Zhao and Liang Advances in Difference Equations (2017) 2017:50 DOI 10.1186/s13662-017-1099-0

RESEARCH

Open Access

Solvability of triple-point integral boundary value problems for a class of impulsive fractional differential equations Kaihong Zhao* and Jiangyan Liang *

Correspondence: [email protected] Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, P.R. China

Abstract This paper is concerned with a class of triple-point integral boundary value problems for impulsive fractional differential equations involving the Riemann-Liouville fractional derivative of order α (2 < α ≤ 3). Some sufficient criteria for the existence of solutions are obtained by applying the contraction mapping principle and the fixed point theorem. As an application, one example is given to demonstrate the validity of our main results. MSC: 34B10; 34B15; 34B37 Keywords: impulsive fractional differential equations; existence of solutions; integral boundary value problems; fixed point theorem

1 Introduction Towards the end of the th century Liouville and Riemann mentioned the definition of the fractional derivative which is the generalization of the traditional integer order differential and integral calculus. The fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The subject of fractional differential equations is gaining much importance and attention because of its extensive applications in many engineering and scientific disciplines such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electroanalytical chemistry, biology, control theory, fitting of experimental data, and so forth. For more details of the basic theory of fractional differential equations, refer to [–] and the references therein. In recent decades, the boundary value problems of fractional differential equations have received a great deal of attention. There are a large number of papers dealing with the existence, nonexistence, multiplicity of solutions of boundary value problem for some nonlinear fractional differential equations (see [–]). As we know, many evolutionary processes experience short-time rapid change after undergoing relatively long smooth variation. In order to describe the dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so on, some authors have used an impulsive differential system to describe these kinds of phenomena since the last century. For the theory of impulsive differential equations, the © The Author(s) 2017. 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.

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reader can refer to [–]. Recently, the boundary value problems of impulsive fractional differential equations have been studied extensively in the literature (see [–]). To the best of our knowledge, there are few articles involving the impulsive fractional order differential equations. Therefore, we will study the existence and uniqueness of solutions for the following impulsive integral boundary value problems (BVPs for short) of fractional order differential equations: ⎧ α  α– ⎪ ⎪ ⎨tk Dt u(t) = f (t, u, u , D u), ⎪ ⎪ ⎩

D

α–

t = tk ,

u(tk ) = Ik (u(tk )), 

u() = u () = ,

k = , . . . , m, η u () =  g(s, u(s)) ds,

(.)



where  < α ≤ , J = [, ], J = [t , t ], Jk = (tk , tk+ ] ⊂ J (k = , , . . . , m). tk Dαt is the Riemann-Liouville fractional derivative of order  < α ≤ . f ∈ C(J × R , R), Ik ∈ C(R, R),  < η < , g ∈ C(J × R, R),  = t < t < · · · < tm < tm+ = . u(tk+ ) = limh→+ u(tk + h) and u(tk– ) = limh→– u(tk + h) represent the right and left limits of u(t) at t = tk , respectively. + + – α– α– u(tk– ) = u(tk ), tk Dα– t u(tk ) = tk Dt u(tk ). The right-hand limits u(tk ) and tk Dt u(tk ) all ex+ – α– ist. Dα– u(tk ) = tk Dα– t u(tk ) – tk– Dt u(tk ). The rest of this paper is organized as follows. In Section , we shall introduce some definitions and lemmas to prove our main results. In Section , we give some sufficient conditions for the existence of single positive solutions for boundary value problem (.). As an application, one interesting example is presented to illustrate the main results in Section . Finally, the conclusion is given to simply recall our studied contents and obtained results in Section .

2 Preliminaries Let C(J, R) be the Banach space of continuous functions from J to R with the norm uC = sup≤t≤ |u(t)|. Now let us to introduce the useful Banach space PC  (J, R) defined by   +  – α– PC  (J, R) = u ∈ C(J, R) : tk Dα– t u tk and tk Dt u tk exist with

 – α– α– tk Dt u(tk ) = tk Dt u tk , k = , , . . . , m

(.)

equipped with the norm uPC  = max{uC , u C , tk Dα– t uC }. Definition . A function u ∈ PC  (J, R) with its Riemann-Liouville derivative of order α existing on J is a solution of (.) if it satisfies (.). For convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature [, , ]. Definition . The Riemann-Liouville fractional integral of order α >  of a function u : (a, +∞) → R is given by α a It u(t) =

 (α)



t

(t – s)α– u(s) ds,

a > ,

a

provided that the right-hand side is point-wise defined on (a, +∞).

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Definition . The Riemann-Liouville fractional derivative of order α >  of a continuous function u : (a, +∞) → R is given by dn  (n – α) dt n

α a Dt u(t) =



t

(t – s)n–α– u(s) ds, a

where a > , n –  < α ≤ n, provided that the right-hand side is point-wise defined on (a, +∞). Lemma . Assume that u ∈ C[a, b], q ≥ p ≥ , then p q q–p a Dt a It u(t) = a It u(t),

t ∈ [a, b].

