Bai et al. Boundary Value Problems (2016) 2016:63 DOI 10.1186/s13661-016-0573-z

RESEARCH

Open Access

Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions Zhanbing Bai1* , Xiaoyu Dong1 and Chun Yin2 *

Correspondence: [email protected] 1 College of Mathematics and System Science, Shandong University of Science and Technology, Qianwangang Road, Qingdao, 266590, P.R. China Full list of author information is available at the end of the article

Abstract In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained. Our results are based on some fixed point theorems. Some examples are also presented to illustrate the main results. MSC: 34B15; 34A08 Keywords: fractional differential equations; impulse; mixed boundary value problem; fixed point theorem

1 Introduction Recently, boundary value problems of nonlinear fractional differential equations have been addressed by several researchers. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, fitting of experimental data, etc. For details, see [–] and the references therein. Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for a better understanding of several real world problems in the applied sciences. Recently, the boundary value problems of impulsive differential equations of integer order have been studied extensively in the literature (see [, –, –]). In [, ], Wang et al. gave a new concept of some impulsive differential equations with fractional derivative, which is a correction of that of piecewise continuous solutions used in [, , –]. This paper is strongly motivated by the above research papers. We investigate the existence and uniqueness of solutions for a mixed boundary value problem of nonlinear impulsive differential equations of fractional order given by ⎧ q C  ⎪ ⎨ D+ u(t) = f (t, u(t)), t ∈ J ,  u(tk ) = Ik (u(tk )), u (tk ) = Jk (u(tk )), ⎪ ⎩  u() + u () = , u() + u () = ,

k = , , . . . , p,

(.)

© 2016 Bai et al. 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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q

where C D+ is the Caputo fractional derivative of order q ∈ (, ), f ∈ C(J × R, R). Ik , Jk ∈ C(R, R), J = [, ], J  = J \ {t , t , . . . , tp }, the {tk } satisfy  = t < t < t < · · · < tp < tp+ = , p ∈ N , u(tk ) = u(tk+ ) – u(tk– ), u (tk ) = u (tk+ ) – u (tk– ), where u(tk+ ) and u(tk– ) represent the right and left limits of u(t) at t = tk . A function u ∈ PC(J, R) is said to be a solution of problem (.) if u(t) = uk (t) for q t ∈ (tk , tk+ ) and uk ∈ C([, tk+ ], R) satisfies C D+ u(t) = f (t, u(t)) a.e. on (, tk+ ) with the restriction that uk (t) on [, tk ) is just uk– (t) and the conditions u(tk ) = Ik (u(tk )), u (tk ) = Jk (u(tk )), k = , , . . . , p with u() + u () = , u() + u () = . The rest of this paper is organized as follows. In Section , we give some notations, recall some concepts and preparation results. In Section , we give the main results, the first result based on Banach contraction principle, the second result based on Krasnoselskii’s fixed point theorem. Two examples are given in Section  to demonstrate the application of our main results.

2 Preliminaries In this section, we introduce preliminary facts which are used throughout this paper. Let J = [, t ], J = (t , t ], . . . , Jp– = (tp– , tp ], Jp = (tp , ]. We have        PC(J) = u : [, ] → R | u ∈ C J  , and u tk+ , u tk– exist, and   u tk– = u(tk ),  ≤ k ≤ p . Obviously, PC(J) is a Banach space with the norm



uPC = sup u(t) . ≤t≤

Definition . The fractional integral of order q of a function f : [, ∞) → R is defined as q

I+ f (t) =

 (q)



t 

f (s) ds, (t – s)–q

t > , q > ,

(.)

provided the right side is point-wise defined on (, ∞), where (·) is the gamma function. Definition . The Caputo derivative of fractional order q for a function f : [, ∞) → R is defined as C

q D+ f (t) =

dn  (n – q) dt n



t 

sk (k) f (s) – n– () k= k! f ds, q–n+ (t – s)

t > , n = –[–q],

(.)

