Existence and uniqueness of positive solution to singular fractional differential equations

Wang Yongqing, Liu Lishan, Wu, Wu Yonghong

Description

In this paper, we discuss the existence and uniqueness of a positive solution to the following singular fractional differential equation with nonlocal boundary value conditions: { D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , D 0 + β u ( 1 ) = ∑ i = 1 m − 2 η i D 0 + β u ( ξ i ) , where 1 < α ≤ 2 , 0 < β < α − 1 , 0 < ξ 1 < ⋯ < ξ m − 2 < 1 with ∑ i = 1 m − 2 η i ξ i α − β − 1 < 1 , D 0 + α is the standard Riemann-Liouville derivative, f may be singular at t = 0 , t = 1 , and u = 0 . MSC: 34B10, 34B15.

Sujets

• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 7
• Langue: English

Extrait

Wangetal.BoundaryValueProblems2012,2012:81
http://www.boundaryvalueproblems.com/content/2012/1/81
RESEARCH OpenAccess
Existenceanduniquenessofapositive
solutiontosingularfractionaldifferential
equations
1* 1,2 2YongqingWang ,LishanLiu andYonghongWu
*Correspondence:
[e-mail] Abstract
1SchoolofMathematicalSciences,
Inthispaper,wediscusstheexistenceanduniquenessofapositivesolutiontotheQufuNormalUniversity,Qufu,
Shandong273165,People’s followingsingularfractionaldifferentialequationwithnonlocalboundaryvalue
RepublicofChina conditions:
Fulllistofauthorinformationis
availableattheendofthearticle
αD u(t)+f(t,u(t))=0, 0<t<1,0+ β m–2 β
u(0)=0, D u(1)= ηD u(ξ),i i0+ i=1 0+
m–2 α–β–1 αwhere1< α ≤2,0< β < α–1,0< ξ <···< ξ <1with η ξ <1,D is1 m–2 ii=1 i 0+
thestandardRiemann-Liouvillederivative,f maybesingularatt=0,t=1,andu=0.
MSC: 34B10;34B15
Keywords: fractionaldifferentialequation;positivesolution;iterativescheme;
singularboundaryvalueproblem
1 Introduction
Inthispaper,weconsiderthefollowingfractionaldifferentialequation:
⎧
⎨ αD u(t)+f(t,u(t))=, <t<,+
(.)β m– β⎩u()=, D u()= η D u(ξ ),i i+ i= +
m– α–β– αwhere  < α ≤,  < β < α–,< ξ < ··· < ξ <with η ξ <, D is the m– i +i= i
standardRiemann-Liouvillederivative,f ∈C((,)×(,+∞) →[,+∞))maybesingular
at t=, t=,and u=.Inthispaper,byapositivesolutionto(.), we mean a function
αu ∈C[,]whichsatisfiesD u ∈L(,),positiveon(,]andsatisfies(.).+
Recently, many results were obtained dealing with the existence of solutions for non-
linear fractional differential equations by using the techniques of nonlinear analysis; see
[–]andreferencestherein.Themulti-pointboundaryvalueproblems(BVPforshort)
haveprovokedagreatdealofattention,forexample[–].In[],theauthorsdiscussed
somepositivepropertiesoftheGreenfunctionforDirechlet-typeBVPofnonlinearfrac-
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tionaldifferentialequation
⎧
⎨ αD u(t)+f(t,u(t))=, <t<,+
(.)
⎩u()=, u()=,
αwhere< α<,D isthestandardRiemann-Liouvillederivative,f ∈C([,]×[,+∞) →+
[,+∞)). By using the Krasnosel’skii fixed point theorem, the existence of positive solu-
tionswereobtainedundersuitableconditionsonf.
In [], the authors investigated the existence and multiplicity of positive solutions by
usingsomefixedpointtheoremsforthefractionaldifferentialequation
⎧
⎨ αD u(t)+f(t,u(t))=, <t<,+