(.)

Lemma . (see [], pp. -) Let α > , n denotes the smallest integer greater than or equal to α. Then the following assertions hold. (i) if λ > –, λ = α – i, i = , , . . . , n + , then for t ∈ [a, b] α a Dt (t

– a)λ =

(λ + ) (t – a)λ–α . (λ – α + )

(.)

(ii) a Dαt (t – a)α–i = , i = , , . . . , n. (iii) a Dαt a Itα u(t) = u(t), for all t ∈ [a, b]. (iv) a Dαt u(t) =  if and only if there exists ci ∈ R (i = , , . . . , n) such that u(t) = c (t – a)α– + c (t – a)α– + · · · + cn (t – a)α–n ,

t ∈ [a, b].

(.)

(v) For all t ∈ [a, b], then α α a It a Dt u(t) = c (t

– a)α– + c (t – a)α– + · · · + cn (t – a)α–n + u(t).

(.)

Lemma . (Schauder fixed point theorem; see []) If U is a closed bounded convex subset of a Banach space X and T : U → U is completely continuous, then T has at least one fixed point in U. Lemma . For a given y ∈ C(J, R), a function u ∈ PC  (J, R) is a solution of BVP (.) ⎧ α ⎪ ⎪ ⎨tk Dt u(t) = y(t), ⎪ ⎪ ⎩

D

α–

t = tk ,  < α ≤ ,

u(tk ) = Ik (u(tk )), 

u() = u () = ,

k = , . . . , m, η u () =  g(s, u(s)) ds,

(.)



if and only if u ∈ PC  (J, R) is a solution of the impulsive fractional integral equation  t α–  (t – s) y(s) ds – ( – s)α– y(s) ds (α)    η t α–  t α–  – Ii u(ti ) + g s, u(s) ds,  ≤ t ≤ . (α) t≤t α– 

 u(t) = (α)



t

α–

i

(.)

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Proof We denote the solution of (.) by u(t) = uk (t) in [tk , tk+ ] (k = , , . . . , m). For t ∈ J = [, t ], by (.), we have u (t) =  Itα y(t) + c t α– + c t α– + c t α– . u() = u () =  implies that c = c = . Applying Lemma ., we get u (t) =  Itα y(t) + c t α– ,

α  α– α– = Dα–  It y(t) + c t t u (t) = Dt



t

y(s) ds + (α)c , 

and    Dα– u t+ = Dα– u t+ = Dα– u (t ) + I u(t )  t  y(s) ds + (α)c + I u(t ) . = 

For t ∈ J = (t , t ], by (.), we get u (t) =  Itα y(t) + c t α– + c t α– + c t α– and  Dα– t u (t) =

t

y(s) ds + (α)c . 

Noting that u() = u () =  and Dα– u (t ) = Dα– u (t+ ), we derive c = c =  and c = (u(t )) c + I(α) . So we can obtain   I (u(t )) α– t u (t) =  Itα y(t) + c + (α) and    Dα– u t+ = Dα– u t+ = Dα– u (t ) + I u(t ) =



t

y(s) ds + (α)c + 

By the recurrent method, for t ∈ Jk = (tk , tk+ ], k = , , . . . , m, we get 

 k α–   c + Ii u(ti ) t (α) i=    t k α–    α– = (t – s) y(s) ds + c + Ii u(ti ) t (α)  (α) i=

uk (t) =  Itα y(t) +

and

  Ii u(ti ) . i=

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+  + = Dα– uk tk+ = Dα– uk (tk+ ) + Ik+ u(tk+ ) Dα– u tk+ 

tk+

y(s) ds + (α)c +

= 

k+  Ii u(ti ) . i=

So, for t ∈ Jm = (tm , tm+ ], we have 

u

 (α – )

(t) = um (t) =

By u () =  

η

η 





t

(t – s)

α–



 m α–   y(s) ds + (α – ) c + Ii u(ti ) t . (α) i=

u(s)ψ(s) ds, we have





 g s, u(s) ds = (α – )







( – s)

α–



 m   y(s) ds + (α – ) c + Ii u(ti ) , (α) i=

which implies that c =

 α–



η

 g s, u(s) ds –



 (α)



  Ii u(ti ) . (α) i= m



( – s)α– y(s) ds – 

Therefore, for t ∈ J = [, ], we have  u(t) = (α)



(t – s)

t – (α)







α–

η

η

 g s, u(s) ds



α–

t y(s) ds – (α)





ti
i=

t





m   ( – s)α– y(s) ds + Ii u(ti ) – Ii u(ti )

(t – s)

t α–

t α– y(s) ds + α–





α–

+

α–



α–

 = (α)

t



( – s)

α–

  y(s) ds + Ii u(ti )



t≤ti

 g s, u(s) ds,



which indicates that u is a solution of (.). Conversely, noting that the above derivations are reversible, we assert that if u is a solution of the impulsive fractional integral equation (.), then u is also the solution of BVP (.). The proof is complete. 

