iti
x„+i = A„x„ 1 5]a,F(x„_.) (2.2)
VOI n G No, trong do a, ^ 0 , i - l,?", 2^,^i ^^t ^^ ^ - ^ .
s6 nguyen duang c6 dinh. Cac gia tri ban dau x,, (j = ^^^^^ la cac s6 dmtng
2.2.1 Tinh hoi tu
Dinh ly sau cho m6t di^u kien du d^ moi nghiem ciia (2.2) hCi tu tdi 0.
Djnh ly 2.1. Gid su ton tgi X* e (0,1) sao cho 0 ^ A„ ^ A* v&i tat cd n G NQ. Khi
do, mgi nghiem cua (22) hgi tu t&i 0 neu F{u) < (1 - X*)u v&i mgi u> 0. Chimg minh. Gia sir F{u) < (1 - X*)u vdi moi u > 0. Goi {xn}n la m6t nghiem
cua (2.2) va M = max-rn^j^ôj- Ta se chung minh rang x„ :^ M vai moi n. That vay, dung phuong phap quy nap, gia sir ring x^ ^ M vcri t^t ca A: ^ n. Tir phuang
trinh sai phan (2.2) ta co
m X n + i = A „ X n + ^a,F{Xn^^) in ^ ÁA/ + ^ a , ( l - A - ) x , _ , 1 = 1 ^ A*A/ + J ^ a , ( l - A - ) A / = A/, i = r l
VI vay Xn ^ M voi moi n. Kf hieu A/i = limsup^^^Xn- Ro rang 0 < Mi < ^ .
Han nua, ton tai day con {?u-} sao cho
A/i = lim Xn^.
Vdi mdi 6 > 0 nhd tuy y, tdn lai sd tu nhien .V = .V(f) sao cho vdi moi n, > -V
ta cd Ml - e ^ x„, ^ Mi + f va vdi moi n > N - m - I : x„ ^ M, - c Mat khac.
TU
„, = A„,__iX,„_, + X^ạ/-\x.„_:-,).
:ho Theo dinh ly gia tri trung binh. ta co the chon mot day con {.v-,} sao c
5 ^ o , F ( x , , . i - J -- F{ỵJ
t = i
trong do
Khi do ta nhan duoc Vi vay < A*x„,_i + F(y„ J < A*x„,_i + (1 - A*)y„, Áx„,_i > x„, - (1 - A*)y„, ^ M i - e - ( l - A ' ) ( A / i + e) x„^_i ^ Ml ——ẹ 2 — A*
Ml -—e ^ x„,_i ^ A/i + e
va
Tuong tu, ta cung cd
va
lim Xn^-] = A/i
1 + A*
A/i —-f ^ y„, ^ A/i + e
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lim yn^ = A/i
fc-+oo
Vi F lien tuc. Ml la mdt nghiem ciia phuong trinh u = Áu + F[u). Nhung do gia thiet F(w) < (1 - Á)u vdi mgi u > 0, ta co Mi = 0 tuc la day {x,.},. hoi tu gia thiet F(w) < (1 - Á)u vdi mgi u > 0, ta co Mi = 0 tuc la day {x,.},. hoi tu
tdi 0. Dinh ly dugc chung minh. ^ Nhan xet 2.1. (i) Trong trudng hgp day {A„}„ dtm dieu tang va bi chan bm hang
sd nhd hon 1, la cd the chgn Á = Um„^ccA,.; trong trudng hgp day {A,.}^ dan dieu giam, ta cd thé chgn A* = AQ.
(ii) Néu A,. = A- e (0,1), Vn e No thi chieu nguac lai cCia dinh ly 2.1 la dung.
(iii) Tir phep Chung minh dinh ly 2.1. ta dC thav rang neu thay ,n bdi day
{m„}„bi chan hoac khdng bi chan nhung neu Hm„^^(n-m„) - o c v a L . ' ' o ' ^ 1 thi dinh ly 2.1 van cdn dung.
2.2.2 Tinh gidi noi
Dinh IS sau cho mdt di^u kien dti d^ mgi nghiem cua (2.2) 1^ gidi ndi ngat. Djnh ly 2.2. Gid su ton tgi Ạ, A* 6 (0,1) sao cho A, ^ A„ ^ A* v&i tat cd n e NQ.
Gid thiet them rdng Fix) = /f(x,x), trong do H : [0, oo) x [0, oo) -> [0, oo) Id hdm lien tuc, dong bien theo bien thilt nhát nhung nghich bién theo bién thu hai vd Hix,y) > 0 neu x,y > 0. Khi do mgi nghiem ciiia (2.2) Id gi&i noi nggt neu dieu kien sau duac thod man
limsup : ^ i ^ < l - A * , (2.3)
(x,y)-^(oo,oo) ^
Hix v)
liminf tll±ll > 1-Ạ. (2.4)
(x,y)-^(0.0) X
Chimg minh. Gia sU {x„}n la mfit nghiSm ciia (2.2). Truoc hfít ta chung minh {xn}n bi chan trSn. That vay, gia su trai lai ring limsup„_^x„ = oọ Vcri m6i
s6' nguyen n ^ - m , ta dat kn :^ maxlp : -rn ^ p ^ n,Xp = nicxx x J . De tháy k_m ^ k^m+i ^ • • • ^ k„-^ oc vh lim xjt„ = 0 0 . •n Vdi kn > 0, ta cd TTl T,-„ = A,„_iX,,.-i + 2 ^ a . F ( x , , _ i - , ) t = i = Afc„_ix^.„_i+F(y^.J ^ A*x,. -I //(y^„,0), trong đ
y^,^ e [inin{x,,.-2---- . J-fc„-i-m}, inH,x{x,„-2.' ' ' .^^.-i —Ii
va do đ
Hm //(!A-.,0) = 00
Tilt day suy ra