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! (a, b), −∞ < a b < +∞ " "!" "
#
$ ! (a, S max {|a|,|b|} (0) $ ! !$ $ b) ⊆ ∞ 1 min a + n , b n=1 ⊂ (a, b) ! a ∈ / (a, b)% &'"
"
" " [a, b] ! (a, b) ( !$ ! " " f
[a, b]
f (t) > f (a), ∀ t ∈ (a, b).
# %
) " ! f !" !' ! !$ {tk }∞ k=1 "" a # ! !" !' ! !$ *! ! (a, b)% ! f (tk ) −−−→ f (a) k→∞ ! *
∀ε > 0 ∃N ∈ N :
n N =⇒ |f (tn ) − f (a)| < ε,
!
f (tn ) < f (a) + ε,
#+%
! !" !' ! !$ {tk }∞ k=1 !' !$ ∞ ! ε " n = n ({tk }k=1, ε) # % #+% ! !
inf f (t) = f (a).
t∈(a,b)
) " ! # % " f * ! , ! $ "
" " ! - " " * !"
' -!$
U "!" "
" R !.$ ! $ U = R # R "!" " % ( !$ " "
. * " / !$ ! !
! [a, +∞) #! (−∞, b]% ' ! -
' " " f (t) !
" +
[a, +∞) ! "" .
! limt→+∞ f (t) = δ ! f (t) > δ, ∀ t a 1 + δ ! f (t) = t−a+1 + ( $ . C[a, b] "!" "
$ U = f (t) ∈ C[a, b] f (t) C[a,b] 1 0
$
U C[a, b] ! " (!" !$ ρ !$ ! !$ $ hk (t) −−−→ h(t) hk (t) ∈ U, ∀ k k→∞
1 ! ∀ε > 0 ∃N ∈ N :
k N =⇒ max |fk (t) − f (t)| < ε, t∈[a,b]
! |fk (t) − f (t)| < ε, ∀ t ∈ [a, b] !
fk (t) 1 ! |f (t)| < 1 + ε, ∀ t ∈ [a, b] - ε ' $ ' !$ ! |f (t)| 1, ∀ t ∈ [a, b] !
f (t) 1 ) " f (t) ! ""
! !$ - ' f (t) ∈ U (!" !$ ! !$ . C[a, b] "" ! !$ $ ! . C[a, b] - ! !$ $ "!" " {fk (t)}∞ k=1 t−a 1 − n b−a , a t < a + b−a , k fk (t) = b−a 0, a + k t b. ! fk (t) −−−→ f (t) ∈ C[a, b] maxt∈[a,b] |f (t)| 1 1 →∞
∀ε > 0 ∃N ∈ N :
N =⇒ max |fk (t) − f (t)| < ε. t∈[a,b]
3! ! {fk (t)}∞ k=1 * b−a ∀t ∈ a + , b =⇒ |fk (t)| < ε ⇐⇒ −ε < fk (t) < ε k |fk (a) − 1| < ε ⇐⇒ 1 − ε < fk (a) < 1 + ε.
#2%
#4% #5%
) " #2% ε < 12 #4% #5% → ∞ ! f (t) a ! f (t) ∈ C[a, b] - ' !'" ! !$ $ {fk (t)}∞ =1 ! ∞ !$ {fk (t)}k=1 ! ! # ! !$ . C[a, b]%
$ U 2
2 ( $ C[a, b] "!" " !
b !" f, g = a f (t)g(t) dt "!" " !$'
(!" !$ C[a, b] 6 ! ' $ !" !" "!" " " ! "
" ! 7 " $ !$ ! ! !$ ! ( f, f = 0 ⇔ f ≡ 0 [a, b] ! 8 $
" " t0 [a, b] |f (t)| ε > 0 - ! !!$ " " δ $ Uδ * # ! $ ! t0 % f (t) > 2δ - 2 b δ δ3 δ 2 2 > 0, f, f = f (t) dt f (t) dt > · = 2 2 8 Uδ a f, f = 0 - ' C[a, b] !$ "!" " !
