thu nhát, chung ta da nghien cuu tinh chát cua nghiem phuong trlnh (1.1) Mot cau hdi tu nhien dugc dat ra la cac két qua nay cua phuong trlnh sai phan ph
1.3.1Phiromg trinh vi phan phi tuyen co cham
Tinh ch^t ciia nghiem phuong trinh (1.2) da dugc nhieu tac gia quan tam ngliien ciiụ Bay gid chung ta khao sat tinh chát ciia nghiem phuong trlnh nay trong trudng hgp cham r = 0 va he thd'ng mdt sd ket qua trong trudng hgp cham r > 0 ciia mdt sd tac gia di trudc. Viec he thd'ng nay se thuan tien cho ta khi so s^nh tinh ch^t ciia nghiem phuong trinh (1.2) va phuong trinh (1.1).
Ta ki hieu {T{t)}t^Q la nira nhdm lien tuc gdm cac don anh lien tuc (phi tuyén)
tilr R*= v^o chinh nd thoa man T(0) = id va T{t + ,s) ^- T(n'/"(>) vdi mgi t,s^Ọ
Cac anh xa ngugc T(0"^ =: T{-t) dugc hieu theo nghla
T{-t)x = {v:T{()v x}.
Ta de ki^m tra dugc
T{t)r{s) = T{t -I- ,s) vdi mgi /, .s G E
Nhu vay cd thé coi hg {T{t)},(:^ la nhom mot tham sọ
Djnh nghla 1.6. Vdi mdi vec to u G K''' ta ggi tap hap {Tu \u : t ^ (J} la quy dao ciia ụ Dat
va ggi nd la tap gidi han ạ- cua ụ
The thi vdi mgi phan tir v G u:{u) deu ton lai mot da\ so .';. -^ oc sao cho
Wmk->ooT{tk)n - r, d day su hoi tu dugc hiOu theo chuan cua khong gian Euclid
da chọ Nhu vay, tap gidi han ú cua ;/ la tap tat ca cac dicni tu cua quy dao
{T{i)u : t ^ 0}.
Dinh ly 1.10. 12il, \3I\ Níu qu\ dao ciia u !u gioi //<'/ ihi up •••n hon ^- cua u la
compact lien thong (khong rong) vd 7(0^4") -•''- '<'' '•"' ' " ""'^"''^ '''' '" cdn CO d{T{t)u,ụiu)) - 0 khi t - ex. (odav d la khoang . ,/. /' ron^ kh.n, ^lan Euclid).
Chiing minh. Vi quy dao ciia u la ti6n compact nen trong bao đng ciia nd nh^t
thi^t phai co di^m tụ Do đ tip gidi han u khdng rdng. Cung tir dinh nghla ta th^y tap nay la compact v^ ro rang la T{t)u;{u) C uj{u) vdi mgi t ^ 0. Cho v G UJ{U) thi V = \imj^ooT{tj)ụ Cho x la mdt di^m tu ciia tap hgp {T{tj - t)u : j e N} thi T{t)x la di^m tu ciia tap hgp {T{tj)u : j e N}. Do đ T{t)x = v. Vay T{t)u(u) = u}{u). Bay gid ta gia sir u{u) = U1UUJ2 vdi uuU2 la hai tap compact cd khoang each la 5 > 0. TOr day suy ra d{T{t)u,u{u)) khdng ih^ tién tdi 0 khi
i -> 00. Ta cd dieu vd lỵ Dinh ly dugc chung minh. D Xet phuong trinh vi phan phi tuyén Xet phuong trinh vi phan phi tuyén
x{t) = -fix{t) i-f{x{t)). (1.19)
vdi x(0) > 0 cho trudc, /x la tham sd duong va / G C([0,oo)). Gia thiet rang bai todn gia tri ban dSu
j±{t) = -^xit) + f{x{t)),
[x(0) = xo
cd nghiem duy nhát tren [0,oo). Ta se xac dinh mdt sd dieu kien de mgi nghiem duong ciia (1.19) la hdi tu ve 0, gidi ndi ngat hay hdi tu tdi trang thai can bang duong duy nhát cua nd. De lien theo doi la cd khai mem sau:
Dinh nghla 1.7. Mdt nghiem x(f) cua (1.2) hay (1.19) duoc goi la gi&i noi ngat
néu
0 < liniinfx(0 ^ liinsupx(0 < oc.
