Zhang and Zhong Boundary Value Problems (2016) 2016:65 DOI 10.1186/s13661-016-0572-0

RESEARCH

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Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations Xingqiu Zhang1,2 and Qiuyan Zhong1* *

Correspondence: [email protected] 1 Department of Information Engineering, Jining Medical College, Jining, Shandong 272067, P.R. China Full list of author information is available at the end of the article

Abstract We consider the existence of multiple positive solutions for the following nonlinear fractional differential equations of nonlocal boundary value problems: 

Dα0+ u(t) + f (t, u(t)) = 0, 0 < t < 1,  β β β D0+ u(1) = ∞ u(0) = 0, D0+ u(0) = 0, i=1 ξi D0+ u(ηi ),

 α –β –1 < 1. Existence where 2 < α ≤ 3, 1 ≤ β ≤ 2, α – β ≥ 1, 0 < ξi , ηi < 1 with ∞ i=1 ξi ηi result of at least two positive solutions is given via fixed point theorem on cones. The nonlinearity f may be singular both on the time and the space variables. MSC: 26A33; 34B15; 34B18 Keywords: fractional differential equations; nonlocal boundary value problem; singularity; multiple positive solutions

1 Introduction The purpose of this paper is to investigate the multiplicity of positive solutions for the following nonlocal boundary value problems of singular fractional differential equations: 

Dα+ u(t) + f (t, u(t)) = ,  < t < ,  β β β D+ u() = ∞ u() = , D+ u() = , i= ξi D+ u(ηi ),

()

 α–β– where  < α ≤ ,  ≤ β ≤ , α – β ≥ ,  < ξi , ηi <  with ∞ < , f ∈ C(J × i= ξi ηi α R++ , R+ ), J = (, ), R+ = [, +∞), R++ = (, +∞), D+ is the standard Riemann-Liouville’s fractional derivative of order α. The nonlinearity f permits singularities at t = ,  and u = . A function u ∈ C[, ] is said to be a positive solution of BVP () if u(t) >  on (, ) and u satisfies () on [, ]. Recently, much attention has been paid on the study of nonlocal boundary value problems of fractional differential equations; see [–] and [–]. By virtue of the contraction map principle and the fixed point index theory, Bai [] investigated the existence and uniqueness of positive solutions for the following fractional differential equation:   Dα+ u(t) + f t, u(t) = ,

 < t < ,  < α ≤ ,

(A)

© 2016 Zhang and Zhong. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Zhang and Zhong Boundary Value Problems (2016) 2016:65

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subject to three point boundary value conditions u() = , u() = μu(ξ ), where  < μξ α– < ,  ≤ μ ≤ ,  < ξ < , f is continuous on [, ] × [, +∞). When f : [, ] × [, +∞) → [, +∞) satisfies Carathéodory type conditions, by using some fixed point theorems, like Leray-Schauder nonlinear alternative and a mixed monotone method, Li et al. [], Xu et al. [, ] obtained the existence and multiplicity results of positive solutions for the fractional differential equation (A) with fractional derivative in boundary conditions u() = ,

β

β

D+ u() = μD+ u(ξ ),

 ≤ β ≤ .

In , Lv [] and Yang et al. [] discussed the existence of minimal and maximal and the uniqueness of positive solutions for fractional differential equation (A) under multi-point boundary value conditions,

u() = ,

β

D+ u() =

m– 

β

ξi D+ u(ηi ),

 ≤ β ≤ .

i=

Motivated by the above papers, when  < α ≤  and f is continuous, Li et al. [] obtained the existence results of at least one and unique solutions for fractional differential equation (A) subject to more general multi-point boundary value conditions

u() = ,

β

D+ u() = ,

β

D+ u() =

m– 

β

ξi D+ u(ηi ).

i=

The tools to obtain the main results are the nonlinear alternative of the Leray-Schauder and the Banach contraction mapping principle. Compared with the existing literature, this paper has the following two new features. First, different from [], infinite-point boundary value conditions are considered in this paper. At the same time, the nonlinearity f in this paper permits singularities with respect to both the time and the space variables which is seldom considered at present. Second, the purpose of this paper is to investigate the existence of multiple positive solutions for BVP (). As to multiple positive solutions, it is worth pointing out that conditions imposed on f are different from that in []. To achieve this goal, first we convert the expression of the unique solution into an integral form and then get the Green function BVP (). After further discussion of the properties of the Green function, a suitable cone is constructed to obtain the main result in this paper by means of the Guo-Krasnoselskii fixed point theorem.

