462YJiWJiangJQiumodelsarefoundtobemoreadequatethanintegerordermodelsfo - PDF document

1 462Y.Ji|W.Jiang|J.Qiumodelsarefoundtobem
462Y.Ji|W.Jiang|J.Qiumodelsarefoundtobemoreadequatethanintegerordermodelsforsomerealworldproblems.Fractionalderivativesprovideanexcellenttoolforthedescrip-tionofmemoryandhereditarypropertiesofvariousmaterialsandprocesses.Thisisthemainadvantageoffractionaldierentialequationsincomparisonwithclassicalintegerordermodels.Inconsequence,thesubjectoffractionaldierentialequationsisgainingmuchimportanceandattention.Fordetails,see[3],[8],[9],[14],[20],[21],[32],[35],[36]andthereferencestherein.In[1],[2],[31],[37]{[39],theauthorshavediscussedtheexistenceofsolutionsforBVPofnonlinearfractionaldierentialequations.Therearealargenumberofpapersdealingwiththesolvabilityofnonlinearfractionaldierentialequations.How-ever,thetheoryofBVPsfornonlinearfractionaldierent

2 ialequationsisstillintheinitialstagesand
ialequationsisstillintheinitialstagesandmanyaspectsofthistheoryneedtobeexplored.Tothebestofourknowledge,thereisfewpapertoinvestigatetheresonancecasewithdimKerL=2ontheintegralboundaryconditions.Ontheniteinterval[0;1],therst-order,second-orderandhigh-ordermulti-pointBVPsatresonancehavebeenstudiedbymanyauthors(see,forexam-ple[4]{[7],[10]{[13],[16],[22]{[28],[30],[34]),wheredimKerL=1.In[18],[40]thesecond-ordermulti-pointBVPsatresonancehavebeendiscussedwhendimKerL=2ontheniteinterval[0;1].Recently,Zhangetal.[40]discussedtheexistenceanduniquenessresultsforthefollowingBVPwithintegralboundaryconditionsatresonanceunderthecasedimKerL=2:8:x00(t)=f(t;x(t);x0(t));t2(0;1);x0(0)=Z10h(t)x0(t)dt;x0(1)=Z10g(t)x0(t)dt;whereh;g2C([0;1];[0;+1))withZ10h(t)dt=1;Z10g(t)dt=1;and

3 f:[0;1]R2!Riscontinuous.In[15],Jiang
f:[0;1]R2!Riscontinuous.In[15],JianginvestigatedtheexistenceofsolutionsforthefollowingBVPoffractionaldierentialequationsatresonancewithdimKerL=2:8&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;:D0+u(t)=f(t;u(t);D�10+u(t));a.e.t2[0;1];u(0)=0;D�10+u(0)=mXi=1aiD�10+u(i);D�20+u(1)=nXj=1bjD�20+u(j);where23,D0+istheRiemann{Liouvillefractionalderivative,01:::m1,01:::n1,mXi=1ai=1;nXj=1bj=1;nXj=1bjj=1; 464Y.Ji|W.Jiang|J.QiuDefinition2.2.TheCaputoderivativeoffractionalorder�0isde-nedascDy(t)=1 �(n�)Zt0y(n)(s) (t�s)�n+1ds;n=[]+1;providedtheright-handsideispointwisedenedon(0;1),where[]denotes

4 theintegerpartoftherealnumber.Thefo
theintegerpartoftherealnumber.Thefollowingareresultsforafractionaldierentialequation(see[19]).Lemma2.3(in[19]).Letu2Cn[0;1]andn�1n,n2Nandv2C1[0;1].Then,fort2[0;1],(a)cDIv(t)=v(t);(b)IcDu(t)=u(t)�n�1Pk=0tk k!u(k)(0).3.PreliminarylemmasInthissection,wepresentthemainresultsinthispaper,whoseproofswillbedonebyusingthefollowingxedpointtheoremduetoMawhin(see[29]).Definition3.1.LetYandZberealBanachspaces,L:domLY!Zisalinearoperator,LissaidtobeaFredholmoperatorofindexzeroprovidedthat(a)ImLisaclosedsubsetofZ,(b)dimKerL=codimImL+1.LetYandZberealBanachspaces,L:domLY!ZbeaFredholmoperatorofindexzeroandP:Y!Y,Q:Z!ZbecontinuousprojectorssuchthatImP=KerL;KerQ=ImL;Y=KerLKerP;Z=ImLImQ:ItfollowsthatLjdomL\KerP:domL\KerP!ImL

