Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 585639 9 pageshttpdxdoiorg1011552013585639

Research ArticlePositive Solutions of a Two-Point Boundary Value Problem forSingular Fractional Differential Equations in Banach Space

Bo Liu and Yansheng Liu

Department of Mathematics Shandong Normal University Jinan 250014 China

Correspondence should be addressed to Yansheng Liu yanshliugmailcom

Received 29 May 2013 Accepted 11 July 2013

Academic Editor William P Ziemer

Copyright copy 2013 B Liu and Y Liu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the existence of positive solutions to a two-point boundary value problem (BVP) for singular fractionaldifferential equations in Banach space and presents a number of new results First by constructing a novel cone and using thefixed point index theory a sufficient condition is established for the existence of at least two positive solutions to the approximateproblem of the considered singular BVP Second using Ascoli-Arzela theorem a sufficient condition is obtained for the existenceof at least two positive solutions to the considered singular BVP from the convergent subsequence of the approximate problemFinally an illustrative example is given to support the obtained new results

1 Introduction

Fractional differential equations have been widely investi-gated recently due to its wide applications [1ndash3] in biologyphysics medicine control theory and so forth As a matterof fact fractional derivatives provide a more excellent toolfor the description of memory and hereditary properties ofvarious materials and processes than integer derivatives Asan important issue for the theory of fractional differentialequations the existence of positive solutions to kinds ofboundary value problems (BVPs) has attracted many schol-arsrsquo attention and lots of excellent results have been obtained[4ndash11] by means of fixed point theorems upper and lowersolutions technique and so forth

It is noted that as a special class of fractional differentialequations the singular fractional differential equations withkinds of boundary values have been studied in a series ofrecent works [7 12 13] In [7] Jiang et al studied a singularnonlinear semipositone fractional differential system withcoupled boundary conditions and presented some sufficientconditions for the existence of a positive solution by using thefixed point theory in cone and constructing some availableintegral operators together with approximating techniqueZhang et al [13] considered a class of two-point BVP forsingular fractional differential equations with a negatively

perturbed term and established some results on the mul-tiplicity of positive solutions by using the approximatingtechnique In [12] Agarwal et al investigated the existence ofpositive solutions for a two-point singular fractional bound-ary value problem and proposed some existence criteriaby using sequential techniques It should be pointed outthat the nonlinearities of [7 13] are singular at 119905 = 0 1while the nonlinearity of [12] is singular at 119909 = 0 Toour best knowledge there are fewer results on two-pointBVPs for singular fractional differential equations with thenonlinearity being singular at both 119905 = 0 1 and 119909 = 0Motivated by this we consider the following two-point BVPof singular fractional differential equations in Banach space

119863120573

0+119909 (119905) + 119891 (119905 119909 (119905)) = 120579 119905 isin 119869

119909 (0) = 119909 (1) = 120579

(1)

where 1 lt 120573 le 2 is a real number 119869 = [0 1] 119891 119869 times E rarr

E is continuous 120579 denotes the null element in the Banachspace E with the norm sdot 119863120573

0+is the standard Riemann-

Liouville fractional derivative and 119891(119905 119909) may be singularat 119905 = 0 1 and 119909 = 120579 Firstly we establish a sufficientcondition for the existence of at least two positive solutions tothe approximate problem of BVP (1) by constructing a novelcone and using the fixed point index theory Secondly using

2 Journal of Function Spaces and Applications

Ascoli-Arzela theorem we obtained a sufficient conditionfor the existence of at least two positive solutions to BVP(1) from the convergent subsequence of the approximateproblem Finally we give an illustrative example to supportthe obtained new results

The main features of this paper are as follows (i) Aclass of fractional-order two-point boundary value problemswith the nonlinearity being singular at both 119905 = 0 1 and119909 = 120579 is firstly studied in this paper which generalizes theexisting singular fractional differential equations [7 12 13]and has wider applications (ii) A sequential-based methodis proposed for singular fractional differential equations withthe nonlinearity being singular at both 119905 = 0 1 and 119909 = 0which enriches the theory of fractional differential equations

The rest of this paper is organized as follows Section 2contains the definition of Riemann-Liouville fractionalderivative and some notationThe main results are presentedin Section 3 which is followed by an illustrative example inSection 4

2 Preliminaries

We first recall some well-known results about Riemann-Liouville derivative For details please refer to [14 15] and thereferences therein

Definition 1 The Riemann-Liouville fractional integral oforder 120573 gt 0 of a function 119910 (0infin) rarr 119877 is given by

1198681205730+119910 (119905) =

1

Γ (120573)int

119905

0(119905 minus 119904)

120573minus1119910 (119904) 119889119904 (2)

provided the right side is pointwise defined on (0infin)

Definition 2 The Riemann-Liouville fractional derivative oforder120573 gt 0 of a continuous function119910 (0infin) rarr 119877 is givenby

1198631205730+119910 (119905) =

1

Γ (119899 minus 120573)(119889

119889119905)

119899

int

119905

0

119910 (119904)

(119905 minus 119904)120573minus119899+1

119889119904 (3)

where 119899 is the smallest integer greater than or equal to 120573provided that the right side is pointwise defined on (0infin)

One can easily obtain the following properties from thedefinition of Riemann-Liouville derivative

Proposition 3 (see [15]) Let 120573 gt 0 if one assumes that119906 isin 119862(0 1) cap 119871(0 1) then the fractional differential equation1198631205730+119906(119905) = 0 has 119906(119905) = 1198621119905

120572minus1+ 1198622119905

120573minus2+ sdot sdot sdot + 119862119899119905

120573minus119899119862119894 isin 119877 119894 = 1 2 119899 as unique solutions where 119899 is thesmallest integer greater than or equal to 120573

Proposition 4 (see [15]) Assume that 119906 isin 119862(0 1) cap 119871(0 1)

with a fractional derivative of order 120573 gt 0 that belongs to119862(0 1) cap 119871(0 1) Then

1198681205730+1198631205730+119906 (119905) = 119906 (119905) + 1198621119905

120573minus1+ 1198622119905120573minus2

+ sdot sdot sdot + 119862119899119905120573minus119899

(4)

for some 119862119894 isin 119877 119894 = 1 2 119899 where 119899 is the smallest integergreater than or equal to 120573

The following lemmas will be used in the proof of themain results

Lemma 5 (see [16]) If 119878 sub 119862[119869E] is bounded and equicon-tinuous then

120572119888 (119878) = sup119905isin119869

(120572119878 (119905)) (5)

where 120572(sdot) and 120572119888(sdot) denote the Kuratowski noncompactnessmeasure of bounded sets in E and 119862[119869E] respectively 119878(119905) =

119909(119905) 119909 isin 119878 (119905 isin 119869) and 119862[119869E] is the Banach space ofall continuous functions 119909 119869 rarr E with the norm 119909119888 =

max119905isin119869119909(119905)

Lemma6 (see [16]) Let119875 be a cone in E and let119875119903 = 119909 isin 119875

119909 lt 119903 Let 119865 119875119903 rarr 119875 be a strict set contraction Assumethat there exist a 1199060 isin 119875 and 1199060 = 120579 such that 119909 minus 119865119909 = 1205821199060 forany 119909 isin 120597119875119903 and 120582 ge 0 Then 119894(119865 119875119903 119875)=0

Lemma 7 (see [16]) Let 119881 = 119909119899 isin 119871[119869E] and there existsa 119892 isin 119871[119869 119877

+] such that 119909119899(119905) le 119892(119905) 119886119890 119905 isin 119869 for all

119909119899 isin 119881 then 120572(int119905

119886119909119899(119904)119889119904 119899 isin 119873) le 2 int

119905

119886120572(119881(119904))119889119904 119905 isin 119869

Lemma 8 (Ascoli-Arzela theorem [16]) 119867 sub 119862[119869E] isrelative compact if and only if119867 is equicontinuous and for any119905 isin 119869119867(119905) is a relatively compact set in E

Lemma 9 (see [17]) Let B sube E be bounded open set 119860 119875 cap

B rarr B is condensing If there exists 1 gt 120583 gt 0 such that119860119909 = 120583119909 where 119909 isin 119875⋂120597B then 119894(119860 119875⋂B 119875) = 1

3 Main Results

Let 119875 = 119909 isin E 119909(119905) ge 120579 forall119905 isin 119869Then one can see that 119875 isa normal solid cone of E Define 119875119889 = 119909 isin 119875 119909 lt 119889 and119875119889 = 119909 isin 119875 119909 le 119889 Let 119875lowast denote the dual cone of 119875 Weconsider BVP (1) in 119862[119869E] 119909 is called a solution to BVP (1)if 119909 satisfies (1) In addition we call 119909(119905) a positive solution toBVP (1) if 119909(119905) gt 120579 forall119905 isin 119869

For convenience let us list the following assumptions

(H1) 119891 isin 119862[(0 1) times 119875 120579 119875] and10038171003817100381710038171003817119891 (119905 119905

120573minus2119909)

10038171003817100381710038171003817le 119896 (119905)

1003817100381710038171003817119902 (119909)

1003817100381710038171003817 119905 isin (0 1) 119909 isin 119875 120579

(6)

where 119896 (0 1) rarr (0 +infin) int10119904(1 minus 119904)

120573minus1119896(119904)119889119904 lt

+infin and 119902 119875 120579 rarr 119875

(H2) 1Γ(120573) int10119904(1minus119904)

120573minus1119896(119904)119902[(120573minus1)119904(1minus119904)

120573minus11199031 1198771]119889119904 lt

+infin for all 1198771 gt 1199031 gt 0 and there exists 119877 gt 0 suchthat 1Γ(120573) int1

0119904(1minus119904)

120573minus1119896(119904)119902[(120573minus1)119904(1minus119904)

120573minus1119877 119877+

1]119889119904 lt 119877 where

119902 [1199031 1198771] = sup119909isin11987511987711198751199031

1003817100381710038171003817119902 (119909)

1003817100381710038171003817lt +infin 1198771 gt 1199031 gt 0 (7)

(H3) 119891(119905 119905120573minus2119909) is uniformly continuous with respect to 119905

on [120574 1 minus 120574] times 1198751198771

1198751199031

where 120574 isin (0 12) and 1198771 gt

1199031 gt 0

Journal of Function Spaces and Applications 3

(H4) There exists a constant 119871 ge 0 such that

120572 (119891 (119905 119905120573minus2

119863)) le 119871120572 (119863)

forall119905 isin (0 1) 119863 sube 1198751198771

1198751199031

(8)

where 119871 lt minΓ(120573)1205731205732(120573 minus 1)120573minus1

Γ(120573)120573(120573 + 1)2and 1198771 gt 1199031 gt 0

(H5) There exists 120593lowast isin 119875lowast with 120593

lowast = 1 and 120593 isin 119871[0 1]

such that

lim119909rarr 0

inf119909isin119875

120593lowast(119891 (119905 119905

120573minus2119909)) ge 120593 (119905) (9)

uniformly in 119905 isin (0 1) where 0 lt int1

0119904(1 minus

119904)120573minus1

120593(119904)119889119904 lt +infin(H6) There exist 120595lowast isin 119875

lowast with 120595lowast = 1 and [119886 119887] sube (0 1)

such that

lim119909rarr+infin

inf119909isin119875

120595lowast(119891 (119905 119905

120573minus2119909))

119909= +infin (10)

uniformly on 119905 isin [119886 119887]

According to [18] BVP (1) is equivalent to

119909 (119905) = int

1

0119866 (119905 119904) 119891 (119904 119909 (119904)) 119889119904 (11)

where

119866 (119905 119904) =

[119905 (1 minus 119904)]120573minus1

minus (119905 minus 119904)120573minus1

Γ (120573) 0 le 119904 le 119905 le 1

[119905 (1 minus 119904)]120573minus1

Γ (120573) 0 le 119905 le 119904 le 1

(12)

Consider the operator 119860 associated with the singularboundary value problem (1) which is defined by

(119860119909) (119905) = int

1

0119866 (119905 119904) 119891 (119904 119909 (119904)) 119889119904 (13)

