Về định lí điểm bất động cho ánh xạ giữa các không gian g metric đầy đủ

G− G− G− G− G− G− G− G− G− G− G− G− G− G− 2− 2− D− G− G− G− G− G− G− G− G− G− G− 2− X R + d : X × X × X → R + x, y ∈ X, z ∈ X d(x, y, z) = 0. d(x, y, z) = 0 x, y, z ∈ X d(x, y, z) = d(x, z, y) = d(y, z, x) x, y, z ∈ X. d(x, y, z) ≤ d(x, y, a) + d(x, a, z) + d(a, y, z), x, y, z, a ∈ X. d 2− X (X, G) 2− X = R 2 , x, y, z ∈ X d(x, y, z) x, y, z. d 2− R 2 . D− D : X × X × X → R + D− D(x, y, z) = 0 x = y = z, D(x, y, z) ≤ D(x, z, z) + D(z, y, y), x, y, z ∈ X. X = R 2 , x, y, z ∈ X d(x, y, z) x, y, z. d D− R 2 . (X, d) (E s ) D s (d)(x, y, z) = 1 3 [d(x, y) + d(y, z) + d(x, z) ] (E m ) D m (d)(x, y, z) = max{d(x, y), d(y, z), d(x, z)} D− X G : X × X × X → R + G(x, y, z) = 0 x = y = z, 0 < G(x, x, y), x, y ∈ X, x = y, G(x, x, y) ≤ G(x, y, z), x, y, z ∈ X, z = y, G(x, y, z) = G(x, z, y) = G(y, z, x) = . . . , , G(x, y, z) ≤ G(x, a, a) + G(a, y, z), x, y, z, a ∈ X G G− X (X, G) G− G− (X, G) G(x, y, y) = G( x, x, y), x, y ∈ X. 2− G− D− G− G : R 3 −→ R + G(x, y, z) = |x − y| + |y −z| + |z − x| x, y, z ∈ R. G G− R (R, G) G− G− G− (X, G) G− x, y, z a ∈ X G(x, y, z) = 0 x = y = z G(x, y, z) ≤ G(x, x, y) + G(x, x, z) G(x, y, y) ≤ 2G(y, x, x) G(x, y, z) ≤ G(x, a, z) + G(a, y, z) G(x, y, z) ≤ 2 3 (G(x, y, a) + G(x, a, z) + G(a, y, z)) G(x, y, z) ≤ (G(x, a, a) + G(y, a, a) + G(z, a, a)) |G(x, y, z) − G(x, y, a)| ≤ max{G(a, z, z), G(z, a, a)} |G(x, y, z) − G(x, y, a)| ≤ G(x, a, z) |G(x, y, z) − G(y, z, z)| ≤ max{G(x, z, z), G(z, x, x)} |G(x, y, y) −G(y, x, x)| ≤ max{G(y, x, x), G(x, y, y)} G(x, y, z) = 0 x = y = z x = y G(x, y, z) ≥ G(x, x, y) > 0 x = y y = z x = y = z G(x, y, z) ≤ G(x, x, y) + G(x, x, z) G(x, y, z) = G(y, x, z) (G5) ≤ G(y, x, x) + G(x, x, z) ( a = x) = G(x, x, y) + G(x, x, z). G(x, y, y) ≤ 2G(y, x, x) z = y G(x, y, z) ≤ G(x, a, z) + G(a, y, z) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) (G3) ≤ G(x, a, z) + G(a, y, z). G(x, y, z) ≤ 2 3 (G(x, y, a) + G(x, a, z) + G(a, y, z)) G(x, y, z) (G5) ≤ G(x, a, a) + G(a, y, z) ≤ G( x, a, y) + G(a, y, z), G(y, z, x) ≤ G(y, a, a) + G(a, z, x) ≤ G(y, a, z) + G(a, z, x), G(z, x, y) ≤ G(z, a, a) + G(a, x, y) ≤ G(z, a, x) + G(a, x, y). 3G(x, y, z) ≤ 2(G(a, y, z) + G(x, y, a) + G(x, a, z)), G(x, y, z) ≤ G(x, a, a) + G(y, a, a) + G(z, a, a) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) G(x, y, z) ≤ G(x, a, a) + G(a, a, y) + G(a, a, z). |G(x, y, z) − G(x, y, a)| ≤ max{G(a, z, z), G(z, a, a)} G(x, y, z) (G5) ≤ G(z, a, a) + G(x, y, a) G(x, y, z) − G(x, y, a) ≤ G(z, a, a). a z G(x, y, a) −G(x, y, z) ≤ G(a, z, z). |G(x, y, z) − G(x, y, a)| ≤ G(x, a, z) G(x, y, z) − G(x, y, a) ≤ G(z, a, a) ≤ G(z, a, x), G(x, y, a) − G(x, y, z) ≤ G(a, z, z) ≤ G(a, z, x). |G(x, y, z) − G(y, z, z)| ≤ max{G(x, z, z), G(z, x, x)} |G(x, y, z) − G(y, z, a)| ≤ max{G(a, x, x), G(x, a, a)}. a = z |G(x, y, y) −G(y, x, x)| ≤ max{G(y, x, x), G(x, y, y)} z = y, a = x G(x, y, y) − G(x, y, x) ≤ G(y, x, x). x y G(y, x, x) − G(y, x, y) ≤ G(x, y, y).  (X, G) G− k > 0 G 1 G 2 G− X G 1 (x, y, z) = min{k, G(x, y, z)} G 2 (x, y, z) = G(x, y, z) k + G(x, y, z) · X = n  i=1 A i X G 3 (x, y, z) =  G(x, y, z), i x, y, z ∈ A i , k + G(x, y, z), G− G 1 (x, y, z) = min{k, G(x, y, z)} G 1 (x, y, z) = min{k, G(x, y, z)} ≤ min{k, G(x, a, a) + G(a, y, z)} ≤ min{k, G(x, a, a)}+ min{k, G(a, y, z)} = G 1 (x, a, a) + G 1 (a, y, z). [...]... g( x) = g( y) = F (y, x) ề ẹề ẻứ (x, y) é ẹ ỉ ễ ửẹ ỉệ ề ề ề ĩ F g á ỉ g( x) = F (x, y) g( y) = F (y, x) ì g( x) = g( y)á ỉ ể ắ à ỉ ể G( g(x), g( y), g( y)) = G( F (x, y), F (y, x), F (y, x)) k (G( g(x), g( y), g( y)) + G( g(y), g( x), g( x))) G( g(x), g( y), g( y)) + G( g(y), g( x), g( x)) 2k (G( g(x), g( y), g( y)) + G( g(y), g( x )g( x))) ẻứ 2k < 1á ỉ G( g(x), g( y), g( y)) + G( g(y), g( x), g( x)) < G( g(x), g( y), g( y)) + G( g(y),... G( g(xn) ,g( xn+1), g( xn+1)) k (G( g(xn1), g( xn), g( xn)) + G( g(yn1), g( yn), g( yn))) 2k 2 (G( g(xn2), g( xn1), g( xn1)) + G( g(yn2), g( yn1), g( yn1))) ããããããããã 1 (2k)n (G( g(x0), g( x1), g( x1)) + G( g(y0), g( y1), g( y1))) 2 ể á ẹ nNỉ G( g(xn ), g( xn+1), g( xn+1)) 1 (2k)n (G( g(x0), g( x1), g( x1)) + G( g(y0 ), g( y1), g( y1))) 2 ể m, n N G ỉệ ỉ m > n è ể ỉ ũề ú à ề ề ỳ ắ ề à ề G( g(xn), g( xm), g( xm)) G( g(xn), g( xn+1), g( xn+1)) + G( g(xn+1),... ắ à G( g(xn1), g( xn), g( xn)) = G( F (xn2, yn2), F (xn1, yn1), F (xn1, yn1)) k (G( g(xn2), g( xn1), g( xn1)) + G( g(yn2), g( yn1), g( yn1))), G( g(yn1), g( yn), g( yn)) = G( F (yn2, xn2), F (yn1, xn1), F (yn1, xn1)) k (G( g(yn2), g( yn1), g( yn1)) + G( g(xn2), g( xn1), g( xn1))), ắ ỉ G( g(xn1) ,g( xn), g( xn)) + G( g(yn1), g( yn), g( yn)) 2k (G( g(xn2), g( xn1), g( xn1)) + G( g(yn2), g( yn1), g( yn1))), ề ẹ n N ể ỉ G( g(xn) ,g( xn+1),... ẻứ g é G ũề ỉ á ỉ (gg(xn )) é G ỉ ú g( x) (gg(yn )) é G ỉ ú gy ẻứ g F ể ể ề ềũề ỉ ề gg(xn+1) = g( F (xn, yn)) = F (g( xn), g( yn)) gg(yn+1 ) = g( F (yn, xn)) = F (g( yn), g( xn)) ể G( gg(xn+1), F (x, y), F (x, y)) = G( F (g( xn), g( yn)), F (x, y), F (x, y)) k (G( gg(xn), g( x), g( x)) + G( gg(yn ), g( y), g( y))) ể n +á ì ề ỉ ụỉ G é é ũề ỉ ỉệũề ụề ề áỉ G( g(x), F (x, y), F (x, y)) k (G( g(x), g( x), g( x)) + G( g(y),... + G( g(xn+1), g( xn+2), g( xn+2)) + + G( g(xm1), g( xm), g( xm)) ẻứ 2k < 1á ỉ ể ắ àỉ G( g(xn ), g( xm), g( xm)) 1 2 m1 (2k)i (G( g(x0), g( x1), g( x1)) + G( g(y0), g( y1), g( y1))) i=n n (2k) (G( g(x0), g( x1), g( x1)) + G( g(y0), g( y1), g( y1))) 