搜索
H41 H3 Vol.41,No.31998T5zACTAMATHEMATICASINICAMay,1998Banach EG P T t ( a 730070)NIW qfe Banachro aX d fZ yq u Xz bc Z n v yq yqa aplhsm jvY gfZ x gfZ a ipw k C0-Xz ω-q ω-qMR(1991)U 34G20,45N05RA O175.15PeriodicSolutionsofSemilinearEvolutionEquationsinBanachSpacesLiYongxiang(DepartmentofMathematics,NorthwestNormalUniversity,Lanzhou730070,China)AbstractInthispaper,thesupersolutionandsubsolutionmethodisappliedtoperiodicsolutionproblemsforsemilinearevolutionequationsinorderedBanachspaces.Theresultsontheexistenceofmaximalperiodicsolutionandminimalperiodicsolutionareobtainedbyusingthecharacteristicsofpositiveoperatorssemigroupsandthemonotoneiterationscheme.Thispapergeneralizesandextendstherelevantresultsinordinarydifferentialequationsandpartialdifferentialequations.KeywordsNormalconvex-cone,PositiveC0-semigroups,ω-supersolution,ω-subsolution1991MRSubjectClassification34G20,45N05ChineseLibraryClassificationO175.151JH X Banach GX ChARY c u(t)+Au(t)=f(t,u(t)),t∈R(1)Cω- C R A X C AR −A AC0- oT(t)(t≥0).f(t,x):R×X→X X wst ω c (1) AVW Y c v jP X Oc c Schr¨odingerc F[1]. Lc C R rKh w!C [2], HLc C HNO(1)Cω- C R p m F s[0,ω] C u(t)+Au(t)=f(t,u(t)),0≤t≤ω,u(0)=u(ω),(2) pu 1997-01-20,Unu 1997-11-26, u 1998-01-08630 ZZ 41 K X C # KnNAX C q W X1 D(A) H ·1=·+A·t CBanach x∃v∈C1([0,ω],X)∩C([0,ω],X1)(2)j F! v(t)+Av(t)≤f(t,v(t)),v(0)≤v(ω),(3) v (1)Cω- x(3) jD C F! v ω- $ VW% jc p N # GCc xpω- v ω- w v≤w, ω- $%oKolesov[3,4]tAmann[5] mPoincar´e C PIc AX Oc C N &K A &Poincare pNd AhARG X ’ LakshmikanthamtLeela[6] AhARG XC jc C N ( %[7]r [6] C )yABanach C jc (1) A≡0CeQ os*f+ tR1p ’C e [7] Cs( C pXVW , tCZ jc Banach CY c , Q p *! GA[1],AVhsARZ j ,[7] C -.m m)C0- oC/* 0!t1.C#LM$WB A % HY c (1) C N &r’Aq(shmC C [3–7]Co#t)y C hmsW% Y c G4$ hmsn+ jP )yA[6]C )hmsqc jJA[3–5] 2ghARG X C3% G2$ v* $ 4K AARY c C 5 N + hAR KNA&, C+2 H G3$ $ 4K A(1)C C N &r’AqVU(1) C, #mC+2s’ W Ht)C0- o H pw)C0- oCR’([8].2DF-O./0M123 K X C # −A CC0- oT(t) ) o[8], Vx≥0,pT(t)x≥0.o $C o$6 ∃M0 7 δ,T(t)≤Meδt,∀t≥0.(4)a(Vsc7 C,−(A+CI)b A)C0- oS(t)=e−CtT(t). X1 D(A) H ·1=·+A·t CBanach o8$C [1],V∀x∈D(A) h(t)∈C1([0,ω],X),ARY c - u(t)+Au(t)=h(t),0≤t≤ω,u(0)=x,(5)p5 uJ u(t)∈C1([0,ω],X)∩C([0,ω],X1),d oT(t).9u(t)=T(t)x+t0T(t−s)h(s)ds,0≤t≤ω.