Set 119866lowast(119905 119904) = 1199052minus120573

119866(119905 119904) As in [19] we have

119866lowast(119905 119904) ge (120573 minus 1) 119905 (1 minus 119905) 119866

lowast(120591 119904)

119866lowast(119905 119904) le

1

Γ (120573)119904(1 minus 119904)

120573minus1

(14)

Define

(119860lowast119909) (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119905

120573minus2119909 (119904)) (15)

Then one can see that 119909(119905) is a fixed point of the operator119860lowast

if and only if 119905120573minus2119909(119905) is a solution of BVP (1)Choose 119890 isin 119875

119900 with 119890 = 1 We consider the followingapproximate problem of (15)

(119860lowast119899119909) (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119905

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119899 isin 119873

(16)

Denote

119876 = 119909 isin 119862 [119869 119875] 119909 (119905) ge (120573 minus 1) 119905 (1 minus 119905) 119909 (119904) forall119905 119904 isin 119869

(17)

It is easy to check that 119876 is a cone in 119862[119869E] Let 119876119903 = 119909 isin

119876 119909 lt 119903 By (H1) and (H2) we can conclude that

1003817100381710038171003817119860lowast1199091003817100381710038171003817le int

1

0

119904(1 minus 119904)120573minus1

Γ (120572)119896 (119904)

times 119902 [(120573minus1) 119904(1 minus 119904)120573minus1

119903 119903] 119889119904 lt +infin forall119909 isin 119876119903

(18)

which implies that the operator 119860lowast is well defined In

addition from the definition of 119876 we can prove that

119909 (119905) ge (120573 minus 1) 119905 (1 minus 119905) 119909119888 forall119909 isin 119876 119905 isin 119869 (19)

By (16) we have (119860lowast119899119909)(119905) ge (120573minus1)119905(1minus119905)(119860lowast119899119909)(119904) forall119905 119904 isin 119869

Thus 119860lowast119899119876 sub 119876In the following we first investigate the existence of

two positive solutions to the approximate problem (16) andthen establish the existence criterion for the existence of twopositive solutions to BVP (1) by using the sequential-basedtechnique and Ascoli-Arzela theorem

Lemma 10 Let (H1) and (H2) be satisfiedThen for any 119903 gt 0the operator119860lowast119899 is a continuous bounded operator from119876119903 into119876

Proof By (H1) we can easily see that 119860lowast119899 is bounded from119876119903

to 119876Next we prove that 119860lowast119899 is continuousLet 119909119898 minus 119909119888 rarr 0 as 119898 rarr +infin (119909119898 119909 isin 119876119903) From

(H1) we have1003817100381710038171003817100381710038171003817119891 (119904 (119909119898 (119904) +

119890

119899) 119904120573minus2

)

1003817100381710038171003817100381710038171003817

le 119896 (119904) 119902 (119909119898 (119904) +119890

119899) le 119896 (119904) 119902 [

1

119899 119903 +

1

119899]

(20)

Hence

1003817100381710038171003817(119860lowast119899119909119898) (119905)

1003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

int

1

0119866lowast(119905 119904) 119891 (119904 (119909119898 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [

1

119899 119903 +

1

119899] 119889119904

lt +infin

(21)

which together with the dominated convergence theoremimply that

lim119898rarr+infin

(119860lowast119899119909119898) (119905) = (119860

lowast119899119909) (119905) (22)

We now show that

lim119898rarr+infin

1003817100381710038171003817119860lowast119899119909119898 minus 119860

lowast119899119909

1003817100381710038171003817119888 = 0 (23)

4 Journal of Function Spaces and Applications

In fact if (23) is not true then there exist a positive number1205760 and a subsequence 119909119898

119894

sub 119909119898 such that

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 119860lowast119899119909

10038171003817100381710038171003817119888ge 1205760 (24)

Since 119860lowast119899119909119898 is relatively compact there is a subsequence of119860lowast119899119909119898119894

which converges to some 119910 isin 119876 Without loss ofgenerality we assume that 119860lowast119899119909119898119894 itself converges to 119910 thatis

lim119894rarrinfin

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 11991010038171003817100381710038171003817119888

= 0 (25)

By virtue of (22) and (25) we have 119910 = 119860lowast119899119909 which

contradicts with (24) Hence (23) holds and the continuityof 119860lowast119899 is proved

Lemma 11 Let (H1)ndash(H4) be satisfied Then for any 119903 gt 0 theoperator 119860lowast119899 is a strict set contraction from 119876119903 to 119876

Proof By virtue of Lemma 10 we know that 119860lowast119899119878 is boundedand equicontinuous on 119869 Thus from Lemma 5 one can seethat

120572119888 (119860lowast119899119878) = sup

119905isin119869

120572 ((119860lowast119899119878) (119905)) 119903 gt 0 119878 sube 119876119903 (26)

where 119860lowast119899119878(119905) = (119860lowast119899119909)(119905) 119909 isin 119878 119905 isin 119869

Set

119863120575 = int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

) 119889119904 119909 isin 119878

120575 isin (01

2)

(27)

Based on (H1) and (H2) we have

100381710038171003817100381710038171003817100381710038171003817

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119888(int

120575

0119904(1 minus 119904)

120573minus1119896 (119904) 119889119904

+int

1minus120575

1119904(1 minus 119904)

120573minus1119896 (119904) 119889119904) 119909 isin 119878 120575 isin (0

1

2)

(28)

where 119888 = 119902[1119899 119903 + (1119899)] Hence

119889119867 (119863120575 119860lowast119899119878) 997888rarr 0 120575 997888rarr 0

+ (29)

where 119889119867(sdot sdot) denotes the Hausdorff metrics which impliesthat

120572 (119860lowast119899119878) = lim

120575rarr0+120572 (119863120575) (30)

Since

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904

isin (1 minus 2120575)

times co 119866lowast (119905 119904) 119891 (119904 119904120573minus2

(119909 (119904)+119890

119899)) 119904 isin [120575 1 minus 120575]

(31)

by Lemmas 5 and 7 we have

120572 (119863120575)

= 120572(int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119909 isin 119878)

le (1 minus 2120575)

times 120572 (co 119866lowast (119905 119904)

times119891(119904 (119909 (119904)+119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le 120572(119866lowast(119905 119904)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

times 120572(119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)119871(120573 minus 1)

120573minus1

120573120573

120572 (119878 (119868120575))

le1

Γ (120573)2119871

(120573 minus 1)120573minus1

120573120573

120572119888 (119878)

(32)

where 119868120575 = [120575 1 minus 120575] By (H4) we can obtain

120572 (119860119899119878) = lim120575rarr0+

120572 (119863120575)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

2119871120572119888 (119878) lt 120572119888 (119878)

(33)

Consequently the operator119860119899 is a strict set contraction from119876119903 into 119876

Theorem 12 Let (H1)ndash(H6) hold Then there exists 119903 isin (0 119877)

such that the operator119860lowast119899 has two fixed points in119876119877119876119903 and119876

119876119877 respectively for arbitrary sufficiently large positive integer119899 where 119877 is given in (H2)

Journal of Function Spaces and Applications 5

Proof For any given 119905 isin (0 1) we have int10119866lowast(119905 119904)120593(119904)119889119904 le

int1

0119904(1 minus 119904)

120573minus1120593(119904)119889119904 Since int1

0119866lowast(119905 119904)120593(119904)119889119904 gt 0 there exists

a 1205761015840 gt 0 such that

1199031015840= int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 gt 0 (34)

Then by (H5) there exists 11990310158401015840 isin (0 119877) such that

120593lowast(119891 (119905 119905

120573minus2119909 (119905))) ge 120593 (119905) minus 120576

1015840 forall119905 isin (0 1) (35)

holds for 119909 isin 119876 and 119909119888 le 11990310158401015840 Letting 0 lt 119903 lt 119897 = min1199031015840

11990310158401015840 we prove that 119909 minus 119860

lowast119899119909 = 120582119890 for any 119909 isin 120597119876119903 120582 ge 0 as

119899 gt 1(119897 minus 119903) In fact suppose that this is false Then thereexist 120582 gt 0 119909 isin 120597119876119903 such that 119909minus119860

lowast119899119909 = 120582119890 It is obvious that

119909 (119905) = (119860lowast119899119909) (119905) + 120582119890 ge (119860

lowast119899119909) (119905)

= int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(36)

Thus

120593lowast(119909 (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904

(37)

This implies that

int

1

0120593lowast(119909 (119905)) 119889119905

ge int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 = 119903

1015840gt 119903

(38)

which is in contradiction with1003816100381610038161003816120593lowast(119909 (119905))

1003816100381610038161003816le 119909 (119905) le 119909119888 = 119903 119905 isin 119869 (39)

Combining with Lemma 6 we obtain that

119894 (119860lowast119899 119876119903 119876) = 0 (40)

Now we show that 119894(119860lowast119899 119876119877 119876) = 1 By Lemma 9 weneed only to prove that 119860lowast119899119909 = 120582119909 for 119909 isin 120597119876119877 and 120582 ge 1 Infact if there exist 119909 isin 120597119876119877 and some 120582 ge 1 such that 119860lowast119899119909 =

120582119909 then

119909 (119905) =1

120582int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(41)

Therefore by (H1) and (H2) we have

119877 = 119909119888

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

1003817100381710038171003817100381710038171003817119902 (119909 (119904) +

119890

119899)

1003817100381710038171003817100381710038171003817119889119904

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

times 119902 [(120573 minus 1) 119904(1 minus 119904)120573minus1

119877 119877 + 1] 119889119904

lt 119877

(42)

which is a contradiction Consequently

119894 (119860lowast119899 119876119877 119876) = 1 (43)

In the following we choose

1198771015840gt ((120573 minus 1) 119886 (1 minus 119887) int

119887

119886max119905isin119869

119866lowast(119905 119904) 119889119904)

minus1

(44)

By (H6) there exists119872 gt 119877 such that120593lowast(119891(119905 119905120573minus2119909)) ge 1198771015840119909

holds for 119909 gt 119872Let

119877 = max

119872+ 1

(120573 minus 1) 119886 (1 minus 119887)

1+1

(120573 minus 1) 1198771015840119886 (1 minus 119887)max119905isin119869 int

119887

119886119866lowast(119905 119904) 119889119904 minus 1

(45)

We now claim that 119909 minus 119860lowast119899119909 = 120582119890 for 119909 isin 120597119876119877 and 120582 ge 0 As a

matter of fact if this is not true then there exists 120582 gt 0 suchthat 119909 minus 119860

lowast119899119909 = 120582119890 that is 119909(119905) gt 119860

lowast119899119909(119905) which implies that

119877 ge 120593lowast(119909 (119905))

ge 120593lowast((119860lowast119899119909) (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) 119877

1015840((120573 minus 1) 119886 (1 minus 119887) (119909 minus 1)) 119889119904

ge (120573 minus 1) 119886 (1 minus 119887) 1198771015840(119877 minus 1)int

119887

119886119866lowast(119905 119904) 119889119904

gt 119877

(46)

which is a contradiction Therefore by Lemma 6 we canobtain that

119894 (119860lowast119899 119876119877 119876) = 0 (47)

6 Journal of Function Spaces and Applications

This together with (40) (43) and (47) implies that

119894 (119860lowast119899 119876119877 119876119903 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119903 119876) = 1

119894 (119860lowast119899 119876119877 119876119877 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119877 119876) = minus1

(48)

Thus there exist 119909 isin 119876119877 119876119903 and 119910 isin 119876119877 119876119877 such that119860lowast119899119909 = 119909 and 119860

lowast119899119910 = 119910 This completes the proof

Theorem 13 Let (H1)ndash(H6) be satisfied Then BVP (1) has atleast two positive solutions in 119862[119869E]

Proof FromTheorem 12 there exists an integer 1198990 gt 0 suchthat the operator119860lowast119899 119899 ge 1198990 has two fixed points in 119909119899 isin 119876119877

119876119903 and119910119899 isin 119876119876119877 Denote119863 = 119909119899 119899 ge 1198990 Obviously119863 isuniformly bounded Next we show that119863 is equicontinuous