where [q] denotes the integer part of the real number q. q

n–q

Remark . In the case f (t) ∈ C n [, +∞), there is C D+ f (t) = I+ f (n) (t). That is to say that Definition . is just the usual Caputo’s fractional derivative. In this paper, we consider an impulsive problem, so Definition . is appropriate. Lemma . ([]) Let M be a closed, convex, and nonempty subset of a Banach space X, and A, B the operators such that () Ax + By ∈ M whenever x, y ∈ M;

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() A is compact and continuous; () B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Lemma . ([]) The set F ⊂ PC([, T], Rn ) is relatively compact if and only if: (i) F is bounded, that is, x ≤ C for each x ∈ F and some C > ; (ii) F is quasi-equicontinuous in [, T]. That is to say that for any  >  there exists δ >  such that if x ∈ F; k ∈ N ; τ , τ ∈ (tk– , tk ], and |τ – τ | < δ, we have |x(τ ) – x(τ )| < . Lemma . ([]) For q > , the general solution of the fractional differential equation q D+ u(t) =  is given by

C

u(t) = c + c t + c t  + · · · + cn– t n– , where ci ∈ R, i = , , , . . . , n – , n = –[–q]. In view of Lemma ., it follows that q

I+

C

q  D+ u (t) = u(t) + c + c t + c t  + · · · + cn– t n– ,

where ci ∈ R, i = , , , . . . , n – , n = –[–q]. Lemma . Let q ∈ (, ) and h : J → R be continuous. A function u given by ⎧

t

  –t q– ⎪ (t – s)q– h(s) ds + (q) h(s) ds ⎪ ⎪  ( – s) (q)  ⎪ ⎪

 ⎪ –t q– ⎪ + ( – s) h(s) ds, t ∈ [, t ]; ⎪ (q–)  ⎪ ⎪

t

 ⎪ ⎪  –t q– q– ⎪ (t – s) h(s) ds + h(s) ds ⎪   ( – s) (q) (q) ⎪ ⎪

⎪  p ⎪ + –t ⎪ ( – s)q– h(s) ds + ( – t) j= Jj (u(tj ))( – tj ) ⎪ (q–)  ⎨ p p p u(t) = + ( – t) j= Ij (u(tj )) – (t – tj ) j=k+ Jj (u(tj )) – j=k+ Ij (u(tj )), ⎪ ⎪ ⎪ ⎪ t ∈ (tk , tk+ ], k = , , . . . , p – ; ⎪ ⎪ ⎪ ⎪

t

 ⎪  –t q– ⎪ (t – s)q– h(s) ds + (q) h(s) ds ⎪   ( – s) (q) ⎪ ⎪

⎪  p ⎪ + –t ⎪ ( – s)q– h(s) ds + ( – t) j= Jj (u(tj ))( – tj ) ⎪ (q–)  ⎪ ⎪ ⎪ ⎩ + ( – t) p I (u(t )), t ∈ (t , t ], j p p+ j= j

(.)

is a unique solution of the following impulsive problem: ⎧ q C  ⎪ ⎪ ⎨ D+ u(t) = h(t), t ∈ J , u(tk ) = Ik (u(tk )), u (tk ) = Jk (u(tk )), ⎪ ⎪ ⎩ u() + u () = , u() + u () = .

(.)

k = , , . . . , p,

q

Proof With Lemma ., a general solution u of the equation C D+ u(t) = h(t) on each interval (tk , tk+ ] (k = , , , . . . , p) is given by u(t) =

 (q)



t

(t – s)q– h(s) ds + ak + bk t, 

for t ∈ (tk , tk+ ],

(.)

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where t =  and tp+ = . Then we have u (t) =

 (q – )



t

for t ∈ (tk , tk+ ].

(t – s)q– h(s) ds + bk ,

(.)



We have u () = b ,

u() = a , 

 u() = (q) u () =



( – s)q– h(s) ds + ap + bp , 

 (q – )





( – s)q– h(s) ds + bp . 

So applying the boundary conditions (.), we have a + b = ,       ( – s)q– h(s) ds + ( – s)q– h(s) ds + ap + bp = . (q)  (q – ) 

(.) (.)