3 Main results According to Lemma ., we obtain the following lemma. Lemma . A function u ∈ PC  (J, R) is a solution of BVP (.) if and only if u ∈ PC  (J, R) is a solution of the impulsive fractional integral equation u(t) =

 t   (t – s)α– f s, u(s), u (s), Dα– u(s) ds (α)       t α– α–  α– ( – s) f s, u(s), u (s), D u(s) ds + Ii u(ti ) – (α)  t≤t α–

+

t α–

 

i

η

 g s, u(s) ds,

 ≤ t ≤ .

(.)

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Define an operator T : PC  (J, R) → PC  (J, R) as follows: (Tu)(t) =

 t   (t – s)α– f s, u(s), u (s), Dα– u(s) ds (α)       t α– ( – s)α– f s, u(s), u (s), Dα– u(s) ds + Ii u(ti ) – (α)  t≤t α–

t α–

+

i



η

 g s, u(s) ds,

 ≤ t ≤ .

(.)



Then BVP (.) has a solution if and only if the operator T exists one fixed point. Lemma . Assume that f ∈ C(J × R , R), and g ∈ C(J × R, R). Then T : PC  (J, R) → PC  (J, R) defined by (.) is completely continuous. Proof Note that T is continuous in view of continuity of f , Ik , and g. Now we show that T is uniformly bounded. In fact, let  ⊂ PC  (J, R) be bounded, then there exist some positive constants li (i = , , ) such that |f (t, u, u , Dα– u)| ≤ l , |g(t, u)| ≤ l , |Ik (u)| ≤ l , for all u ∈ . Thus for u ∈ , we have   (Tu)(t) ≤

 t     (t – s)α– f s, u(s), u (s), Dα– u(s)  ds (α)           t α– α–   α–    + f s, u(s), u (s), D u(s) ds + Ii u(ti ) ( – s) (α)  t≤t α–

t α–

+



i

η

  g s, u(s)  ds



l η l + ml +  M , ≤ (α) α–   t      (Tu) (t) =  (t – s)α– f s, u(s), u (s), Dα– u(s) ds  (α – )    α–  t – ( – s)α– f s, u(s), u (s), Dα– u(s) ds (α – )    α– η   t α–  – Ii u(ti ) + t g s, u(s) ds (α – ) t≤ti





≤ 





 ( – s)α–   f s, u(s), u (s), Dα– u(s)  ds (α – )

 ( – s)α–   f s, u(s), u (s), Dα– u(s)  ds (α – )   η        Ii u(ti )  + g s, u(s)  ds + (α – ) t≤t  

+

i

≤ and

l + ml + l η  M (α – )

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  α– D (Tu)(t)  t      ( – s)α– f s, u(s), u (s), Dα– u(s) ds =  f s, u(s), u (s), Dα– u(s) ds – 



  Ii u(ti ) + (α – ) – 

t≤ti



η

  g s, u(s) ds 

t

  f s, u(s), u (s), Dα– u(s)  ds +

≤







   ( – s)α– f s, u(s), u (s), Dα– u(s)  ds



 η       Ii u(ti )  + (α – ) g s, u(s)  ds + 

t≤ti

≤ l + ml + l η(α – )  M , which means that uPC  ≤ max{M , M , M }, that is, T is uniformly bounded. Next, we should prove that T is equicontinuous on J = [, ]. Indeed, for all t¯ , t¯ ∈ [, ] with t¯ ≤ t¯ , we have    t¯    t¯     (Tu) (s) ds ≤ (Tu)(t¯ ) – (Tu)(t¯ ) =  (Tu) (s) ds  ¯  ¯ t

t

≤ M (t¯ – t¯ ) → , as t¯ → t¯ ,   (Tu) (t¯ ) – (Tu) (t¯ )  t¯

    (t¯ – s)α– – (t¯ – s)α– f s, u(s), u (s), Dα– u(s)  ds ≤ (α – )   t¯     (t¯ – s)α– f s, u(s), u (s), Dα– u(s)  ds + (α – ) t¯     t¯α– – t¯α–  ( – s)α– f s, u(s), u (s), Dα– u(s)  ds + (α – )      t¯α– – t¯α–   t¯α– Ii u(ti )  +  Ii u(ti )  + (α – ) ¯ (α – ) ¯ t ≤ti t ≤ti
l  α– α– l  α– α– l t¯ – t¯ t¯ – t¯ ≤ (t¯ – t¯ )α– + + (α) (α) (α)   l ml  α– α– + t¯ – t¯ (t¯ – t¯ ) + l η t¯α– – t¯α– → , + (α – ) (α – ) as t¯ → t¯ , and   α–   D (Tu)(t¯ ) – Dα– (Tu)(t¯ ) =    ≤

t¯ t¯





f s, u(s), u (s), D

α–



u(s) ds +

t¯ ≤ti
   Ii u(ti ) 

     f s, u(s), u (s), Dα– u(s)  ds + Ii u(ti ) 

t¯ 

t¯

≤ l (t¯ – t¯ ) + l (t¯ – t¯ ) → ,

t¯ ≤ti
as t¯ → t¯ .