Firstly we prove that

lim119905rarr0+

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (49)

lim119905rarr1minus

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (50)

On one hand by the absolute continuity of integration[19] for any 120576 gt 0 there exists 1205751 gt 0 such thatint119905

0(1Γ(120573))119904(1 minus 119904)

120573minus1119891(119904 (119909(119904) + 119890119899)119904

120573minus2)119889119904 lt 1205762 forall0 lt 119905 le

1205751On the other hand from (H2) we set

119872 = int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

(51)

and 120575 = min1205751 1205751120576119872 For 0 lt 119905 lt 120575 we have

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

= int

1205751

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1205751

119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

lt120576

2+ int

1

1205751

119905(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 (119909 (119904) +

119890

119899) 119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

le 120576

(52)

which implies that (49) holds Similarly one can prove that(50) holds

Next we show that119863 is equicontinuous for 119905 isin [120574 1 minus 120574]0 lt 120574 lt 12

From (H3) we define

= max119905isin[1205741minus120574]

119891(119905 (119909 (119905) +119890

119899) 119905120573minus2

) (53)

By the absolute continuity of integration for 120574 isin (0 12)

and the previous 120576 gt 0 there exists 0 lt 1205752 lt 120574 such that

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

lt120576

6 (54)

Since ((1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732 )Γ(120573))119891(119904 (119909(119904) +

119890119899)119904120573minus2

) is bounded for 119904 isin [1199051 1199052] there exists 1205753 gt 0 suchthat

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1199051

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205753

(55)

For |1199052 minus 1199051| lt 1205754 = 120574120576(6 int1

0(119904(1 minus 119904)

120573minus1Γ(120573))119896(119904)119902[(120573 minus

1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) one can see that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(56)

Similarly for |1199052 minus 1199051| lt 1205755 = 1205752120576(6 int1

0(119904(1 minus

119904)120573minus1

Γ(120573))119896(119904)119902[(120573 minus 1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) we have

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(57)

Since

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816

1

1205752(1 minus 1199051)120573minus1

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119904(1 minus 119904)120573minus1

119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

(58)

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

Ascoli-Arzela theorem we obtained a sufficient conditionfor the existence of at least two positive solutions to BVP(1) from the convergent subsequence of the approximateproblem Finally we give an illustrative example to supportthe obtained new results

The main features of this paper are as follows (i) Aclass of fractional-order two-point boundary value problemswith the nonlinearity being singular at both 119905 = 0 1 and119909 = 120579 is firstly studied in this paper which generalizes theexisting singular fractional differential equations [7 12 13]and has wider applications (ii) A sequential-based methodis proposed for singular fractional differential equations withthe nonlinearity being singular at both 119905 = 0 1 and 119909 = 0which enriches the theory of fractional differential equations

The rest of this paper is organized as follows Section 2contains the definition of Riemann-Liouville fractionalderivative and some notationThe main results are presentedin Section 3 which is followed by an illustrative example inSection 4

2 Preliminaries

We first recall some well-known results about Riemann-Liouville derivative For details please refer to [14 15] and thereferences therein

Definition 1 The Riemann-Liouville fractional integral oforder 120573 gt 0 of a function 119910 (0infin) rarr 119877 is given by

1198681205730+119910 (119905) =

1

Γ (120573)int

119905

0(119905 minus 119904)

120573minus1119910 (119904) 119889119904 (2)

provided the right side is pointwise defined on (0infin)

Definition 2 The Riemann-Liouville fractional derivative oforder120573 gt 0 of a continuous function119910 (0infin) rarr 119877 is givenby

1198631205730+119910 (119905) =

1

Γ (119899 minus 120573)(119889

119889119905)

119899

int

119905

0

119910 (119904)

(119905 minus 119904)120573minus119899+1

119889119904 (3)

where 119899 is the smallest integer greater than or equal to 120573provided that the right side is pointwise defined on (0infin)

One can easily obtain the following properties from thedefinition of Riemann-Liouville derivative

Proposition 3 (see [15]) Let 120573 gt 0 if one assumes that119906 isin 119862(0 1) cap 119871(0 1) then the fractional differential equation1198631205730+119906(119905) = 0 has 119906(119905) = 1198621119905

120572minus1+ 1198622119905

120573minus2+ sdot sdot sdot + 119862119899119905

120573minus119899119862119894 isin 119877 119894 = 1 2 119899 as unique solutions where 119899 is thesmallest integer greater than or equal to 120573

Proposition 4 (see [15]) Assume that 119906 isin 119862(0 1) cap 119871(0 1)

with a fractional derivative of order 120573 gt 0 that belongs to119862(0 1) cap 119871(0 1) Then

1198681205730+1198631205730+119906 (119905) = 119906 (119905) + 1198621119905

120573minus1+ 1198622119905120573minus2

+ sdot sdot sdot + 119862119899119905120573minus119899

(4)

for some 119862119894 isin 119877 119894 = 1 2 119899 where 119899 is the smallest integergreater than or equal to 120573

The following lemmas will be used in the proof of themain results

Lemma 5 (see [16]) If 119878 sub 119862[119869E] is bounded and equicon-tinuous then

120572119888 (119878) = sup119905isin119869

(120572119878 (119905)) (5)

where 120572(sdot) and 120572119888(sdot) denote the Kuratowski noncompactnessmeasure of bounded sets in E and 119862[119869E] respectively 119878(119905) =

119909(119905) 119909 isin 119878 (119905 isin 119869) and 119862[119869E] is the Banach space ofall continuous functions 119909 119869 rarr E with the norm 119909119888 =

max119905isin119869119909(119905)

Lemma6 (see [16]) Let119875 be a cone in E and let119875119903 = 119909 isin 119875

119909 lt 119903 Let 119865 119875119903 rarr 119875 be a strict set contraction Assumethat there exist a 1199060 isin 119875 and 1199060 = 120579 such that 119909 minus 119865119909 = 1205821199060 forany 119909 isin 120597119875119903 and 120582 ge 0 Then 119894(119865 119875119903 119875)=0

Lemma 7 (see [16]) Let 119881 = 119909119899 isin 119871[119869E] and there existsa 119892 isin 119871[119869 119877

+] such that 119909119899(119905) le 119892(119905) 119886119890 119905 isin 119869 for all

119909119899 isin 119881 then 120572(int119905

119886119909119899(119904)119889119904 119899 isin 119873) le 2 int

119905

119886120572(119881(119904))119889119904 119905 isin 119869

Lemma 8 (Ascoli-Arzela theorem [16]) 119867 sub 119862[119869E] isrelative compact if and only if119867 is equicontinuous and for any119905 isin 119869119867(119905) is a relatively compact set in E

Lemma 9 (see [17]) Let B sube E be bounded open set 119860 119875 cap

B rarr B is condensing If there exists 1 gt 120583 gt 0 such that119860119909 = 120583119909 where 119909 isin 119875⋂120597B then 119894(119860 119875⋂B 119875) = 1

3 Main Results

Let 119875 = 119909 isin E 119909(119905) ge 120579 forall119905 isin 119869Then one can see that 119875 isa normal solid cone of E Define 119875119889 = 119909 isin 119875 119909 lt 119889 and119875119889 = 119909 isin 119875 119909 le 119889 Let 119875lowast denote the dual cone of 119875 Weconsider BVP (1) in 119862[119869E] 119909 is called a solution to BVP (1)if 119909 satisfies (1) In addition we call 119909(119905) a positive solution toBVP (1) if 119909(119905) gt 120579 forall119905 isin 119869

For convenience let us list the following assumptions

(H1) 119891 isin 119862[(0 1) times 119875 120579 119875] and10038171003817100381710038171003817119891 (119905 119905

120573minus2119909)

10038171003817100381710038171003817le 119896 (119905)

1003817100381710038171003817119902 (119909)

1003817100381710038171003817 119905 isin (0 1) 119909 isin 119875 120579

(6)

where 119896 (0 1) rarr (0 +infin) int10119904(1 minus 119904)

120573minus1119896(119904)119889119904 lt

+infin and 119902 119875 120579 rarr 119875

(H2) 1Γ(120573) int10119904(1minus119904)

120573minus1119896(119904)119902[(120573minus1)119904(1minus119904)

120573minus11199031 1198771]119889119904 lt

+infin for all 1198771 gt 1199031 gt 0 and there exists 119877 gt 0 suchthat 1Γ(120573) int1

0119904(1minus119904)

120573minus1119896(119904)119902[(120573minus1)119904(1minus119904)

120573minus1119877 119877+

1]119889119904 lt 119877 where

119902 [1199031 1198771] = sup119909isin11987511987711198751199031

1003817100381710038171003817119902 (119909)

1003817100381710038171003817lt +infin 1198771 gt 1199031 gt 0 (7)

(H3) 119891(119905 119905120573minus2119909) is uniformly continuous with respect to 119905

on [120574 1 minus 120574] times 1198751198771

1198751199031

where 120574 isin (0 12) and 1198771 gt

1199031 gt 0

Journal of Function Spaces and Applications 3

(H4) There exists a constant 119871 ge 0 such that

120572 (119891 (119905 119905120573minus2

119863)) le 119871120572 (119863)

forall119905 isin (0 1) 119863 sube 1198751198771

1198751199031

(8)

where 119871 lt minΓ(120573)1205731205732(120573 minus 1)120573minus1

Γ(120573)120573(120573 + 1)2and 1198771 gt 1199031 gt 0

(H5) There exists 120593lowast isin 119875lowast with 120593

lowast = 1 and 120593 isin 119871[0 1]

such that

lim119909rarr 0

inf119909isin119875

120593lowast(119891 (119905 119905

120573minus2119909)) ge 120593 (119905) (9)

uniformly in 119905 isin (0 1) where 0 lt int1

0119904(1 minus

119904)120573minus1

120593(119904)119889119904 lt +infin(H6) There exist 120595lowast isin 119875

lowast with 120595lowast = 1 and [119886 119887] sube (0 1)

such that

lim119909rarr+infin

inf119909isin119875

120595lowast(119891 (119905 119905

120573minus2119909))

119909= +infin (10)

uniformly on 119905 isin [119886 119887]

According to [18] BVP (1) is equivalent to

119909 (119905) = int

1

0119866 (119905 119904) 119891 (119904 119909 (119904)) 119889119904 (11)

where

119866 (119905 119904) =

[119905 (1 minus 119904)]120573minus1

minus (119905 minus 119904)120573minus1

Γ (120573) 0 le 119904 le 119905 le 1

[119905 (1 minus 119904)]120573minus1

Γ (120573) 0 le 119905 le 119904 le 1

(12)

Consider the operator 119860 associated with the singularboundary value problem (1) which is defined by

(119860119909) (119905) = int

1

0119866 (119905 119904) 119891 (119904 119909 (119904)) 119889119904 (13)

Set 119866lowast(119905 119904) = 1199052minus120573

119866(119905 119904) As in [19] we have

119866lowast(119905 119904) ge (120573 minus 1) 119905 (1 minus 119905) 119866

lowast(120591 119904)

119866lowast(119905 119904) le

1

Γ (120573)119904(1 minus 119904)

120573minus1

(14)

Define

(119860lowast119909) (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119905

120573minus2119909 (119904)) (15)

Then one can see that 119909(119905) is a fixed point of the operator119860lowast

if and only if 119905120573minus2119909(119905) is a solution of BVP (1)Choose 119890 isin 119875

119900 with 119890 = 1 We consider the followingapproximate problem of (15)

(119860lowast119899119909) (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119905

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119899 isin 119873

(16)

Denote

119876 = 119909 isin 119862 [119869 119875] 119909 (119905) ge (120573 minus 1) 119905 (1 minus 119905) 119909 (119904) forall119905 119904 isin 119869

(17)

It is easy to check that 119876 is a cone in 119862[119869E] Let 119876119903 = 119909 isin

