Existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance

Ouyang Zigen, Li, Li Gangzhao

Description

A class of nonlinear fractional order differential equation is investigated in this paper, where D 0 + α is the standard Riemann-Liouville fractional derivative of order 1 < α ≤ 2 , 0 < η < 1 , f ∈ C ( [ 0 , 1 ] × R , R ) . Using intermediate value theorem, we obtain a sufficient condition for the existence of the solutions for the above fractional order differential equations.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 2
• Langue: English

Extrait

OuyangandLiBoundaryValueProblems2012,2012:68
http://www.boundaryvalueproblems.com/content/2012/1/68
RESEARCH OpenAccess
Existenceofthesolutionsforaclassof
nonlinearfractionalorderthree-point
boundaryvalueproblemswithresonance
*ZigenOuyang andGangzhaoLi
*Correspondence:
[e-mail] Abstract
SchoolofMathematicsandPhysics,
AclassofnonlinearfractionalorderdifferentialequationSchoolofNuclearScienceand
Technology,UniversityofSouth
China,Hengyang421001,P.R.China αD u(t)+f(t,u(t))=0, 0<t<1,0+
1
u(0)=0, u(1)= u(η)
α–1η
αisinvestigatedinthispaper,whereD isthestandardRiemann-Liouvillefractional0+
derivativeoforder1< α ≤2,0< η<1,f ∈C([0,1] ×R,R).Usingintermediatevalue
theorem,weobtainasufficientconditionfortheexistenceofthesolutionsforthe
abovefractionalorderdifferentialequations.
1 Introduction
Considerthefollowingboundaryvalueproblem

αD u(t)+f t,u(t) =, <t<, (.)+

u()=, u()= u(η), (.)
α–η
αwhere D is the standard Riemann-Liouville fractional derivative of order < α ≤, <+
η<andf ∈C([,]
×R,R).
Inthelastfewdecades,manyauthorshaveinvestigatedfractionaldifferentialequations
whichhavebeenappliedinmanyfieldssuchasphysics,mechanics,chemistry,engineer-
ingetc.(seereferences[,,,–]).Especially,manyworkshavebeendevotedtothe
studyofinitialvalueproblemsandboundedvalueproblemsforfractionalorderdifferentialequations[,,,].
Recently, the existence of positive solutions of fractional differential equations has
attracted many authors’ attention [–, , , , , –,, ]. Using some fixed
point
theorems,theyobtainedsomeniceexistenceconditionsforpositivesolutions.
Inmorerecentworks,JiangandYuan[]consideredthefollowingboundaryvalueproblemoffractionaldifferentialequations

αD u(t)+f t,u(t) =, <t<, (.)+
© 2012 Ouyang and Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
inanymedium,providedtheoriginalworkisproperlycited.OuyangandLiBoundaryValueProblems2012,2012:68 Page2of13
http://www.boundaryvalueproblems.com/content/2012/1/68
u()=u()=, (.)
αwhere D is the standard Riemann-Liouville fractional derivative of order < α<and+
f :[,] ×R → R is continuous. Using some properties of the Green function G(t,s),+ +
they obtain some new sufficient conditions for the existence of positive solutions for the
aboveproblem.
Further, Li, Luo, and Zhou [] investigated the following fractional order three-point
boundaryvalueproblems