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Thus, for any ε >  (small enough), there exists δ = δ(ε) >  with independence of t¯ , t¯ and u such that (Tu)(t¯ ) – (Tu)(t¯ )PC  < ε, whenever |t¯ – t¯ | < δ. Therefore, T is equicontinuous on J = [, ]. According to the Arzela-Ascoli theorem, it follows that T : PC  (J, R) → PC  (J, R) is completely continuous.  Theorem . Assume that the conditions (B )-(B ) hold. ¯ v¯ , w) ¯ ∈ J × R , there exist some functions ψi ∈ (B ) f ∈ C(J × R , R), for all (t, u, v, w), (t, u, L([, ]) (i = , , ) such that     f (t, u, v, w) – f (t, u, ¯ v¯ , w) ¯  ≤ ψ (t)|u – u| ¯     ¯ + ψ (t)|v – v¯ | + ψ (t)|w – w|. (B ) Ik ∈ C(R, R), for all u, v ∈ R, there exist some constants Lk >  such that   Ik (u) – Ik (v) ≤ Lk |u – v|,

k = , , . . . , m.

(B ) g ∈ C(J, R), for all (t, u), (t, v) ∈ J × R, there exists a function ψ ∈ L([, ]) such that     g(t, u) – g(t, v) ≤ ψ(t)|u – v|.  η  If ρ =   [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds + m k= Lk + (α – )  |ψ(s)| ds < , then BVP (.) has a unique solution on J. Proof Let M = supt∈J |f (t, , , )| + supt∈J |g(t, )| and Br = {u ∈ PC(J, R) : uPC  ≤ r},   where r ≥ –ρ [( + (α – )η)M + m k= |Ik ()|]. Define an operator T : Br → PC(J, R) as (.). It is obvious that T is jointly continuous and maps bounded subsets of J × R to bounded subsets of R. We will prove Theorem . through the following two steps. Step . We show that T(Br ) ⊂ Br . In fact, noting that u() = u () = Dα– u() = , we have, for u ∈ Br , t ∈ J = [, ],   (Tu)(t)  η       t α–  α– g s, u(s)  ds  f s, u(s), u (s), D u(s) ds + (t – s) α –     α–         t Ii u(ti )  + ( – s)α– f s, u(s), u (s), Dα– u(s)  ds + (α)  t≤t

 ≤ (α)

≤

 (α)



t



t



α–

+

t (α) α–

+

+

t (α) t α– α–

α– 

i

  

  (t – s)α– f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds





  

  ( – s)α– f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds



       Ii u(ti ) – Ii u()  + Ii u()  t≤ti



    g s, u(s) – g(s, ) + g(s, ) ds

η  

Zhao and Liang Advances in Difference Equations (2017) 2017:50

≤

r

Page 9 of 19



 [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds + M

(α) +

 (α)

m

  Lk r + Ik () +

k=

r α–



+

r



 [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds + M

(α)

η

 ψ(s) ds + ηM α–



    η m     [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds     + ψ(s) ds r = Lk + (α) (α) α–  k=

 m  η    + M+ Ik () + (α) α –  (α) k=     η m          ψ (s) + ψ (s) + ψ (s) ds + ψ(s) ds r Lk + (α – ) ≤  





k=

m  

 Ik () +  + (α – )η M + k=

≤ ρr + ( – ρ)r = r,   (Tu) (t) ≤

 (α – )

 

α–

+

t (α – ) α–

+ 

t (α – ) t

≤ 



t

   (t – s)α– f s, u(s), u (s), Dα– u(s)  ds





   ( – s)α– f s, u(s), u (s), Dα– u(s)  ds



 η       Ii u(ti )  + t α– g s, u(s)  ds 

t≤ti

   (t – s)α–   f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds (α – )

   [t( – s)]α–   f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds (α – )  α–        t Ii u(ti ) – Ii u()  + Ii u()  + (α – ) t≤t i  η   

  g s, u(s) – g(s, ) + g(s, ) ds + t α– 

+

≤

r





 [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds + M

(α – )    Lk r + Ik () + r (α – ) m

+

k=



+

r



 [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds + M

(α – )

η

 ψ(s) ds + ηM



    η m     [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds    Lk + ψ(s) ds r = + (α – ) (α – )  k=

 +



  +η M+ (α – ) (α – )

m

  Ik ()

k=

    η m          ψ (s) + ψ (s) + ψ (s) ds + ψ(s) ds r Lk + (α – ) ≤  

k=



Zhao and Liang Advances in Difference Equations (2017) 2017:50

Page 10 of 19

m  

 Ik () +  + (α – )η M + k=

≤ ρr + ( – ρ)r = r and   α– D (Tu)(t)    t        α–   ( – s)α– f s, u(s), u (s), Dα– u(s)  ds f s, u(s), u (s), D u(s) ds + ≤ 