5 isinvertible.Wedenotetheinverseofthatmap
isinvertible.WedenotetheinverseofthatmapbyKP.Definition3.2.LetL:domLY!ZbeaFredholmoperatorofindexzero.If isanopenboundedsubsetofYsuchthatdomL\ 6=;,themapN:Y!ZwillbecalledL-compacton ifQN( )isboundedandinverseKP(I�Q)N: !Yiscompact.ThetheoremweuseistheTheorem2.4of[8]ortheTheoremIV.13of[29].Theorem3.3.LetL:domLY!ZbeaFredholmoperatorofindexzeroandletN:Y!ZbeL-compacton .Assumethatthefollowingconditionsaresatised:(a)Lx6=Nxforevery(x;)2[(domLnKerL)\@ ](0;1); 466Y.Ji|W.Jiang|J.QiuThentheBVP(1.1)canbewrittenasLx=Nx,x2domL.Forconvenience,wedenoteT1y=Z10h(t)Zt0(t�s)�2 �(�1)y(s)dsdt;T2y=T21y�T22y;whereT21y=Z10(1�s)�2 �(�1)y(s)ds;T22y=Z10g(t)Zt0(t�s)�2 �(�1)y(s)dsdt:Lemma3.4.I

6 fconditions(C1){(C3)hold,thenL:domL
fconditions(C1){(C3)hold,thenL:domLY!ZisaFredholmoperatorofindexzero.Furthermore,thelinearcontinuousprojectoroperatorQ:Z!ZcanbedenedbyQy=Q1y+(Q2y)
t;whereQ1y=(411T1y+412T2y)=4,Q2y=(421T1y+422T2y)=4,4ijisthealgebraiccofactorofaij(i;j=1;2),andthelinearoperatorKP:ImL!domL\KerPcanbewrittenby(KPy)(t)=Zt0(t�s)�1 �()y(s)ds;y2ImL:Moreover,(3.1)kKPykAkykp;y2ImL;where(3.2)A=maxfA1;A2g;A1=1 �()((�1)q+1)1=q;A2=1 �(�1)((�2)q+1)1=q:Proof.ItisclearthatKerL=fa+bt:a;b2R;t2[0;1]g.Moreover,wehave(3.3)ImL=y2Z:T1y=Z10h(t)Zt0(t�s)�2 �(�1)y(s)dsdt=0;T2y=Z10(1�s)�2 �(�1)y(s)ds�Z10g(t)Zt0(t�s)�2 �(�1)y(s)dsdt=0:Infact,Ify2ImL,thenthereexistsx2d

7 omLsuchthatcDx(t)=y(t).Integratingi
omLsuchthatcDx(t)=y(t).Integratingitfrom0tot,weknow(3.4)x0(t)=Zt0(t�s)�2 �(�1)y(s)ds+b0:Substitutingboundaryconditionx0(0)=R10h(t)x0(t)dtintothe(3.4),wehavex0(t)=Z10h(t)x0(t)dt+Zt0(t�s)�2 �(�1)y(s)ds: 468Y.Ji|W.Jiang|J.QiuitisobviousthatdimImQ=2.AgainfromQ1(Q1y)=1 4(411T1(Q1y)+412T2(Q1y))=1 4(411a11+412a12)(Q1y)=Q1y;Q1((Q2y)
t)=1 4(411T1((Q2y)
t)+412T2((Q2y)
t))=1 4(411a21+412a22)(Q2y)=0;Q2(Q1y)=1 4(421T1(Q1y)+422T2(Q1y))=1 4(421a11+422a12)(Q1y)=0;Q2((Q2y)
t)=1 4(421T1((Q2y)
t)+422T2((Q2y)
t))=1 4(421a21+422a22)(Q2y)=Q2y;wehaveQ2y=Q(Q1y+(Q2y)
t)=Q1((Q1y)+((Q2y)
t))+Q2((Q1y)+((Q2y)
t))
t=Q1(Q1y)+Q1((Q2y)
t)+Q2(Q1y)
t+Q2((Q2y)
t)
t=Q1y+(Q2y)
t=Qy;whichimpliestheoperatorQisaline