2 Preliminaries and several lemmas Definitions and useful lemmas from fractional calculus theory can be found in the recent literature [–], we omit them here.

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Lemma  ([]) Assume that u ∈ C(, ) ∩ L(, ) with a fractional derivative of order α >  that belongs to C(, ) ∩ L(, ). Then α Dα+ u(t) = u(t) + C t α– + C t α– + · · · + CN t α–N I+

for some Ci ∈ R, i = , , . . . , N , where N is the smallest integer greater than or equal to α. Lemma  Let y ∈ L [, ] and  < α ≤ , the unique solution of 

Dα+ u(t) + y(t) = , β u() = , D+ u() = ,

∞

()

p(s)( – s)α–β– t α– – p()(t – s)α– ,  ≤ s ≤ t ≤ ,  ≤ t ≤ s ≤ , p(s)( – s)α–β– t α– ,

()

β

D+ u() =

β i= ξi D+ u(ηi ),

is 



G(t, s)y(s) ds,

u(t) = 

where  G(t, s) = p()(α) here p(s) =  –





ηi –s α–β– . s≤ηi ξi ( –s )

Proof By Lemma , we get u(t) = –Iα+ y(t) + C t α– + C t α– + C t α– .

()

It follows from the condition u() =  that C = . Considering the relation Dα+ t γ = (γ +) γ –α t , we have (γ –α+)  β β D+ u(t) = D+ –

t 

 (t – s)α– y(s) ds (α)



(α) α–β– (α – ) α–β– t t + C . + C (α – β) (α – β – ) Since  < α ≤ ,  ≤ β ≤ , and α ≥ β + , we have – ≤ α – β –  ≤ . Thus, C = . As  β β deduced by the boundary value condition D+ u() = ∞ i= ξi D+ u(ηi ), we have –

 (α – β) =–

∞  i=





( – s)α–β– y(s) ds + C 

 ξi (α – β)



ηi

(α) (α – β) ∞ 

(ηi – s)α–β– y(s) ds + C



i=

ξi

(α) α–β– η , (α – β) i

which implies that 



C = 

 = 



∞

 ( – s)α–β– ξi y(s) ds – ∞ α–β– (α)( – i= ξi ηi ) i= ( – s)α–β– p(s) y(s) ds, (α)p()

 

ηi

(ηi – s)α–β– y(s) ds  α–β– (α)( – ∞ ) i= ξi ηi

Zhang and Zhong Boundary Value Problems (2016) 2016:65

where p(s) =  – 



t

 t

p(s)( – s)

= 



+  =

t

Thus, we have

 (t – s)α– y(s) ds + (α)

u(t) = – 

ηi –s α–β– . s≤ηi ξi ( –s )

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







( – s)α–β– p(s) y(s) ds · t α– (α)p()

α–β– α–

t – p()(t – s)α– y(s) ds (α)p()

p(s)( – s)α–β– t α– y(s) ds (α)p()



G(t, s) ds. 



Lemma  The function p(s) > , s ∈ [, ], and p is nondecreasing on [, ]. Proof By direct computation, we have p (s) =



ξi (α – β – )(ηi – s)α–β– ( – s)β+–α

s≤ηi

+ ξi (β +  – α)(ηi – s)α–β– ( – s)β–α   = ξi (ηi – s)α–β– ( – s)β–α (α – β – )( – s) + (β +  – α)(ηi – s) s≤ηi

=



ξi (ηi – s)α–β– ( – s)β–α (α – β – )( – ηi ) ≥ .