!" $ C[a, b] "!" " !$' !$
!" " (!" * !$ ! ∞ !$
{fk (t)}k=1 ⊂ C[a, b] ! ρ(f, g) = f − g, f − g ! " ' a = −1, b = 1 ⎧ 1 ⎪ −1, t ∈ −1, − , ⎨ 1 1 n fn (t) = nt, t ∈ − n , n , ⎪ ⎩ 1, t ∈ n1 , 1 . ( !$ $ * ! !$ $ "!" " !$ 8 1 1 n 2 ρ(fn , fn+m (t)) = |fn (t) − fn+m (t)| dt = |fn (t) − fn+m (t)|2 dt
=
1 −n
−1
1 n+m
1 − n+m
2
(nt − (n + m)t) dt + 2
1 n 1 n+m
2
(1 − nt) dt =
1 1 − −−−−→ 0. n n + m n,m→∞
) " ! ! !$ ρ "!" " " " # !$ !$ " "
" ! !$ " !$ 4
!$ " " ' $ ⎧ ⎪ ⎨−1, f (t) = 0, ⎪ ⎩ 1,
%
t ∈ [−1, 0) , t = 0, t ∈ (0, 1] .
1 C[a, b] "!" " ! !$
. ! !$' - ' C[a, b] "!" " !$' !$
4 ( $ 9 !!! : L2 (a, b) "!" "
7 9 !!! : U = u(t) ∈ L2 (0, 2π) −1 u(t) 1,
t ∈ (0, wπ) .
; !$ " ! !$ $ {cos nt}∞ n=1 ! U !$ " ! $ "" L2 ! !$ $ ( !$ $ {cos nk t}∞ k=1 " L2 - L2 ! * !$ " ! !$
$ {cos nk t}∞ k=1 "!" " !$ !" !' m, ∈ N, m = ! " " # " !$ $ cos nm t cos n t% 2π ρ(cos nm t, cos n t) = (cos nm t − cos n t)2 dt 0
2π
= 0
cos2
nm t dt +
2π
cos2
0
2π
= 0
n t dt − 2
2π
cos nm t cos n t dt
0
cos2
nm t dt +
2π 0
cos2 n t dt =
√
2π > 0.
- ' * ! !$ *! ! 9 !!! : L2 (0, 2π) !$ " ! $ "" ! !$ 9 !!! : L2 (a, b) ' "
"!" " 5
!" x ! ! 1 f (x + h) = f (x) + f1 · h + f2 · h2 + o h2 , 2
" "!" " * 5
7
x3 sin x1 , x = 0, f (x) = 0, x = 0.
0 $ ! "
" 3x2 sin x1 − x cos x1 , x = 0, f (x) = 0, x = 0. ) " ! ! . ! #h = 0%
g(h) =
3h2 sin h1 − h cos h1 f (h) − f (0) 1 1 = = 3h sin − cos h h h h
h → 0 ! (!" * $ ' 1 ! ! !$ $ {hk }∞ k=1 , hk = πk 8
g(hk ) = − cos
1 = (−1)k+1 c, k→∞ hk
∀ c ∈ R.
(!" . " !$ $ !" f (x) !
! ( !$
0 f (h) = o h2 = 0 + 0 · h + · h2 + o h2 . 2
0, q(x) 0 6
! " k(x) q(x) # % ' ! ' . ! B $ y u ! " $ y(x; u) = L(x)[u] L(x) 6 ! L2 (0, ) ) * ! ' ! J(u) = |y(x; u) − z(x)|2 dx = L(x)[u] − z(x) 2L2 (0,) . 0
C J (u) = 2L∗ (x)[L(x)[u] − z(x)] = L(x) L∗ (x) C '
!$
!"
. !$ 7 !$ ⎧ ⎪ G(x, s) x !'
s, 0 x , 0 < s < , ⎪ ⎪ ⎪ ⎨(k(x)G (x, s)) − q(x)G(x, s) = 0, x ∈ [0, ] \ {s}, x ⎪ G(0, s) = G(, s) = 0, s ∈ [0, ], ⎪ ⎪ ⎪ ⎩G (s + 0, s) − G (s − 0, s) = 1 , s ∈ (0, ). k(s) . !