Furthermore, using the impulsive condition u (tk ) = u (tk+ ) – u (tk– ) = Jk (u(tk )), we derive   bk = bk– + Jk u(tk ) , bk = bp –

p    Jj u(tj )

(.) (k = , , . . . , p – ).

(.)

j=k+

In the same way, using the impulsive condition u(tk ) = u(tk+ ) – u(tk– ) = Ik (u(tk )), we derive   ak + bk tk = ak– + bk– tk + Ik u(tk ) ,

(.)

which by (.) implies that     ak = ak– – Jk u(tk ) tk + Ik u(tk ) .

(.)

Thus ak = ap +

p p       Jj u(tj ) tj – Ij u(tj ) (k = , , , . . . , p – ). j=k+

(.)

j=k+

Combining (.), (.), (.) with (.) yields  ap = (q) –





( – s) 

q–

 h(s) ds + (q – )





( – s)q– h(s) ds 

p p       Jj u(tj ) (tj – ) +  Ij u(tj ) , j=

j=

(.)

Bai et al. Boundary Value Problems (2016) 2016:63

bp = –



 (q)

+

Page 5 of 11



( – s)q– h(s) ds – 



 (q – )



( – s)q– h(s) ds 

p p       Jj u(tj ) (tj – ) – Ij u(tj ) . j=

(.)

j=

Furthermore, by (.), (.), (.), (.) we have  ak = (q) –





( – s)

q–



 h(s) ds + (q – )

( – s)q– h(s) ds 

j=

      Jj u(tj ) tj – Ij u(tj ) (k = , , , . . . , p – ), p

p

j=k+

j=k+

 bk = – (q) +



p p       Jj u(tj ) (tj – ) +  Ij u(tj ) j=

+







( – s)

q–



 h(s) ds – (q – )



(.)



( – s)q– h(s) ds 

p p       Jj u(tj ) (tj – ) – Ij u(tj ) j=

j=

   Jj u(tj ) p

–

(k = , , , . . . , p – ).

(.)

j=k+

Hence for k = , , , . . . , p – , (.) and (.) imply a k + bk t =

–t (q)





( – s)q– h(s) ds + 

–t (q – )





( – s)q– h(s) ds 

p p       Jj u(tj ) ( – tj ) + ( – t) Ij u(tj )

+ ( – t)

j=

– (t – tj )

j=

p p       Jj u(tj ) – Ij u(tj ) . j=k+

(.)

j=k+

For k = p, (.) and (.) imply –t a k + bk t = (q)





( – s) 

+ ( – t)

q–

–t h(s) ds + (q – )





( – s)q– h(s) ds 

p p       Jj u(tj ) ( – tj ) + ( – t) Ij u(tj ) . j=

(.)

j=

Now it is clear that (.), (.), (.) imply that (.) holds. Conversely, assume that u satisfies (.). By a direct computation, it follows that the solution given by (.) satisfies (.). 

3 Main results This section deals with the existence and uniqueness of solutions to problem (.).

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Theorem . Let f : J × R → R be a continuous function. Suppose there exist positive constants L , L , L , M , M such that (A) |f (t, x) – f (t, y)| ≤ L |x – y|, for all t ∈ J, x, y ∈ R; (A) |Ik (x) – Ik (y)| ≤ L |x – y|, |Jk (x) – Jk (y)| ≤ L |x – y|, |Ik (x)| ≤ M , |Jk (x)| ≤ M , x, y ∈ R, k = , , . . . , p, with L ≤



(q + ) , ( + q)

L

   + + p(L + L ) < . (q + ) (q)

Then problem (.) has a unique solution on J. Proof Define an operator T : PC(J) → PC(J) 

     –t  (t – s)q– f s, u(s) ds + ( – s)q– f s, u(s) ds (q)     p      –t ( – s)q– f s, u(s) ds + ( – t) Jj u(tj ) ( – tj ) + (q – )  j=

 (q)

(Tu)(t) :=

t

+ ( – t)

p p p          Ij u(tj ) – (t – tj ) Jj u(tj ) – Ij u(tj ) , j=

j=k+

j=k+

t ∈ (tk , tk+ ], k = , , , . . . , p. Let supt∈J |f (t, )| = M, and Br = {u ∈ PC(J, R) | uPC ≤ r}, where  r≥