s≤ηi

Then we have p is a nondecreasing function on [, ], which implies that p(s) ≥ p() = ∞ α–β– > , s ∈ [, ].  i= ξi ηi Lemma  The function G(t, s) defined by () admits the following properties: (i) G(t, s) > , ∂t∂ G(t, s) > ,  < t, s < ;  (ii) maxt∈[,] G(t, s) = G(, s) = p()(α) [p(s)( – s)α–β– – p()( – s)α– ],  ≤ s ≤ ; (iii) G(t, s) ≥ t α– G(, s),  ≤ t, s ≤ . Proof (i) For  < s ≤ t < , noticing that  < α ≤ , by Lemma , is easy to see that   p(s)( – s)α–β– t α– – p()(t – s)α– p()(α)   p()( – s)α– t α– – p()(t – s)α– ≥ p()(α)



s α–  α– α– t ( – s) –  – ≥ . = (α) t

G(t, s) =

It is clear that for  < t ≤ s < , G(t, s) > . By direct computation, we have ⎧ α–β– α– ⎪ t ⎨ (α – )p(s)( – s) ∂  G(t, s) = – (α – )p()(t – s)α– ,  ≤ s ≤ t ≤ , ∂t p()(α) ⎪ ⎩ (α – )p(s)( – s)α–β– t α– ,  ≤ t ≤ s ≤ .

Zhang and Zhong Boundary Value Problems (2016) 2016:65

∂ G(t, s) ∂t

It is clear that β ≤ , we get

Page 5 of 11

is continuous on [, ] × [, ]. For  < s ≤ t < , noticing that  ≤

  ∂ G(t, s) = (α – )p(s)( – s)α–β– t α– – (α – )p()(t – s)α– ∂t p()(α)

 s α– (α – ) α– p(s)( – s)α–β– – p()  – t = p()(α) t

α–  (α – )p(s) α– s ≥ ( – s)α– –  – ≥ . t p()(α) t It is clear that for  < t ≤ s < , ∂t∂ G(t, s) > . (ii) By (i), we know that G(t, s) is increasing with respect to t, thus we have maxt∈[,] G(t,  [p(s)( – s)α–β– – p()( – s)α– ],  ≤ s ≤ . s) = G(, s) = p()(α) (iii) For  ≤ s ≤ t ≤ , we get  α–  t p(s)( – s)α–β– – p()(t – s)α– p()(α)

 s α–  α– α–β– p(s)( – s) t – p()  – = p()(α) t   ≥ t α– p(s)( – s)α–β– – p()( – s)α– p()(α)

G(t, s) =

= t α– G(, s). For  ≤ t ≤ s ≤ , we have G(t, s) =

 t α– p(s)( – s)α–β– p()(α)

= t α– ·

 p(s)( – s)α–β– p()(α)

≥ t α– G(, s).



We make the following assumptions: (H ) f : (J × R++ , R+ ) is continuous. (H ) There exist a, b ∈ C(J, R+ ), g ∈ C(R++ , R+ ) such that f (t, u) ≤ a(t)g(u) + b(t),

∀t ∈ J, u ∈ R++

and a∗r =





a(t)gr (t) dt < +∞ 

for any r > , where   gr (t) = max g(u) : t α– r ≤ u ≤ r

Zhang and Zhong Boundary Value Problems (2016) 2016:65

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and b∗ =





b(t) dt < +∞. 

(H ) There exists c ∈ C(J, R+ ) such that f (t, u) → +∞ as u → +∞ c(t)u uniformly for t ∈ J, and c∗ =





c(t) dt < +∞. 

(H ) There exists d ∈ C(J, R+ ) such that f (t, u) → +∞ as u → + d(t) uniformly for t ∈ J, and d∗ =





d(t) dt < +∞. 

Let E = C[, ] be the Banach space equipped with the maximum norm u = max≤t≤ |u(t)| and let P be the cone of nonnegative functions in C[, ] with the following form:   P = u ∈ E|u(t) ≥ t α– u, t ∈ [, ] . Denote P+ = {u ∈ P : u > } and Pmn = {u ∈ P, m ≤ u ≤ n} for any n > m > . Define the operator T as follows:  (Tu)(t) =



  G(t, s)f s, u(s) ds,

 ≤ t ≤ .

()



Clearly, T : P\{} → C[, ]. Lemma  Suppose that (H ) and (H ) hold, then for any j > i > , T : Pij → P is completely continuous. Proof For any u ∈ Pij , then i ≤ u ≤ j. By the definition of cone P, we have it α– ≤ u(t) ≤ j,

∀t ∈ [, ].