αD u(t)+f t,u(t) =, <t<, (.)+
β β
u()=, D u()=aD u(ξ), (.)+ +
αwhereD isthestandardRiemann-Liouvillefractionalderivativeoforder< α ≤, ≤+
α–β–β ≤ ,  ≤ a ≤ , ξ ∈ (,), aξ ≤– β, ≤ α – β–,and f :[,] × R → R is+ +
continuous.
Inthispaper,wediscusstheboundaryvalueproblem(.)-(.).Usingsomeproperties
oftheGreenfunctionG(t,s)andintermediatevaluetheorem,weestablishsomesufficient
conditionsfortheexistenceofthepositivesolutionsoftheproblem(.)-(.).
The paper is arranged as follows: In Section , we introduce some definitions for
frac-
tionalorderdifferentialequationsandgiveourmainresultsfortheboundaryvalueproblem (.)-(.). We give some lemmas for our results in Section .InSection ,weprove
ourmainresult;andfinally,wegiveanexampletoillustrateourresults.
2 Mainresults
Inthissection,weintroducesomedefinitionsandpreliminaryfactswhichareusedinthis
paper.
Definition . ([, ]) The fractional integral of order α with the lower limit t for a
functionf isdefinedas
t  f(s)αI f(t) = ds, t>t ,α>,t + –α(α) (t–s)t
provided that the integral on the right-hand side is point-wise defined on [t ,∞), where
istheGammafunction.
Definition.([,]) Riemann-Liouvillederivativeoforder αwiththelowerlimitt for
afunctionf :[,∞) →Rcanbewrittenas
n t  d f(s)αD f(t) = ds, t>t ,n–< α ≤n,t + n α+–n(n– α)dt (t–s)t
wherenisapositiveinteger.
We call the function u(t) a solution of (.)-(.)if u(t) ∈ C[,] ∩ L[,] with a
fractionalderivativeoforder α belongstoC[,] ∩L[,]andsatisfiesEquation(.)andthe
boundarycondition(.).OuyangandLiBoundaryValueProblems2012,2012:68 Page3of13
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We also need to introduce some lemmas as follows, which will be used in the proof of
ourmaintheorems.
Lemma.([]) Assumethath(t) ∈C(,)∩L(,)withafractionalderivativeoforder
α>belongstoC(,) ∩L(,).Then,thefractionalequation

αD h(t) = (.)t +
hassolutions
α– α– α–nh(t)=c t +c t +···+c t , c ∈R,i=,,...,n,n=[α]+. (.)  n i
Lemma.([]) Assumethath(t) ∈C(,)∩L(,)withafractionalderivativeoforder
α>belongstoC(,) ∩L(,).Then
α α α– α– α–nI D h(t)=h(t)+c t +c t +···+c t (.)  nt + t + 
forsomec ∈R,i=,,...,n,n=[α]+.i
Lemma.([]) SupposethatX beaBanachspace,C ⊂X isclosedandconvex.Assume
thatUisarelativelyopensubsetofCwith ∈U,andT :U →Cisacompletelycontinuous
operator.Then,either
(i) T hasafixedpointinU,or
(ii) thereexistu ∈ ∂U and γ ∈(,)withu= γTu.
Throughoutthispaper,weassumethatf(t,u)satisfiesthefollowing:
(H) f(t,u) ∈C([,] ×R,R),andthereexisttwopositivefunctions a(t) ∈C([,],R ) and+
b(t) ∈C([,],R )suchthat+

α– p f t,t u ≤a(t)+b(t)|u| , t ∈[,], (.)
where ≤p ≤.Furthermore,
α–lim f t,t u = ±∞ (.)
u→±∞
foranyt ∈(,).
Wehaveourmainresults:
Theorem. Supposethat(H)holds.If

*G (s,s)b(s)ds<, (.)

thentheboundaryvalueproblem(.)-(.)hasatleastonesolution,where
⎧
α– α–⎨ (–s) –(η–s) , ≤s ≤ η,*G (s,s)=
α– ⎩ α–(α)(– η ) (–s) , η ≤s ≤.OuyangandLiBoundaryValueProblems2012,2012:68 Page4of13
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3 Somelemmas
Let =C[,],u ∈ equippedthenorm

u= sup u(t),(.)
≤t≤
then isaBanachspace.
Wefirstgivesomelemmasasfollows:
Lemma. Problem(.)-(.)isequivalenttothefollowingintegralequation

α–u(t)= G(t,s)f s,u(s) ds+u()t,(.)

where
⎧
α– α– α– α– α– α–t (–s) –t (η–s) –(– η )(t–s)⎪⎪ , α–⎪ (α)(– η )⎪⎪⎪⎪