119876 119909 lt 119903 By (H1) and (H2) we can conclude that

1003817100381710038171003817119860lowast1199091003817100381710038171003817le int

1

0

119904(1 minus 119904)120573minus1

Γ (120572)119896 (119904)

times 119902 [(120573minus1) 119904(1 minus 119904)120573minus1

119903 119903] 119889119904 lt +infin forall119909 isin 119876119903

(18)

which implies that the operator 119860lowast is well defined In

addition from the definition of 119876 we can prove that

119909 (119905) ge (120573 minus 1) 119905 (1 minus 119905) 119909119888 forall119909 isin 119876 119905 isin 119869 (19)

By (16) we have (119860lowast119899119909)(119905) ge (120573minus1)119905(1minus119905)(119860lowast119899119909)(119904) forall119905 119904 isin 119869

Thus 119860lowast119899119876 sub 119876In the following we first investigate the existence of

two positive solutions to the approximate problem (16) andthen establish the existence criterion for the existence of twopositive solutions to BVP (1) by using the sequential-basedtechnique and Ascoli-Arzela theorem

Lemma 10 Let (H1) and (H2) be satisfiedThen for any 119903 gt 0the operator119860lowast119899 is a continuous bounded operator from119876119903 into119876

Proof By (H1) we can easily see that 119860lowast119899 is bounded from119876119903

to 119876Next we prove that 119860lowast119899 is continuousLet 119909119898 minus 119909119888 rarr 0 as 119898 rarr +infin (119909119898 119909 isin 119876119903) From

(H1) we have1003817100381710038171003817100381710038171003817119891 (119904 (119909119898 (119904) +

119890

119899) 119904120573minus2

)

1003817100381710038171003817100381710038171003817

le 119896 (119904) 119902 (119909119898 (119904) +119890

119899) le 119896 (119904) 119902 [

1

119899 119903 +

1

119899]

(20)

Hence

1003817100381710038171003817(119860lowast119899119909119898) (119905)

1003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

int

1

0119866lowast(119905 119904) 119891 (119904 (119909119898 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [

1

119899 119903 +

1

119899] 119889119904

lt +infin

(21)

which together with the dominated convergence theoremimply that

lim119898rarr+infin

(119860lowast119899119909119898) (119905) = (119860

lowast119899119909) (119905) (22)

We now show that

lim119898rarr+infin

1003817100381710038171003817119860lowast119899119909119898 minus 119860

lowast119899119909

1003817100381710038171003817119888 = 0 (23)

4 Journal of Function Spaces and Applications

In fact if (23) is not true then there exist a positive number1205760 and a subsequence 119909119898

119894

sub 119909119898 such that

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 119860lowast119899119909

10038171003817100381710038171003817119888ge 1205760 (24)

Since 119860lowast119899119909119898 is relatively compact there is a subsequence of119860lowast119899119909119898119894

which converges to some 119910 isin 119876 Without loss ofgenerality we assume that 119860lowast119899119909119898119894 itself converges to 119910 thatis

lim119894rarrinfin

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 11991010038171003817100381710038171003817119888

= 0 (25)

By virtue of (22) and (25) we have 119910 = 119860lowast119899119909 which

contradicts with (24) Hence (23) holds and the continuityof 119860lowast119899 is proved

Lemma 11 Let (H1)ndash(H4) be satisfied Then for any 119903 gt 0 theoperator 119860lowast119899 is a strict set contraction from 119876119903 to 119876

Proof By virtue of Lemma 10 we know that 119860lowast119899119878 is boundedand equicontinuous on 119869 Thus from Lemma 5 one can seethat

120572119888 (119860lowast119899119878) = sup

119905isin119869

120572 ((119860lowast119899119878) (119905)) 119903 gt 0 119878 sube 119876119903 (26)

where 119860lowast119899119878(119905) = (119860lowast119899119909)(119905) 119909 isin 119878 119905 isin 119869

Set

119863120575 = int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

) 119889119904 119909 isin 119878

120575 isin (01

2)

(27)

Based on (H1) and (H2) we have

100381710038171003817100381710038171003817100381710038171003817

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119888(int

120575

0119904(1 minus 119904)

120573minus1119896 (119904) 119889119904

+int

1minus120575

1119904(1 minus 119904)

120573minus1119896 (119904) 119889119904) 119909 isin 119878 120575 isin (0

1

2)

(28)

where 119888 = 119902[1119899 119903 + (1119899)] Hence

119889119867 (119863120575 119860lowast119899119878) 997888rarr 0 120575 997888rarr 0

+ (29)

where 119889119867(sdot sdot) denotes the Hausdorff metrics which impliesthat

120572 (119860lowast119899119878) = lim

120575rarr0+120572 (119863120575) (30)

Since

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904

isin (1 minus 2120575)

times co 119866lowast (119905 119904) 119891 (119904 119904120573minus2

(119909 (119904)+119890

119899)) 119904 isin [120575 1 minus 120575]

(31)

by Lemmas 5 and 7 we have

120572 (119863120575)

= 120572(int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119909 isin 119878)

le (1 minus 2120575)

times 120572 (co 119866lowast (119905 119904)

times119891(119904 (119909 (119904)+119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le 120572(119866lowast(119905 119904)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

times 120572(119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)119871(120573 minus 1)

120573minus1

120573120573

120572 (119878 (119868120575))

le1

Γ (120573)2119871

(120573 minus 1)120573minus1

120573120573

120572119888 (119878)

(32)

where 119868120575 = [120575 1 minus 120575] By (H4) we can obtain

120572 (119860119899119878) = lim120575rarr0+

120572 (119863120575)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

2119871120572119888 (119878) lt 120572119888 (119878)

(33)

Consequently the operator119860119899 is a strict set contraction from119876119903 into 119876

Theorem 12 Let (H1)ndash(H6) hold Then there exists 119903 isin (0 119877)

such that the operator119860lowast119899 has two fixed points in119876119877119876119903 and119876

119876119877 respectively for arbitrary sufficiently large positive integer119899 where 119877 is given in (H2)

Journal of Function Spaces and Applications 5

Proof For any given 119905 isin (0 1) we have int10119866lowast(119905 119904)120593(119904)119889119904 le

int1

0119904(1 minus 119904)

120573minus1120593(119904)119889119904 Since int1

0119866lowast(119905 119904)120593(119904)119889119904 gt 0 there exists

a 1205761015840 gt 0 such that

1199031015840= int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 gt 0 (34)

Then by (H5) there exists 11990310158401015840 isin (0 119877) such that

120593lowast(119891 (119905 119905

120573minus2119909 (119905))) ge 120593 (119905) minus 120576

1015840 forall119905 isin (0 1) (35)

holds for 119909 isin 119876 and 119909119888 le 11990310158401015840 Letting 0 lt 119903 lt 119897 = min1199031015840

11990310158401015840 we prove that 119909 minus 119860

lowast119899119909 = 120582119890 for any 119909 isin 120597119876119903 120582 ge 0 as

119899 gt 1(119897 minus 119903) In fact suppose that this is false Then thereexist 120582 gt 0 119909 isin 120597119876119903 such that 119909minus119860

lowast119899119909 = 120582119890 It is obvious that

119909 (119905) = (119860lowast119899119909) (119905) + 120582119890 ge (119860

lowast119899119909) (119905)

= int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(36)

Thus

120593lowast(119909 (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904

(37)

This implies that

int

1

0120593lowast(119909 (119905)) 119889119905

ge int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 = 119903

1015840gt 119903

(38)

which is in contradiction with1003816100381610038161003816120593lowast(119909 (119905))

1003816100381610038161003816le 119909 (119905) le 119909119888 = 119903 119905 isin 119869 (39)

Combining with Lemma 6 we obtain that

119894 (119860lowast119899 119876119903 119876) = 0 (40)

Now we show that 119894(119860lowast119899 119876119877 119876) = 1 By Lemma 9 weneed only to prove that 119860lowast119899119909 = 120582119909 for 119909 isin 120597119876119877 and 120582 ge 1 Infact if there exist 119909 isin 120597119876119877 and some 120582 ge 1 such that 119860lowast119899119909 =

120582119909 then

119909 (119905) =1

120582int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(41)

Therefore by (H1) and (H2) we have

119877 = 119909119888

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

1003817100381710038171003817100381710038171003817119902 (119909 (119904) +

119890

119899)

1003817100381710038171003817100381710038171003817119889119904

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

times 119902 [(120573 minus 1) 119904(1 minus 119904)120573minus1

119877 119877 + 1] 119889119904

lt 119877

(42)

which is a contradiction Consequently

119894 (119860lowast119899 119876119877 119876) = 1 (43)

In the following we choose

1198771015840gt ((120573 minus 1) 119886 (1 minus 119887) int

119887

119886max119905isin119869

119866lowast(119905 119904) 119889119904)

minus1

(44)

By (H6) there exists119872 gt 119877 such that120593lowast(119891(119905 119905120573minus2119909)) ge 1198771015840119909

holds for 119909 gt 119872Let

119877 = max

119872+ 1

(120573 minus 1) 119886 (1 minus 119887)

1+1

(120573 minus 1) 1198771015840119886 (1 minus 119887)max119905isin119869 int

119887

119886119866lowast(119905 119904) 119889119904 minus 1

(45)

We now claim that 119909 minus 119860lowast119899119909 = 120582119890 for 119909 isin 120597119876119877 and 120582 ge 0 As a

matter of fact if this is not true then there exists 120582 gt 0 suchthat 119909 minus 119860

lowast119899119909 = 120582119890 that is 119909(119905) gt 119860

lowast119899119909(119905) which implies that

119877 ge 120593lowast(119909 (119905))

ge 120593lowast((119860lowast119899119909) (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) 119877

1015840((120573 minus 1) 119886 (1 minus 119887) (119909 minus 1)) 119889119904

ge (120573 minus 1) 119886 (1 minus 119887) 1198771015840(119877 minus 1)int

119887

119886119866lowast(119905 119904) 119889119904

gt 119877

(46)

which is a contradiction Therefore by Lemma 6 we canobtain that

119894 (119860lowast119899 119876119877 119876) = 0 (47)

6 Journal of Function Spaces and Applications

This together with (40) (43) and (47) implies that

119894 (119860lowast119899 119876119877 119876119903 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119903 119876) = 1

119894 (119860lowast119899 119876119877 119876119877 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119877 119876) = minus1

(48)

Thus there exist 119909 isin 119876119877 119876119903 and 119910 isin 119876119877 119876119877 such that119860lowast119899119909 = 119909 and 119860

lowast119899119910 = 119910 This completes the proof

Theorem 13 Let (H1)ndash(H6) be satisfied Then BVP (1) has atleast two positive solutions in 119862[119869E]

Proof FromTheorem 12 there exists an integer 1198990 gt 0 suchthat the operator119860lowast119899 119899 ge 1198990 has two fixed points in 119909119899 isin 119876119877

119876119903 and119910119899 isin 119876119876119877 Denote119863 = 119909119899 119899 ge 1198990 Obviously119863 isuniformly bounded Next we show that119863 is equicontinuous

Firstly we prove that

lim119905rarr0+

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (49)

lim119905rarr1minus

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (50)

On one hand by the absolute continuity of integration[19] for any 120576 gt 0 there exists 1205751 gt 0 such thatint119905

0(1Γ(120573))119904(1 minus 119904)

120573minus1119891(119904 (119909(119904) + 119890119899)119904

120573minus2)119889119904 lt 1205762 forall0 lt 119905 le

1205751On the other hand from (H2) we set

119872 = int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

(51)

and 120575 = min1205751 1205751120576119872 For 0 lt 119905 lt 120575 we have

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

= int

1205751

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1205751

119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

lt120576

2+ int

1

1205751

119905(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 (119909 (119904) +

119890

119899) 119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

le 120576

(52)

which implies that (49) holds Similarly one can prove that(50) holds

Next we show that119863 is equicontinuous for 119905 isin [120574 1 minus 120574]0 lt 120574 lt 12

From (H3) we define

= max119905isin[1205741minus120574]

119891(119905 (119909 (119905) +119890

119899) 119905120573minus2

) (53)

By the absolute continuity of integration for 120574 isin (0 12)

and the previous 120576 gt 0 there exists 0 lt 1205752 lt 120574 such that

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

lt120576

6 (54)

