➢ ♥❤❛✉ t➵✐ ❜✃t ❦ú ➤✐Ó♠ ¯ ♥➭♦✳ ❚❛ ❝ã t❤Ó ➤Þ♥❤ ♥❣❤Ü❛ ❜❐❝ ❝❤♦ ➳♥❤ ①➵ f ∈ C (Ω; Rn ) ➤è✐ ✈í✐ t❐♣ Ω t➵✐ ➤✐Ĩ♠ y ∈ f (∂Ω) ❙❛r❞✮✱ ❜❐❝ ❝ñ❛ ♥❤➢ s❛✉✳ n n ❈❤♦ Ω ⊂ R ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥✱ ✈➭ y ∈ R \f (∂Ω), ρ0 = ρ(y, f (∂Ω)) ¯ Rn ) ❇❐❝ ❝đ❛ ➳♥❤ ①➵ f ➤è✐ ✈í✐ ề t ể y ợ ị s C (Ω; ➜Þ♥❤ ♥❣❤Ü❛ ✷✳ 0, f ∈ > deg(f, Ω, y) = deg(f, Ω, z), tr♦♥❣ ➤ã✱ z ∈ B(y, ρ0 ) \ f (S) deg(f, Ω, y) ❧➭ ❤➭♠ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ ➤è✐ ✈í✐ y tr➟♥ t❐♣ Rn \ f () n r ỗ t t❤➠♥❣ A ⊂ R \ f (∂Ω) ❜❐❝ deg(f, Ω, y) ❧➭ ❦❤➠♥❣ t❤❛② ➤æ✐✳ ◆❤❐♥ ①Ðt✳ ✶✳✶✳✸ ✶✳ ❇❐❝ ❳➞② ❞ù♥❣ ❜❐❝ ❝đ❛ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ❱✐Ư❝ ①➞② ❞ù♥❣ ❜❐❝ ❝❤♦ ➳♥❤ ①➵ f t❤✉é❝ ❧í♣ ¯ Rn ) C(Ω; ¯ Rn ) C(Ω; ➤è✐ ✈í✐ t❐♣ Ω t➵✐ ➤✐Ĩ♠ y ❦❤➠♥❣ ∂Ω ❝ò♥❣ ❝➬♥ ➤Õ♥ ♠ét ▼Ư♥❤ ➤Ị✱ ❣✐è♥❣ ♥❤➢ ▼Ư♥❤ ➤Ị ✷✱ ❝❤♦ t❛ t❤✃② ¯ tÝ♥❤ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ❜❐❝ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ f t❤✉é❝ ❧í♣ C (Ω; Rn ) ♥➺♠ tr♦♥❣ ➯♥❤ ❝ñ❛ ❜✐➟♥ ✻ ¯ Rn ) ✈➭ y Rn ✱ f, g ❧➭ ❝➳❝ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ C (Ω; ❧➭ ♠ét ➤✐Ó♠ ❦❤➠♥❣ ♥➺♠ tr➟♥ ➯♥❤ ❝ñ❛ ❜✐➟♥ ∂Ω ❝ñ❛ ➳♥❤ ①➵ f ✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ ✭♣❤ô t❤✉é❝ ✈➭♦ f, g, Ω✮ s❛♦ ❝❤♦ ❈❤♦ ▼Ư♥❤ ➤Ị ✸✳ Ω ❧➭ t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣ deg(f + tg, Ω, y) = deg(f, Ω, y), ❈❤ø♥❣ ♠✐♥❤✳ ❑❤✐ ∀0 < |t| < ||g||∞ = supx∈Ω¯ |g(x)| = t❤× t❛ ❞Ơ ❞➭♥❣ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ||g||∞ > 0, ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ▼Ư♥❤ ➤Ị ♥➭②✱ t❛ ❝❤✐❛ t❤➭♥❤ ❜❛ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ¯ ❤❛② ρ = ρ(y, f (Ω)) ¯ > ❚❍✶✿ y ∈ f (Ω) ρ ❱í✐ = t❤× 2||g||∞ ❑❤✐ ¯ ≥ ρ(y, f (Ω)) ¯ − t||g||∞ ≥ ρ > 0, ρ(y, (f + tg)(Ω)) ∀0 < |t| < ♥➟♥ deg(f + tg, Ω, y) = = deg(f, Ω, y), ∀0 < |t| < ¯ \ (f (S) ∪ f (∂Ω)) ❝ã f −1 (y) = {x1 , , xm }, Jf (xi ) = 0, i = 1, , m y ∈ f (Ω) ❳Ðt ➳♥❤ ①➵ h(t, x) = f (x) − tg(x) − y ❝ã ❚❍✷✿ h(0, xi ) = 0, Dx h(0, xi ) = f (xi ), ¯ Rn ), f (xi ) ❦❤➠♥❣ s✉② ❜✐Õ♥✱ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ❍➭♠ ➮♥ f ∈ C (Ω; ❧➞♥ ❝❐♥ ♠ë Ui ❝ñ❛ xi ✈➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝ ϕi : (− i , i ) → Ui s❛♦ ❝❤♦ ♠➭ ϕi (0) = xi , tå♥ t➵✐ sè ❞➢➡♥❣ i, h(t, ϕi (t)) = ∀t ∈ (− i , i ), ∀i = 1, , m, (t, ϕi (t)) ❧➭ ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ h(t, x) = tr♦♥❣ (− i , i ) × Ui ¯ ❉♦ f, g ∈ C (Ω; Rn ) ♥➟♥ t❛ ❝ã t❤Ó t❤✉ ♥❤á (− i , i ) ì Ui s Ui ♠ét rê✐ ♥❤❛✉✱ • sgn Jf +tg (x) = sgn Jf (x) = sgn Jf (xi ) ∀(t, x) ∈ (− i , i ) × Ui , ¯ \ (∪m Ui )) ∀t ∈ (− i , i ) • y ∈ (f + tg)(Ω i=1 ➜➷t = 1≤i≤m i , ✈í✐ < |t| < : (f + tg)−1 (y) = {ϕ1 (t), , ϕm (t)} sgn Jf +tg (ϕi (t)) = sgn Jf (ϕi (t)) = sgn Jf (xi ) = ∀t ∈ (− , ), y ∈ (f + tg)(Ω) \ ((f + tg)(S) ∪ (f + tg)(∂Ω)) ❑❤✐ ➤ã✱ m deg(f +tg, Ω, y) = m sgn Jf +tg ((ϕi (t))) = i=1 sgn Jf (xi ) = deg(f, Ω, y), i=1 ∀0 < |t| < ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ❚❍✸✿ ✼ y ∈ f (S) \ f (∂Ω) ❝ã z ∈ B(y, ρ3 ) \ f (S) s❛♦ ❝❤♦ deg(f, Ω, y) = deg(f, Ω, z) ▼➭ z ∈ B(y, ρ3 ) \ f (S) ❤❛② ¯ \ (f (S) ∪ f (∂Ω)) z ∈ f (Ω) ✭✶✳✹✮ ♥➟♥ tõ ❚❍✷ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ s❛♦ ❝❤♦ deg(f + tg, Ω, z) = deg(f, Ω, z), ❈❤ä♥ sè ❞➢➡♥❣ s❛♦ ❝❤♦ ∀0 < |t| < (∂Ω)) } ❱í✐ < |t| < < min{ , ρ(y,f 3||g||∞ ρ(y, (f + tg)(∂Ω)) ≥ ρ(y, f (∂Ω)) − t||g||∞ ≥ ρ(y, z) ≤ ✭✶✳✺✮ ❝ã 2ρ(y, f (∂Ω)) , ρ(y, f (∂Ω)) ♥➟♥ ρ(y, (f + tg)(∂Ω)) ≥ 2ρ(y, z) ❞♦ ➤ã✱ tõ ✭✶✳✹✮✱ ✭✶✳✺✮ ❝ã deg(f + tg, Ω, y) = deg(f + tg, Ω, z) = deg(f, Ω, z) = deg(f, , y) ỗ Rn ), y f (∂Ω) ♥Õ✉ g0 , g1 ∈ C (Ω; ¯ Rn ), ||f − gi ||∞ ≤ f ∈ C(Ω; ρ(y,f (∂Ω)) ,i = 0, t❤× deg(g1 , Ω, y) = deg(g2 , Ω, y) ❚❤❐t ✈❐②✱ ✈í✐ ≤ t ≤ ❝ã ||f − (g0 + t(g1 − g0 ))||∞ ≤ (1 − t)||f − g0 ||∞ + t||f − g1 ||∞ ≤ ρ(y, f (∂Ω)) , ¯ Rn ) ♥➟♥ t❤❡♦ ▼Ư♥❤ ➤Ị ✸ t❤× ❤➭♠ deg(g0 + t(g1 − g0 ), Ω, y) g0 , g0 + t(g1 − g0 ) ∈ C (Ω; ❧➭ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ t❤❡♦ t tr➟♥ t❐♣ ❝♦♠♣❛❝t [0, 1], ❞♦ ➤ã ❧➭ ❤➺♥❣ tr➟♥ [0, 1] ❤❛② ♠➭ deg(g0 , Ω, y) = deg(g1 , Ω, y) ❉♦ Ω ❧➭ t❐♣ ❜Þ ❝❤➷♥ ♥➟♥ t❐♣ g tộ rỗ từ ó t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ ❜❐❝ ❝❤♦ ➳♥❤ ①➵ ➤✐Ĩ♠ y ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝ñ❛ ❜✐➟♥ f ¯ Rn ) C (Ω; t❤✉é❝ ❧í♣ ρ(y,f (∂Ω)) t❐♣ Ω t➵✐ ∂Ω ♥❤➢ s❛✉✳ n n ❈❤♦ Ω ⊂ R ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥✱ ✈➭ y ∈ R \f (∂Ω), ρ0 = ρ(y, f (∂Ω)) ¯ Rn ) ❇❐❝ ❝đ❛ ➳♥❤ ①➵ f ➤è✐ ✈í✐ ♠✐Ị♥ Ω t➵✐ ể y ợ ị s C (; ➜Þ♥❤ ♥❣❤Ü❛ ✸✳ 0, f ∈ ||f − g||∞ < ¯ C(Ω; Rn ) ➤è✐ ✈í✐ ♠➭ deg(f, Ω, y) = deg(g, Ω, y), tr♦♥❣ ➤ã✱ ¯ Rn ), ||f − g||∞ ≤ g ∈ C (Ω; ρ0 > ✽ ❈❤ó ý✳ ➜è✐ ✈í✐ tr➢ê♥❣ ❤ỵ♣ ➳♥❤ ①➵ ¯ Rn ), ➤✐Ó♠ y ∈ f (Ω) \ (f (∂Ω) ∪ f (S)) t❛ ❝ã f inC (; ị s deg(f, , y) ị deg(f, , y) t❤➠♥❣ q✉❛ ❝➯ ❜❛ ❜➢í❝✱ ➤➬✉ t✐➟♥ ①✃♣ ①Ø ❜ë✐ ➳♥❤ ①➵ t❤✉é❝ ❧í♣ ¯ Rn ), C (Ω; s❛✉ ➤ã ①➳❝ ➤Þ♥❤ t❤❡♦ ❜➢í❝ ✷✳ ❚✉② ♥❤✐➟♥✱ ❤❛✐ ❝➳❝❤ ♥➭② ➤Ò✉ ❝❤♦ t❛ ❝ï♥❣ ♠ét ❦Õt q✉➯ ✳ ➜✐Ị✉ ♥➭② ➤➢ỵ❝ ❦✐Ĩ♠ tr❛ ♥❤➢ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ ë ❚❍✷ ❝đ❛ ▼Ư♥❤ ➤Ị ✸✳ ✶✳✷ Ω ▼ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❜❐❝ ◆❤➢ ✈❐② t❛ ➤➲ ➤Þ♥❤ ♥❣❤Ü❛ ➤➢ỵ❝ ❦❤➳✐ ♥✐Ư♠ ❜❐❝ ❝❤♦ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ f tõ ♠ét t❐♣ ❜Þ ❝❤➷♥ n n tr♦♥❣ R ✈➭♦ R ➤è✐ ✈í✐ t❐♣ Ω t➵✐ ♠ét ➤✐Ĩ♠ y ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝ñ❛ ❜✐➟♥ ∂Ω ❍❛② ♥ã✐ ❝➳❝❤ ❦❤➳❝✱ t❛ ➤➲ ①➞② ❞ù♥❣ ➤➢ỵ❝ ♠ét ❤➭♠ tõ t❐♣ ❝➳❝ ❜é ❜❛ (f, Ω, y)✱ tr♦♥❣ ➤ã Ω ❧➭ ♠ét n n t❐♣ ❜Þ ❝❤➷♥ tr♦♥❣ R ✱ f ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝ tõ Ω ✈➭♦ R ✱ y ❧➭ ♠ét ➤✐Ĩ♠ ❦❤➠♥❣ ♥➺♠ tr♦♥❣ ➯♥❤ ❝đ❛ ❜✐➟♥ ∂Ω✱ ✈➭♦ t❐♣ ❝➳❝ sè ♥❣✉②➟♥✿ ❜Þ ❝❤➷♥ ❧✐➟♥ tơ❝ deg : {(f, Ω, y)|f : Ω −→ Rn , Ω ⊂ Rn , y ∈ Rn \ f (∂Ω)} → Z ❚õ ✈✐Ö❝ ①➞② ❞ù♥❣ ❜❐❝ t❛ t❤➞ý ❜❐❝ ❝ã ♠ét sè tÝ♥❤ ❝❤✃t s❛✉✳ ➜Þ♥❤ ❧ý ✹✳ ✭❞✶✮ deg(id, Ω, y) = 1, ♥Õ✉ y ∈ Ω ✭❞✷✮ deg(f, Ω, y) = deg(f, Ω1 , y) + deg(f, Ω2 , y), tr♦♥❣ ➤ã Ω1 ∩ Ω2 = ∅, Ω1 ∪ Ω2 ⊂ Ω, y ∈ ¯ \ (Ω1 ∪ Ω2 )) f (Ω ✭❞✸✮ ¯→ deg(h(t, ), Ω, y(t)) ❧➭ ❤➭♠ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ t tr➟♥ [0, 1], tr♦♥❣ ➤ã h : [0, 1]× Ω n n R , y : [0, 1] → R ❧➭ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tô❝✱ y(t) ∈ h(t, ∂Ω), ∀t ∈ [0, 1] ✭❞✹✮ ♥Õ✉ deg(f, Ω, y) = t❤× f −1 (y) = ∅ ✭❞✺✮ deg(., Ω, y) ❧➭ ❤➭♠ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ tr➟♥ t❐♣ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ f : Ω → Rn , ♠➭ y ∈ f (∂Ω) deg(f, Ω, ) ❧➭ ❤➭♠ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ tr➟♥ t❐♣ Rn \ f (∂Ω) ❉♦ ➤ã✱ deg(f, Ω, ) ❧➭ ❤➺♥❣ n sè tr➟♥ tõ♥❣ t❤➭♥❤ ♣❤➬♥ ❧✐➟♥ t❤➠♥❣ ❝ñ❛ t❐♣ R \ f (∂Ω) ✭❞✻✮ deg(g, Ω, y) = deg(f, Ω, y) ♥Õ✉ y ∈ f (∂Ω), f |∂Ω = g|∂Ω ✭❞✼✮ deg(f, Ω, y) = deg(f, Ω1 , y) ♥Õ✉ Ω1 ❈❤ó ý✳ ❧➭ t❐♣ ♠ë tr♦♥❣ ¯ \ Ω1 ) Ω✱ y ∈ f (Ω ◆❣➢ê✐ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ❝ã ❞✉② ♥❤✃t ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ❜Þ ❝❤➷♥ deg : {(f, Ω, y)|f : Ω −→ Rn , Ω ⊂ Rn , y ∈ Rn \ f (∂Ω)} → Z, ♠➭ t❤♦➯ ♠➲♥ ❜❛ tÝ♥❤ ❝❤✃t (d1), (d2), (d3) ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ✾ (d1), (d4) ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛✳ ❚Ý♥❤ ❝❤✃t (d7) ➤➢ỵ❝ s✉② r❛ tõ tÝ♥❤ ❝❤✃t (d2) ❜➺♥❣ ❝➳❝❤ ❧✃② Ω2 = ∅ ✈➭ deg(f, ∅, y) = ❚Ý♥❤ ❝❤✃t (d6) ➤➢ỵ❝ s✉② r❛ tõ tÝ♥❤ ❝❤✃t (d3) ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ❝➳❝ ➳♥❤ ①➵ h, y ♥❤➢ s❛✉ ❈❤ø♥❣ ♠✐♥❤✳ ❈➳❝ tÝ♥❤ ❝❤✃t ¯ → Rn , h(t, x) = tf (x) + (1 − t)g(x) h : [0, 1] × Ω y : [0, 1] → Rn , y(t) = y d(3) ➤➢ỵ❝ s✉② r❛ từ tí t (d6) s ỗ t0 ∈ [0, 1], ❞♦ deg(., Ω, y(t0 )) ❧➭ ❤➭♠ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ tr➟♥ t❐♣ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ f : Ω → Rn ♠➭ y(t0 ) ∈ f (∂Ω), ✈➭ h, y ❧➭ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tô❝✱ y(t0 ) ∈ h(t0 , ∂Ω), ♥➟♥ tå♥ t➵✐ ♠ét ❧➞♥ ❝❐♥ wt0 ❝ñ❛ t0 tr♦♥❣ [0, 1] ♠➭ ❚Ý♥❤ ❝❤✃t deg(h(t, ), Ω, y(t0 )) = deg(h(t0 , ), Ω, y(t0 )) ∀t ∈ wt0 ❇➺♥❣ ❝➳❝❤ t❤✉ ♥❤á ❧➞♥ ❝❐♥ wt0 s❛♦ ❝❤♦ ✭✶✳✻✮ ρ(y(t), y(t0 )) < ρ(y(t0 ), h(t, ∂Ω)) t❛ ❝ã deg(h(t, ), Ω, y(t)) = deg(h(t, ), Ω, y(t0 )) ∀t ∈ wt0 ✭✶✳✼✮ ❚õ ✭✶✳✻✮✱ ✭✶✳✼✮ t❛ ❝ã deg(h(t, ), Ω, y(t)) = deg(h(t0 , ), Ω, y(t0 )) ∀t ∈ wt0 ❤❛② deg(h(t, ), Ω, y(t)) ❧➭ ❤➭♠ ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ tr➟♥ t❐♣ ❝♦♠♣❛❝t [0, 1], ❤❛② ❧➭ ❤➭♠ ❤➺♥❣ tr➟♥ ➤ã✳ (d2), (d5) ¯ ❚❛ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t (d2) ❚õ ❣✐➯ t❤✐Õt y ∈ f (Ω \ (Ω1 ∪ Ω2 )) ❝ã ◆❤➢ ✈❐② t❛ ❝❤Ø ❝ß♥ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ ❤❛✐ tÝ♥❤ ❝❤✃t ¯ \ (Ω1 ∪ Ω2 )))), ρ(y, f (∂Ω1 )) ≥ ρ0 , ρ(y, f (∂Ω2 )) ≥ ρ0 ρ(y, f (∂Ω)) ≥ ρ0 (= ρ(y, f (Ω ❑❤✐ ➤ã✱ tå♥ t➵✐ ¯ ||f − g||∞ ≤ g ∈ C (Ω), ρ0 , ρ1 ¯ \ (Ω1 ∪ Ω2 )) > ♠➭ = ρ(y, g(Ω deg(f, Ω, y) = deg(g, Ω, y), deg(f, Ωi , y) = deg(g, Ωi , y), ❚❤❡♦ ➜Þ♥❤ ♥❣❤Ü❛✱ tå♥ t➵✐ ✭✶✳✽✮ i = 1, z ∈ B(y, ρ1 ) \ g(S) deg(g, Ω, y) = deg(g, Ω, z), deg(g, Ωi , y) = deg(g, Ωi , z), ❉♦ ✭✶✳✾✮ ✭✶✳✶✵✮ i = 1, ✭✶✳✶✶✮ ¯ \ (Ω1 ∪ Ω2 )) > ♥➟♥ z ∈ (g(Ω ¯ \ (Ω1 ∪ Ω2 ) ∪ g(S)), ❞♦ ➤ã tõ ➜Þ♥❤ ♥❣❤Ü❛ ρ1 = ρ(y, g(Ω ❝ã deg(g, Ω, z) = deg(g, Ω1 , z) + deg(g, Ω2 , z) ❚õ ✭✶✳✽✮✲✭✶✳✶✷✮ t❛ ❝ã deg(f, Ω, z) = deg(f, Ω1 , z) + deg(f, Ω2 , z) ✭✶✳✶✷✮ ✶✵ ❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t (d5) ▲✃② ¯ y ∈ f (∂Ω)(ρ0 = ρ(y, f (∂Ω)) > 0) f ∈ C(Ω), ❈❤ä♥ ❧➞♥ ❝❐♥ ¯ ||f − g||∞ < ρ0 } U (f ) = {g ∈ C(Ω)| ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ❧➞♥ ❝❐♥ U (f ) ❜❐❝ deg(., Ω, y) ❧➭ ❦❤➠♥❣ ➤ỉ✐✳ ❚❤❐t ✈❐②✱ tõ ➜Þ♥❤ ♥❣❤Ü❛ ¯ ❝ã ♠ét ➳♥❤ ①➵ g0 ∈ U (f ) ∩ C (Ω; Rn ) s❛♦ ❝❤♦ deg(f, Ω, y) = deg(g0 , Ω, y) ❱í✐ ✭✶✳✶✸✮ g ∈ U (f ) ❝ã 3ρ0 , 3ρ0 ≥ , ||g − g0 ||∞ ≤ ||f − g||∞ + ||f − g0 ||∞ ≤ ρ(y, g(∂Ω)) ≥ ρ(y, f (∂Ω)) − ||f − g||∞ ♥➟♥ ||g − g0 ||∞ ≤ 21 ρ(y, g(∂Ω)), ❞♦ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✸ ❝ã deg(g, Ω, y) = deg(g0 , Ω, y) ✭✶✳✶✹✮ ❚õ ✭✶✳✶✸✮✱ ✭✶✳✶✹✮ ❝ã deg(f, Ω, y) = deg(g, Ω, y) deg(., Ω, y) ❧➭ ❤➭♠ ❤➺♥❣ tr♦♥❣ ❧➞♥ ❝❐♥ U (f ) ❈✉è✐ ❝ï♥❣✱ t❛ ❝❤ø♥❣ ♠✐♥❤ deg(f, Ω, ) ❧➭ ❤➭♠ ❤➺♥❣ ➤Þ❛ n tõ♥❣ t❤➭♥❤ ♣❤➬♥ ❧✐➟♥ t❤➠♥❣ ❝đ❛ t❐♣ R \ f (∂Ω)✳ ¯ ❤❛② ρ = ρ(y, f (Ω)) ¯ > t❛ ❝❤ä♥ ❧➞♥ ❝❐♥ ◆Õ✉ y ∈ f (Ω) ❤❛② ♣❤➢➡♥❣ t❤❡♦ y ✈➭ ❧➭ ❤➺♥❣ sè tr➟♥ U (y) = {z ∈ Rn |ρ(y, z) < ρ} ❑❤✐ ➤ã✱ z ∈ U (y) t❤× f −1 (z) = ∅ ❤❛② deg(f, Ω, y) = deg(f, Ω, z), ◆Õ✉ ∀z ∈ U (y) y ∈ f (Ω) \ f () tì t ị ĩ tồ t y0 ∈ B(y, ρ20 ) \ f (S) s❛♦ ❝❤♦ deg(f, Ω, y) = deg(f, Ω, y0 ) ❱í✐ z ∈ B(y, ρ40 ) \ f (∂Ω) ❝ã ρ(z, f (∂Ω)) ≥ ρ(y, f (∂Ω)) − ρ(y, z) ≥ ρ(z, y0 ) ≤ ρ(z, y) + ρ(y, y0 ) ≤ ♥➟♥ ✭✶✳✶✺✮ 3ρ0 , 3ρ0 , ρ(z, f (∂Ω)) ≥ ρ(z, y0 ) ❞♦ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✷ ❝ã deg(f, Ω, z) = deg(f, Ω, y0 ) ✭✶✳✶✻✮ ❚õ ✭✶✳✶✺✮✱ ✭✶✳✶✻✮ ❝ã deg(f, Ω, z) = deg(f, Ω, y), ❱✐Ö❝ ❝❤ø♥❣ ♠✐♥❤ ∀z ∈ B(y, ρ0 ) \ f (∂Ω) deg(f, Ω, ) ❧➭ ❤➺♥❣ sè tr➟♥ tõ♥❣ t❤➭♥❤ ♣❤➬♥ ❧✐➟♥ t❤➠♥❣ ❝ñ❛ t❐♣ Rn \ f (∂Ω) ➤➢ỵ❝ s✉② r❛ ❞Ơ ❞➭♥❣ tõ tÝ♥❤ ❝❤✃t ❤➺♥❣ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♥ã✳ ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ✶✳✸ ✶✶ ❈➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ❧ý t❤✉②Õt ❜❐❝ ➜➬✉ t✐➟♥✱ t❛ sÏ ❞ï♥❣ ❧ý t❤✉②Õt ❜❐❝ ➤Ó ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➜Þ♥❤ ❧ý ❝❤➻♥❣ ❤➵♥ ➜Þ♥❤ ❧ý ❝♦ rót✱ ➜Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❇r♦✇❡r✱ ➜Þ♥❤ ❧ý ▼✐r❛♥❞❛✲P♦✐♥❝❛r❡✱ ✈➭ ➤➷❝ ❜✐Ưt ➜Þ♥❤ ❧ý ❇♦rs✉❦✳ ➜Þ♥❤ ❧ý ❇♦rs✉❦ ❝ã ♥❤✐Ị✉ ➳♣ ❞ơ♥❣✱ ♥❤➢ ➜Þ♥❤ ❧ý ✈Ị ◗✉➯ ❜ã♥❣ tã❝ ✭❍❛✐r② ❜❛❧❧✮✱ ➜Þ♥❤ ❧ý ❜➳♥❤ ❙❛♥❞✇✐❝❤✱ ✈➭ ♠ét sè ➜Þ♥❤ ❧ý ❤✃♣ ❞➱♥ ❦❤➳❝✳ ❚r➢í❝ ❤Õt✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ◗✉➯ ❝➬✉ tã❝✳ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ♥➭② t❛ ❝➬♥ ❇ỉ ➤Ị s❛✉✳ ❇ỉ ➤Ị ✺✳ ❱í✐ n ❧➭ ♠ét sè ❧❰✱ ❦❤➠♥❣ t❤Ó ❝ã ➤å♥❣ ❧✉➞♥ H : [0, 1] × Sn−1 → Sn−1 ♠➭ H(., 0) = id, H(., 1) = −id ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❝ã ♠ét ➤å♥❣ ❧✉➞♥ H : [0, 1] × Sn−1 → Sn−1 ♠➭ H(0, ) = id, H(1, ) = −id ❚õ ➜Þ♥❤ ❧ý t❤➳❝ t✐Ĩ♥ ❚✐❡t③❡✱ t❛ ❝ã t❤Ĩ t❤➳❝ tr✐Ó♥ ➤å♥❣ ❧✉➞♥ tr➟♥ t❤➭♥❤ ➤å♥❣ ❧✉➞♥ H : [0, 1] × Bn → Rn ▼➭ H(0, x) = x, H(1, x) = −x ❦❤✐ x ∈ Sn−1 ♥➟♥ t❤❡♦ tÝ♥❤ ❝❤✃t (d3), (d6) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ ❝ã = deg(id, Bn , 0) = deg(H(0, ), Bn , 0) n ❧❰ = deg(H(1, ), Bn , 0) = deg(−id, Bn , 0) = (−1)n = −1 ➜✐Ò✉ ♥➭② ❧➭ ✈➠ ❧ý✳ ✭➜Þ♥❤ ❧ý ◗✉➯ ❝➬✉ tã❝✮ ❱í✐ n ❧➭ ♠ét sè ❧❰✳ ❱í✐ ♠ét tr➢ê♥❣ ✈❡❝t➡ ❜✃t ❦ú tr➟♥ ♠➷t n−1 n−1 ❝➬✉ ➤➡♥ ✈Þ S ➤Ị✉ ó tể tì ợ tr t S ột ể ♠➭ t➵✐ ➤ã tr➢ê♥❣ ✈❡❝t➡ ➜Þ♥❤ ❧ý ✻✳ ❝ã ❣✐➳ trÞ ❧➭ ✈❡❝t➡ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ♥➭② ❜➺♥❣ ♣❤➯♥ ❝❤ø♥❣✳ ●✐➯ sö ❝ã tr➢➡♥❣ ✈❡❝t➡ Sn−1 ♠➭ ♥ã ❦❤➳❝ t➵✐ ♠ä✐ ➤✐Ó♠ tr➟♥ ♠➷t ❝➬✉ Sn−1 , ♥❣❤Ü❛ ❧➭ ❧✐➟♥ tô❝ ϕ : Sn−1 −→ Rn , (ϕ(x), x) = 0, ∀x ∈ Sn−1 , ϕ(x) = 0, ∀x ∈ Sn−1 ϕ tr➟♥ ✶✷ ❳Ðt ➤å♥❣ ❧✉➞♥ s❛✉ H : [0, 1] × Sn−1 → Rn , H(t, x) = cos(πt)x + sin(πt) ||x|| ϕ(x), ||ϕ(x)|| t❤× ||H(t, x)|| = ||x|| = 1, ∀(t, x) ∈ [0, 1] × Sn−1 , H(0, x) = x, H(1, x) = −x, ∀x ∈ Sn−1 ➜✐Ị✉ ♥➭② tr➳✐ ✈í✐ ❇ỉ ➤Ị ✺✳ ✶✳✸✳✶ ➜Þ♥❤ ❧ý ❇r♦✇❡r ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ♠ét sè ❞➵♥❣ t➢➡♥❣ ➤➢➡♥❣ ❝đ❛ ♥ã ➜Þ♥❤ ❧ý ✼✳ ✭➜Þ♥❤ ❧ý ❝♦ rót✮ ì ó ị n B tr Rn ❦❤➠♥❣ ❧➭ t❐♣ ❝♦ ¯ n ❧➟♥ ♠➷t tõ ì ó B rút ợ ó t❤Ĩ ❝ã ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ f n−1 n−1 ❝➬✉ S ♠➭ ❤➵♥ ❝❤Õ ❝ñ❛ ♥ã tr➟♥ ♠➷t ❝➬✉ S ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö t❛ ❝ã t❤Ĩ ①➞② ❞ù♥❣ ➤➢ỵ❝ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ f tõ ❤×♥❤ ❝➬✉ ➤ã♥❣ n−1 n−1 ❧➟♥ ♠➷t ❝➬✉ S ♠➭ ❤➵♥ ❝❤Õ ❝ñ❛ ♥ã tr➟♥ ♠➷t ❝➬✉ S ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t✳ ¯n B ❑❤✐ ➤ã✱ t❛ ①Ðt ➤å♥❣ ❧✉➞♥ s❛✉ ¯ n → Rn h :[0, 1] × B h(t, x) = tx + (1 − t)f (x) ❈ã h(0, x) = f (x), h(1, x) = x, h(t, x) = tx + (1 − t)f (x) = tx + (1 − t)x = x = 0, ∀x ∈ Sn−1 , ∀t ∈ [0, 1], ♥➟♥ t❤❡♦ tÝ♥❤ ❝❤✃t (d3), (d1) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ ❝ã ¯ n , 0) = deg(id, B ¯ n , 0) = deg(f, B (d4) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ t❤× f −1 (0) = ∅ ¯ n ✈➭♦ Sn−1 ❍❛② ➤✐Ị✉ ❣✐➯ sư ❧➭ s❛✐✱ ♥❣❤Ü❛ ❧➭ ➜✐Ị✉ ♥➭② ❧➭ tr➳✐ ✈í✐ ❣✐➯ t❤✐Õt f ❧➭ ➳♥❤ ①➵ tõ B ¯ n ❧➟♥ ♠➷t ❝➬✉ Sn−1 ♠➭ ❤➵♥ ❝❤Õ ❝đ❛ ❦❤➠♥❣ ❝ã ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ f ♥➭♦ tõ ❤×♥❤ ❝➬✉ ➤ã♥❣ B n−1 ♥ã tr➟♥ ♠➷t ❝➬✉ S ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t✳ ❞♦ ➤ã✱ t❤❡♦ tÝ♥❤ ❝❤✃t ➜Þ♥❤ ❧ý ✽✳ ♥ã✳ ❑❤✐ ➤ã✱ ✭➜Þ♥❤ ❧ý ❇r♦✇❡r✮ ❈❤♦ f f ❧➭ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ tõ ❤×♥❤ ❝➬✉ ➤ã♥❣ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ♥❣❤Ü❛ ❧➭ ¯ n : f (x) = x ∃x ∈ B ¯n B ✈➭♦ ❝❤Ý♥❤ ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ✶✸ ❈❤ø♥❣ ♠✐♥❤✳ ❈ã ♥❤✐Ị✉ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ♥➭②✳ ë ➤➞② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❤❛✐ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤✳ ❚❤ø ♥❤✃t✱ ❞ï♥❣ ❧ý t❤✉②Õt ❜❐❝ ➤Ó ❝❤ø♥❣ ♠✐♥❤✳ ❚❤ø ❤❛✐✱ ♥ã ❧➭ ❍Ư q✉➯ ❝đ❛ ➜Þ♥❤ ❧ý ❝♦ rót✳ n−1 ❈➳❝❤ ✶✳ ◆Õ✉ ❝ã ♠ét ➤✐Ĩ♠ x S f (x) = x tì ị ý ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ n−1 ◆Õ✉ ✈í✐ ♠ä✐ x ∈ S ♠➭ f (x) = x ❤❛② x − tf (x) = 0, ∀0 ≤ t ≤ ❳Ðt ➤å♥❣ ❧✉➞♥ s❛✉ ¯ n → Rn h :[0, 1] × B h(t, x) = x − tf (x) ❈ã h(0, x) = x, h(1, x) = x − f (x), h(t, x) = x − tf (x) = 0, ∀x ∈ Sn−1 , ∀t ∈ [0, 1], ♥➟♥ t❤❡♦ tÝ♥❤ ❝❤✃t (d3), (d1) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ ❝ã ¯ n , 0) = deg(id, B ¯ n , 0) = deg(id − f, B ❞♦ ➤ã✱ t❤❡♦ tÝ♥❤ ❝❤✃t (d4) tr ị ý tì (id f )1 (0) = ∅ ❍❛② ¯ n : f (x) = x ∃x ∈ B ❈➳❝❤ ✷✳ ●✐➯ sö ➳♥❤ ①➵ ❧✐➟♥ tô❝ f tõ ¯n B ✈➭♦ ❝❤Ý♥❤ ♥ã ❦❤➠♥❣ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣✱ ♥❣❤Ü❛ ❧➭ ¯ n f (x) = x, ∀x ∈ B f (x) t❤➭♥❤ ♠ét t✐❛ Tx = {tf (x) + (1 − t)x|t ≥ 0} ❝ã ❣è❝ n−1 t➵✐ x✳ ❚✐❛ Tx ♥➭② sÏ ❝➽t ♠➷t ❝➬✉ S t➵✐ ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ϕ(x) ◆❤➢ ✈❐②✱ t❛ ➤➲ ①➞② ❞ù♥❣ ❑❤✐ ➤ã✱ t❛ ❧✉➠♥ ❝ã t❤Ó ♥è✐ x ✈➭ ➤➢ỵ❝ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ¯ n → Sn−1 ϕ:B ❝ã tÝ♥❤ ❝❤✃t ϕ(x) = x ♥Õ✉ x ∈ Sn−1 ➜✐Ị✉ ♥➭② tr➳✐ ✈í✐ ➜Þ♥❤ ❧ý ❝♦ rót✳ ❉♦ ➤ã✱ ➤✐Ị✉ ❣✐➯ sư s❛✐ ❤❛② ♠ét ➳♥❤ ①➵ ❧✐➟♥ tô❝ f tõ ¯n B ✈➭♦ ❝❤Ý♥❤ ♥ã ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❚Ý♥❤ ❝❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ❜✃t ❜✐Õ♥ ➤è✐ ✈í✐ ♠ét ♣❤Ð♣ ➤å♥❣ ♣❤➠✐✱ ♥➟♥ ♠ét ➳♥❤ ¯ n ✈➭♦ ❝❤Ý♥❤ ♥ã ➤Ò✉ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ①➵ ❧✐➟♥ tơ❝ tõ ♠ét t❐♣ ➤å♥❣ ♣❤➠✐ ✈í✐ ❤×♥❤ ❝➬✉ ➤ã♥❣ B ◆❤❐♥ ①Ðt✳ ❈❤➻♥❣ ❤➵♥✱ ➳♥❤ ①➵ ❧✐➟♥ tô❝ f tõ ♠ét t❐♣ ❧å✐✱ ❝♦♠♣❛❝t✱ rỗ D í ó ó ể t ộ ❚❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ❦Õt q✉➯ ♥➭② ❜➺♥❣ ➜Þ♥❤ ❧ý t❤➳❝ tr✐Ó♥ ❚✐❡t③❡✱ ♠➭ ❦❤➠♥❣ ❝➬♥ ¯ n ❧➟♥ t t rỗ D s ự é từ ì ó B ị ❧ý t❤➳❝ tr✐Ó♥ ❚✐❡t③❡✱ tå♥ t➵✐ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tô❝ f˜ : Rn → Rn , f˜|D = f, f (Rn ) ⊂ conv(f (D)) ⊂ D ✶✹ ▼➭ D ❧➭ t❐♣ ❝♦♠♣❛❝t ♥➟♥ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ R ➤Ó D ⊂ B(0, R)✳ ❑❤✐ ➤ã✱ f˜|B(0,R) : B(0, R) → (D ⊂)B(0, R)✳ ❉♦ ➤ã✱ f˜|B(0,R) ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❤❛② ∃x ∈ B(0, R) : x = f˜|B(0,R) (x) ♠➭ f˜|B(0,R) (B(0, R)) ⊂ D ♥➟♥ x ∈ D : x = f˜|B(0,R) (x) = f (x) ❚❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ n (i) ✭➜Þ♥❤ ❧ý ▼✐r❛♥❞❛✲P♦✐♥❝❛r❡✮ ❑ý ❤✐Ö✉ [a, b] = {x ∈ R |a 1, , n}, tr♦♥❣ ➤ã a = (a(1) , , a(n) ), b = (b(1) , , b(n) ), a(i) < b(i) , i n ❧➭ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ tõ [a, b] ✈➭♦ R t❤♦➯ ♠➲♥ ➜Þ♥❤ ❧ý ✾✳ ≤ x(i) ≤ b(i) , i = = 1, , n ❈❤♦ f fi (x(1) , , x(i−1) , a(i) , x(i+1) , , x(n) ) ≥ 0, fi (x(1) , , x(i−1) , b(i) , x(i+1) , , x(n) ) ≤ ❑❤✐ ➤ã✱ tå♥ t➵✐ ❈❤ó ý✳ x ∈ [a, b] ♠➭ f (x) = ➜➞② ❧➭ ♠ë ré♥❣ tù ♥❤✐➟♥ ủ ị ý ị ý trị tr ì ❧➟♥ ♥❤✐Ị✉ ❝❤✐Ị✉✳ ❚✉② ♥❤✐➟♥ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ t❤× ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ♠ë ré♥❣ t➬♠ t❤➢ê♥❣✳ ❈ã t❤Ó ❝❤ø♥❣ ♠✐♥❤ trù❝ t✐Õ♣ ➜Þ♥❤ ❧ý ♥➭② ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ ♥❤➢♥❣ ♣❤➯✐ sư ❞ơ♥❣ t❤➭♥❤ t❤➵♦ ❝➳❝ ➜Þ♥❤ ❧ý ❍➭♠ ♥❣➢ỵ❝✱ ❍➭♠ ➮♥✱ ❙❛r❞ ✈➭ ♣❤Ð♣ ➤å♥❣ ❧✉➞♥✳ ë ➤➞②✱ ❝❤ó♥❣ t➠✐ sư ❞ơ♥❣ ❧ý t❤✉②Õt ❜❐❝ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤✳ x0 = 12 (a + b) ●✐➯ sö (1 − t)f (x) = t(x − x0 ), ✈í✐ t ∈ [0, 1], x ∈ ∂[a, b]✳ (i) ❉♦ x ∈ ∂[a, b] ♥➟♥ ❝ã ♠ét ❝❤Ø sè i s❛♦ ❝❤♦ ❤♦➷❝ x = a(i) ❤♦➷❝ x(i) = b(i) (i) (i) ◆Õ✉ x = a(i) t❤× x(i) − x0 < 0, fi (x) ≥ ♥➟♥ t = ❞♦ ➤ã f (x) = (i) (i) ◆Õ✉ x = b(i) t❤× x(i) − x0 > 0, fi (x) ≤ ♥➟♥ t = ❞♦ ➤ã f (x) = ◆❤➢ ✈❐②✱ ♥Õ✉ ❝ã x ∈ ∂[a, b] ♠➭ f (x) = t❤× t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❈ß♥ ♥Õ✉ ❦❤➠♥❣ ♣❤➯✐ ✈❐② t❤× (1 − t)f (x) − t(x − x0 ) = 0, ∀t ∈ [0, 1], ∀x ∈ ∂[a, b] ❑❤✐ ➤ã✱ ①Ðt ➤å♥❣ ❧✉➞♥ s❛✉ ❳Ðt ➤å♥❣ ❧✉➞♥ s❛✉ ¯ n → Rn h :[0, 1] × B h(t, x) = (1 − t)f (x) − t(x − x0 ) ❈ã h(0, x) = f (x), h(1, x) = x0 − x, h(t, x) = (1 − t)f (x) − t(x − x0 ) = 0, ∀x ∈ ∂[a, b], ∀t ∈ [0, 1], ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ♥➟♥ t❤❡♦ tÝ♥❤ ❝❤✃t ✶✺ (d3), (d1) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ ❝ã deg(f, [a, b], 0) = deg(x0 − id, [a, b], 0) = (−1)n = (d4) tr♦♥❣ ị ý tì f (0) = tå♥ t➵✐ x ∈ [a, b] ♠➭ f (x) = ❞♦ ➤ã✱ t❤❡♦ tÝ♥❤ ❝❤✃t ◆❤❐♥ ①Ðt✳ ◆❣➢ê✐ t❛ ứ ợ r ị ý rr ị ❧ý ▼✐r❛♥❞❛✲P♦✐♥❝❛r❡ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ♥❤❛✉✳ ë ➤➞②✱ ❝❤ó♥❣ t➠✐ sÏ ❞ï♥❣ ➜Þ♥❤ ❧ý ▼✐r❛♥❞❛✲P♦✐♥❝❛r❡ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ❇r♦✇❡r ♥❤➢ s❛✉✳ ¯n f ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝ từ ì ị ó B ỗ m > 1✱ ①Ðt ➳♥❤ ①➵ s❛✉ ●✐➯ sö ✭t❤❡♦ ❝❤✉➮♥ ♠❛①✮ ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❱í✐ ¯ n → Rn , g m (x) = fi (x) − gm : B i ◆Õ✉ x = (x(1) , , x(i−1) , 1, x(i+1) , , x(n) ) t❤× gim (x) < − ◆Õ✉ m xi m−1 m < m−1 x = (x(1) , , x(i−1) , −1, x(i+1) , , x(n) ) t❤× gim (x) > −1 + m > m−1 ¯ n ♠➭ g m (xm ) = 0✳ ❉♦ ➤ã t❤❡♦ ➜Þ♥❤ ❧ý ▼✐r❛♥❞❛✲P♦✐♥❝❛r❡ tå♥ t➵✐ xm ∈ B ¯ n✳ ¯ n ❧➭ t❐♣ ❝♦♠♣❛❝t ♥➟♥ ❞➲② {xm }∞ ❝ã ❞➲② ❝♦♥ ❤é✐ tô ➤Õ♥ x0 ∈ B ▼➭ B m=2 m ¯ n ❦❤✐ m t✐Õ♥ r❛ ✈➠ ❝ï♥❣✳ ▲➵✐ ❝ã✱ g ❤é✐ tơ ➤Ị✉ ➤Õ♥ ➳♥❤ ①➵ f (x) − x tr➟♥ B ❉♦ ➤ã✱ ✶✳✸✳✷ f (x0 ) = x0 ➜Þ♥❤ ❧ý ❇♦rs✉❦ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ♥ã ✭➜Þ♥❤ ❧ý ❇♦rs✉❦✮ ❈❤♦ Ω ❧➭ ♠ét t❐♣ ♠ë✱ ❜Þ ❝❤➷♥✱ ➤è✐ ①ø♥❣ n ¯ Rn ) ❧➭ ➳♥❤ ①➵ ❧❰ (f (−x)) ✈➭ ❝❤ø❛ ❣è❝ t♦➵ ➤é tr♦♥❣ R ✱ f ∈ C(Ω; ➜Þ♥❤ ❧ý ✶✵✳ ❜✐➟♥ (x ∈ Ω ↔ (−x) ∈ Ω) = −f (x) ✈➭ ➯♥❤ ❝ñ❛ f (∂Ω) ❦❤➠♥❣ ❝❤ø❛ ❣è❝ t♦➵ ➤é✳ ❑❤✐ ➤ã✱ deg(f, Ω, 0) ❧➭ sè ❧❰✳ f ❜✃t ❦ú tr➟♥ ♠ét t❐♣ ➤è✐ ①ø♥❣ ❝ã ♥❤✐Ị✉ ❝➳❝❤ ➤Ĩ t➵♦ r❛ ♠ét ➳♥❤ ①➵ ❧❰ ❝❤➻♥❣ ❤➵♥ (f (x)−f (−x)) ❧➭ ♠ét ➳♥❤ ①➵ ❧❰✱ ❧✐➟♥ tô❝✳ ◆❣♦➭✐ r❛✱ ♥Õ✉ ❜✐Õt ❜❐❝ ❧➭ ♠ét sè ❧❰✱ ♥❣❤Ü❛ ❧➭ ♥ã ❦❤➳❝ 0✱ t❤× ♣❤➢➡♥❣ ❈❤ó ý✳ ❚r➢í❝ ❦❤✐ ➤✐ ✈➭♦ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ❇♦rs✉❦✱ t❛ ➤Ĩ ý r➺♥❣ ✈í✐ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ tr×♥❤ ❝ã ♥❣❤✐Ư♠✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ ❦❤➠♥❣ t❛ ❝ã t❤Ĩ ①➞② ❞ù♥❣ ♠ét ➳♥❤ ¯ Rn ), ∈ f (∂Ω) ✈➭ Jf (0) = ❱× ♥Õ✉ f ∈ C (Ω; ¯ ①➵ ❧❰ f˜ ∈ C (Ω; Rn ), ∈ f˜(∂Ω) ✈➭ Jf (0) = ♠➭ deg(f, Ω, 0) = deg(f˜, Ω, 0) ♥❤➢ s❛✉✳ ¯ ▲✃② ♠ét ➳♥❤ ①➵ g1 ∈ C (Ω; Rn ) s❛♦ ❝❤♦ ||f − g1 ||∞ ❧➭ ♠ét sè ➤ñ ♥❤á ✭➜✐Ị✉ ♥➭② ❧➭♠ ➤➢ỵ❝ ¯ ❧➭ ❝♦♠♣❛❝t ✮✳ ➜➷t ❞♦ Ω g2 (x) = (g1 (x) − g1 (−x)) ✶✻ ❈ã ¯ Rn ) g2 ∈ C (Ω; ❝❤ä♥ ♠ét sè ❞➢➡♥❣ ❧➭ ➳♥❤ ①➵ ❧❰ ✈➭ g2 (0) ❝ã ➤ó♥❣ n ❣✐➳ trÞ r✐➟♥❣ ✭❦Ĩ ❝➯ ❜é✐✮ ♥➟♥ t❛ ❝ã t❤Ĩ λ ➤đ ♥❤á ♠➭ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❣✐➳ trÞ r✐➟♥❣ ❝đ❛ g2 (0) ❚❛ ➤➷t f˜(x) = g2 (x) − λx, ❝ã ¯ Rn ) ❧➭ ➳♥❤ ①➵ ❧❰, J ˜(0) = Jg2 −λid (0) = 0, f˜ ∈ C (Ω; f ||f˜ − f ||∞ ≤ sup ||(g1 (x) − f (x)) − (g1 (−x) − f (−x)) + λx|| x∈Ω¯ ≤ ||g1 − f ||∞ + λ sup ||x|| ¯ x∈Ω ♠➭ ¯ ❧➭ ❝♦♠♣❛❝t✱ ♥➟♥ ✈í✐ ||g1 − f ||∞ Ω ➤đ ♥❤á✱ λ ➤đ ♥❤á t❤× deg(f˜, Ω, 0) = deg(f, Ω, 0) Jf (0) = ♥❤➢♥❣ ✈➱♥ ❝ã t❤Ĩ ❧➭ ➤✐Ĩ♠ ♥Õ♣ ❝đ❛ f r trờ ợ f (Sf ) tì f ❧➭ ➳♥❤ ①➵ ❧❰ ♥➟♥ ♥Õ✉ f (x) = 0, Jf (x) = f (−x) = 0, Jf (−x) = (−1)n Jf (x) ❉♦ ➤ã✱ ❝ã ❈❤ó ý r➺♥❣✱ ♠➷❝ ❞ï deg(f, Ω, 0) = sgn Jf (0) + t❤× sgn Jf (x) x∈f −1 (0)\{0} ❧➭ sè ❧❰✳ ❚❛ sÏ ①➞② ❞ù♥❣ ♠ét ➳♥❤ ①➵ ❧❰ ¯ Rn ) ♠➭ ∈ (g(Sg ) ∪ g(∂Ω)) ✈➭ g ∈ C (Ω; deg(f, Ω, 0) = deg(g, Ω, 0) ϕ ∈ C (R; R) ❤➵♥ ϕ(t) = t ✮✳ ➜➷t ❈❤ä♥ ♠➭ ϕ(−t) = −ϕ(t)∀t ∈ R, ϕ (0) = 0, ϕ(t) = ❝❤Ø ❦❤✐ t=0 ✭❝❤➻♥❣ Ωk = {x ∈ Ω|∃i ∈ {1, , k} : x(i) = 0}, f (x) f˜1 (x) = (f˜1 : Ω1 → Rn ) ϕ(x(1) ) f˜1 ∈ C (Ω1 ; Rn ) ❧➭ ➳♥❤ ①➵ ❧❰✳ ❉♦ m(f˜1 (Sf˜ )) = ♥➟♥ t❛ ❝ã t❤Ó ❝❤ä♥ y1 ❈ã ❣➬♥ ❣è❝ t♦➵ ➤é t✉② ý s❛♦ ❝❤♦ ➜➷t g1 (x) = f (x) − ϕ(x(1) )y1 ❉♦ ❦❤✐ g1 (x) = 0, x(1) = t❤× g1 (x) = ϕ(x(1) )f˜1 (x) ♥➟♥ ¯ Rn ) ❧➭ ➳♥❤ ①➵ ❧❰✱ • g1 ∈ C (Ω; • ∈ g1 (Sg1 ∩ Ω1 ), y1 ∈ f˜1 (Sf˜1 ) ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r • g1 ➤đ ❣➬♥ f ✭❦❤✐ ✶✼ ||y1 || ➤đ ♥❤á✮✳ ●✐➯ sư t❛ ➤➲ ①➞② ❞ù♥❣ ➤➢ỵ❝ ➳♥❤ ①➵ ❧❰ ¯ Rn )(1 ≤ k < n) gk ∈ C (Ω; ➤ñ ❣➬♥ f ✈➭ ∈ gk (Sgk ∩ Ωk ) ➜➷t f˜k+1 (x) = gk (x) (f˜k+1 : {x ∈ Ωk+1 |x(k+1) = 0} → Rn ) ϕ(x(k+1) ) m(f˜k+1 (Sf˜k+1 )) = f˜k+1 (Sf˜k+1 ) ❉♦ ♥➟♥ t❛ ❝ã t❤Ó ❝❤ä♥ yk+1 ❣➬♥ ❣è❝ t♦➵ ➤é tï② ý s❛♦ ❝❤♦ yk+1 ∈ ➜➷t gk+1 (x) = gk (x) − ϕ(x(k+1) )yk+1 ❉♦ ❦❤✐ gk+1 (x) = 0, x(k+1) = t❤× gk+1 (x) = gk (x) − (0, , ϕ (x(k+1) )yk+1 , , 0)t = ϕ(x(k+1) )f˜k+1 (x) ♥➟♥ ¯ Rn ) ❧➭ ➳♥❤ ①➵ ❧❰✱ • gk+1 ∈ C (Ω; • ∈ gk+1 (Sgk+1 ∩ {x ∈ Ωk+1 | x(k+1) = 0}), x ∈ Ωk+1 , x(k+1) = t❤× gk+1 (x) = gk (x) ✈➭ gk+1 (x) = gk (x), ❞♦ ➤ã ∈ gk+1 (Sgk+1 ∩ Ωk+1 ), • gk+1 ➤đ ❣➬♥ f ✭❦❤✐ ||y1 ||, , ||yk+1 || ➤ñ ♥❤á✮✳ ◆❤➢ ✈❐②✱ ❜➺♥❣ q✉② ♥➵♣ t❛ sÏ ①➞② ❞ù♥❣ ➤➢ỵ❝ ➳♥❤ ①➵ ¯ Rn ) s❛♦ ❝❤♦ g = gn ∈ C (Ω; ¯ Rn ) ❧➭ ➳♥❤ ①➵ ❧❰✱ • gn ∈ C (Ω; • ∈ gn (Sgn ∩ Ωn ), x ∈ Ω \ Ωn , ♥❣❤Ü❛ ❧➭ x = t❤× gn (0) = gk (0) = f (0), ❞♦ ➤ã ∈ gn (Sgn ∩ Ω), • gn ➤đ ❣➬♥ f ✭❦❤✐ ||y1 ||, , ||yn || ➤ñ ♥❤á✮ ❤❛② deg(gn , Ω, 0) = deg(f, Ω, 0) ❙❛✉ ➤➞② ❧➭ ♠ét ✈➭✐ ø♥❣ ❞ơ♥❣ ❝đ❛ ➜Þ♥❤ ❧ý ❇♦rs✉❦✳ ➜Þ♥❤ ❧ý ✶✶✳ t➵✐ ¯n x∈B ✭➜Þ♥❤ ❧ý ❇♦rs✉❦✲ ❯❧❛♠✮ ❈❤♦ ♠➭ f (−x) = f (x) ¯ n → Rn f :B ❧➭ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tô❝✳ ❑❤✐ ➤ã✱ tå♥ ✶✽ ❈❤ø♥❣ ♠✐♥❤✳ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ❇♦rs✉❦ ❝❤♦ ➳♥❤ ①➵ s❛✉ ¯ n → Rn , g:B g(x) = (f (x) − f (−x)), ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝✱ ❧❰ tr➟♥ t❐♣ ❜Þ ❝❤➷♥✱ ➤è✐ ①ø♥❣✱ ❝❤ø❛ ❣è❝ t♦➵ ➤é✳ n−1 ◆Õ✉ ∈ g(S ) t❤× deg(g, Bn , 0) ❧➭ sè ❧❰✱ ❦❤➳❝ 0✱ ♥➟♥ tå♥ t➵✐ x ∈ Bn ♠➭ f (−x)) = ❤❛② f (x) = f (−x) ¯n ∈ g(Sn−1 ) ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ x ∈ B f (−x) ◆Õ✉ ♠➭ g(x) = 12 (f (x) − g(x) = 12 (f (x) − f (−x)) = ❤❛② f (x) = A1 , , An ❧➭ ❝➳❝ t ị ợ tr R ó tå♥ t➵✐ ♠ét s✐➟✉ ♣❤➻♥❣ H = {y ∈ Rn | (y, a) = b}, tr♦♥❣ ➤ã a ∈ Rn , b ∈ R ❧➭ ❝è ➤Þ♥❤✱ ❝❤✐❛ ➤Ị✉ ❝➳❝ Ai , i = 1, , n t❤❡♦ ➤é ➤♦✱ ♥❣❤Ü❛ ❧➭ ➜Þ♥❤ ❧ý ✶✷✳ ✭➜Þ♥❤ ❧ý ❜➳♥❤ ❙❛♥❞✇✐❝❤✮ ❈❤♦ n m(Ai ∩ H + ) = m(Ai ∩ H − ), ∀i = 1, , n, tr♦♥❣ ➤ã✱ H + = {y ∈ Rn | (y, a) ≥ b}, H − = {y ∈ Rn | (y, a) ≤ b} ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã ♥❤❐♥ ①Ðt s❛✉ ¯ n = {x ∈ Rn | ∃x(n+1) ≥ : (x, x(n+1) ) ∈ Sn } B ❧➭ ♠ét t❐♣ ❝♦♠♣❛❝t✱ ➤è✐ ①ø♥❣ ✈➭ ❝❤ø❛ ố t ộ n (n+1) ỗ x B ❝ã ❞✉② ♥❤✃t ♠ét x ≥ ♠➭ (x, x(n+1) ) ∈ Sn , t❛ ➤➷t Hx = {y ∈ Rn | (y, x) = x(n+1) }, Hx+ = {y ∈ Rn | (y, x) ≥ x(n+1) } ❚❛ ①➞② ❞ù♥❣ ➤➢ỵ❝ ➳♥❤ ①➵ s❛✉ ¯ n → Rn , f :B fi (x) = m(Ai ∩ Hx+ ) ❉Ô t❤✃② f t❤♦➯ ♠➲♥ ➜Þ♥❤ ❧ý ❇♦rs✉❦✲ ❯❧❛♠✱ ♥➟♥ tå♥ t➵✐ ¯n x0 ∈ B ♠➭ fi (−x0 ) = fi (x0 ), ∀i = 1, , n ❤❛② m(Ai ∩ Hx+0 ) = m(Ai ∩ Hx−0 ), ∀i = 1, , n ❚r♦♥❣ ý tết P trì t ó ột ề tú ị s❛✉✳ ❚r♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❝ã sè ❝❤✐Ị✉ ❧í♥ ❤➡♥ 2✱ t♦➳♥ tư ✈✐ ♣❤➞♥ aα Dα ❧➭ ❡❧❧✐♣t✐❝ t❤× ❜❐❝ ❝đ❛ ♥ã m ♣❤➯✐ ❧➭ sè ❝❤➼♥✳ |α|≤m ➜✐Ị✉ ♥➭② ♥❣❤Ü❛ ❧➭ ✈í✐ n ≥ 3, ♥Õ✉ ➤❛ t❤ø❝ s❛✉ aα ξ α , aα ∈ C P (ξ) = |α|=m ❇➭✐ ✶✳ ▲ý t❤✉②Õt ❜❐❝ ❇r♦✇❡r ✶✾ ξ ∈ Rn \ {0} t❤× ❜❐❝ m ❝đ❛ ♥ã ❧➭ sè ❝❤➼♥✳ ❈❤ó ý r➺♥❣✱ ♥Õ✉ ❤Ư sè aα ❧➭ ❝➳❝ sè t❤ù❝ t❤× ❞Ơ ❞➭♥❣ ❝❤ø♥❣ ♠✐♥❤✳ ❚✉② ♥❤✐➟♥✱ ë ➤➞② ❝➳❝ ❤Ư sè aα ❧➭ ❝➳❝ sè ♣❤ø❝ t❤× ❦❤➠♥❣ q✉➳ ❤✐Ĩ♥ ♥❤✐➟♥✳ ❇ë✐ ✈×✱ ❦❤✐ n = ➤✐Ị✉ ♥➭② ❦❤➠♥❣ ❝ß♥ ➤ó♥❣✱ ❦❤➠♥❣ ❝ã ♥❣❤✐Ư♠ ❝❤➻♥❣ ❤➵♥ ➤❛ t❤ø❝ s❛✉ P2 (ξ1 , ξ2 ) = ξ1 + (−1)1/2 ξ2 ➜Þ♥❤ ❧ý ✶✸✳ ❱í✐ n ≥ 3, ♥Õ✉ ➤❛ t❤ø❝ s❛✉ aα ξ α , aα ∈ C P (ξ) = |α|=m ❦❤➠♥❣ ❝ã ♥❣❤✐Ö♠ ξ ∈ Rn \ {0} t❤× ❜❐❝ m ❝đ❛ ♥ã ❧➭ sè ❝❤➼♥✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ❜➺♥❣ ♣❤➯♥ ❝❤ø♥❣✳ ●✐➯ sö m ❧➭ sè ❧❰✳ ❳Ðt ➳♥❤ ①➵ f : R2 → R , f (ξ1 , ξ2 ) = (ReP (ξ1 , ξ2 , 0, , 0), ImP (ξ1 , ξ2 , 0, , 0)) ❈ã m ❧❰ • f (−ξ1 , −ξ2 ) = (−1)m f (ξ1 , ξ2 ) = −f (ξ1 , ξ2 ), • f (ξ1 , ξ2 ) = 0, ∀||ξ1 ||2 + ||ξ2 ||2 = 0, deg(f, B(0, 1), 0) ❧➭ sè ❧❰✳ h > ①Ðt ➳♥❤ ①➵ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý rs ỗ fh : R2 R2 , fh (ξ1 , ξ2 ) = (ReP (ξ1 , ξ2 , h, 0, , 0), ImP (ξ1 , ξ2 , h, 0, , 0)) ❱í✐ h > ➤đ ♥❤á t❤× fh ➤đ ❣➬♥ f ♥➟♥ t❤❡♦ tÝ♥❤ ❝❤✃t (d5) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ ❝ã deg(fh , B(0, 1), 0) = deg(f, B(0, 1), 0) ❧➭ ♠ét sè ❧❰✱ ❦❤➳❝ 0✱ ♥➟♥ tå♥ t➵✐ (ξ1 , ξ2 ) ∈ B(0, 1) s❛♦ ❝❤♦ fh (ξ1 , ξ2 ) = ❤❛② P (ξ1 , ξ2 , h, 0, , 0) = ➜✐Ị✉ ♥➭② tr➳✐ ✈í✐ ❣✐➯ t❤✐Õt ❤❛② ✈í✐ (ξ1 , ξ2 , h, 0, , 0) ∈ Rn \ {0} m ❧➭ sè ❝❤➼♥✳ ➜Þ♥❤ ❧ý ❞➢í✐ ➤➞② ❧➭ ♠ét ➜Þ♥❤ ❧ý t❤ó ✈Þ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♥❤✐Ị✉ ❝❤✐Ị✉✳ ❑❤✐ sè ❝❤✐Ị✉ ❜➺♥❣ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ❦❤➠♥❣ ❦❤ã✳ ◆❤➢♥❣ ❦❤✐ sè ❝❤✐Ị✉ ❧í♥ ❤➡♥ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ❦❤➠♥❣ ❝ß♥ ➤➡♥ ❣✐➯♥✳ f : Rn → Rn ❧➭ ➳♥❤ ||x|| → ∞ ❑❤✐ ➤ã✱ f ❧➭ ♠ét ➤å♥❣ ♣❤➠✐✳ ➜Þ♥❤ ❧ý ✶✹✳ ❈❤♦ ①➵ ❧✐➟♥ tơ❝✱ ➤➡♥ ➳♥❤ t❤♦➯ ♠➲♥ ||f (x)|| → ∞ ❦❤✐ ✷✵ ❧➭ t♦➭♥ ➳♥❤✳ ❑❤✐ ➤ã✱ tõ ❣✐➯ t❤✐Õt✱ f ❝ã ➳♥❤ ①➵ ♥❣➢ỵ❝✳ n n ❧➭ t♦➭♥ ➳♥❤✱ ➤Ĩ ý r➺♥❣ R ❧➭ t❐♣ ❧✐➟♥ t❤➠♥❣✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ f (R ) ✈õ❛ ❈❤ø♥❣ ♠✐♥❤✳ ➜➬✉ t✐➟♥✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ➜Ó ❝❤ø♥❣ ♠✐♥❤ f f ➤ã♥❣✱ ✈õ❛ ♠ë✳ n ∞ n ▲✃② y ∈ R s❛♦ ❝❤♦ ❝ã ❞➲② {xn }n=1 tr♦♥❣ R ♠➭ f (xn ) → y ❦❤✐ n → ∞ ∞ ❉➲② {xn }n=1 ❧➭ ❞➲② ị ì ế ó ó ột xnk → ∞✱ t❤❡♦ ❣✐➯ t❤✐Õt t❤× f (xnk ) → ∞ ❦❤✐ nk → ∞ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ✈✐Ư❝ f (xn ) → y ❦❤✐ n → ∞ n ∞ ❑❤✐ ➤ã✱ {xn }n=1 ❝ã ♠ét ❞➲② ❝♦♥ xnk → x ∈ R , ♠➭ f ❧✐➟♥ tô❝✱ f (xnk ) → f (x) ❦❤✐ nk → ∞ ❉♦ ➤ã✱ y = f (x) ❤❛② y ∈ f (Rn ) n ❉♦ ➤ã✱ f (R ) ➤ã♥❣✳ n n ❚❛ ❝❤ø♥❣ ♠✐♥❤ f (R ) ♠ë✱ ♥❣❤Ü❛ ỗ x0 R t ỉ r ❝➳❝ sè ❞➢➡♥❣ r, R s❛♦ ❝❤♦ B(f (x0 ), R) ⊂ f (B(x0 , r)) ❚❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ♥➭② ➤è✐ ✈í✐ ➤✐Ĩ♠ x0 = ✈➭ ❤➭♠ f ♠➭ f (0) = ❱× ♥Õ✉ ❦❤➠♥❣ t❛ ①Ðt ❤➭♠ f˜(x) = f (x + x0 ) f (x0 ) tì lt,11 f : Rn −→ Rn , ||f˜(x)|| → ∞ ❦❤✐ ||x|| → ∞, f˜(0) = 0, • B(0, r) + x0 = B(x0 , r), B(0, R) = B(f (x0 ), R) − f (x0 ) ¯ n = {x ∈ Rn | ||x|| ≤ 1} r = 1, ❦ý ❤✐Ö✉ B ¯ n → f (B ¯ n ) ❧➭ s♦♥❣ ➳♥❤✳ ❉♦ f ❧➭ ➤➡♥ ➳♥❤ ♥➟♥ f |B ¯n : B ❈❤ä♥ ❳Ðt ➤å♥❣ ❧✉➞♥ ¯ n → Rn , H : [0, 1] × B −t H(t, x) = f |B¯ n ( x) − f |B¯ n ( x) 1+t 1+t ❈ã ✶✮H(0, x) = f |B¯ n (x), H(1, x) = f |B¯ n ( x2 ) − f |B¯ n ( −x ), −t H(t, x) = 0, (t, x) ∈ [0, 1] × Sn−1 t❤× f |B¯ n ( 1+t x) = f |B¯ n ( 1+t x) ❞♦ ➤ã 1+t x= n−1 n−1 ❤❛② x = ✭✈➠ ❧ý ✈× x ∈ S ✮✱ ♥➟♥ H(t, x) = 0, ∀(t, x) ∈ [0, 1] × S , ✷✮ ♥Õ✉ ✸✮ H(1, x) ❧➭ ➳♥❤ ①➵ ❧❰ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ❇♦rs✉❦ deg(H(1, ), Bn , 0) ❧➭ sè ❧❰✳ ❑❤✐ ➤ã✱ t❤❡♦ tÝ♥❤ ❝❤✃t (d3), (d5) tr♦♥❣ ➜Þ♥❤ ❧ý ✹ ❝ã ♠ét sè ❞➢➡♥❣ R s❛♦ ❝❤♦ deg(f |Bn , Bn , y) = deg(f |Bn , Bn , 0) = deg(H(1, ), Bn , 0), ∀y ∈ B(0, R), B(0, R) ⊂ f (Bn ) −1 ❱✐Ö❝ ❝❤ø♥❣ ♠✐♥❤ f ❧✐➟♥ tô❝ ❦❤➠♥❣ ❦❤ã✳ ❧➭ sè ❧❰ ❤❛② −t x 1+t