 η       Ii u(ti )  + (α – ) g s, u(s)  ds + 

t≤ti



    f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds

t 

≤ 





+

  

  ( – s)α– f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds





    g s, u(s) – g(s, ) + g(s, ) ds

η 

+ (α – ) 

       Ii u(ti ) – Ii u()  + Ii u()  + 

t≤ti

     ψ (s) + ψ (s) + ψ (s) ds + M + r

 

≤r 

+



     ψ (s) + ψ (s) + ψ (s) ds + M

  

m

  η          Lk r + Ik () + (α – ) r ψ(s) ds + ηM 

k=

    η m                  ψ (s) + ψ (s) + ψ (s) ds + Lk + (α – ) ψ(s) ds r =  

k=



m  

 Ik () +  + (α – )η M + k=

≤ ρr + ( – ρ)r = r, which imply that TuPC  ≤ r, that is, T(Br ) ⊂ Br . Step . We show that T is a contraction mapping. Indeed, for all u, v ∈ Br , for each t ∈ J = [, ], we obtain   (Tu)(t) – (Tv)(t)  t      ≤ (t – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds (α)       t α–  + ( – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds (α)   η       t α– t α–   g s, u(s) – g s, v(s)  ds + Ii u(ti ) – Ii v(ti )  + (α) t≤t α–  i

Zhao and Liang Advances in Difference Equations (2017) 2017:50

Page 11 of 19

  m   [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds  + ≤ Lk (α) (α) k=   η      ψ(s) ds u – vPC  + α–      η m                  ψ (s) + ψ (s) + ψ (s) ds + ≤  Lk + (α – ) ψ(s) ds u – vPC  

k=



= ρu – vPC  ,   (Tu) (t) – (Tv) (t)  t   (t – s)α–   f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds ≤  (α – )       t α– ( – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds + (α – )   η       t α–   g s, u(s) – g s, v(s)  ds + Ii u(ti ) – Ii v(ti )  + t α– (α – ) t≤t  i     η m     [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds    ≤ + Lk + ψ(s) ds u – vPC  (α – ) (α – )  k=     η m                  ψ(s) ds u – vPC  ψ (s) + ψ (s) + ψ (s) ds + ≤  Lk + (α – ) 

k=



= ρu – vPC  , and   α– D (Tu)(t) – Dα– (Tv)(t)  t     f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds ≤ 





+

    ( – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds



 η         Ii u(ti ) – Ii v(ti )  + (α – ) g s, u(s) – g s, v(s)  ds + t≤ti



    η m                  ≤  ψ (s) + ψ (s) + ψ (s) ds + ψ(s) ds u – vPC  Lk + (α – ) 

k=



= ρu – vPC  ,  which indicates Tu – TvPC  ≤ ρu – vPC , where ρ =   [|ψ (s)| + |ψ (s)| + |ψ (s)|] ds + η m  k= Lk + (α – )  |ψ(s)| ds < . Therefore T is a contraction mapping on PC (J, R). According to the contraction mapping principle, we conclude that T has a unique fixed point u(t) ∈ PC  (J, R), which is the unique solution of BVP (.). The proof is complete.  Now we give a simple and easily verifiable result as follows.

Zhao and Liang Advances in Difference Equations (2017) 2017:50

Page 12 of 19

Corollary . Assume that the conditions (B ), (B ), and (B ) hold. ¯ v¯ , w) ¯ ∈ J × R , there exist some constants Ni >  (B ) f ∈ C(J × R , R), for all (t, u, v, w), (t, u, (i = , , ) such that   f (t, u, v, w) – f (t, u, ¯ v¯ , w) ¯  ≤ N |u – u| ¯ + N |v – v¯ | + N |w – w|. ¯ If ρ = on J.

α(N +N +N ) α–

+

m

k= Lk

+ (α – )

η 

|ψ(s)| ds < , then BVP (.) has a unique solution

Proof Let M = supt∈J |f (t, , , )| + supt∈J |g(t, )| and Br = {u ∈ PC(J, R) : uPC  ≤ r},   α where r ≥ –ρ [( α– + (α – )η)M + m k= |Ik ()|]. Define an operator T : Br → PC(J, R) as (.). It is obvious that T is jointly continuous and maps bounded subsets of J × R to bounded subsets of R. Similarly, we will prove Corollary . through the following two steps. Step . We show that T(Br ) ⊂ Br . In fact, noting that u() = u () = Dα– u() = , we have, for u ∈ Br , t ∈ J = [, ],   (Tu)(t)  η  t     t α– (t – s)α–    α– g s, u(s)  ds  f s, u(s), u (s), D u(s) ds + ≤ (α) α –     α–         t Ii u(ti )  + ( – s)α– f s, u(s), u (s), Dα– u(s)  ds + (α)  t≤t ≤

 (α)





α–

+

t (α) α–

+

+

t (α) t α– α–

i

t

  