()

It is not difficult to see that condition (H ) implies that a∗ij =





a(t)gij (t) dt < +∞ 

()

Zhang and Zhong Boundary Value Problems (2016) 2016:65

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for any j > i > , where   gij (t) = max g(u) : it α– ≤ u ≤ j .

()

By (H ), (H ), (), and Lemma (ii), we get   f t, u(t) ≤ a(t)gij (t) + b(t),

∀t ∈ J

()

and 



(Tu)(t) =

  G(t, s)f s, u(s) ds



≤

 p()(α)

 ≤ p()(α)





  p(s)( – s)α–β– – p()( – s)α– f s, u(s) ds







  p(s)( – s)α–β– f s, u(s) ds







a(s)gij (s) + b(s) ds

≤

 p()(α)

≤

  a∗ij + b∗ , p()(α)





∀t ∈ [, ],

()

which implies that T is well defined. By Lemma (iii), we have 



  G(t, s)f s, u(s) ds



  G(, s)f s, u(s) ds,

(Tu)(t) = 

 ≤

∀t ∈ [, ]

()



and 



(Tu)(t) =

  G(t, s)f s, u(s) ds



≥t





α–

  G(, s)f s, u(s) ds,

∀t ∈ [, ],

()



which means that T maps Pij into P. Next, we are in a position to show that T is completely continuous. Let un , u¯ ∈ Pij , un – ¯ ¯ →  (n → ∞), then limn→∞ un (t) = u(t), t ∈ [, ]. Let u   (T u)(t) = f t, u(t) ,  < t < , u ∈ Pij ,   (T u)(t) = G(t, s)u(s) ds,  < t < , u ∈ L [, ]. 

By (H ),     ¯ , lim f t, un (t) = f t, u(t)

n→∞

 < t < .

()

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Similar to () and (), for un , u¯ ∈ Pij , we have   f t, un (t) ≤ a(t)gij (t) + b(t),

  ¯ f t, u(t) ≤ a(t)gij (t) + b(t),

∀t ∈ J.

Thus, we have        ≤  a(t)gij (t) + b(t) = σ (t) ∈ L [, ]. f t, un (t) – f t, u(t) ¯

()

It follows from (), (), and the Lebesgue dominated convergence theorem that  ¯ dt = , which implies that T : Pij → L [, ] is continulimn→∞  |(T un )(t) – (T u(t))| ous. By the Arzela-Ascoli theorem, we know that T : L [, ] → C[, ] is completely continuous. As a consequence, T = T ◦ T : Pij → C[, ] is completely continuous.  In order to prove the main theorem, we state the following Guo-Krasnoselskii fixed point theorem. Lemma  ([]) Let  and  be two bounded open sets in Banach space E such that θ ∈

 and  ⊂  , A : P ∩ (  \  ) → P a completely continuous operator, where θ denotes the zero element of E and P a cone of E. Suppose that one of the two conditions holds: (i) Au ≤ u, ∀u ∈ P ∩ ∂  ; Au ≥ u, ∀u ∈ P ∩ ∂  ; (ii) Au ≥ u, ∀u ∈ P ∩ ∂  ; Au ≤ u, ∀u ∈ P ∩ ∂  . Then A has a fixed point in P ∩ (  \  ).

3 Main result Theorem  Let conditions (H )-(H ) be satisfied. Assume in addition that there exists r >  such that  ∗   ar + b∗ < r, p()(α)

()

where a∗r and b∗ are defined in condition (H ). Then the boundary value problem () has at least two positive solutions u∗ and u∗∗ with  < u∗  < r < u∗∗ . Proof By Lemma , the operator T defined by () is completely continuous from Pmn into P for any n > m > . We need only to prove that T has two fixed points u∗ and u∗∗ ∈ P+ with  < u∗  < r < u∗∗ . By condition (H ), there exists r >  such that  f (t, u) ≥ α–



  

G



–  , s c(s) ds c(t)u, 

∀t ∈ J, u ≥ r .

()

Choose   r > max α– r , r .

()

For u ∈ P, u = r , we have, by the construction of cone P, u(t) ≥ t α– r ≥

α–  r > r , 

∀t ∈

  , . 

()

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So, we get from () and ()  

    = G , s f s, u(s) ds (Tu)     

–  

  α– ≥ G , s c(s) ds G , s c(s)u(s) ds        

–  

α–    ≥ α– G , s c(s) ds G , s c(s) ds · r = r .       