!" *
. D 3 $ . !" " y(x; u) = − G(x, s)u(s) ds = L(x)[u]. 0
& $ L(x) 6 "
1 ! D
! "!" " '
"
M(x)[y] = (k(x)y (x)) − q(x)y(x) - ' J (u) = 2L(x)[L(x)[u] − z(x)] = 2 G(x, t) G(t, s)u(s) ds + z(t) dt
=2
0
0
G(x, t)G(t, s)u(s) ds dt + 2 0
0
G(x, t)z(t) dt, 0
G(x, s) !" " !$ E
E $ H 6 !$' L 6 ( $ " H L "!" " !
"
# !$
" H L%
0' !$ " H
L P - L !" ' ! # ! ! !"% # !% " P h !
!" !' h ∈ H 7 ! " H "
L !$ ! " L⊥ ) * ' " !' *! h ∈ H H 6 !$' ! " h = h1 + h2 , h1 ∈ L, h2 ∈ L⊥ *
arg min ρ(u, h) = arg min ρ (u, h1 + h2 ) = arg min u − h1 − h2 , u − h1 − h2 u∈L
u∈L
u∈L
= arg min u − h1 − h2 , u − h1 − h2 u∈L = arg min u 2 − 2 u, h1 + h1 2 + h2 2 . u∈L
B
u 2 − 2 u, h1 + h1 2 + h2 2 = u − h1 2 + h2 2 h2 2 ,
" u = h1 - '
h1 = arg min ρ(u, h). u∈L
0 ! $ ! $ " P 0
$ P ! " '! ' 6 "
$ P 7 !" 8
P u, v = u, P ∗ v ,
∀ u, v ∈ H.
) !$ " ! *! !$' L !$ ! 8 u = u1 +u2 , v = v1 +v2 u1 , v1 ∈ L, u2 , v2 ∈ L⊥ - !$ " !
!$
P u, v = u1 , v1 + v2 = u1 , v1 = u1 + u2 , v1 = u, P v . - ' !$ " "!" " !
"
$ x0 6
*! H x0 + L 6
' ( $ p = prx0 +L h !$ ! p ∈ x0 + L, p − h, H = 0, ∀ ∈ L
3 ! h x0 + L (!" *
min u − h, u − h = min x0 + g − h, x0 + g − h .
u∈x0 +L
g∈L
7 !" ' $" * " !$ "
f (g) = g 2 + 2 g, x0 − 2 g, h , g ∈ L. ) !$ " !" x0 h ! L !$ ! 8 h = h1 + h2 , x0 = x1 + x2 , h1 , x1 ∈ L, h2 , x2 ∈ L⊥ ! *
f (g) = g 2 − 2 g, h1 − x1 = g − (h1 − x1 ) 2 − h1 − x1 2 h1 − x1 2 ,
! ! " g = h1 − x1 ∈ L - ' " x0 + L *! !$' H
p = x0 + h1 − x1 = x2 + h1 . 0 ' $
. !"8 !$
p − h, = x2 − h2 , = 0, x2 − h2 ∈ L⊥ ( $ " 8 !" * $ *! p h ! ! !$ ! !$ L8
p − h, = x0 + g − h, = x1 + x2 + g − h1 − h2 , = x1 − h1 + g, = 0, ! " " !" ∈ L 1 x1 −h1 +g ⊥ L x1 −h1 +g ∈ L L 6 ! H 1 g = h1 − x1 + ) ! $ ! $ {u ∈ H | x, u H = β} !$' H !!! Rn
{u = (u1 , . . . , un ) ∈ Rn | αi ui βi , i = 1, . . . , n} .