  (t – s)α– f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds





  

  ( – s)α– f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds



       Ii u(ti ) – Ii u()  + Ii u()  t≤ti



    g s, u(s) – g(s, ) + g(s, ) ds

η  

 η m      r [(N + N + N )r + M] ψ(s) ds + ηM Lk r + Ik () + + (α + ) (α) α–  α– k=    η m     (N + N + N )   ψ(s) ds r + Lk + = (α + ) (α) α–  ≤

k=

 +



 η    Ik () + M+ (α + ) α –  (α) k=



α(N + N + N ) + ≤ α–  +

m

m

 Lk + (α – ) 

k=

 m   α Ik () + (α – )η M + α– k=

≤ ρr + ( – ρ)r = r,

  ψ(s) ds r

η

Zhao and Liang Advances in Difference Equations (2017) 2017:50

Page 13 of 19

  (Tu) (t)  η  t     (t – s)α–   g s, u(s)  ds f s, u(s), u (s), Dα– u(s)  ds + t α– ≤  (α – )   α–         t α–   α–    f s, u(s), u (s), D u(s) ds + Ii u(ti ) ( – s) + (α – )  t≤t i

 ≤ 

    (t – s) f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds (α – ) α– 

t



   [t( – s)]α–   f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds (α – )       t α–   Ii u(ti ) – Ii u()  + Ii u()  + (α – ) t≤t i  η   

  g s, u(s) – g(s, ) + g(s, ) ds + t α– 

+

 m  

 [(N + N + N )r + M] Lk r + Ik () + r + (α) (α – ) k=    η m    (N + N + N )   ψ(s) ds r + Lk + = (α) (α – ) 



≤

η

 ψ(s) ds + ηM



k=

 +



  +η M+ (α) (α – )

m k=



α(N + N + N ) + (α – ) ≤ α–  +

α α–



  Ik ()  m  ψ(s) ds + Lk r

η 

k=

 m   Ik () + (α – )η M + k=

≤ ρr + ( – ρ)r = r, and   α– D (Tu)(t)    t       f s, u(s), u (s), Dα– u(s)  ds + ( – s)α– f s, u(s), u (s), Dα– u(s)  ds ≤ 



 η         g s, u(s)  ds Ii u(ti ) + (α – ) + 

t≤ti



    f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds

t 

≤ 





+

  

  ( – s)α– f s, u(s), u (s), Dα– u(s) – f (s, , , ) + f (s, , , ) ds



       Ii u(ti ) – Ii u()  + Ii u()  + t≤ti



    g s, u(s) – g(s, ) + g(s, ) ds

η 

+ (α – ) 

Zhao and Liang Advances in Difference Equations (2017) 2017:50

≤r



r

Ni + M +



i= Ni

α–

i=

+M

+

Page 14 of 19

m

  Lk r + Ik ()

k=

  η    ψ(s) ds + ηM + (α – ) r 



α(N + N + N ) + Lk + (α – ) = α– m

k=

 +

  ψ(s) ds r



η 

 m   α Ik () + (α – )η M + α– k=

≤ ρr + ( – ρ)r = r, which imply that TuPC  ≤ r, that is, T(Br ) ⊂ Br . Step . We show that T is a contraction mapping. Indeed, for all u, v ∈ Br , for each t ∈ J = [, ], we obtain   (Tu)(t) – (Tv)(t)  t      ≤ (t – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds (α)       t α–  ( – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds + (α)   η       t α– t α–   g s, u(s) – g s, v(s)  ds  + Ii u(ti ) – Ii v(ti ) + (α) t≤t α–  

i

  η m   N + N  + N  N  + N  + N    ψ(s) ds u – vPC  ≤ + + Lk + (α + ) (α + ) (α) α–  k=    η m   α(N + N + N )   ψ(s) ds u – vPC  = ρu – vPC  , + ≤ Lk + (α – ) α–  k=

  (Tu) (t) – (Tv) (t)  t   (t – s)α–   f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds ≤  (α – )       t α– ( – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds + (α – )   η       t α–   g s, u(s) – g s, v(s)  ds Ii u(ti ) – Ii v(ti )  + t α– + (α – ) t≤t  i

  η m   (N + N + N )    ψ(s) ds u – vPC  ≤ + Lk + (α) (α – )  k=    η m   α(N + N + N ) ψ(s) ds u – vPC  = ρu – vPC  , + ≤ Lk + (α – ) α–  

k=

and

Zhao and Liang Advances in Difference Equations (2017) 2017:50

Page 15 of 19

  α– D (Tu)(t) – Dα– (Tv)(t)  t     f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds ≤ 