()

Therefore,            ≥ r = u, Tu = max (Tu)(t) ≥ (Tu) t∈[,]  

∀u ∈ P, u = r .

()

By condition (H ), there exists r >  such that  f (t, u) ≥

  





–  G , s d(s) ds d(t)r, 

∀t ∈ J,  < u < r .

()

Choose  < r < min{r , r}.

()

For u ∈ P, u = r , we have r > r = u ≥ r t α– > ,

∀t ∈ J.

So, we get (Tu)

 

    G , s f s, u(s) ds =   

–  

 

  G , s d(s) ds G , s d(s)r ds ≥      

–  

 

  G , s d(s) ds G , s d(s) ds · r = r > r . ≥      

()

Therefore,         > r = u, (Tu) Tu = max (Tu)(t) ≥   t∈[,]  

∀u ∈ P, u = r .

()

On the other hand, for u ∈ P, u = r, similar to (), by (H ), Lemma (iii) and (), we get  (Tu)(t) =



  G(t, s)f s, u(s) ds



≤

 p()(α)

 



  p(s)( – s)α–β– – p()( – s)α– f s, u(s) ds

Zhang and Zhong Boundary Value Problems (2016) 2016:65

≤ ≤

 p()(α)  p()(α)



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

  p(s)( – s)α–β– f s, u(s) ds



  ( – s)α–β– f s, u(s) ds







 ∗   ≤ ar + b∗ < r, p()(α)

∀t ∈ [, ].

()

Thus, from (), we get Tu < u,

∀u ∈ P, u = r.

()

We know from (), (), (), and Lemma  that T has two fixed points u∗ , u∗∗ ∈ Pr r such that  < r < u∗  < r < u∗∗  ≤ r . 

4 An example Example  Consider the following infinite-point boundary value problems: ⎧  ⎨ D  u(t) + + ⎩

u() = ,

 √ (u   (–t)

+

 

  √ ) +  √  t(–t)  u

D+ u() = ,

= ,  < t < ,   –   D+ u() = ∞ i=  i D+ u( i ).

()

 

Conclusion BVP () has at least two positive solutions u∗ and u∗∗ with  < u∗  < r < u∗∗ .     √ Proof In this problem, α =  , β =  , ξi =  i– , ηi = i , f (t, u) =  √  (–t) (u +  u ) +  √  t(–t) .  α–β– ∞ = . < , p() = .. By simple computation, we have (  ) = ., i= ξi ηi    √ For any r > , it is easy to see that (H ) holds for a(t) =  √  (–t) , g(u) = u +  u , b(t) =  √ ,   t(–t)

a∗r

and 

=





a(t)gr (t) dt < 





   r +  √ dt < +∞, √   ( – t) t  r

  b∗ =   √  t(–t) dt = .. Obviously, (H ) and (H ) hold for c(t) = d(t) = ∗ ∗ c = d = .. Take r = , we have, by (),

()  √ ,   (–t)

and

  ∗   a + b∗ < × (. + . + .) p()(α) . × . = . <  = r. Consequently, () holds, and our conclusion follows from Theorem .

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors, XZ and QZ, contributed to each part of this work equally and read and approved the final version of the manuscript. Author details 1 Department of Information Engineering, Jining Medical College, Jining, Shandong 272067, P.R. China. 2 School of Mathematics, Liaocheng University, Liaocheng, Shandong 252059, P.R. China.



Zhang and Zhong Boundary Value Problems (2016) 2016:65

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Acknowledgements The project is supported financially by the Natural Science Foundation of Shandong Province of China (ZR2015AL002), a Project of Shandong Province Higher Educational Science and Technology Program (J15LI16), the Foundations for Jining Medical College Natural Science (JYQ14KJ06, JY2015KJ019, JY2015BS07), and the National Natural Science Foundation of China (11571197, 11571296, 11371221, 11071141). Received: 24 January 2016 Accepted: 7 March 2016 References 1. Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916-924 (2010) 2. Li, C, Luo, X, Zhou, Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363-1375 (2010) 3. Xu, X, Fei, X: The positive properties of Green’s function for three point boundary value problems of nonlinear fractional differential equations and its applications. Commun. Nonlinear Sci. Numer. 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