( - • 0 0 ( - k O C
Dinh ly 1.11. Dieu kien cdn vd du de tat cd cac nghiem i{t) cuăL19) hoi tu ten
0 khi t tien ra v6 cung Id J[i) < //x v&i moi x > 0.
Chimg minlị Gia su /(x) < fix khdng dugc thda man NOI moi x ;• 0. Khi đ c6 (adsbygoogle = window.adsbygoogle || []).push({});
Trudng hap 1: 3x > 0 sao cho /(x) > /ix. Ta chiing minh rang khi do x{t) > x(0) vdi moi t > 0. That vay ta cd x(t) = -^ix{t) + f{x{t)) > 0 nen x{t) la
h ^ don dieu tang do vay nd khdng hdi tu tdi 0 khi f ra vd cdng.
Trucmg hcrp 2: f{a) = /za vdi mdt sd a > 0 nao đ. Khi đ x{t) = a, Vt e
[0, oo) nen limt_>oo ^{t) = a > 0.
Bay gid di chutig minh phSn dao lai ta gia sir /(x) < fix vdi mgi x > 0. Khi đ x{t) - -fix{t) + f{x{t)) < 0 nen x{t) la ham don dieu giam. Do đ tdn tai gidi han limt-^oôit) = ọ. L^y tich phan hai v6 ciia phuong trinh vi phan (1.19)
tiJr T tdi T + 1 rdi ap dung dinh ly trung binh ta dugc
r + i
x ( T + l ) - x ( r ) = f [-fix{t) + f{x{t))]dt
= -px(^(T)) +/(x(C(T)))
vdi ^(T) nam giira T va T + 1. Cho T đn ra vd cung, ve trai lien tdi 0 cdn ve phai tién tdi /(u) - fia = 0. Tir gia thiet /(x) < fix vdi mgi x > 0 ta suy ra a = 0.
v a y Umt^oo x{t) = 0. Dinh ly dugc chixng minh. •
Dinh ly 1.12. Neu
hm inf ^^-^-^ > fi > hm sup
x-^O X i->oo I
thi tát cd cac nghiem cua (1.19) Id gi&i noi nggt.
Chimg minh. Trudc het ta gia sir trai lai rang nghiem j[t) cd liminf,^:^ x{t) = 0.
Vdi mdi / > 0 la dinh nghla
oft) = minfr : x(r) = min x{s)}.
Vi liminf,^oox(0 = 0 nen ta phai cd a{t) -^ oc khi f ^ oc. Ngoai ra ta con cd lim,_oox(ă0) = 0 va x{a{t)) ^ 0. Tu phuong trlnh vi ph^ln (1.19) ta su>
/ ( X ( Q ( 0 ) ) ^ /ix(ă0) do đ
hminf ^^-^— ^ liminf—. .... ^ Z^-
I-.0 X '-•<» x ( a ( 0 )
Di^u nay trai vdi gia t h i ^ ciia ta nen ]iminit-^^x{t) > 0. Tuong ni nhu vay
lim supf_,oo x{t) < oọ Dinh 1^ dugc chung minh. •
Nh^n xet 1.5. Xet hg cac anh xa
T(t):[0,+oo) - ^ [0,+oo)
x(0) H-). x{t).
Khi đ hg {T{t)}t^o lap thanh mdt nira nhdm. That vay, ta cd
+ T{t)x{0) = x{t) nen T(0)x(0) = x(0) = idx(O). Do đ T{0) = id.
+ Vf, s ^ 0, ta cd T{t + s)x(O) = x{t + s) = T{t)x{s) = T{t)T{.s)x{0), suy ra (adsbygoogle = window.adsbygoogle || []).push({});
Tit + .s) = T(/)T(.s).