8 arprojector.Obviously,Qiscontinuous.Now,
arprojector.Obviously,Qiscontinuous.Now,wewillshowthatKerQ=ImL.Ify2KerQ,fromQy=0,weget(411T1y+412T2y=0;421T1y+422T2y=0:Since411412421422=46=0;soT1y=T2y=0,whichyieldsy2ImL.Ontheotherhand,ify2ImL,thenT1y=T2y=0,fromthedenitionsofoperatorQ,itisobviousthatQy=0,thusy2KerQ.Hence,KerQ=ImL.Fory2Z,y=(y�Qy)+Qy,wehaveQy2ImQandQ(y�Qy)=0.ItfollowsfromQ(y�Qy)=0,thedenitionsofQ,Q1,Q2andcondition(C3),thatT1(y�Qy)=T2(y�Qy)=0,i.e.y�Qy2ImL.So,Z=ImL+ImQ.Takey2ImL\ImQ,theny=Qy=0,i.e.Z=ImLImQ,So,wehavedimKerL=dimImQ=codimImL=2,thusLisaFredholmoperatorwithindexzero. 470Y.Ji|W.Jiang|J.QiuForx2 ,sincefisaS-Caratheodoryfunction,bythedenitionsofQ1and(C3),wegetjQ1Nx(s)j=1 4(411T1Nx(s)+412T2Nx(s)

9 )1 4[a22jT1Nx(s
)1 4[a22jT1Nx(s)j+a21jT2Nx(s)j]1 4[a22T1('r(s)+je(s)j)+a21(T21('r(s)+je(s)j)+T22('r(s)+je(s)j))]l1:Similarly,weobtainjQ2Nx(s)j=1 4(421T1Nx(s)+422T2Nx(s))1 4[a12jT1Nx(s)j+a11jT2Nx(s)j]1 4[a12T1('r(s)+je(s)j)+a11(T21('r(s)+je(s)j)+T22('r(s)+je(s)j))]l2:Thus,(3.5)kQNxkp=Z10jQNx(s)jpds1=pZ10(jQ1Nx(s)j+jQ2Nx(s)j)pds1=pl1+l2:So,QN( )isbounded.Now,wewillprovethatKP(I�Q)N( )iscompact.(a)Obviously,KP(I�Q)N: !Yiscontinuous.Forx2 ,since(3.6)kNxkp=Z10jf(s;x(s);x0(s))+e(s)jpds1=pZ10('r(s)+je(s)j)pds1=p:=l3;wehavejKP(I�Q)Nx(t)j=1 �()Zt0(t�s)�1(I�Q)Nx(s)ds1 �

10 ()Zt0j(t�s)�1(I�Q)Nx(
()Zt0j(t�s)�1(I�Q)Nx(s)jds1 �()((�1)q+1)1=q(l1+l2+l3):Similarly,weobtainj[KP(I�Q)Nx]0(t)j=1 �(�1)Zt0(t�s)�2(I�Q)Nx(s)ds 472Y.Ji|W.Jiang|J.Qiu(l1+l2+l3) �()((�1)q+1)1=q((t(�1)q+12�t(�1)q+11)1=q+(t2�t1)(�1)+1=q)(l1+l2+l3) �()((�1)q+1)1=q([(�1)q+1]1=q(t2�t1)1=q+(t2�t1)1=q)(l1+l2+l3) �()((�1)q+1)1=q([(�1)q+1]1=q+1)(t2�t1)1=q":Next,wewillprovethatift1;t22[0;1]aresuchthat0t2�t12,then(3.8)j[KP(I�Q)Nx]0(t2)�[KP(I�Q)Nx]0(t1)j":Infact,j[KP(I�Q)Nx]0(t2)�[KP(I�Q)Nx]0(t1)j=1 �(�1)Zt20(t2�s)�2

11 (I�Q)Nx(s)ds�1 �(�1)Zt1
(I�Q)Nx(s)ds�1 �(�1)Zt10(t1�s)�2(I�Q)Nx(s)ds1 �(�1)Zt10[(t2�s)�2�(t1�s)�2](I�Q)Nx(s)ds+1 �(�1)Zt2t1j(t2�s)�2(I�Q)Nx(s)jds(kNxkp+kQNxkp) �(�1)Zt10j(t1�s)�2�(t2�s)�2jqds1=q+Zt2t1jt2�sj(�2)qds1=q(l1+l2+l3) �(�1)Zt10[(t1�s)(�2)q�(t2�s)(�2)q]ds1=q+Zt2t1jt2�sj(�2)qds1=q=(l1+l2+l3) �(�1)((�2)q+1)1=q((t(�2)q+11+(t2�t1)(�2)q+1�t(�2)q+12)1=q+((t2�t1)(�2)q+1)1=q)(l1+l2+l3) �(�1)((�2)q+1)1=q2