Since ((1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732 )Γ(120573))119891(119904 (119909(119904) +

119890119899)119904120573minus2

) is bounded for 119904 isin [1199051 1199052] there exists 1205753 gt 0 suchthat

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1199051

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205753

(55)

For |1199052 minus 1199051| lt 1205754 = 120574120576(6 int1

0(119904(1 minus 119904)

120573minus1Γ(120573))119896(119904)119902[(120573 minus

1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) one can see that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(56)

Similarly for |1199052 minus 1199051| lt 1205755 = 1205752120576(6 int1

0(119904(1 minus

119904)120573minus1

Γ(120573))119896(119904)119902[(120573 minus 1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) we have

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(57)

Since

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816

1

1205752(1 minus 1199051)120573minus1

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119904(1 minus 119904)120573minus1

119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

(58)

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

(H4) There exists a constant 119871 ge 0 such that

120572 (119891 (119905 119905120573minus2

119863)) le 119871120572 (119863)

forall119905 isin (0 1) 119863 sube 1198751198771

1198751199031

(8)

where 119871 lt minΓ(120573)1205731205732(120573 minus 1)120573minus1

Γ(120573)120573(120573 + 1)2and 1198771 gt 1199031 gt 0

(H5) There exists 120593lowast isin 119875lowast with 120593

lowast = 1 and 120593 isin 119871[0 1]

such that

lim119909rarr 0

inf119909isin119875

120593lowast(119891 (119905 119905

120573minus2119909)) ge 120593 (119905) (9)

uniformly in 119905 isin (0 1) where 0 lt int1

0119904(1 minus

119904)120573minus1

120593(119904)119889119904 lt +infin(H6) There exist 120595lowast isin 119875

lowast with 120595lowast = 1 and [119886 119887] sube (0 1)

such that

lim119909rarr+infin

inf119909isin119875

120595lowast(119891 (119905 119905

120573minus2119909))

119909= +infin (10)

uniformly on 119905 isin [119886 119887]

According to [18] BVP (1) is equivalent to

119909 (119905) = int

1

0119866 (119905 119904) 119891 (119904 119909 (119904)) 119889119904 (11)

where

119866 (119905 119904) =

[119905 (1 minus 119904)]120573minus1

minus (119905 minus 119904)120573minus1

Γ (120573) 0 le 119904 le 119905 le 1

[119905 (1 minus 119904)]120573minus1

Γ (120573) 0 le 119905 le 119904 le 1

(12)

Consider the operator 119860 associated with the singularboundary value problem (1) which is defined by

(119860119909) (119905) = int

1

0119866 (119905 119904) 119891 (119904 119909 (119904)) 119889119904 (13)

Set 119866lowast(119905 119904) = 1199052minus120573

119866(119905 119904) As in [19] we have

119866lowast(119905 119904) ge (120573 minus 1) 119905 (1 minus 119905) 119866

lowast(120591 119904)

119866lowast(119905 119904) le

1

Γ (120573)119904(1 minus 119904)

120573minus1

(14)

Define

(119860lowast119909) (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119905

120573minus2119909 (119904)) (15)

Then one can see that 119909(119905) is a fixed point of the operator119860lowast

if and only if 119905120573minus2119909(119905) is a solution of BVP (1)Choose 119890 isin 119875

119900 with 119890 = 1 We consider the followingapproximate problem of (15)

(119860lowast119899119909) (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119905

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119899 isin 119873

(16)

Denote

119876 = 119909 isin 119862 [119869 119875] 119909 (119905) ge (120573 minus 1) 119905 (1 minus 119905) 119909 (119904) forall119905 119904 isin 119869

(17)

It is easy to check that 119876 is a cone in 119862[119869E] Let 119876119903 = 119909 isin

119876 119909 lt 119903 By (H1) and (H2) we can conclude that

1003817100381710038171003817119860lowast1199091003817100381710038171003817le int

1

0

119904(1 minus 119904)120573minus1

Γ (120572)119896 (119904)

times 119902 [(120573minus1) 119904(1 minus 119904)120573minus1

119903 119903] 119889119904 lt +infin forall119909 isin 119876119903

(18)

which implies that the operator 119860lowast is well defined In

addition from the definition of 119876 we can prove that

119909 (119905) ge (120573 minus 1) 119905 (1 minus 119905) 119909119888 forall119909 isin 119876 119905 isin 119869 (19)

By (16) we have (119860lowast119899119909)(119905) ge (120573minus1)119905(1minus119905)(119860lowast119899119909)(119904) forall119905 119904 isin 119869

Thus 119860lowast119899119876 sub 119876In the following we first investigate the existence of

two positive solutions to the approximate problem (16) andthen establish the existence criterion for the existence of twopositive solutions to BVP (1) by using the sequential-basedtechnique and Ascoli-Arzela theorem

Lemma 10 Let (H1) and (H2) be satisfiedThen for any 119903 gt 0the operator119860lowast119899 is a continuous bounded operator from119876119903 into119876

Proof By (H1) we can easily see that 119860lowast119899 is bounded from119876119903

to 119876Next we prove that 119860lowast119899 is continuousLet 119909119898 minus 119909119888 rarr 0 as 119898 rarr +infin (119909119898 119909 isin 119876119903) From

(H1) we have1003817100381710038171003817100381710038171003817119891 (119904 (119909119898 (119904) +

119890

119899) 119904120573minus2

)

1003817100381710038171003817100381710038171003817

le 119896 (119904) 119902 (119909119898 (119904) +119890

119899) le 119896 (119904) 119902 [

1

119899 119903 +

1

119899]

(20)

Hence

1003817100381710038171003817(119860lowast119899119909119898) (119905)

1003817100381710038171003817=

100381710038171003817100381710038171003817100381710038171003817

int

1

0119866lowast(119905 119904) 119891 (119904 (119909119898 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [

1

119899 119903 +

1

119899] 119889119904

lt +infin

(21)

which together with the dominated convergence theoremimply that

lim119898rarr+infin

(119860lowast119899119909119898) (119905) = (119860

lowast119899119909) (119905) (22)

We now show that

lim119898rarr+infin

1003817100381710038171003817119860lowast119899119909119898 minus 119860

lowast119899119909

1003817100381710038171003817119888 = 0 (23)

4 Journal of Function Spaces and Applications

In fact if (23) is not true then there exist a positive number1205760 and a subsequence 119909119898

119894

sub 119909119898 such that

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 119860lowast119899119909

10038171003817100381710038171003817119888ge 1205760 (24)

Since 119860lowast119899119909119898 is relatively compact there is a subsequence of119860lowast119899119909119898119894

which converges to some 119910 isin 119876 Without loss ofgenerality we assume that 119860lowast119899119909119898119894 itself converges to 119910 thatis

lim119894rarrinfin

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 11991010038171003817100381710038171003817119888

= 0 (25)

By virtue of (22) and (25) we have 119910 = 119860lowast119899119909 which

contradicts with (24) Hence (23) holds and the continuityof 119860lowast119899 is proved

Lemma 11 Let (H1)ndash(H4) be satisfied Then for any 119903 gt 0 theoperator 119860lowast119899 is a strict set contraction from 119876119903 to 119876

Proof By virtue of Lemma 10 we know that 119860lowast119899119878 is boundedand equicontinuous on 119869 Thus from Lemma 5 one can seethat

120572119888 (119860lowast119899119878) = sup

119905isin119869

120572 ((119860lowast119899119878) (119905)) 119903 gt 0 119878 sube 119876119903 (26)

where 119860lowast119899119878(119905) = (119860lowast119899119909)(119905) 119909 isin 119878 119905 isin 119869

Set

119863120575 = int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

) 119889119904 119909 isin 119878

120575 isin (01

2)

(27)

Based on (H1) and (H2) we have

100381710038171003817100381710038171003817100381710038171003817

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119888(int

120575

0119904(1 minus 119904)

120573minus1119896 (119904) 119889119904

+int

1minus120575

1119904(1 minus 119904)

120573minus1119896 (119904) 119889119904) 119909 isin 119878 120575 isin (0

1

2)

(28)

where 119888 = 119902[1119899 119903 + (1119899)] Hence

119889119867 (119863120575 119860lowast119899119878) 997888rarr 0 120575 997888rarr 0

+ (29)

where 119889119867(sdot sdot) denotes the Hausdorff metrics which impliesthat

120572 (119860lowast119899119878) = lim

120575rarr0+120572 (119863120575) (30)

Since

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904

isin (1 minus 2120575)

times co 119866lowast (119905 119904) 119891 (119904 119904120573minus2

(119909 (119904)+119890

119899)) 119904 isin [120575 1 minus 120575]

(31)

by Lemmas 5 and 7 we have

120572 (119863120575)

= 120572(int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119909 isin 119878)

le (1 minus 2120575)

times 120572 (co 119866lowast (119905 119904)

times119891(119904 (119909 (119904)+119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le 120572(119866lowast(119905 119904)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

times 120572(119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)119871(120573 minus 1)

120573minus1

120573120573

120572 (119878 (119868120575))

le1

Γ (120573)2119871

(120573 minus 1)120573minus1

120573120573

120572119888 (119878)

(32)

where 119868120575 = [120575 1 minus 120575] By (H4) we can obtain

120572 (119860119899119878) = lim120575rarr0+

120572 (119863120575)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

2119871120572119888 (119878) lt 120572119888 (119878)

(33)

Consequently the operator119860119899 is a strict set contraction from119876119903 into 119876

Theorem 12 Let (H1)ndash(H6) hold Then there exists 119903 isin (0 119877)

such that the operator119860lowast119899 has two fixed points in119876119877119876119903 and119876

119876119877 respectively for arbitrary sufficiently large positive integer119899 where 119877 is given in (H2)

Journal of Function Spaces and Applications 5

Proof For any given 119905 isin (0 1) we have int10119866lowast(119905 119904)120593(119904)119889119904 le

int1

0119904(1 minus 119904)

120573minus1120593(119904)119889119904 Since int1

0119866lowast(119905 119904)120593(119904)119889119904 gt 0 there exists

a 1205761015840 gt 0 such that

1199031015840= int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 gt 0 (34)

Then by (H5) there exists 11990310158401015840 isin (0 119877) such that

120593lowast(119891 (119905 119905

120573minus2119909 (119905))) ge 120593 (119905) minus 120576

1015840 forall119905 isin (0 1) (35)

holds for 119909 isin 119876 and 119909119888 le 11990310158401015840 Letting 0 lt 119903 lt 119897 = min1199031015840

11990310158401015840 we prove that 119909 minus 119860

lowast119899119909 = 120582119890 for any 119909 isin 120597119876119903 120582 ge 0 as

119899 gt 1(119897 minus 119903) In fact suppose that this is false Then thereexist 120582 gt 0 119909 isin 120597119876119903 such that 119909minus119860

lowast119899119909 = 120582119890 It is obvious that

119909 (119905) = (119860lowast119899119909) (119905) + 120582119890 ge (119860

lowast119899119909) (119905)

= int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(36)

Thus

120593lowast(119909 (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904

(37)

This implies that

int

1

0120593lowast(119909 (119905)) 119889119905

ge int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 = 119903

1015840gt 119903

(38)

which is in contradiction with1003816100381610038161003816120593lowast(119909 (119905))

1003816100381610038161003816le 119909 (119905) le 119909119888 = 119903 119905 isin 119869 (39)

Combining with Lemma 6 we obtain that

119894 (119860lowast119899 119876119903 119876) = 0 (40)

Now we show that 119894(119860lowast119899 119876119877 119876) = 1 By Lemma 9 weneed only to prove that 119860lowast119899119909 = 120582119909 for 119909 isin 120597119876119877 and 120582 ge 1 Infact if there exist 119909 isin 120597119876119877 and some 120582 ge 1 such that 119860lowast119899119909 =

120582119909 then

119909 (119905) =1

120582int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(41)

Therefore by (H1) and (H2) we have

119877 = 119909119888

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

1003817100381710038171003817100381710038171003817119902 (119909 (119904) +

119890

119899)