2(1 2k) ể n, m +á ỉ lim n,m+ G( g(xn), g( xm), g( xm)) = 0 ẻ í (g( xn)) é G í ỉệểề g( X) è ề ỉ á ỉ ỷ ệ (g( yn)) é G í ỉệểề g( X) ẻứ g( X) é G í á ỉ (g( xn)) g( yn ) é G ỉ ú... g( y))) ể n + ề ỉ ụỉ G é é ũề ỉ ỉệũề ụề ề áỉ G( x, g( x), g( x)) k (G( x, g( x), g( x)) + G( y, g( y), g( y))) è ề ỉ áỉ ỷệ G( y, g( y), g( y)) k (G( x, g( x), g( x)) + G( y, g( y), g( y))) ể G( x, g( x), g( x)) + G( y, g( y), g( y)) 2k (G( x, g( x), g( x)) + G( y, g( y), g( y))) ẻứ 2k < 1á ỉ ề ỉ ỷ ĩ í ệ G( x, g( x), g( x)) = 0 G( y, g( y), g( y)) = 0 ậí ệ x = g( x) y = g( y) ể ỉ g( x) = F (x, x) = x ... G( g(y), g( y), g( y))) = 0 ậí ệ g( x) = F (x, y) è ề ỉ á ỉ ỷ ệ ệ ề g( y) = F (y, x) è ể ú ắẵá (x, y) é ẹ ỉ ễ ửẹ ỉ ề ề ĩ F g ể g( x) = F (x, y) = F (y, x) = g( y) ẻứ (g( xn+1)) é í ểề (g( xn)) ềũề (g( xn+1)) é G ỉ ú x ể í G( g(xn+1), g( x), g( x)) = G( g(xn+1), F (x, y), F (x, y)) = G( F (xn, yn ), F (x, y), F (x, y)) k (G( g(xn), g( x), g( x)) + G( g(yn), g( y), g( y))) ể n + ề ỉ ụỉ G é é ũề ỉ ỉệũề ụề ề áỉ G( x, g( x),... + G( g(y), g( x), g( x)), ẵ ú ề í ẹ ỉ ề ẻ í g( x) = g( y) ìí ệ F (x, y) = g( x) = g( y) = F (y, x) ề é ắ ẵẵà ể (X, G) é ẹ ỉ X ìX X g: X X é ề ề ề ĩ ì ể ể G ỉệ ể F : G( F (x, y), F (u, v), F (z, w)) k (G( g(x), g( u), g( z)) + G( g(y), g( y), g( w))), ắ à ì F g ỉ ẹ ề ú ữề ì ẹ x, y, z, u, v, w X ẵà F (X ì X) g( X)á í á ể ể ề à g é G ũề ỉ ắà g( X) é G F ặụ k 0, 1 ỉ ứ í ề ỉ x ỉệểề X ì ể ể g( x) = F (x,... ề é ề í G Gẹ í ỉệ ỉ G í ề ề G ỉệ (X, G) é í G í ỉệểề (X, G) é G ỉ ỉệểề (X, G) ề ề ẹề ẹữề ú ì ữề ú ẵẵẳ ỉ ềụ (X, dG) é ẹ ỉ ề ề ề G ỉệ (X, G) é ề ẹ ỉệ í G ữ ế ẵ ặụ Y é ẹ ỉ ỉ ễ ểề ệ ề ẹ ỉ ề í ềụ ỷ ề G ỉệ í (X, G) ỉ ứ (Y, G| Y ) é í ỷ Y é G ề ỉệểề (X, G) á ỉệểề G| Y é ỉ ễ ẹ ì ề ỉ ễ G ỉệũề Y ữ ế ẵ ể (X, G) é ẹ ỉ ề ề G ỉệ (Fn) é ẹ ỉ í ệề Xì ể ẹ (F1 F2 F3 ) ỉ ễ ểề G ề ể sup {G( x, y, z)... (xn) é G n ỉ ể G( x, xn, xn) 0 ỉ ỉ xá ỉ é lim G( x, xn, xm) = 0á ể m,n ể dG (xn, x) = G( xn, x, x) + G( xn, xn, x) = G( x, xn, x) + G( x, xn, xn) G( x, xn, xn) + G( xn, xn, x) + G( x, xn, xn) n = 3G( x, xn, xn) 0 ậí ệ xn x è ể ẹ ỉệ dG ắà à ửề ề ũề G( xn , xn, x) G( xn , x, x) + G( xn , xn, x) = dG (xn, x)á ềũề ềụ dG (xn, x) 0 ỉ ứ G( xn , xn, x) 0 à à è G( xn , x, x) = G( x, xn, x) G( x, xn, xn) + G( xn, . (x n ) G  G G G G G (X, G) G G G (X, G) G G (X, G) G (X, G) G (X, G) G (X, d G ) Y G (X, G) (Y, G| Y ) Y G (X, G) G| Y G Y (X, G) G (F n ) (F 1 ⊇ F 2 ⊇ F 3 ⊇ . . .) G X sup {G( x,. G G G G G G G G G G G G G G 2− 2− D− G G G G G G G G G G 2− X R + d : X × X × X → R + x, y ∈ X, z ∈ X d(x, y, z). (X, G) G ∞  n=1 F n G (X, G) G ε > 0 A ⊆ X ε− (X, G) x ∈ X a ∈ A x ∈ B G (a, ε) A A ε− (X , G) A ε− X =  a∈A B G (a, ε) G (X, G) G ε > 0 ε− G (X, G) G G G G (X, G) (X, G) G (X,