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Wangetal.
AdvancesinDierenceEquations2013,2013:280http://www.
advancesindifferenceequations.
com/content/2013/1/280RESEARCHOpenAccessExistenceresultsfornonlinearfractionaldifferentialequationsinvolvingdifferentRiemann-LiouvillefractionalderivativesGuotaoWang1,SanyangLiu1*,DumitruBaleanu2,3,4andLihongZhang5*Correspondence:liusanyang@126.
com1DepartmentofAppliedMathematics,XidianUniversity,Xi'an,Shaanxi710071,People'sRepublicofChinaFulllistofauthorinformationisavailableattheendofthearticleAbstractByapplyinganiterativetechnique,anecessaryandsucientconditionisobtainedfortheexistenceoftheuniquesolutionofnonlinearfractionaldierentialequationsinvolvingtwoRiemann-Liouvillederivativesofdierentfractionalorders.
Finally,anexampleisalsogiventoillustratetheavailabilityofourmainresults.
Keywords:dierentfractional-order;nonlinearfractionaldierentialequations;Riemann-Liouvillederivative;monotoneiterativetechnique1IntroductionRecently,thestudyoffractionaldierentialequationshasacquiredpopularity,seebooks[–]formoreinformation.
Inthispaper,weconsiderthefollowingnonlinearfractionaldierentialequations:Dαu(t)=f(t,Dαu(t),Dβu(t),u(t)),Dβu()=,u()=,(.
)wheret∈J=[,T](Itisworthwhiletoindicatethatthenonlineartermfinvolvestheunknownfunction'sRiemann-Liouvillefractionalderivativeswithdierentorders.
Themethodofupperandlowersolutionscoupledwiththemonotoneiterativetech-niqueisaninterestingandpowerfulmechanism.
Theimportanceandadvantageofthemethodneedsnospecialemphasis[,].
TherehaveappearedsomepapersdealingwiththeexistenceofthesolutionofnonlinearRiemann-Liouville-typefractionaldier-entialequations[–]ornonlinearCaputo-typefractionaldierentialequations[–]byusingthemethod.
Forexample,byemployingthemethodoflowerandupperso-lutionscombinedwiththemonotoneiterativetechnique,LakshmikanthanandVatsala[],McRae[]andZhang[]successfullyinvestigatedtheinitialvalueproblemsofRiemann-LiouvillefractionaldierentialequationDαu(t)=f(t,u(t)),where<α≤.
However,intheexistingliterature[–],onlyonecasewhenα∈(,]isconsidered.
Theresearch,involvingRiemann-Liouvillefractionalderivativeoforder<α≤,pro-ceedsslowlyandthereappearsomenewdicultiesinemployingthemonotoneiterativemethod.
Toovercomethesediculties,weapplyasubstitutionDαu(t)=y(t).
Notethat2013Wangetal.
;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-tionLicense(http://creativecommons.
org/licenses/by/2.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Wangetal.
AdvancesinDierenceEquations2013,2013:280Page2of7http://www.
advancesindifferenceequations.
com/content/2013/1/280thetechniquehasbeendiscussedforfractionalproblemsinpapers[,].
Tothebestofourknowledge,itistherstpaper,inwhichthemonotoneiterativemethodisappliedtononlinearRiemann-Liouville-typefractionaldierentialequations,involvingtwodier-entfractionalderivativesDαandDβ.
Weorganizetherestofthispaperasfollows.
InSection,byusingthemonotoneiter-ativetechniqueandthemethodofupperandlowersolutions,theminimalandmaximalsolutionsofanequivalentproblemof(.
)areinvestigatedandtwoexplicitmonotoneit-erativesequences,convergingtothecorrespondingminimalandmaximalsolution,aregiven.
Inaddition,theuniquenessofthesolutionforfractionaldierentialequations(.
)isdiscussed.
InSection,anexampleisgiventoillustrateourresults.
2ExistenceresultsLemma.
Foragivenfunctiony∈C(J,R),thefollowingproblemDαu(t)=y(t),Dβu()=u()=,(.
)hasauniquesolutionu(t)=Iαy(t),whereIisthefractionalintegralandIαy(t)=t(t–s)α–(α)y(s)ds,<α≤,<β≤and<α–β≤.
ProofOnecanreduceequationDαu(t)=y(t)toanequivalentintegralequationu(t)=Iαy(t)+ctα–+ctα–(.
)forsomec,c∈R.
Byu()=,itfollowsc=.
Consequently,thegeneralsolutionof(.
)isu(t)=Iαy(t)+ctα–.
(.
)Thus,wehaveDβu(t)=Iα–βy(t)+c(α)(α–β)tα–β–=t(t–s)α–β–(α–β)y(s)ds+c(α)(α–β)tα–β–.
(.
)BytheconditionDβu()=,itfollowsthatc=.
Therefore,wehaveu(t)=Iαy(t).
Conversely,byadirectcomputation,wecangetDαu(t)=y(t)andDβu(t)=Iα–βy(t).
Itiseasytoverifyu(t)=Iαy(t)satises(.