1003817100381710038171003817100381710038171003817119889119904

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

times 119902 [(120573 minus 1) 119904(1 minus 119904)120573minus1

119877 119877 + 1] 119889119904

lt 119877

(42)

which is a contradiction Consequently

119894 (119860lowast119899 119876119877 119876) = 1 (43)

In the following we choose

1198771015840gt ((120573 minus 1) 119886 (1 minus 119887) int

119887

119886max119905isin119869

119866lowast(119905 119904) 119889119904)

minus1

(44)

By (H6) there exists119872 gt 119877 such that120593lowast(119891(119905 119905120573minus2119909)) ge 1198771015840119909

holds for 119909 gt 119872Let

119877 = max

119872+ 1

(120573 minus 1) 119886 (1 minus 119887)

1+1

(120573 minus 1) 1198771015840119886 (1 minus 119887)max119905isin119869 int

119887

119886119866lowast(119905 119904) 119889119904 minus 1

(45)

We now claim that 119909 minus 119860lowast119899119909 = 120582119890 for 119909 isin 120597119876119877 and 120582 ge 0 As a

matter of fact if this is not true then there exists 120582 gt 0 suchthat 119909 minus 119860

lowast119899119909 = 120582119890 that is 119909(119905) gt 119860

lowast119899119909(119905) which implies that

119877 ge 120593lowast(119909 (119905))

ge 120593lowast((119860lowast119899119909) (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) 119877

1015840((120573 minus 1) 119886 (1 minus 119887) (119909 minus 1)) 119889119904

ge (120573 minus 1) 119886 (1 minus 119887) 1198771015840(119877 minus 1)int

119887

119886119866lowast(119905 119904) 119889119904

gt 119877

(46)

which is a contradiction Therefore by Lemma 6 we canobtain that

119894 (119860lowast119899 119876119877 119876) = 0 (47)

6 Journal of Function Spaces and Applications

This together with (40) (43) and (47) implies that

119894 (119860lowast119899 119876119877 119876119903 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119903 119876) = 1

119894 (119860lowast119899 119876119877 119876119877 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119877 119876) = minus1

(48)

Thus there exist 119909 isin 119876119877 119876119903 and 119910 isin 119876119877 119876119877 such that119860lowast119899119909 = 119909 and 119860

lowast119899119910 = 119910 This completes the proof

Theorem 13 Let (H1)ndash(H6) be satisfied Then BVP (1) has atleast two positive solutions in 119862[119869E]

Proof FromTheorem 12 there exists an integer 1198990 gt 0 suchthat the operator119860lowast119899 119899 ge 1198990 has two fixed points in 119909119899 isin 119876119877

119876119903 and119910119899 isin 119876119876119877 Denote119863 = 119909119899 119899 ge 1198990 Obviously119863 isuniformly bounded Next we show that119863 is equicontinuous

Firstly we prove that

lim119905rarr0+

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (49)

lim119905rarr1minus

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (50)

On one hand by the absolute continuity of integration[19] for any 120576 gt 0 there exists 1205751 gt 0 such thatint119905

0(1Γ(120573))119904(1 minus 119904)

120573minus1119891(119904 (119909(119904) + 119890119899)119904

120573minus2)119889119904 lt 1205762 forall0 lt 119905 le

1205751On the other hand from (H2) we set

119872 = int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

(51)

and 120575 = min1205751 1205751120576119872 For 0 lt 119905 lt 120575 we have

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

= int

1205751

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1205751

119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

lt120576

2+ int

1

1205751

119905(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 (119909 (119904) +

119890

119899) 119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

le 120576

(52)

which implies that (49) holds Similarly one can prove that(50) holds

Next we show that119863 is equicontinuous for 119905 isin [120574 1 minus 120574]0 lt 120574 lt 12

From (H3) we define

= max119905isin[1205741minus120574]

119891(119905 (119909 (119905) +119890

119899) 119905120573minus2

) (53)

By the absolute continuity of integration for 120574 isin (0 12)

and the previous 120576 gt 0 there exists 0 lt 1205752 lt 120574 such that

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

lt120576

6 (54)

Since ((1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732 )Γ(120573))119891(119904 (119909(119904) +

119890119899)119904120573minus2

) is bounded for 119904 isin [1199051 1199052] there exists 1205753 gt 0 suchthat

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1199051

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205753

(55)

For |1199052 minus 1199051| lt 1205754 = 120574120576(6 int1

0(119904(1 minus 119904)

120573minus1Γ(120573))119896(119904)119902[(120573 minus

1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) one can see that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(56)

Similarly for |1199052 minus 1199051| lt 1205755 = 1205752120576(6 int1

0(119904(1 minus

119904)120573minus1

Γ(120573))119896(119904)119902[(120573 minus 1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) we have

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(57)

Since

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816

1

1205752(1 minus 1199051)120573minus1

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119904(1 minus 119904)120573minus1

119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

(58)

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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4 Journal of Function Spaces and Applications

In fact if (23) is not true then there exist a positive number1205760 and a subsequence 119909119898

119894

sub 119909119898 such that

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 119860lowast119899119909

10038171003817100381710038171003817119888ge 1205760 (24)

Since 119860lowast119899119909119898 is relatively compact there is a subsequence of119860lowast119899119909119898119894

which converges to some 119910 isin 119876 Without loss ofgenerality we assume that 119860lowast119899119909119898119894 itself converges to 119910 thatis

lim119894rarrinfin

10038171003817100381710038171003817119860lowast119899119909119898119894

minus 11991010038171003817100381710038171003817119888

= 0 (25)

By virtue of (22) and (25) we have 119910 = 119860lowast119899119909 which

contradicts with (24) Hence (23) holds and the continuityof 119860lowast119899 is proved

Lemma 11 Let (H1)ndash(H4) be satisfied Then for any 119903 gt 0 theoperator 119860lowast119899 is a strict set contraction from 119876119903 to 119876

Proof By virtue of Lemma 10 we know that 119860lowast119899119878 is boundedand equicontinuous on 119869 Thus from Lemma 5 one can seethat

120572119888 (119860lowast119899119878) = sup

119905isin119869

120572 ((119860lowast119899119878) (119905)) 119903 gt 0 119878 sube 119876119903 (26)

where 119860lowast119899119878(119905) = (119860lowast119899119909)(119905) 119909 isin 119878 119905 isin 119869

Set

119863120575 = int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

) 119889119904 119909 isin 119878

120575 isin (01

2)

(27)

Based on (H1) and (H2) we have

100381710038171003817100381710038171003817100381710038171003817

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381710038171003817100381710038171003817100381710038171003817

le 119888(int

120575

0119904(1 minus 119904)

120573minus1119896 (119904) 119889119904

+int

1minus120575

1119904(1 minus 119904)

120573minus1119896 (119904) 119889119904) 119909 isin 119878 120575 isin (0

1

2)

(28)

where 119888 = 119902[1119899 119903 + (1119899)] Hence

119889119867 (119863120575 119860lowast119899119878) 997888rarr 0 120575 997888rarr 0

+ (29)

where 119889119867(sdot sdot) denotes the Hausdorff metrics which impliesthat

120572 (119860lowast119899119878) = lim

120575rarr0+120572 (119863120575) (30)

Since

int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904

isin (1 minus 2120575)

times co 119866lowast (119905 119904) 119891 (119904 119904120573minus2

(119909 (119904)+119890

119899)) 119904 isin [120575 1 minus 120575]

(31)

by Lemmas 5 and 7 we have

120572 (119863120575)

= 120572(int

1minus120575

120575119866lowast(119905 119904) 119891 (119904 119904

120573minus2(119909 (119904) +

119890

119899)) 119889119904 119909 isin 119878)

le (1 minus 2120575)

times 120572 (co 119866lowast (119905 119904)

times119891(119904 (119909 (119904)+119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le 120572(119866lowast(119905 119904)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

times 120572(119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

) 119904 isin [120575 1 minus 120575] 119909 isin 119878)

le1

Γ (120573)119871(120573 minus 1)

120573minus1

120573120573

120572 (119878 (119868120575))

le1

Γ (120573)2119871

(120573 minus 1)120573minus1

120573120573

120572119888 (119878)

(32)

where 119868120575 = [120575 1 minus 120575] By (H4) we can obtain

120572 (119860119899119878) = lim120575rarr0+

120572 (119863120575)

le1

Γ (120573)

(120573 minus 1)120573minus1

120573120573

2119871120572119888 (119878) lt 120572119888 (119878)

(33)

Consequently the operator119860119899 is a strict set contraction from119876119903 into 119876

Theorem 12 Let (H1)ndash(H6) hold Then there exists 119903 isin (0 119877)

such that the operator119860lowast119899 has two fixed points in119876119877119876119903 and119876

119876119877 respectively for arbitrary sufficiently large positive integer119899 where 119877 is given in (H2)

Journal of Function Spaces and Applications 5

Proof For any given 119905 isin (0 1) we have int10119866lowast(119905 119904)120593(119904)119889119904 le

int1

0119904(1 minus 119904)

120573minus1120593(119904)119889119904 Since int1

0119866lowast(119905 119904)120593(119904)119889119904 gt 0 there exists

a 1205761015840 gt 0 such that

1199031015840= int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 gt 0 (34)

Then by (H5) there exists 11990310158401015840 isin (0 119877) such that

120593lowast(119891 (119905 119905

120573minus2119909 (119905))) ge 120593 (119905) minus 120576

1015840 forall119905 isin (0 1) (35)

holds for 119909 isin 119876 and 119909119888 le 11990310158401015840 Letting 0 lt 119903 lt 119897 = min1199031015840

11990310158401015840 we prove that 119909 minus 119860

lowast119899119909 = 120582119890 for any 119909 isin 120597119876119903 120582 ge 0 as

119899 gt 1(119897 minus 119903) In fact suppose that this is false Then thereexist 120582 gt 0 119909 isin 120597119876119903 such that 119909minus119860

lowast119899119909 = 120582119890 It is obvious that

119909 (119905) = (119860lowast119899119909) (119905) + 120582119890 ge (119860

lowast119899119909) (119905)

= int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(36)

Thus

120593lowast(119909 (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904

(37)

This implies that

int

1

0120593lowast(119909 (119905)) 119889119905

ge int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 = 119903

1015840gt 119903

(38)

which is in contradiction with1003816100381610038161003816120593lowast(119909 (119905))

1003816100381610038161003816le 119909 (119905) le 119909119888 = 119903 119905 isin 119869 (39)

Combining with Lemma 6 we obtain that

119894 (119860lowast119899 119876119903 119876) = 0 (40)

Now we show that 119894(119860lowast119899 119876119877 119876) = 1 By Lemma 9 weneed only to prove that 119860lowast119899119909 = 120582119909 for 119909 isin 120597119876119877 and 120582 ge 1 Infact if there exist 119909 isin 120597119876119877 and some 120582 ge 1 such that 119860lowast119899119909 =

120582119909 then

119909 (119905) =1

120582int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(41)

Therefore by (H1) and (H2) we have

119877 = 119909119888

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

1003817100381710038171003817100381710038171003817119902 (119909 (119904) +

119890

119899)

1003817100381710038171003817100381710038171003817119889119904

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

times 119902 [(120573 minus 1) 119904(1 minus 119904)120573minus1

119877 119877 + 1] 119889119904

lt 119877

(42)

which is a contradiction Consequently

119894 (119860lowast119899 119876119877 119876) = 1 (43)

In the following we choose

1198771015840gt ((120573 minus 1) 119886 (1 minus 119887) int

119887

119886max119905isin119869

119866lowast(119905 119904) 119889119904)

minus1

(44)

By (H6) there exists119872 gt 119877 such that120593lowast(119891(119905 119905120573minus2119909)) ge 1198771015840119909

holds for 119909 gt 119872Let

119877 = max

119872+ 1

(120573 minus 1) 119886 (1 minus 119887)

1+1

(120573 minus 1) 1198771015840119886 (1 minus 119887)max119905isin119869 int

119887

119886119866lowast(119905 119904) 119889119904 minus 1

(45)