(6)Vx∈X h∈C([0,ω],X),o(6)!nNCu∈C([0,ω],X),u N /m (5)C %yd mild [1].3 )kE0Banach*+1BSZ4d2D5 ,631:; VhARc (1),xu∈C([0,ω],X)I6-jc u(t)=T(t)u(0)+t0T(t−s)f(s,u(s))ds,0≤t≤ω,(7) u (1) [0,ω] Cmild a((7) C323% VOq7 C,pu(t)=S(t)u(0)+t0S(t−s)(f(s,u(s))+Cu(s))ds,(8) S(t)=e−CtT(t) −(A+CI) CC0- o VARY c pv 782.1 −A CC0- oT(t)9 N ∃M,ν0,T(t)≤Me−νt,∀t≥0,(9) ARY c u(t)+Au(t)=h(t),0≤t≤ω,u(0)=u(ω),(10)V∀h∈C([0,ω],X) 5 mild P(h)∈C([0,ω],X),di)xh∈C1([0,ω],X), P(h)∈C1([0,ω],X)∩C([0,ω],X1) uJ ii)xh(t)≥0, P(h)(t)≥0.Q:o oR’ (T(ω))n=T(nω)≤Me−nν,v. /r(T(ω))≤e−ν1,# I−T(ω)pp!S (I−T(ω))−1=∞n=0T(nω).(11)FB(h)=(I−T(ω))−1ω0T(ω−s)h(s)ds, u(t)=T(t)B(h)+t0T(t−s)h(s)dsP(h)(12) - (5)I6u(0)=B(h)Cmild du(0)=u(ω)=B(h).a(P(h) (10)C5 mild h∈C1([0,ω],X), v(t)=t0T(t−s)h(s)ds (5)I6v(0)=0CuJ v∈C1([0,ω],X)∩C([0,ω],X1)./4v(ω)=ω0T(ω−s)h(s)ds∈D(A).oC0- oR’ T(t)v(ω)∈D(A),∀t≥0.Fxm=mn=0T(nω)v(ω)(m=1,2,···).o(11)$ m→∞,xm→(I−T(ω))−1v(ω),Axm=mn=0T(nω)Av(ω)→(I−T(ω))−1Av(ω),oAC R$ B(h)∈D(A).s o(12)nNC(5)CI6u(0)=B(h)C P(h) uJ h(t)≥0, o oT(t)C)R v(ω)=ω0T(ω−s)h(s)ds≥0,# B(h)=∞n=0T(nω)v(ω)≥0.s (12),P(h)(t)≥0.632 ZZ 41 3;DF-O./0S= K )z#[9],−A C oT(t) )C0- o f(t,x):R×X→X X wst ω a(C([0,ω],X) u=maxu(t)t Banach Y,Y #K={u∈Y:u(t)≥0,0≤t≤ω}t pWBanach Kb )z# C0- oT(t)(t≥0) 0 o[1] 9 t0,T(t) X C0 V0 oeQp783.1 −A CC0- oT(t) 0 o xhARY c (1) ω- v0 ω- w0,v0(t)≤w0(t)(0≤t≤ω),hARGf(t,x) Wi [v0,w0] I6 (P)∃C0,V∀u,v∈[v0,w0],u≤v, pf(t,v)−f(t,u)≥−C(v−u), (1) [v0,w0] 5mildω- u Kmildω- u:v0(t)≤u(t)≤u(t)≤w0(t)(0≤t≤ω).Q: f Cδ, −(A+CI) AX 9 NC)C0- oS(t)=e−CtT(t). D=[v0,w0],oN 2.1,V∀h∈D,X CARY c u(t)+Au(t)+Cu(t)=f(t,h(t))+Ch(t),0≤t≤ω,u(0)=u(ω),(13) 5 mild (Qh)(t)=S(t)B1(h)+t0S(t−s)(f(s,h(s))+Ch(s))ds,(14) B1(h)=(1−S(ω))−1ω0S(ω−s)(f(s,h(s))+Ch(s))ds.a(QjDwY X &Q = h1,h2∈D,h1≤h2, o (P):f(t,h1(t))+Ch1(t)≤f(t,h2(t))+Ch2(t)(0≤t≤ω).f S(t) )C0- o vω0S(ω−s)(f(s,h1(s))+Ch1(s))ds≤ω0S(ω−s)(f(s,h2(s))+Ch2(s))ds.o !(11),B1(h1)≤B1(h2).-o(14)!pQh1≤Qh2.-& v0≤Qv0,Qw0≤w0.f v0∈C1([0,ω],X1)∩C([0,ω],X1) ω- v0(t)+Av0(t)≤f(t,v0(t)),vv0(t)+Av0(t)+Cv0(t)≤f(t,v0(t))+Cv0(t),0≤t≤ω. !?T g(t), v0(t)=S(t)v0(0)+t0S(t−s)g(s)ds≤S(t)v0(0)+t0S(t−s)(f(s,v0(s))+Cv0(s))ds.(15)/4v0(ω)≤S(ω)v0(0)+ω0S(ω−s)(f(s,v0(s))+Cv0(s))ds.f v0(0)≤v0(ω),vv0(0)≤(I−S(ω))−1ω0S(ω−s)(f(s,v0(s))+Cv0(s))ds=B1(v0).E cL o(14)p(Qv0)(t)=S(t)B1(v0)+t0S(t−s)(f(s,v0(s))+Cv0(s))ds,(16)(16)t(15)D1p (Qv0)(t)−v0(t)≥S(t)(B1(v0)−v0(0))≥0.# v0≤Qv0.: &Qw0≤w0.s Q(D)⊂D.&Q:D→Dl X 6F(Wh)(t)=t0S(t−s)(f(s,h(s))+Ch(s