12 4;(�2)+1=q2":By(3.7)and(3.8),weg
4;(�2)+1=q2":By(3.7)and(3.8),wegetthatfunctionsfromKP(I�Q)N( )areequi-conti-nuouson[0;1].TheArzela{AscoliTheoremimpliesthatNisL-compacton . 474Y.Ji|W.Jiang|J.QiuThus,fromthis,by(4.3)and(4.4),weobtain(4.5)kPxk=maxmaxt2[0;1]jx(0)+x0(0)tj;jx0(0)jjx(0)j+jx0(0)jc1+2c2+2A2kNxkp:Againfromallx2 1;(I�P)x2domL\KerP,LPx=0,thusfromLemma3.4,weget(4.6)k(I�P)xk=kKPL(I�P)xkAkL(I�P)xkp=AkLxkpAkNxkp:Hence,from(4.5)and(4.6),wehave(4.7)kxkkPxk+k(I�P)xkc1+2c2+(A+2A2)kNxkp:If(H1)holds,thenfrom(4.1)and(4.7),weget(4.8)kxk(A+2A2)(k 1kpkxk1+k 2kpkx0k1+ky3kpkx0k1+krkp+kekp)+c1+2c2:Thus,fromkxk1kxkand(4.8),weobtain(4.9)kxk1A+2A2 1�(A+2A2)k 1kp
k 2kpkx0k1+k 3kpkx0k1+krkp+kekp+c1+2c2 A+2A2&#

13 19;:Againfrom(4.8),(4.9),onehas(4:10)kx0
19;:Againfrom(4.8),(4.9),onehas(4:10)kx0k1(A+2A2)k 3kp 1�(A+2A2)(k 1kp+k 2kp)kx0k1+A+2A2 1�(A+2A2)(k 1kp+k 2kp)krkp+kekp+c1+2c2 A+2A2:Since2(0;1),fromtheabovelastinequality,thereexistsaconstantK1�0suchthat(4.11)kx0k1K1;thusfrom(4.9)and(4.11),thereexistsaconstantK2�0suchthatkxk1K2,hencekxk=maxfkxk1;kx0k1gmaxfK1;K2g.Therefore 1isbounded.If(4.2)holds,similartotheaboveargument,wecanprovethat 1isboundedtoo.TheproofofStep1isnished.Step2.Let 2=fx2KerL:Nx2ImLg.Nowweshowthat 2isbounded.Infact,x2 2impliesx=a+bt,a;b2RandQNx=0.Thus,Q1N(a+bt)=Q2N(a+bt)=0.By(H2),thereexistt0;t12[0;1]suchthatjx(t0)jc1,jx0(t1)jc2,thenkx0k1=jbjc2: 476Y.Ji|W.Jiang|J.Qiu5.ExampleExample5.1.ConsiderthefollowingBVP(5.1)8�

14 0;�����
0;�����&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;:cD3=2x(t)=m(t)t2+sinx(t) 12+(1+t)x0(t) 14+3sin(x0(t))1=3+5+cos2t;0t1;x0(0)=Z10x0(t)dt;x0(1)=Z10x0(t)dt;where(5.2)m(t)=8&#x]TJ/;ñ 9;&#x.963;&#x Tf ;'.3;— 0;&#x Td[;&#x]TJ/;ñ 9;&#x.963;&#x Tf ;'.3;— 0;&#x Td[;&#x]TJ/;ñ 9;&#x.963;&#x Tf ;'.3;— 0;&#x Td[;&#x]TJ/;ñ 9;&#x.963;&#x Tf ;'.3;— 0;&#x Td[;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.93; Td;&#x[000;&#x]TJ ;� -1;.9

