一类非自治非线性时滞微分方程的全局吸引性new

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1、82582EAEo,,Vol.25No.22002A4RACTAMATHEMATICAEAPPLICATAESINICAApr.,2002;+(;'
y

2、!9?(je!-Qp-M;!-0Qp-0v
100080)8A>(]i})-BFp-
]i200240)@:=()-p-
yP030006)%//OohO#dfRK
x
(t)=r(t)1+x(t)f(xt),t≥0,1.F2P=34"R{;
b/[1]3KEf@DK&b:dfLo-gisticK;w1./4!3f;wW/4

3、7`fzOohO#dfRK;P=#;[>;O[>1)
ar,g∈C[0,∞),R,r(t)>0,g(t)

4、ϕ(s)

5、.As∈[g(t),t]−1(t)=sup{s:g(s)=t},−1−1:]t≥0,5gWg(t)k6]T@-*lqIg(g(t))≥−1(t)>t.t,gJaf∈C(Ct,R).6QpiQ%egTM!
x(t)=r(t)1+x(t)f(x),t≥0.(

6、1)taτ=−tinf≥0g(t),WM!(1)5$h

7、Yx(θ)=ϕ(θ),θ∈[−τ,0],
(2)ϕ∈C[−τ,0],[−1,∞)Iϕ(0)>−1.Gx(t;0,ϕ)jM!(1)9r$h

8、(2)5
x(t).2000B1S17Wl/2001B7S17Wl/&XZ2EU64&OohO#dfRK3P=#217V[1]kct7_%0r(t)=rIegIeM!(1)R?5g[2]10M!dN(t)N(g(t))α=r(t)N(t)1−,t≥0(∗)dtK5R?%Gkr(t)kq5[1]

9、5g@CGuT[3]j0(BegN(t)TM!,HLM!(∗)vx(t)=−1qw2M!9r[3]5

10、2,a[3]K5g3@CGL^mM!(∗).5m2gk<5=)1,-r(t)=rI2LegIe=)10[1]k5m2g~1atr(s)ds≤M,(3)g(t)∞r(s)ds=∞.(4)0Ygf9rY[]

11、(i)f∈C(Ct,R),f(−1)>0,∀u∈[0,∞)}ϕ∈Ct,-ϕ−uV[g(t),t]^me,7i
(ϕ−u)f(ϕ)−f(u)≤07m-ϕ(s)≡u,g(t)≤s≤te,;(

12、ii)M!f(x)=0I64x=0;(iii)M!su=eMf(−v)−1,v=1−eMf(u)VOD=(u,v):u≥0,0≤v<1kI6u=v=0,WM!(1),(2)54k6]R?nZ*f(−1)jJlfVlq−15Nb8(s<V9D_%=)15_=V9[_%G6CG2)}{&7&16B0,W-t≥gex(t)=r(t)(1+x(t))f(xt)≤r(t)(1+x(t))f(0)=0,

13、ax(t)Iyac,9ac≥0.M!(1)"x(t)=r(t)f(xt),(5)1+x(t)t'T0txT^i2ln(1+x(t))−ln(1+x(T))=Tr(s)f(xs)ds.Yc>0,H(i),(ii)"∞2f(c)<0,Ut→∞,H

14、(4),WIln(1+c)−ln(1+x(T))=f(c)r(s)ds=−∞,TZ"@LkcIc=0.-x(t)tlVe)"_t2x(t)?&eH

15、(3)a(Vt0≥0,-t≥g(t0)eg(t)r(s)ds≤M.N{tn}k6]+<:XIMLK5fq)19rg(tn)≥t0,

16、x(tn)=0,Ix(t)≥0,-t∈[t2n−1,t2n],x(t)≤0,-t∈[t2n,t2n+1],218@Dn++25∗∗IZ
tn∈(t2n−1,t2n),sn∈(t2n,t2n+1)g2x(t∗)=max{x(t):t≤t≤t},x(s∗)=min{x(t):t≤t≤t},n2n−12nn2n2n+1∗∗∗∗WALn=1,2,···,Ix(tn)>0,x(tn)=0,x(sn)<0,x(sn)=0,H&}(1)ia∗∗∗∗f(xt∗)=0,f(xs∗)=0,'C"%x(t)V[g(tn),tn]n[g(sn),sn]^^e_I6

17、]4nn∈[g(t∗),t∗]∗∗∗=Nδnnng9rx(δn)=0,x(t)>0,t∈(δn,tn].