We now claim that 119909 minus 119860lowast119899119909 = 120582119890 for 119909 isin 120597119876119877 and 120582 ge 0 As a

matter of fact if this is not true then there exists 120582 gt 0 suchthat 119909 minus 119860

lowast119899119909 = 120582119890 that is 119909(119905) gt 119860

lowast119899119909(119905) which implies that

119877 ge 120593lowast(119909 (119905))

ge 120593lowast((119860lowast119899119909) (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) 119877

1015840((120573 minus 1) 119886 (1 minus 119887) (119909 minus 1)) 119889119904

ge (120573 minus 1) 119886 (1 minus 119887) 1198771015840(119877 minus 1)int

119887

119886119866lowast(119905 119904) 119889119904

gt 119877

(46)

which is a contradiction Therefore by Lemma 6 we canobtain that

119894 (119860lowast119899 119876119877 119876) = 0 (47)

6 Journal of Function Spaces and Applications

This together with (40) (43) and (47) implies that

119894 (119860lowast119899 119876119877 119876119903 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119903 119876) = 1

119894 (119860lowast119899 119876119877 119876119877 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119877 119876) = minus1

(48)

Thus there exist 119909 isin 119876119877 119876119903 and 119910 isin 119876119877 119876119877 such that119860lowast119899119909 = 119909 and 119860

lowast119899119910 = 119910 This completes the proof

Theorem 13 Let (H1)ndash(H6) be satisfied Then BVP (1) has atleast two positive solutions in 119862[119869E]

Proof FromTheorem 12 there exists an integer 1198990 gt 0 suchthat the operator119860lowast119899 119899 ge 1198990 has two fixed points in 119909119899 isin 119876119877

119876119903 and119910119899 isin 119876119876119877 Denote119863 = 119909119899 119899 ge 1198990 Obviously119863 isuniformly bounded Next we show that119863 is equicontinuous

Firstly we prove that

lim119905rarr0+

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (49)

lim119905rarr1minus

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (50)

On one hand by the absolute continuity of integration[19] for any 120576 gt 0 there exists 1205751 gt 0 such thatint119905

0(1Γ(120573))119904(1 minus 119904)

120573minus1119891(119904 (119909(119904) + 119890119899)119904

120573minus2)119889119904 lt 1205762 forall0 lt 119905 le

1205751On the other hand from (H2) we set

119872 = int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

(51)

and 120575 = min1205751 1205751120576119872 For 0 lt 119905 lt 120575 we have

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

= int

1205751

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1205751

119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

lt120576

2+ int

1

1205751

119905(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 (119909 (119904) +

119890

119899) 119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

le 120576

(52)

which implies that (49) holds Similarly one can prove that(50) holds

Next we show that119863 is equicontinuous for 119905 isin [120574 1 minus 120574]0 lt 120574 lt 12

From (H3) we define

= max119905isin[1205741minus120574]

119891(119905 (119909 (119905) +119890

119899) 119905120573minus2

) (53)

By the absolute continuity of integration for 120574 isin (0 12)

and the previous 120576 gt 0 there exists 0 lt 1205752 lt 120574 such that

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

lt120576

6 (54)

Since ((1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732 )Γ(120573))119891(119904 (119909(119904) +

119890119899)119904120573minus2

) is bounded for 119904 isin [1199051 1199052] there exists 1205753 gt 0 suchthat

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1199051

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205753

(55)

For |1199052 minus 1199051| lt 1205754 = 120574120576(6 int1

0(119904(1 minus 119904)

120573minus1Γ(120573))119896(119904)119902[(120573 minus

1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) one can see that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(56)

Similarly for |1199052 minus 1199051| lt 1205755 = 1205752120576(6 int1

0(119904(1 minus

119904)120573minus1

Γ(120573))119896(119904)119902[(120573 minus 1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) we have

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(57)

Since

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816

1

1205752(1 minus 1199051)120573minus1

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119904(1 minus 119904)120573minus1

119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

(58)

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

Proof For any given 119905 isin (0 1) we have int10119866lowast(119905 119904)120593(119904)119889119904 le

int1

0119904(1 minus 119904)

120573minus1120593(119904)119889119904 Since int1

0119866lowast(119905 119904)120593(119904)119889119904 gt 0 there exists

a 1205761015840 gt 0 such that

1199031015840= int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 gt 0 (34)

Then by (H5) there exists 11990310158401015840 isin (0 119877) such that

120593lowast(119891 (119905 119905

120573minus2119909 (119905))) ge 120593 (119905) minus 120576

1015840 forall119905 isin (0 1) (35)

holds for 119909 isin 119876 and 119909119888 le 11990310158401015840 Letting 0 lt 119903 lt 119897 = min1199031015840

11990310158401015840 we prove that 119909 minus 119860

lowast119899119909 = 120582119890 for any 119909 isin 120597119876119903 120582 ge 0 as

119899 gt 1(119897 minus 119903) In fact suppose that this is false Then thereexist 120582 gt 0 119909 isin 120597119876119903 such that 119909minus119860

lowast119899119909 = 120582119890 It is obvious that

119909 (119905) = (119860lowast119899119909) (119905) + 120582119890 ge (119860

lowast119899119909) (119905)

= int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(36)

Thus

120593lowast(119909 (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904

(37)

This implies that

int

1

0120593lowast(119909 (119905)) 119889119905

ge int

1

0int

1

0119866lowast(119905 119904) (120593 (119904) minus 120576

1015840) 119889119904 119889119905 = 119903

1015840gt 119903

(38)

which is in contradiction with1003816100381610038161003816120593lowast(119909 (119905))

1003816100381610038161003816le 119909 (119905) le 119909119888 = 119903 119905 isin 119869 (39)

Combining with Lemma 6 we obtain that

119894 (119860lowast119899 119876119903 119876) = 0 (40)

Now we show that 119894(119860lowast119899 119876119877 119876) = 1 By Lemma 9 weneed only to prove that 119860lowast119899119909 = 120582119909 for 119909 isin 120597119876119877 and 120582 ge 1 Infact if there exist 119909 isin 120597119876119877 and some 120582 ge 1 such that 119860lowast119899119909 =

120582119909 then

119909 (119905) =1

120582int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

(41)

Therefore by (H1) and (H2) we have

119877 = 119909119888

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

1003817100381710038171003817100381710038171003817119902 (119909 (119904) +

119890

119899)

1003817100381710038171003817100381710038171003817119889119904

lt1

Γ (120573)int

1

0119904(1 minus 119904)

120573minus1119896 (119904)

times 119902 [(120573 minus 1) 119904(1 minus 119904)120573minus1

119877 119877 + 1] 119889119904

lt 119877

(42)

which is a contradiction Consequently

119894 (119860lowast119899 119876119877 119876) = 1 (43)

In the following we choose

1198771015840gt ((120573 minus 1) 119886 (1 minus 119887) int

119887

119886max119905isin119869

119866lowast(119905 119904) 119889119904)

minus1

(44)

By (H6) there exists119872 gt 119877 such that120593lowast(119891(119905 119905120573minus2119909)) ge 1198771015840119909

holds for 119909 gt 119872Let

119877 = max

119872+ 1

(120573 minus 1) 119886 (1 minus 119887)

1+1

(120573 minus 1) 1198771015840119886 (1 minus 119887)max119905isin119869 int

119887

119886119866lowast(119905 119904) 119889119904 minus 1

(45)

We now claim that 119909 minus 119860lowast119899119909 = 120582119890 for 119909 isin 120597119876119877 and 120582 ge 0 As a

matter of fact if this is not true then there exists 120582 gt 0 suchthat 119909 minus 119860

lowast119899119909 = 120582119890 that is 119909(119905) gt 119860

lowast119899119909(119905) which implies that

119877 ge 120593lowast(119909 (119905))

ge 120593lowast((119860lowast119899119909) (119905))

ge int

1

0119866lowast(119905 119904) 120593

lowast(119891(119904 (119909 (119904) +

119890

119899) 119904120573minus2

))119889119904

ge int

1

0119866lowast(119905 119904) 119877

1015840((120573 minus 1) 119886 (1 minus 119887) (119909 minus 1)) 119889119904

ge (120573 minus 1) 119886 (1 minus 119887) 1198771015840(119877 minus 1)int

119887

119886119866lowast(119905 119904) 119889119904

gt 119877

(46)

which is a contradiction Therefore by Lemma 6 we canobtain that

119894 (119860lowast119899 119876119877 119876) = 0 (47)

6 Journal of Function Spaces and Applications

This together with (40) (43) and (47) implies that

119894 (119860lowast119899 119876119877 119876119903 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119903 119876) = 1

119894 (119860lowast119899 119876119877 119876119877 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119877 119876) = minus1

(48)

Thus there exist 119909 isin 119876119877 119876119903 and 119910 isin 119876119877 119876119877 such that119860lowast119899119909 = 119909 and 119860

lowast119899119910 = 119910 This completes the proof

Theorem 13 Let (H1)ndash(H6) be satisfied Then BVP (1) has atleast two positive solutions in 119862[119869E]

Proof FromTheorem 12 there exists an integer 1198990 gt 0 suchthat the operator119860lowast119899 119899 ge 1198990 has two fixed points in 119909119899 isin 119876119877

119876119903 and119910119899 isin 119876119876119877 Denote119863 = 119909119899 119899 ge 1198990 Obviously119863 isuniformly bounded Next we show that119863 is equicontinuous

Firstly we prove that

lim119905rarr0+

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (49)

lim119905rarr1minus

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (50)

On one hand by the absolute continuity of integration[19] for any 120576 gt 0 there exists 1205751 gt 0 such thatint119905

0(1Γ(120573))119904(1 minus 119904)

120573minus1119891(119904 (119909(119904) + 119890119899)119904

120573minus2)119889119904 lt 1205762 forall0 lt 119905 le

1205751On the other hand from (H2) we set

119872 = int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

(51)

and 120575 = min1205751 1205751120576119872 For 0 lt 119905 lt 120575 we have

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

= int

1205751

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1205751

119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

lt120576

2+ int

1

1205751

119905(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 (119909 (119904) +

119890

119899) 119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

le 120576

(52)

which implies that (49) holds Similarly one can prove that(50) holds

Next we show that119863 is equicontinuous for 119905 isin [120574 1 minus 120574]0 lt 120574 lt 12

From (H3) we define

= max119905isin[1205741minus120574]

119891(119905 (119909 (119905) +119890

119899) 119905120573minus2

) (53)

By the absolute continuity of integration for 120574 isin (0 12)

and the previous 120576 gt 0 there exists 0 lt 1205752 lt 120574 such that

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

lt120576

6 (54)

Since ((1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732 )Γ(120573))119891(119904 (119909(119904) +

119890119899)119904120573minus2

) is bounded for 119904 isin [1199051 1199052] there exists 1205753 gt 0 suchthat

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1199051

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205753

(55)

For |1199052 minus 1199051| lt 1205754 = 120574120576(6 int1

0(119904(1 minus 119904)

120573minus1Γ(120573))119896(119904)119902[(120573 minus

1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) one can see that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(56)

Similarly for |1199052 minus 1199051| lt 1205755 = 1205752120576(6 int1

0(119904(1 minus

119904)120573minus1

Γ(120573))119896(119904)119902[(120573 minus 1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) we have

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(57)

Since

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816

1

1205752(1 minus 1199051)120573minus1

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119904(1 minus 119904)120573minus1

119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

(58)

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces and Applications

This together with (40) (43) and (47) implies that

119894 (119860lowast119899 119876119877 119876119903 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119903 119876) = 1

119894 (119860lowast119899 119876119877 119876119877 119876) = 119894 (119860

lowast119899 119876119877 119876) minus 119894 (119860

lowast119899 119876119877 119876) = minus1

(48)

Thus there exist 119909 isin 119876119877 119876119903 and 119910 isin 119876119877 119876119877 such that119860lowast119899119909 = 119909 and 119860

lowast119899119910 = 119910 This completes the proof

Theorem 13 Let (H1)ndash(H6) be satisfied Then BVP (1) has atleast two positive solutions in 119862[119869E]