15 3; Td;&#x[000;:�1;t2[0;1=3];3t�
3; Td;&#x[000;:�1;t2[0;1=3];3t�2;t2[1=3;2=3];0;t2[2=3;4=5];5t�4;t2[4=5;1]:Let=3=2,p=3,q=3=2,h(t)=1,g(t)=1,andf(t;x(t);x0(t))=m(t)!(t;x(t);x0(t))=m(t)t2+4+sinx(t) 12+(1+t)x0(t) 14+3sin(x0(t))1 3;e(t)=m(t)(t)=m(t)[1+cos2t]:ItisnotdiculttoseethatZ10h(t)dt=1;Z10g(t)dt=1;A=2p 2 p and4=1 �(3 2)Z10t3=2�1dt1 �(3 2)1�Z10t3=2�1dt1 �(3 2+1)Z10t3=2dt1 �(3 2+1)1�Z10t3=2dt=4 3p 2 3p 8 15p 4 5p 6=0:Nowweprove(H1){(H3)aresatised.Let 1(t)=1=12, 2(t)=1=7, 3(t)=3,r(t)=5,=1=3.Thenwehave(A+2A2)(k 1kp+k 2kp)=19 14r 2 1andj

16 f(t;u;v)j=jm(t)j
j!(t)j=jm(t)j


f(t;u;v)j=jm(t)j
j!(t)j=jm(t)j
t2+4+sinu 12+(1+t)v 14+3sin(v1=3) 1(t)juj+ 2(t)jvj+ 3(t)jvj+r(t):Thus(H1)issatised. 478Y.Ji|W.Jiang|J.Qiu[14]R.W.IbrahimandM.Darus,Subordinationandsuperordinationforunivalentsolu-tionsforfractionaldierentialequations,J.Math.Anal.Appl.345(2008),871{879.[15]W.Jiang,Theexistenceofsolutionstoboundaryvalueproblemsoffractionaldieren-tialequationsatresonance,NonlinearAnal.74(2011),1987{1994.[16]G.L.KarakostasandP.Ch.Tsamatos,Onanonlocalboundaryvalueproblematresonance,J.Math.Anal.Appl.259(2001),209{218.[17]A.A.Kilbas,H.M.SrivastavaandJ.J.Trujillo,TheoryandApplicationsofFrac-tionalDierentialEquations,North-HollandMathematicsStudies,vol.204,ElsevierScienceB.V.,Amsterdam,20

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18 boundaryvalueproblemsforordinarydif-fere
boundaryvalueproblemsforordinarydif-ferentialequationsofhigherorder,NonlinearAnal.57(2004),435{458.[25]S.LuandW.Ge,Ontheexistenceofm-pointboundaryvalueproblematresonanceforhigherorderdierentialequation,J.Math.Anal.Appl.287(2003),522{539.[26]R.Ma,Multiplicityresultsforathirdorderboundaryvalueproblematresonance,Non-linearAnal.32(1998),493{499.[27] ,Multiplicityresultsforathree-pointboundaryvalueproblematresonance,Non-linearAnal.53(2003),777{789.[28] ,Existenceresultsofam-pointboundaryvalueproblematresonance,J.Math.Anal.Appl.294(2004),147{157.[29]J.Mawhin,Topologicaldegreemethodsinnonlinearboundaryvalueproblems,NS-FCBMSRegionalConferenceSeriesinMathematics,Amer.Math.Soc.,Providence,RI,1979.[30]R.K.NagleandK.L.Pothoven,Onathird-ordernonlinearboundaryvalueproblematres

19 onance,J.Math.Anal.Appl.195(1995),148{15
onance,J.Math.Anal.Appl.195(1995),148{159.[31]A.M.Nakhushev,TheSturm{Liouvilleproblemforasecondorderordinarydierentialequationwithfractionalderivativesinthelowerterms,Dokl.Akad.NaukSSSR234(1977),308{311.[32]I.Podlubny,FractionalDierentialEquations,AcademicPress,SanDiego,1999.[33] ,FractionalDierentialEquations,MathematicsinScienceandEngineering,vol.198,AcademicPress,NewYork,London,Toronto,1999.[34]B.PrezeradzkiandR.Stanczy,Solvabilityofamulti-pointboundaryvalueproblematresonance,J.Math.Anal.Appl.264(2001),253{261.[35]S.Z.Rida,H.M.El-SherbinyandA.A.M.Arafa,OnthesolutionofthefractionalnonlinearSchrodingerequation,Phys.Lett.A372(2008),553{558.[36]S.G.Samko,A.A.KilbasandO.I.Marichev,FractionalIntegralsandDerivatives,TheoryandApplications,GordonandBreach,Yv