).
Thiscompletestheproof.
CombinedwithLemma.
,weseethat(.
)canbetranslatedintothefollowingsystemy(t)=ft,y(t),Iα–βy(t),Iαy(t),(.
)wherey(t)=Dαu(t),t∈JandIα,Iα–βarethestandardfractionalintegrals.
Wangetal.
AdvancesinDierenceEquations2013,2013:280Page3of7http://www.
advancesindifferenceequations.
com/content/2013/1/280Now,welistforconveniencethefollowingcondition:(H)Thereexisty,z∈C(J,R)satisfyingy≤zsuchthaty(t)≤f(t,y(t),Iα–βy(t),Iαy(t)),z(t)≥f(t,z(t),Iα–βz(t),Iαz(t)).
(H)ThereexistsafunctionM∈C(J,(–,+∞))suchthatft,u(t),Iα–βu(t),Iαu(t)–ft,v(t),Iα–βv(t),Iαv(t)≥–M(t)(u–v)(t),wherey≤v≤u≤z,t∈J.
(H)ThereexistfunctionsN,K,L∈C(J,[,+∞))suchthatft,u(t),Iα–βu(t),Iαu(t)–ft,v(t),Iα–βv(t),Iαv(t)≤N(t)(u–v)(t)+K(t)Iα–β(u–v)(t)+L(t)Iα(u–v)(t),wherey≤v≤u≤z,t∈J.
Theorem.
Assumethat(H)and(H)hold.
Thenproblem(.
)hastheminimalandmaximalsolutiony,zintheorderedinterval[y,z].
Moreover,thereexistex-plicitmonotoneiterativesequences{yn},{zn}[y,z]suchthatlimn→∞yn(t)=y(t)andlimn→∞zn(t)=z(t),whereyn(t),zn(t)aredenedasyn(t)=+M(t)ft,yn–(t),Iα–βyn–(t),Iαyn–(t)+M(t)yn–(t),t∈J,n=,,.
.
.
,zn(t)=+M(t)ft,zn–(t),Iα–βzn–(t),Iαzn–(t)+M(t)zn–(t),t∈J,n=,,.
.
.
,(.
)andy≤y≤···≤yn≤···≤y≤z≤···≤zn≤···≤z≤z.
(.
)ProofDeneanoperatorQ:[y,z]→C(J,R)byx=Qη,wherexistheuniquesolutionofthecorrespondinglinearproblemcorrespondingtoη∈[y,z]andQη=+M(t)ft,η(t),Iα–βη(t),Iαη(t)+M(t)η(t).
(.
)Then,theoperatorQhasthefollowingproperties:(a)y≤Qy,Qz≤z;(b)Qh≤Qh,h,h∈[y,z],h≤h.
(.
)Wangetal.
AdvancesinDierenceEquations2013,2013:280Page4of7http://www.
advancesindifferenceequations.
com/content/2013/1/280Firstly,weshowthat(a)holds.
Lety=Qy,p=y–y.
By(H)andthedenitionofQ,weknowthatp(t)=+M(t)ft,y(t),Iα–βy(t),Iαy(t)+M(t)y(t)–y(t)≥+M(t)y(t)+M(t)y(t)–y(t)=.
Thus,wecanobtainp(t)≥,t∈J.
Thatis,y≤Qy.
Similarly,wecanprovethatQz≤z.
Then,(a)holds.
Secondly,letq=Qh–Qh,by(.
)and(H),wehaveq(t)=+M(t)ft,h(t),Iα–βh(t),Iαh(t)+M(t)h(t)–+M(t)ft,h(t),Iα–βh(t),Iαh(t)+M(t)h(t)≥+M(t)–M(t)(h–h)(t)+M(t)(h–h)(t)=.
Hence,wehaveq(t)≥,t∈J.
Thatis,Qh≥Qh.
Then,(b)holds.
Now,putyn=Qyn–,zn=Qzn–,n=,,.
.
.
.
(.
)By(.
),wecangety≤y≤···≤yn≤···≤zn≤···≤z≤z.
Obviously,yn,znsatisfyyn(t)=ft,yn–(t),Iα–βyn–(t),Iαyn–(t)–M(t)(un–yn–)(t),zn(t)=ft,zn–(t),Iα–βzn–(t),Iαzn–(t)–M(t)(zn–zn–)(t).
(.
)EmployingthesameargumentsusedinRef.
[],weseethat{yn},{zn}convergetotheirlimitfunctionsy,z,respectively.
Thatis,limn→∞yn(t)=y(t)andlimn→∞zn(t)=z(t).
Moreover,y(t),z(t)aresolutionsof(.
)in[y,z].
(.
)istrue.
Finally,weprovethaty(t),z(t)aretheminimalandthemaximalsolutionof(.
)in[y,z].
Letw∈[y,z]beanysolutionof(.
),thenQw=w.
Byy≤w≤z,(.
)and(.
),wecanobtainyn≤w≤zn,n=,,.
.
.
.
(.
)Thus,takinglimitin(.
)asn→+∞,wehavey≤w≤z.
