Ta CO
F(x) < (1 - A)x <^ hf{x) < [1 - (1 - fih)]x ^ fix) < fix, V I > 0.
K^t luan: Di6u kien hdi tu tdi 0 la tuong thich. Dieu kien gidi noi ng^t
Mgi nghiem x{t) ciia (1.2) la gidi ndi ngat néu /(x) > 0 vdi tát ca x > 0 va
umsup < fi,
I—>cx) X
fix)
liminf > ^.
X-+0 X
Moi nghiSm {xn}n cua (1.1) la gidi n6i ngat neu
hmsup < 1 — A, X-+00 X hmmi > 1 — Ạ a:->0 X Ta CO •t > hmsup < 1 — A X-+00 X hf(x)
limsup ^^-^^ < 1 - (1 -/i/i) =/i/i
X — • o o hmsup < /ị X Mat khac, X—fOO h m ini > 1 — A x - * 0 X x-»0 X Hmmf^^^ > fị i-->0 X
Két luan: Dieu kien gidi ndi ngat la tuctng thích.
Sir hoi tu tdi trang thai can bang duong vdi tat ca cac cham * Két qua 1.
D6i vdi (1.2), / don dieu tang, Umsup^^^ ^ < M < lim inf ^_o ^ .
Ddi vdi (1.1), F don dieu tang, Umsup,_^ £M < i _ ^ < Uminf.^o ^ Tac6
F(x) fix)
lim sup < 1 - A o lim sup ^ - ^ < fi
X—foo X X—voo X
va
liminf ^ ^ > 1 - A ^ liminf ^ > /ị
x - f O X X-+0 r
Két luan: Két qua 1 la tuong thich. * Két qua 2.
D6'i vdi (1.2), / dofn dieu giam va he 2 phucmg trinh a - ^ , /^ = ^ co
nghiem duy nhát a = p.
D6i voi (1.1), F dcm dieu giam va he hai phucmg trinh a - | ^ , /^ ^ T ^ co
nghiem duy nhát a = ^. Ta CO F{p) hfip) hfiP) ^_ /(;3) a — 1 ^ A 1 - A 1 - ( 1 - fih) /i v a ^ " " 1 - A " 1 - A l-{l-^ih) fi
Ket luan: Ket qua 2 la tuomg thich. * Ket qua 3.
Doi VOI (1.2), f{yo) ^ fnJo va dieu kien sau duoc thoa man
fir) /l-^^ (adsbygoogle = window.adsbygoogle || []).push({});
lim sup ^^-^^ < //, liminf > fị
X—>oc -^
nwo) ^ ,,„ ii„, inf _ p £i£l > 1 - A va dieu kien sau duoc thoa Dd'i vdi (1.1), ^ ^ yo, liminfx
man
0 < liminfx,, ^ lim sup x„
T i - * c x : T i - » > :
Ta CO
nyo) < ,. ^ _JiZM ^ ^ /(,„) <; ;.y„
1 - A ^- 1 - ( 1 - / ^ M
Hieu suát cua do tri doi vdi sir hoi tu tdi trang thai can bang duong
Dd'i vdi (1.2): Gia sir tdn lai cac hang sd duong Li,L2 sao cho ham / thoa mdt cap di^u kien Lipschitz mdt phia:
0 ^ fir) -fif^ Li(f - r) vdi ê^^f ^r <f,
0^ fxf - fir) ^ Liir -f)v6\f<r^D, (0 < /(r) ^ D).
Khi do mgi nghiem gidi ndi ngat hdi tu den trang thai can bang duang f néu
cham T thoa man
1 - ẽ"^ <
Dd'i vdi (1.1): Gia sir tdn tai cac hfing sd duong A-.L- sao cho ham / (vdi
f{^) = f ^ ) ^hoa mdt cap dieu kien Lipschitz mot phia:
0 ^ fix) - X ^ L:)(x-xj vdi mgi x G [A"'^'x,x],
0 ^ X - fix) ^ L.iix - 7) vdi mgi x G [x, /(yo)].
Khi đ mgi nghiem gidi noi ngat {,(„}„ hgi tu dC-'n J neu cham m ihoa man
1 A"*+' > 1 -
Két luan: Tuong thich ve hieu suát cua do ire, cu thC^ la vdi do tre du nho thi mgi nghiem gidi ndi ngat cua (1.1) hay (1.2) hoi lu tdi irang thai can bang x. Hieu suát ciia do trd doi voi su tuan hoan
Xet (1.2) vdi // -^ 1: Neu mot nghiem .v.n khong Jao Jong xung quanh ^ ihi
x(0 lien tdi f khi / tien ra vo cung. Do Jo moi nghiem tuan hoan Jaong khfic hang phai dao dong .xung quanh f.
Gia sir / kha vi tai f va r > r. ^ - 7 ^ ^ arccos ;rV Kh. Jo ton tai nghiem tu5n hoan khac hang cua (1.2) Idn hmi /Ạ va Jao Jong cham xung quanh r.
Xet (1.1) vdi A - 1 - h: Neu mot nehicm {.r..}.. k!^>ng Jao Jong xung quanh X Ihl {x„}„ lien tdi X kh. n ticn ra vo cimg. Do Jo moi nghiem luan hoan Juong
Gia sir F kha vi tai x v^ m > " ^ ^ ^ , w6i D = F'(x) e [-1 - A, - 1 + A] u [1 - A, 1 + A]. Khi đ (1.1) nhan mdt nghiem tuSn hoan khdng l ^ thudng, xnSil ph^t t£r X vh. dao đng cham xung quanh x.
K^t luan: Tuong thich vi hieu suSft cua đ tre, cu ihi la vdi đ tre dii Idn thi
tdn tai Idi giai tuSn hoan khdng iSm thudng trong (1.1) va (1.2). (adsbygoogle = window.adsbygoogle || []).push({});
KET LUAN CHUONG 1
Ndi dung chinh ciia chuong nay la nghien cuu tfnh chát ciia nghiem phucmg trlnh (1.1) va ap dung két qua nhan dugc cho bai loan md la sir phat tridn cua
mdt sd quSn thi sinh hgc. Ddng thdi cung nghien cuu mdi lien he vS tfnh chat
cua nghiem phuong trinh (1.1) va (1.2). Chiing tdi da nhan dugc mot sd két qua nhu sau:
1. Mdt sd dieu kien di mgi nghiem ciia phuong trinh (1.1) la hgi tti tdi 0, gidi ndi ngat, hdi lu tdi nghiem ducmg duy nhát ciia phucmg trinh x - Ax + Fix).
Dac biet la chi ra sir tdn lai nghiem dao đng cham va nghiem tuAn hoan khCng tSm thudng cua phuong trinh naỵ
2. Rut ra mdt sd ket luan cd y nghia cho su diet vong, trudng t6n, phat triCn ben vung va tuSn hoan cua mdt sd quan the sinh hgc dugc mo ta bdi phucmg trlnh sai phan phi tuyen cd cham (1.1). Dac biet da khao sat chi tiet su phat tnen cua qucin ih^ chim ciit d bang Wisconsin hgp chung qudc Hoa Ky va quan the rudi xanh Nicholson.
3. Chi ra sir tuong thích ve tfnh chát cua nghiem phuang trinh sai phan phi tuyen cd cham (1.1) va phuong trinh vi phan phi tuyen co chAm (1.2). Ngoai ra cdn nhan dugc mdt sd dieu kien đ moi nghiem cua phuang trinh vi phan phi tuyén (1.19) la hdi lu tdi 0, gidi ndi ngat hay hoi tu tọ nghiem duang duy nhat