Djnh ly 1.13. Cho x{t) Id mot nghiem gic'ri noi ngat ciia (1.19) xuat phat tit x(0).
The thi tap cac diem tu u{x) ciia x(0) la cd doan dong
[liminf x(t), limsupx(/)].
Han nita, tat cd cac nghiem xudt phat tit uj{x) deu nam trong u-ix) va co the trdi rong ra v&i tát cd t eR. Ta luon co
- i n f / ( x ) ^ Hminfx(0 ^ limsupx(0 ^ - s u p / ( x ) .
H x > 0 t-*x (_>cc f^ J>0
Chimg minh. Vi tap gidi han u{x) la compact lien thong nen nd phai láy tát ca
cac didm cua doan [Hminf,^oc ^(Ôlimsup,,.^., x(0]- Xet nua nhom {r(0}r^o cac
anh xa tu [0, foe) vao chinh nd xac dinh bdi T{t) : x(0^' ^ x(0. Xet tren tap gidi han a-(x) thi T{t) la song anh vdi mdi t ^ 0. Vi vay ta co the xac dinh
T{t) - [ r ( - / ) ] ' ' vdi moi f < 0
va nhu thé la nhan dugc nhdm mdt tham so {r(f)},. s cac anh xa tu . i x vao chinh nd. Do đ mgi nghiem xuai phat tu doan nay co the irai rong ra vd, tat
ngay u(0) = maxtu(t) nSn u{0) = 0. Tilt phucmg trinh vi phan (1.19) ta suy ra
fj,u{0) = f{u{0)) hay u(0) = ^/(u(0)). Do do ta cd
Umsupx(f) = w(0) = -/(w(0)) ^ - s u p / ( x ) .
t->oo / i fi x>0
Tuong tu nhu vay xet nghiem v{t) cd xu^t phdt diim la u(0) = liminf^^oo x{t)
thi cd ngay v{0) = mintt;(t) nen i;(0) = 0. Tilr phuong trinh vi phan (1.19) ta suy
ra
liminfx(0 = t^(0) = -f{v{0)) ^ - m f / ( x ) . f->oo fx fi i>0
Dinh 15^ dugc chting minh. D
Djnh ly 1.14. Néu phuong trinh /(x) = fix co nghiem duang duy nhát x = x thi
moi nghiem gi&i noi nggt ciia (1.19) tién t&i x khi t tien ra v6 ciing. Tuc Id lap giai hgn u{x) chi co mot diem duy nhat la x.
Chimg minh. Ta xet nghiem u{t) vdi xi(0) = limsup,_3^x(0. Khi đ u[t) ^ u(0)
vdi mgi t G M. Do đ ham u{t) dat gia tri Idn nhát tai 0. Vay dao ham ciia u tai
0 bang 0 nen lir phuong trlnh vi phan (1.19) ta suy ra u(0) phai la nghiem cua
phuong trinh dai sd /(x) = fix. Theo gia thiél u(0) = x. Tuong tu nhu vay xet nghiem v{t) vdi y(0) = liminft^ocx(0 la cung suy ra í(0) = x. Do do tap gidi
han uj{x) chi cd mdt didm duy nhát la x. Dinh ly dugc chung minh. D
Nhan xet 1.6. Dinh ly 1.14 kh6ng la trudng hgp neng cua bd de 1.3 trong [18].
B 6 de nay chi suy ra két luan cua dinh ly 1.14 trong trudng hop /i = 1. Hon nuạ (adsbygoogle = window.adsbygoogle || []).push({});
trong bd de 1.3 cac tac gia da dat them nhieu gia thiet ve ham / va tham sd ^L.
