一类非线性无界时滞差分方程正解的不存在性

1、27(410003)(axn+1+bxn)(cxn)+kku󰀉i=1pi(n)xki(n)=0(1.1)(axn1+bxn)(cxn)+a,b,c(0,),k(0,),i(n)<n,i(n)>nkku󰀉i=1pi(n)xki(n)=0,(1.2)nupi(n):N0=0,1,2,···limi(n)=,limi(n)=,i=1,2,···,u.n(1.1)i(n)=ni,k(axn+1+bxn)(cxn)+1ku󰀉i=1pi(n)xkni=0.(1.3)n1u1𘀀

2、9;󰀉k1(k1)limpi(j)>akc,n(+1)j=ni=1(1.4)(1.3)=mini|1iu,=(cb),0<1.2200222420037281163(1.4)N1n1󰀊j=nu󰀈i=1󰀆E(),nNsup󰀄u󰀅󰀇(1k)󰀉<1,pi(j)1i=1(1.5)󰀂E()=|>0;1(1k)󰀃pi(n)0.k=a=c=1,b=0,i(n)=ni,xn+1xn+(1.6)k=a=c=u=1,b=0,(1.1

3、)u󰀉i=1pi(n)xni=0.36(1.6).(1.1)xn+1xn+Pnx(n)=0.(1.7)(1.7)12nlim(n(n)=,nµ=liminfnn1󰀈i=(n)3,715.pi>,(1.7)13limsupn󰀈i=(n)n󰀈i=(n)pi>1,(1.7)nlim(n(n)=,0µ,nlimsuppi>1+In(µ),(µ)(µ)=eµ(µ)(1.7)(1.1)󰀂󰀃󰀃(n)=maxi(

4、n)|1iu,(n)=mini(n)|1iu,n=0,1,2,···,=,0<1.un1󰀉󰀉1pi(j)µ=liminf,(k1)(ji(j)nk1i=1j=(n)un󰀉󰀉1pi(j)=limsup.ini=1j=(n)2.12xn(1.1)xn+1xnxn,xi(n)i(n)(n)x(n).(2.1)(2.2)2.215x,yx=y,rxr1(xy)>xryr>ryr1(xy),r<0r>11642.3xn(1.1)x󰀉un+1xn+(1k)(

5、ni(n)xi(n)0,i=1pi(nx󰀉un+1xn+k(ni(n)i=1pi(n)xn0.xki(n)󰀄xxn=󰀄xi(n)󰀅k1nxi(n)󰀅ki(n)=nxn.2.1xki(n)xn(1k)(ni(n)xi(n),xki(n)k(ni(n)xxnn.(1.1)2.20k(cxn)k1(axn+1(cb)xn)+󰀉upi(n)xki(n),i=10axn+1(cb)x󰀉uxkn+i(n)n).i=1kck1xpni(2.5),(2.6)(2.7),(2.3),(2.4)2.4,

6、nlim(n(n)=xn(1.1)n10µ=nliminf1󰀉󰀉uPi(j)1k1(k1)(j,j=(n)i=1i(j)x(n)nliminfn(µ),(µ)(µ)(µ)=eµxn(1.1)2.1xn(2.9)µ=00<µ,(0,µ)1n󰀉1󰀉upi(j)k1(k1)(ji(j)>µ.j=(n)i=127(2.3)(2.4)(2.5)(2.6)(2.7)(2.8)(2.9)x(n)n1,n(2.10)1165(2.3),

7、u󰀉xi(n)xn+11xn+1pi(n)1nnk1i=1(k1)(ni(n)nu󰀉xi(n)1pi(n)1,i=1(k1)(ni(n)n󰀇n1n1󰀆u󰀊󰀊󰀉xi(j)1xnxj+1pi(j)1=(n)ji=1(k1)(ji(j)j󰀆1dj=(n)j=(n)󰀆󰀇󰀇n(n)un1󰀉󰀉xi(j)1pi(j)1.iji=1j=(n)(2.11)x(n)n>1=1,(2.11),(2.10)

8、,󰀄1(µ)󰀅n(n)xn<1.(n)󰀁󰀋xy=1(b),xn<e1(µs),(n)x(n)d>e1(µs)=2.nx(n)d>em(µ)=m+1,nm=1,2,···.1=1<2<···<m<m+1<e,m=1,2,···.(2.12)i)1=1<e,m<e,(2.8),m(µ)<1.m+1=em(µ)<

9、e.m<em=1,2,···ii)1=1<e1(µ)=2,m1<m,m=em1(µ)<em(µ)=m+1,m<m+1,m=1,2,···i),ii)(2.12)limm=e(µ).nliminfx(n)nm(µ,).0,(2.9),(µ)=eµ(µ).33.1un11󰀄n(n)+1󰀅n(n)+1󰀉󰀉pi(j)µliminf>1,(k1)(ji(j)

10、nk1i=1j=(n)16627(1.1)1n(2.3)14lim(n(n)=,un1󰀉󰀉11pi(j),µ=liminf>(k1)(ji(j)ni=1j=(n)(1.1)2i(n)=ni,n1u1󰀉󰀉k1(k1)liminfpi(j)>akc,nj=ni=1(+1)+1(1.3)1=mini|1iu,=(cb),0<1.3.2nu󰀉󰀉1pi(j)=limsup>1,(k1)(ji(j)nk1i=1j=(n)(1.1)(1.1)xn,(2.3)x(n)+jx(n)+