+

    ( – s)α– f s, u(s), u (s), Dα– u(s) – f s, v(s), v (s), Dα– v(s)  ds



 η         Ii u(ti ) – Ii v(ti )  + (α – ) g s, u(s) – g s, v(s)  ds + 

t≤ti

  η m   N + N + N   ψ(s) ds u – vPC  + Lk + (α – ) ≤ N + N  + N  + α–  k=    η m   α(N + N + N )   ψ(s) ds u – vPC  = ρu – vPC  , + = Lk + (α – ) α–  

k=

  +N ) which indicates Tu – TvPC  ≤ ρu – vPC  , where ρ = α(N +N + m k= Lk + (α – α– η  )  |ψ(s)| ds < . Therefore T is a contraction mapping on PC (J, R). According to the contraction mapping principle, we conclude that T has a unique fixed point u(t) ∈  PC  (J, R), which is the unique solution of BVP (.). The proof is complete. For some fixed r > , considering BVP (.) on the cylinder R = [, ] × B(, r), we obtain the following theorem. Theorem . Assume that conditions (B )-(B ) hold. Then BVP (.) has at least one soη  lution in J, provided that = m k= Lk + (α – )  |ψ(s)| ds < . (B ) f ∈ C(J ×R , R), for all (t, u, u , Dα– u) ∈ J ×R , there exist p ∈ (, ), h ∈ L/p ([, ], R+ ) such that |f (t, u, u , Dα– u)| ≤ h(t), where L/p ([, ], R+ ) denotes space /p-Lebesgue   measurable functions from [, ] to R+ with the norm v/p = (  |v(s)| p ds)p , for v ∈ L/p ([, ], R+ ). (B ) Ik ∈ C(R, R), for all u ∈ R, there exist some constants Lk >  such that |Ik (u)| < Lk |u|, k = , , . . . , m. (B ) g ∈ C(J, R), for all (t, u) ∈ (J, R), there exists ψ ∈ L[, ] such that |g(t, u)| ≤ |ψ(t)||u|. Proof Let Bλ be a closed bounded convex subset of PC  ([, ], R) defined by Bλ = {u : –p –p A , A = [ + ( α–p– ) ]h/p . u ≤ λ}, λ ≥ – Define the operator T : Bλ → PC  ([, ], R) as (.). For u ∈ ∂Bλ , we have   (Tu)(t)  η  t     (t – s)α–   t α– g s, u(s)  ds f s, u(s), u (s), Dα– u(s)  ds + ≤ (α) α–    α–         t α–   α–    + ( – s) f s, u(s), u (s), D u(s) ds + Ii u(ti ) (α)  t≤t  ≤



α–



i



α–

( – s) ( – s) h(s) ds + h(s) ds (α) (α)    η        ψ(s)u(s) ds + Li u(ti ) + (α) t≤t α–  i

Zhao and Liang Advances in Difference Equations (2017) 2017:50

≤

 (α) + +





–p 

α–



( – s) –p ds

 (α)

p

p

ds

h(s)

p

h(s)



 (α)

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





–p 

α–



( – s) –p ds  m

p ds



Lk uPC  +

k=

uPC  α–



η

 ψ(s) ds



   η m   h/p     ψ(s) ds ≤ +λ Lk + (α) (α) α–  k=  m    –p   η   –p   ≤ + h/p + λ Lk + (α – ) ψ(s) ds = A + λ ≤ λ, α–p–  

–p

–p α–p



–p + α–p–

–p 

k=

  (Tu) (t)  η  t     (t – s)α–   g s, u(s)  ds f s, u(s), u (s), Dα– u(s)  ds + t α– ≤  (α – )   α–         t α–   α–    f s, u(s), u (s), D u(s) ds + Ii u(ti ) ( – s) + (α – )  t≤t  ≤

≤





α–

i



α–

( – s) ( – s) h(s) ds + h(s) ds (α – )   (α – )  η       ψ(s)u(s) ds + Li u(ti ) + (α – ) t≤t 

α– (α) + +

i



α– (α) α– (α)



α–

–p 



( – s) –p ds 

p

p

ds

h(s)

p

h(s)







α–

–p 



( – s) –p ds  m

p ds



 Lk uPC  + uPC 

k=

η

 ψ(s) ds



   η m   h/p α–   ψ(s) ds +λ Lk + (α) (α)  k=   m   –p   η   –p   ψ(s) ds = A + λ ≤ λ, ≤ + h/p + λ Lk + (α – ) α–p–  

–p ≤ (α – ) α–p–

–p

k=

and  α–  D (Tu)(t)    t       f s, u(s), u (s), Dα– u(s)  ds + ( – s)α– f s, u(s), u (s), Dα– u(s)  ds ≤ 



 η       Ii u(ti )  + (α – ) g s, u(s)  ds + t≤ti





≤







( – s)α– h(s) ds +

h(s) ds + 



 η      ψ(s)u(s) ds Li u(ti ) + (α – ) t≤ti



Zhao and Liang Advances in Difference Equations (2017) 2017:50





≤

h(s)

p

p



–p 

α–

h(s)



 + (α – )uPC  

–p = + α–p–



( – s) –p ds







+

ds

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p

p +

ds



m

Lk uPC 

k=

η

 ψ(s) ds



–p  h/p + λ

 m

  ψ(s) ds = A + λ ≤ λ.