Thatis,y,zaretheminimalandmaximalsolutionof(.
)intheorderedinterval[y,z],respectively.
Thiscompletestheproof.
Wangetal.
AdvancesinDierenceEquations2013,2013:280Page5of7http://www.
advancesindifferenceequations.
com/content/2013/1/280Theorem.
LetN(t)≥–M(t).
Assumeconditions(H)-(H)hold.
Ifλ(t)=N(t)+K(t)tα–β(α–β+)+L(t)tα(α+)<,thenproblem(.
)hasauniquesolutionx(t)∈[y,z].
ProofByTheorem.
,wehaveprovedthaty,zaretheminimalandmaximalsolutionof(.
)andy(t)≤y(t)≤z(t)≤z(t),t∈J.
Now,wearegoingtoshowthatproblem(.
)hasauniquesolutionx,i.
e.
,y(t)=z(t)=x(t).
Letp(t)=z(t)–y(t),by(H),wehave≤p(t)≤ft,z(t),Iα–βz(t),Iαz(t)–ft,y(t),Iα–βy(t),Iαy(t)≤N(t)z–y(t)+K(t)Iα–βz–y(t)+L(t)Iαz–y(t)=N(t)p(t)+K(t)t(t–s)α–β–(α–β)p(s)ds+L(t)t(t–s)α–(α)p(s)ds≤N(t)+K(t)tα–β(α–β+)+L(t)tα(α+)maxt∈Jp(t)λ(t)maxt∈Jp(t),whichimpliesthatmaxt∈Jp(t)≤.
Sincep(t)≥,thenitholdsp(t)=.
Thatis,y(t)=z(t).
Therefore,problem(.
)hasauniquesolutionx∈[y,z].
Letx(t)betheuniquesolutionof(.
).
Notingthatx∈[y,z]andu(t)=Iαx(t),wecaneasilyobtainthefollowingtheorem.
Theorem.
LetallconditionsofTheorem.
hold.
Thenproblem(.
)hasauniquesolutionu∈[Iαy,Iαz],t∈J.
3ExampleConsiderthefollowingproblem:Du(t)=t[–Du(t)]+tDu(t)+t[–Du(t)]+tu(t),Du()=,u()=,(.
)wheret∈[,].
LetDu(t)=y(t),thenDu(t)=Iy(t),u(t)=Iy(t).
So,(.
)canbetranslatedintothefollowingproblemy(t)=t–y(t)+ty(t)+t–Iy(t)+tIy(t),(.
)Wangetal.
AdvancesinDierenceEquations2013,2013:280Page6of7http://www.
advancesindifferenceequations.
com/content/2013/1/280Notingthatα=,β=,thenft,y,Iα–βy,Iαy=t[–y]+ty+t–Iy+tIy.
Takey(t)=,z(t)=,wehavey(t)=≤t+t=f(t,y(t),Iα–βy(t),Iαy(t)),z(t)=≥ty+t(–t)+tπ=f(t,z(t),Iα–βz(t),Iαz(t)).
Hence,condition(H)holds.
Fory≤y≤z≤z,wehaveft,z,Iα–βz,Iαz–ft,y,Iα–βy,Iαy=t(–z)–(–y)+t(z–y)+t–Iz––Iy+tIz–Iy≥–t–t(z–y)andft,z,Iα–βz,Iαz–ft,y,Iα–βy,Iαy≤–t(z–y)+tI(z–y)+t√πI(z–y).
TakeM(t)=t–t,N(t)=K(t)=t,L(t)=t√π.
Throughasimplecalculation,wehaveλ(t)=t+t+tπ<.
Then,allconditionsofTheorem.
aresatised.
Inconsequence,theproblem(.
)hasauniquesolutionu∈[,t√π].
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsAllauthorshaveequalcontributions.
Authordetails1DepartmentofAppliedMathematics,XidianUniversity,Xi'an,Shaanxi710071,People'sRepublicofChina.
2DepartmentofMathematics,FacultyofArtandSciences,Balgat,06530,Turkey.
3InstituteofSpaceSciences,Magurele-Bucharest,Romania.
4DepartmentofChemicalandMaterialsEngineering,FacultyofEngineering,KingAbdulazizUniversity,P.
O.
Box80204,Jeddah,21589,SaudiArabia.
5SchoolofMathematicsandComputerScience,ShanxiNormalUniversity,Linfen,Shanxi041004,People'sRepublicofChina.
AcknowledgementsTheauthorswouldliketothanktherefereesfortheirusefulcommentsandremarks.
ThisworkissupportedbytheNNSFofChina(No.
61373174)andtheNaturalScienceFoundationforYoungScientistsofShanxiProvince,China(No.
2012021002-3).
Received:9May2013Accepted:15August2013Published:4October2013Wangetal.
AdvancesinDierenceEquations2013,2013:280Page7of7http://www.
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:ExistenceresultsfornonlinearfractionaldifferentialequationsinvolvingdifferentRiemann-Liouvillefractionalderivatives.
AdvancesinDierenceEquations20132013:280.