Proof FromTheorem 12 there exists an integer 1198990 gt 0 suchthat the operator119860lowast119899 119899 ge 1198990 has two fixed points in 119909119899 isin 119876119877

119876119903 and119910119899 isin 119876119876119877 Denote119863 = 119909119899 119899 ge 1198990 Obviously119863 isuniformly bounded Next we show that119863 is equicontinuous

Firstly we prove that

lim119905rarr0+

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (49)

lim119905rarr1minus

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904 = 0 (50)

On one hand by the absolute continuity of integration[19] for any 120576 gt 0 there exists 1205751 gt 0 such thatint119905

0(1Γ(120573))119904(1 minus 119904)

120573minus1119891(119904 (119909(119904) + 119890119899)119904

120573minus2)119889119904 lt 1205762 forall0 lt 119905 le

1205751On the other hand from (H2) we set

119872 = int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

(51)

and 120575 = min1205751 1205751120576119872 For 0 lt 119905 lt 120575 we have

int

1

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

= int

1205751

0119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1205751

119866lowast(119905 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

lt120576

2+ int

1

1205751

119905(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 (119909 (119904) +

119890

119899) 119889119904

le120576

2+

119905

1205751

int

1

1205751

119904(1 minus 119904)120573minus1

Γ (120573)119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119903 119877] 119889119904

le 120576

(52)

which implies that (49) holds Similarly one can prove that(50) holds

Next we show that119863 is equicontinuous for 119905 isin [120574 1 minus 120574]0 lt 120574 lt 12

From (H3) we define

= max119905isin[1205741minus120574]

119891(119905 (119909 (119905) +119890

119899) 119905120573minus2

) (53)

By the absolute continuity of integration for 120574 isin (0 12)

and the previous 120576 gt 0 there exists 0 lt 1205752 lt 120574 such that

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

lt120576

6 (54)

Since ((1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732 )Γ(120573))119891(119904 (119909(119904) +

119890119899)119904120573minus2

) is bounded for 119904 isin [1199051 1199052] there exists 1205753 gt 0 suchthat

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1199051

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205753

(55)

For |1199052 minus 1199051| lt 1205754 = 120574120576(6 int1

0(119904(1 minus 119904)

120573minus1Γ(120573))119896(119904)119902[(120573 minus

1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) one can see that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(56)

Similarly for |1199052 minus 1199051| lt 1205755 = 1205752120576(6 int1

0(119904(1 minus

119904)120573minus1

Γ(120573))119896(119904)119902[(120573 minus 1)119904(1 minus 119904)120573minus1

119903 119877]119889119904) we have

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

(57)

Since

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816

1

1205752(1 minus 1199051)120573minus1

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119904(1 minus 119904)120573minus1

119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

(58)

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 7

119891(119905 (119909(119905)+ 119890119899)119905120573minus2

) is bounded on 119905 isin [1205752 1199051] and (1199052minus1205732 (1199052 minus

119904)120573minus1

minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)Γ(120573) rarr 0 as 1199051 rarr 1199052 there exists

1205756 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6

10038161003816100381610038161199052 minus 1199051

1003816100381610038161003816lt 1205756

(59)

We choose 1205751015840 = min120575119894 119894 = 2 3 4 5 6 For |1199052minus1199051| lt 1205751015840

we have

1003816100381610038161003816119909 (1199052) minus 119909 (1199051)

1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

1

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

+ int

1

1199052

119866lowast(1199052 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minus int

1199051

1205752

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

minusint

1

1199051

119866lowast(1199051 119904) 119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

le 2

100381610038161003816100381610038161003816100381610038161003816

int

1205752

0

119904(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199052

1205752

1199052(1 minus 119904)120573minus1

minus (1199052 minus 119904)120573minus1

1199052minus1205732

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

1199051(1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1199052

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052 minus 1199051) (1 minus 119904)120573minus1

Γ (120573)119891 (119904 (119909 (119904) +

119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int

1199051

1205752

(1199052minus1205732 (1199052 minus 119904)

120573minus1minus 1199052minus1205731 (1199051 minus 119904)

120573minus1)

Γ (120573)

times 119891(119904 (119909 (119904) +119890

119899) 119904120573minus2

)119889119904

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt 120576

(60)

This together with (49) and (50) implies that119863 is equicontin-uous for 119905 isin [0 1]

Now we show that 119863(119905) is relatively compact ByLemma 7 we have

120572 (119863 (119905))

le 120572(int

1

0119866lowast(119905 119904) 119891 (119904 (119909119899 (119904) +

119890

119899) 119904120573minus2

119889119904) 119899 ge 1198990)

le 2int

1

0119866lowast(119905 119904)

times 120572 (119891(119904 (119909119899 (119904) +119890

119899) 119904120573minus2

) 119899 ge 1198990)119889119904

le 2119871int

1

0119866lowast(119905 119904) 120572 (119863 (119904)) 119889119904

(61)

which together with Lemma 5 and (H4) implies that

120572119888 (119863) le 21198711

Γ (120573)(int

1

0119904(1 minus 119904)

120573minus1119889119904) 120572119888 (119863) (62)

Thus 120572119888(119863) = 0 It follows from Lemma 8 that there is aconvergent subsequence of 119909119899 Without loss of generalitywe assume that 119909119899 itself converges to some 119909 isin 119876Then thedominated convergence theorem and (16) imply that

119909 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119909 (119904)) 119889119904 (63)

Thus the singular BVP (1) has a positive solution 119905120573minus2

119909(119905)Similarly 119910119899 has a convergent subsequence which con-verges to119910(119905)Then 119905120573minus2119910(119905) is also a positive solution to BVP(1)

Finally we show 119909 = 119910 We only need to prove that theoperator 119860lowast has no fixed point in 120597119876119877

In fact if it is not true then we assume that 119911 is a fixedpoint of the operator 119860lowast in 120597119876119877 Then

119911 (119905) = int

1

0119866lowast(119905 119904) 119891 (119904 119904

120573minus2119911 (119904)) 119889119904 (64)

with (120573 minus 1)119905(1 minus 119905)120573minus1

119877 le 119911(119905) le 119877 + 1

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces and Applications

By (H1) and (H2) one can see that

119877 = max119905isin119869

119911 (119905)

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877] 119889119904

le int

1

0119904(1 minus 119904)

120573minus1119896 (119904) 119902 [(120573 minus 1) 119904(1 minus 119904)

120573minus1119877 119877 + 1] 119889119904

lt 119877

(65)

which is a contradiction The proof of this theorem iscompleted

Remark 14 It is noted that the singularity of119891(119905 119909) at 119905 = 0 1

is overcome by (H2) while the singularity at 119909 = 120579 is handledby solving the approximate problem and using the sequential-based technique

4 Example

Consider the following BVP

minus 119863320+

119909119899 (119905)

=1

2120587radic119905 (1 minus 119905)

times (1

1003816100381610038161003816sum119898119894=1 119909119894 (119905)

100381610038161003816100381611990512

+ radic119905119909119899+1 (119905) + 1199051199092119899+2 (119905))

119905 isin (0 1)

119909119899 (0) = 119909119899 (1) = 0 119899 = 1 2 119898

(66)

where 119909119898+119899 = 119909119899 Then BVP (66) has at least two positivesolutions

Proof We consider the problem (66) in 119898-dimensionalEuclidean space 119877

119898= 119909 = (1199091 1199092 119909119898) 119909119899 isin 119877 119899 =

1 2 119898 with the norm 119909 = sum119898119899=1 |119909119899| Then BVP (66)

has the form of (1) with

119909 (119905) = (1199091 (119905) 1199092 (119905) 119909119898 (119905))

119891119899 (119905 119909) =1

2120587radic119905 (1 minus 119905)

(1

1003816100381610038161003816sum119898119894=1 119909119894

100381610038161003816100381611990512

+ radic119905119909119899+1 + 1199051199092119899+2)

(67)

It is easy to see that119891(119905 119909) is singular at 119905 = 0 1 and 119909 = 120579Set 119896(119905) = 1(2120587radic119905(1 minus 119905)) and 119902(119909) = (1| sum

119898119894=1 119909119894|) + 119909119899+1 +

1199092119899+2 One can easily see that (H1) holdsWe choose 119877 = 1 and 120593

lowast= 120601lowast

= (1 1 1)Considering that

int

1

0

radic119904 (1 minus 119904)119889119904 =120587

8 (68)

we can easily check that (H2)ndash(H6) hold FromTheorem 13 BVP (66) has at least two positive solutions119909 = (1199091 1199092 119909119898) and (1199101 1199102 119910119898) which satisfy0 lt 119909 lt 1 lt 119910 lt +infin

Acknowledgments

The authors wish to thank referees and Dr Haitao Li fortheir valuable suggestions The project was supported by theNational Natural Science Foundation of China (11171192)Graduate Educational Innovation Foundation of ShandongProvince (SDYY1005) and the Promotive Research Fund forExcellent Young and Middle-Aged Scientists of ShandongProvince (BS2010SF025)

References

[1] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems Irdquo Applicable Analysisvol 78 no 1-2 pp 153ndash192 2001

[2] A A Kilbas and J J Trujillo ldquoDifferential equations of fraction-al order methods results and problems IIrdquoApplicable Analysisvol 81 no 2 pp 435ndash493 2002

[3] I Podlubny Fractional Differential Equations Mathematics inScience and Engineering Academic Press New York NY USA1999

[4] R P Agarwal and D OrsquoRegan ldquoSingular boundary value prob-lemsrdquo Nonlinear Analysis Theory Methods amp Applications Avol 27 pp 645ndash656 1996

[5] R P Agarwal and D OrsquoRegan ldquoNonlinear superlinear singularand nonsingular second order boundary value problemsrdquoJournal of Differential Equations vol 143 no 1 pp 60ndash95 1998

[6] H H Alsulami S K Ntouyas and B Ahmad ldquoExistence resultsfor a Riemann-Liouvilletype fractional multivalued problemwith integral boundary conditionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 676045 7 pages 2013

[7] J Jiang L Liu and Y Wu ldquoPositive solutions to singular frac-tional differential system with coupled boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 18 no 11 pp 3061ndash3074 2013

[8] H Li X Kong and C Yu ldquoExistence of three non-negativesolutions for a three-point boundary-value problem of non-linear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 88 pp 12ndash1 2012

[9] A Shi and S Zhang ldquoUpper and lower solutions method and afractional differential equation boundary value problemrdquo Elec-tronic Journal ofQualitativeTheory ofDifferential Equations vol30 pp 1ndash13 2009

[10] Z L Wei ldquoPositive solutions of singular boundary value prob-lems for negative exponent Emden-Fowler equationsrdquo ActaMathematica Sinica Chinese Series vol 41 no 3 pp 655ndash6621998 (Chinese)

[11] S Zhang ldquoPositive solutions for boundary-value problems ofnonlinear fractional differential equationsrdquo Electronic Journal ofDifferential Equations vol 36 pp 1ndash12 2006

[12] R P Agarwal D OrsquoRegan and S Stanek ldquoPositive solutions forDirichlet problems of singular nonlinear fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 371 no 1 pp 57ndash68 2010

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 9

[13] X Zhang L Liu and Y Wu ldquoMultiple positive solutionsof a singular fractional differential equation with negativelyperturbed termrdquo Mathematical and Computer Modelling vol55 no 3-4 pp 1263ndash1274 2012

[14] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[15] S G Samko A A Kilbas and O I Marichev Fractional In-tegrals and Derivatives (Theory and Applications) Gordon andBreach Yverdon Switzerland 1993

[16] D Guo V Lakshmikantham and X Liu Nonlinear IntegralEquations in Abstract Spaces vol 373 Kluwer Academic Pub-lishers Dordrecht The Netherlands 1996

[17] D Guo Nonlinera Function Analysis Shandong Science andTechnology Publishing House Jinan China 1985

[18] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[19] D Jiang and C Yuan ldquoThe positive properties of the Greenfunction for Dirichlet-type boundary value problems of non-linear fractional differential equations and its applicationrdquoNonlinear Analysis Theory Methods amp Applications A vol 72no 2 pp 710ndash719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of