(.)
β β⎩u()=, D u()=aD u(ξ),+ +
α–β–where  < α ≤ ,  ≤ β ≤ ,  < ξ<, ≤ a ≤with aξ <– β, ≤ α – β–, f :
[,] ×[,+∞) →[,+∞)satisfiedCarathéodorytypeconditions.
In[,],theauthorsconsideredthefractionaldifferentialequationgivenby
⎧
⎨ α (n–)D u(t)+f(t,u,u,...,u )=, <t<,n–< α ≤n,n ≥,+
(.)
⎩ (n–) (n–)u()=u()=···=u ()=, u ()=.
Inordertoobtaintheexistenceofpositivesolutionsof(.),theyconsideredthefollowing
fractionaldifferentialequation:
⎧
⎨ α–n+ n– n– D v(t)+f(t,I v(t),I v(t),...,I v(t),v(t))=, <t<,+ + + +
(.)
⎩v()=v()=.
In [], f = q(t)(g +h), and g, h have different monotone properties. By using the fixed
pointtheoremforthemixedmonotoneoperator,Zhangobtained(.)andhadaunique
n– α–n+ α–n+positive solution u(t)= I v(t)with v ∈ Q =: {x(t): t ≤ x(t) ≤ Mt }.Butthe+ M
resultsarenottruesincev(t)isapositivesolutionof(.),andv()=.Whatcausesitlies
intheunsuitableusingofpropertiesoftheGreenfunction.
n–In[],f ∈C([,]×[,+∞)×R →[,+∞)),f(t,y ,y ,...,y )isincreasingfory ≥  n– i
,i=,,...,n–.ByusingthepositivepropertiesoftheGreenfunctionobtainedin[]
and fixed point theory for the u concave operator, the authors obtained the uniqueness
ofapositivesolutionfortheBVP(.).
Motivatedbytheworksmentionedabove,inthispaperweaimtoestablishtheexistence
and uniqueness of a positive solution to the BVP (.). Our work presented in this paper
has the following features. Firstly, the BVP (.) possesses singularity, that is, f may be
singularatt=,t=,andu=.Secondly,weimposeweakerpositivityconditionsonthe
nonlocal boundary term, that is, some of the coefficients η can be negative. Thirdly, thei
uniquepositivesolutioncanbeapproximatedbyaniterativescheme.
Therestofthepaperisorganizedasfollows.InSection,wepresentsomepreliminaries
andlemmasthatwillbeusedtoproveourmainresults.Wealsodevelopsomenewpositive
propertiesoftheGreenfunction.InSection,wediscusstheexistenceanduniquenessofWangetal.BoundaryValueProblems2012,2012:81 Page3of12
http://www.boundaryvalueproblems.com/content/2012/1/81
apositivesolutionoftheBVP(.),wealsogiveanexampletodemonstratetheapplication
ofourtheoreticalresults.
2Preliminaries
For the convenience of the reader, we present here the necessary definitions from frac-
tionalcalculustheory.Thesedefinitionscanbefoundinrecentliterature.
Definition. Thefractionalintegraloforder α>ofafunctionu:(,+∞) →Risgiven
by
tα α–I u(t)= (t–s) u(s)ds+ (α) 
providedtheright-handsideisdefinedpointwiseon(,+∞).
Definition . The fractional derivative of order α >  of a continuous function u :
(,+∞) →Risgivenby
n t dα n–α–D u(t)= (t–s) u(s)ds,+ (n– α) dt 
where n=[α]+,[α] denotestheintegralpartofthenumber α,providedtheright-hand
sideispointwiselydefinedon(,+∞).


Definition. Byu ∈L(,),wemean |u(t)|dt< ∞.

αLemma.([]) Let α>.Thenthefollowingequalityholdsforu ∈L(,),D u ∈L(,),+
α α α– α– α–nI D u(t)=u(t)+c t +c t +···+c t ,  n+ +
wherec ∈R,i=,,...,n,n–< α ≤n.i
Set
⎧
⎨ α– α–β–t (–s), ≤t ≤s ≤,
G (t,s)= (.)
⎩ α– α–β– α–(α) t (–s) –(t–s) , ≤s ≤t ≤,
α–β– ξ –si
p(s)=– η , (.)i
–s
s≤ξi
α–G(t,s)=G (t,s)+q(s)t,(.)
where
m–p(s)–p() α–β–α–β–q(s)= (–s) , p()=– η ξ.(.)i i
(