η

Lk + (α – ) 

k=

Therefore, T(Bλ ) ⊂ Bλ . By Lemma ., we see that T: Bλ → Bλ is completely continuous. Thus BVP (.) has at least one solution by Lemma .. The proof is complete. 

4 Illustrative example As an application of the main results, we consider the following impulsive fractional differential equation with integral boundary conditions: ⎧   ⎪   ⎪  ⎪ ⎨tk Dt u(t) = f (t, u, u , D u),

t = tk ,

 

(.)

D u(tk ) = Ik (u(tk ), k = , . . . , m, ⎪ ⎪ η ⎪ ⎩u() = u () = , u () =  g(s, u(s)) ds, here t ∈ J = [, ], tk =  –

 k

(k = , , . . . , m), α =  , η =  .

Case  Let 

f t, u, u , D u =



 + h (u(t), u (t), D  u(t)) +

 Ik u(tk ) =



t  u(t)

 



+

e–t D  u(t) 

 + h (u(t), u (t), D  u(t))

u (t) 

( + t) ( + h (u(t), u (t), D  u(t))

|u(tk )| + , k  + |u(tk )|

g(t, u) =

,

t  u (t) ,  + u (t)



where hi (u, u , D  u) ≥  (i = , , ). By simple computation, we have ψ (s) =     , ψ (s) = e t , Lk = k , ψ(s) = s , (+s) 

m      ψ (s) + ψ (s) + ψ (s) ds + Lk + (α – )

 

ρ= 

   – e– + +    √  +  π < . <  =

k=



s , 

ψ (s) =

η

 ψ(s) ds



  (  )   +  – m+ +    

Thus, all the assumptions of Theorem . are satisfied. Hence BVP (.) has a unique solution on J = [, ]. Case  Take 

 t  (u + u + D  u)  . f t, u, u , D  u =   + (u + u + D  u)

Zhao and Liang Advances in Difference Equations (2017) 2017:50

Page 18 of 19



Ik (u(tk )) and g(t, u) are the same as Case . It is clear that |f (t, u, u , D  u)| ≤ t   h(t) and √ η m + π = k= Lk + (α – )  |ψ(s)| ds <  < . Thus, BVP (.) has at least one solution in J = [, ] by Theorem ..

5 Conclusions Compared with previous papers involving impulsive fractional order differential equations, the impulse of our boundary value problem (.) is related to the fractional order derivative, namely, Dα– u(tk ) = Ik (u(tk )). It is difficult and challenging to find the Green function of (.). Our results are new and interesting. Our methods can be used to study the existence of positive solutions for the high order or multiple-point boundary value problems of nonlinear fractional differential equation with the impulses involving the fractional order derivative. However, there exist some difficulties and complexities to address the structure of the Green function for these boundary value problems. Competing interests The authors declare to have no competing interests. Authors’ contributions The authors read and approved the final manuscript. Acknowledgements The authors would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the National Natural Sciences Foundation of Peoples Republic of China under Grant (No. 11161025; No. 11661047), and the Yunnan Province natural scientific research fund project (No. 2011FZ058). Received: 10 November 2016 Accepted: 30 January 2017 References 1. Kilbas, AA, Srivastava, H, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) 2. Podlubny, I: Fractional Differential Equation. Academic Press, San Diego (1999) 3. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993) 4. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) 5. Tarasov, VE: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Berlin (2010) 6. Diethelm, K: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) 7. Feng, M, Ge, W: Existence results for a class of nth order m-point boundary value problems in Banach spaces. Appl. Math. Lett. 22, 1303-1308 (2009) 8. Chang, Y, Nieto, JJ: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605-609 (2009) 9. Goodrich, C: Existence of a positive solution to a class of fractional differential equations. Comput. Math. Appl. 59, 3489-3499 (2010) 10. Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916-924 (2010) 11. Zhang, X, Liu, L, Wu, Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55(3), 1263-1274 (2012) 12. 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) 13. Zhang, S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59, 1300-1309 (2010) 14. Wang, Y, Liu, Y, Wu, Y: Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Anal., Theory Methods Appl. 74(17), 6434-6441 (2011) 15. Goodrich, C: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050-1055 (2010) 16. Salem, H: On the existence of continuous solutions for a singular system of nonlinear fractional differential equations. Appl. Math. Comput. 198, 445-452 (2008) 17. Jafari, H, Gejji, V: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180, 700-706 (2006) 18. Jiang, D, Yuan, C: The positive properties of the green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. TMA 72, 710-719 (2010) 19. Zhao, KH, Gong, P: Existence of positive solutions for a class of higher-order Caputo fractional differential equation. Qual. Theory Dyn. Syst. 14(1), 157-171 (2015) 20. Zhao, KH, Gong, P: Positive solutions of Riemann-Stieltjes integral boundary problems for the nonlinear coupling system involving fractional-order differential. Adv. Differ. Equ. 2014, 254 (2014)

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