Tren day chiing ta da khao sal tinh chat ciia nghiem phuong trinh (1.2) trong trudng hgp đ tre r = 0. Viec nghien cuu (1.2) trong trudng hgp r > 0. dac biet
1^ T du nhd hay du km da dugc de cap trong [17i va ilSj. Bay gid, ta he thdng mdt sd két qua chinh ciia cac tac gia di trudc v<5 vftn ,\c naỵ
Dinh ly 1.15. [77] Dieu kien cdn vd dii détat cd cac nghiem .r[t)cua (1.2) hoi tu
Dinh ly 1.16. [77] Gid sit f(x) > 0 vol tat ca x > 0 va V / W hmsup—^-^ < /i, I-+0O X hm mf —^—^ > ụ a:-^0 X
The thi mgi nghiem x{t) cua (12) Id gicn noi nggt.
Djnh ly 1.17. [77] Gid sijt phuong trinh /xx = /(x) co duy nhat nghiem x e (0, oo)
vd / Id hdm dcm dieu tang. The thi mgi nghiem giai noi nggt x{i) cua (1.2) hoi
tu den X.
Djnh ly 1.18. [77] Gid su phucmg trinh fix = f{x) co duy nhat nghiem x e (0, oo),
/ Id hdm dcm dieu gidm vd he hai phuctng trinh sau
CO nghiem duy nhat a = p. The thi mgi nghiem gi&i noi nggt x{t) cua (1.2) hoi tu dén X.
Tiip theo, gia sir / la ham hinh chudng, tuc la t6n tai yo > 0 sao cho
/(yo) - mâ/(x)
va / la ham don dieu tang trong [0,yo], dm dieu giam trong (yo, x).
Dmh ly 1.19. [17] Gid sUphuang trinh fix = /(x) co dux nhat nghiem 1 G (0. oc).
/(yo) ^ yo vd
lim inf ^^^^ > fị
I-+0 X
Khi do mgi nghiem giui not aggi x\i) cua [1.2) hgi tu den x.
Bo d^ 1.3. [18] Gid sit rdng
(i) feC{[0,^));
(ii) Ton tgi trgng thdi cdn bang r > 0, titc la fif = f{r) > 0, (Hi) f{r) > fir v&i 0 <r <f va f{r) < fir v&i r >f.
Khi do vdi mSi nghiem khong tam thucmg x{t) cua (12) ta co fir) e ^'^r ^ liminf x(t) ^ r ^ limsupx(t) ^ max =-^.
Djnh ly 1.20. [78] Gid si( cac dieu kien (i)-(iii) cua bo de 1.3 dugc thod man vd
ton tgi cac hang so duang Li,L2 sao cho hdm f thod man dieu kien:
0 ^ / ( r ) — pr ^ Li(f — r) v&i é^'^f ^ r <f, 0 ^ /if — f{r) ^ L2(r — f) v&i f < r ^ By
trong do B Id mot can tren cua f{r). Gid thiet them cham r thod man
1 - e"''" <
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Khi do mgi nghiem khdng tam thu&ng cua (12) hoi tu den trgng thdi cdn hang f.
Wi su t6n tai nghiem tuSn hoan khOng tim thuong, trong [18], cac tac gia da
diing nguyen ly re nhanh Hopf va dinh ly diem bát dong Browder de chung minh sir t6n tai nghiem tuln hoan khac hang ciia phuomg trinh
i ( f ) = - x ( 0 + / ( x ( f - T ) )
voi cham r du Ion. Cu ihi la, voi
1 1
r -> T^ =: — — arccos
trong đ f'{f) < - 1 . Hon nua, Ỵ Cao [6] da chimg minh rang vdi r ^ r. thi khdng ton tai nghiem tuan hoan Idn hon yo (vdi /(yo) = luiix^^ọfi-n) va dao đng cham xung quanh trang thai can bang duong f. Vdi r > r. thi cd nhiSu nhát mdt nghiem lufm hoan Ion hon yo va dao đng cham xung quimh ?. Nhac lai rang nghiem T-tụ'in hoan duoc goi la dao dong cham .xung quanh trang thai
bang ducmg neu T > r. x(0) = x(T) ^ f va tdn tạ ^o e (0, T ^ r) sao cho
x{to) = f, x{t)>f v d i ^ G ( 0 , f o ) v a x ( n < f v d H G i ^ o . 7 - ) .