= h ! $ !$' $ c = 0 - ! ! βc
! p = h − h, c c c 2 + c 2 ( !$
p, c = h, c − h, c
+
c 2
c 2 + β = β.
c 2
c 2
) "
" "
p − h, u − p = 0 !" !' u ! 3! !$
! # "!" " ! $% * p 6 " h ! $ 7 !!! Π Rn 3! ! !" " Rn h "!" "
arg min u∈Π
n
(ui − hi )2
i=1
0
" !8 $ h " s, s n Π #' " ' ' $ * s % ) * ! " # !$ %
p=
h1 , . . . , hs ,
arg min us+1 ∈{αs+1 ,βs+1 }
|us+1 − hs+1 | , . . . , arg min |un − hn | . un ∈{αn ,βn }
2 ) $ " !" . αk . ! J(u) = 12 Au, u H − f, u H ! A∗ = A 0
0' . !" ! ' 8
αk = arg min J(uk − αJ (uk )). α0
D # >% J (u) = 12 (A + A∗ )u − f " ! "
J (u) = Au − f - ' $ ! !8
F (α) = J(uk − αAuk + αf ) 1 = A(uk − αAuk + αf ), uk − αAuk + αf − f, uk − αAuk + αf . 2 ! " ' !
F (α) = ! " = α2 ( 12 A3 uk , uk − Auk , Af + 12 Af, f )−α Auk − f 2 + 12 Auk , uk − f, uk . 2
1 $ α !$ ! " 1 ! $ ! ' 8 ! ' ! !$ $ #
Auk − f 2 . αk = arg min F (α) = max 0, 3 A uk , uk + Af, f − 2 Auk , Af α0 F! ' !$ αk = 0 F! ' ! $ ! !$ * * ! αk = 0 4 = ! U = {u ∈ R4 | u 0, Au = b} 1 1 3 1 3 A= , b= , 1 −1 1 2 1 ! $
$
(!"
! "
!$ " ! 0 rg A = 2 * ' < !' A 7. " "!" " ! . $ 8 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 0 5 0 0 0 ⎜1⎟ ⎜0⎟ ⎜0⎟ ⎜0⎟ ⎜5⎟ ⎜0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜3⎟ ⎜ ⎟ u1 = ⎜ ⎝0⎠ , u2 = ⎝1⎠ , u3 = ⎝ 0 ⎠ , u4 = ⎝1⎠ , u5 = ⎝0 ⎠ , u6 = ⎝1⎠ . 4 0 0 −2 0 0 3 ) ' ! 8 u1 , u2 , u3 , u5 - u3 " !$ 0 !$ # ! % !" ' ! U *
"!" " u1 , u5 !$ ' ! !$ u2 6
" 5 ( $ ! ! .
J(u) → inf,
u∈U
"!" " !'
=' $ ! "8
#G% J∗ > −∞ #H% U∗ = ∅ 4
#I% !'" !'" !$ " !' !$ ! !$
! U∗ (
" #G% C $ ! U ⊆ U0
J∗ = inf J(u) inf J(u) = J0 > −∞. U
U0
#H% U∗ = ∅ (!" * !$ !$
! !$ $ {uk }∞ k=1 ) !$ . ' ! " " δ > 0 !" k '!$. K ∈ N ! " " uk ∈ U(δ) = u ∈ U0 | gi+ δ, i = 1, m + s . 0' ! !$ $ {vk }∞ k=1 ) !
U(δ) H 6 !$' ! !$ {vk }∞ k=1 $ !' !$ $ $ {vk }∞ 6 "
! =1 !$ $ {vk }∞ k=1
lim J(uk ) = lim J(uk ) ∈ U(δ) →∞
k→∞
u0 6 !' !$ {vk }∞ =1 J " !' ! +
$ J(u), gi (u)
J(u0 ) lim J(uk ) = J∗ , →∞
"
0 gi+ (u0 ) lim gi+ (uk ) = 0. →∞
) ! !' U0 ! u0 ∈ U0 ! " "
$ u0 ∈ U - ' !.$ " J(u0 ) = J∗ $ u0 ∈ U∗ = ∅ #I% 1 ! " .
! J(uk ) −−−→ J∗ !' !$
{uk }∞ k=1 " U∗
k→∞