11、ju󰀉1pi(n)+j)x(n)+j+1+x(n)+j),i=1iij=0,1,2,···,n(n).j=0n(n)x(n)2.1xnnn󰀉󰀉1pi(j)xn+1+x.ii(j)i=1j=(n)(3.1)(3.1)x(n)x(n),nn󰀅󰀉󰀉1pi(j)x(n)(k1)(ji(j)k1i=1j=(n)󰀄un󰀉󰀉1pi(j)=limsup1.(k1)(ji(j)ni=1j=(n)3.1,3.211673.3nlim(n(n)

12、=,0µ,nu󰀉󰀉11+In(µ)pi(j),=limsup>(k1)(ji(j)ni=1j=(u)(3.2)(µ)=eµ(µ).(1.1)0<µ,µ=0,(1.1)(0)=1,xn,(3.2)>1,3.2,0<<minµ,1n,(2.10)x(n)>.nx(n)n>>1,x(n)(n)=1,(n)<ln1,x(n).lxl(n)1,x(n)>,l+11xl+1,<(n)l+1(n)(n)=,0<1,0,1,x(

13、n)=,1+(xl+1l)(3.3)t0,1xj+1xji=1j+t(xj+1j)j+t(xj+1j)u󰀉x(j)1pi(j),j=(n),···,l.k1i=1(k1)(ji(j)ju󰀈pi(j)xii(j)(3.4)0,1(3.4),u󰀄1󰀅󰀉pi(j)xjxjInIn(),j+1j+1k1i=1(k1)(ji(j)j=(n),···,l1.(3.5)(3.4)j=10u󰀄1󰀅󰀉xlpi(l)In(),1

14、+(xl+1l)i=1(k1)(li(l)(3.6)16827(3.5)j=(n)Inj=l1(3.6)x(n)1+(xl+1l)󰀆l1u1󰀄󰀉󰀉k1j=(n)i=1pi(j)(k1)(ji(j)+u󰀄󰀉i=1pi(l)(k1)(li(l)󰀅󰀅󰀇(),x(n)x(n)Inl+(xl+1l)l+(xl+1l)󰀆l1uu󰀄󰀉󰀅󰀅󰀇pi(j)pi(l)1ϗ

15、044;󰀉󰀉().+(k1)(ji(j)(k1)(li(l)k1i=1i=1Inj=(n)(3.3)1k1󰀆󰀉l1󰀉uj=(n)i=1pi(j)(k1)(ji(j)+u󰀄󰀉i=1pi(j)(k1)(ji(l)󰀅󰀇x(n)11In=In().l+(xl+1l)(3.7)0<n1,xn+1xnxn+1xn,n=xn+1xnn+1n,n1,2.3,u󰀉1pi(n)x(n).xn+1xni=1iin=xn+1i=1xnpi(n)ixi(n

16、),n1,nxl+(xl+1xl)xn+1=n󰀉j=ln󰀉=(xj+1xj)+(xl+1xl)i=l1(xj+1xj)+(xl+1xl)jlnuu󰀅pi(j)pi(l)l󰀉1󰀄󰀉󰀉ixx=(k1)(ji(j)i(j)(k1)(li(l)i(l)k1j1i=1j=li=1unu󰀅pi(j)pi(l)1󰀄󰀉󰀉jl󰀉=x+(1)x(k1)(ji(j)i(j)(k1)(li(l)i(l)k1jli=1i=l+1i

17、=11169nuu1󰀉󰀉jl󰀉pi(j)pi(l)x+(1)x(k1)(ji(j)(n)(k1)(1i(l)(n)jli=1j=l+1i=1nuu󰀅󰀉1󰀄󰀉󰀉pi(j)pi(l)+(1)x(n)(k1)(jI(j)(k1)(li(l)k1i=1j=l+1i=1󰀆󰀉n󰀉uu󰀄󰀉󰀅󰀇1pi(j)pi(l)=x(n),(k1)(ji(j)(k1)(li(l)k1i

18、=1i=1j=l1(3.3)󰀆󰀉n󰀉uj=li=1pi(j)iu󰀄󰀉i=1pi(l)i󰀅󰀇xl+(xl+lxl),(n)1k1(3.7)(3.8),󰀆󰀉n󰀉uj=li=1pi(j)(k1)(ji(j)u󰀄󰀉i=1pi(l)(k1)(li(l)󰀅󰀇1,(3.8)un󰀉󰀉1+In()1pi(j).<ii=1j=(n)0,nu󰀉

19、;󰀉11+In(µ)pi(j),limsup(k1)(ji(j)nk1i=1j=(n)(axn+1+bxn)(cxn)+kku󰀉i=1pi(n)xki(n)0(axn+1+bxn)(cxn)+kku󰀉i=1pi(n)xki(n)04(1.2)(1.2),(1.1)17027(n)u󰀉󰀉11pi(j),H1:µ=liminf>nk1j=n+1i=1(k1)(i(j)j)(n)u󰀉󰀉pi(j)1H2:=limsup>1.nj=ni=1(k1)(i(j)j)

20、1H3:0µ,(n)u󰀉󰀉11+In(µ)pi(j),=limsup>nk1j=ni=1(k1)(i(j)j)(µ)=eµ(µ).4.1lim(n)n)=,nH1,H2,H3H1,H2,H3u󰀉i=1(1.2)4.2nlim(n)n)=,k(axn1+bxn)(cxn)+kpi(n)xki(n)0(axn1+bxn)(cxn)+kku󰀉i=1pi(n)xki(n)01HardGH,LittlewoodJE,PolyaG.Inequatities,Inded,CambridgeUniversityPress,19522,2000,4:562567(YangJ,GuanXP.TheNon-existenceofPositiveSolutionofaNonlinearDierenceEquation.ActaMathematicaeApplicataeSinica,2000,4:562567)3ErbeLH,ZhangBG.OscillationofDiscreteAnaloguesofDel