 +q M + p(M + M ) . (q + )

Step . We show that TBr ⊂ Br . For u ∈ Br , t ∈ J, we have



(Tu)(t)

 ≤ (q) +



t

(t – s)

 

f s, u(s) ds +

q–



 (q – )







 

( – s)q– f s, u(s) ds



        

Ij u(tj ) +

Jj u(tj ) +

Ij u(tj )

 ≤ (q)

+



j=

j=

+



p 



 



Jj u(tj )

( – s)q– f s, u(s) ds + 

p

+

 (q)



 (q)

t

(t – s) 

 

 (q – )

p

p

j=k+

j=k+



 f s, u(s) – f (s, ) ds +

q–



t

(t – s)



f (s, ) ds

q–



 



 ( – s)q– f s, u(s) – f (s, ) ds +

 

 





 ( – s)q– f s, u(s) – f (s, ) ds





( – s)q– f (s, ) ds



Bai et al. Boundary Value Problems (2016) 2016:63

+



 (q – )

+



p 



 

Jj u(tj )

( – s)q– f (s, ) ds + 



j=

p p p 

       

Ij u(tj ) +

Jj u(tj ) +

Ij u(tj )

j=

≤

Page 7 of 11

j=k+

j=k+

M L r M L r M L r + + + + + (q + ) (q + ) (q + ) (q + ) (q) (q) + pM + pM + pM + pM

= L

+p +p r+ M + p(M + M ). (q + ) (q + )

Since 

 +q r≥ M + p(M + M ) , (q + )

(q + ) L ≤ , ( + q) we have



(Tu)(t) ≤ r,

TBr ⊂ Br .

Step . T is a contraction mapping. For x, y ∈ Br and t ∈ J, we have



(Tx)(t) – (Ty)(t)



  t     –t 

q– =

(t – s) f s, x(s) ds + ( – s)q– f s, x(s) ds

(q)  (q)  

–t + (q – ) + ( – t)  –

+



     ( – s)q– f s, x(s) ds + ( – t) Jj x(tj ) (tj – ) p



j=

p          Ij x(tj ) – (t – tj ) Jj x(tj ) – Ij x(tj ) p

p

j=

j=k+



 (q)

t



–t (q – )

  –t (t – s)q– f s, y(s) ds + (q)





j=k+





  ( – s)q– f s, y(s) ds



     ( – s)q– f s, y(s) ds + ( – t) Jj y(tj ) (tj – ) p



j=



p p p         

Ij y(tj ) – (t – tj ) Jj y(tj ) – Ij y(tj )

+ ( – t)

j=

≤

 (q)



t

j=k+

   

(t – s)q– f s, x(s) – f s, y(s) ds



 + (q) +

j=k+







 (q – )

   

( – s)q– f s, x(s) – f s, y(s) ds  



   

( – s)q– f s, x(s) – f s, y(s) ds

Bai et al. Boundary Value Problems (2016) 2016:63

+

Page 8 of 11

p p  



  

 

 

Jj x(tj ) – Jj y(tj ) + 

Ij x(tj ) – Ij y(tj )

j=

+

j=

p p 

        

Jj x(tj ) – Jj y(tj ) +

Ij x(tj ) – Ij y(tj )

j=k+

≤

j=k+

L L L x – yPC + x – yPC + x – yPC (q + ) (q + ) (q) +

p 

L x – yPC + 

p 

j=

+

p 

L x – yPC +

j=k+

≤

L x – yPC

j= p 

L x – yPC

k=j+

L L x – yPC + x – yPC + pL x – yPC (q + ) (q)

+ pL x – yPC + pL x – yPC + pL x – yPC       + + p(L + L ) x – yPC . = L (q + ) (q) Since  L

   + + p(L + L ) < , (q + ) (q)

T is a contraction mapping. Thus, the conclusion follows by the contraction mapping principle.  Theorem . Assume that |f (t, u)| ≤ μ(t) for all (t, u) ∈ J × R where μ ∈ L/σ (J, R) and σ ∈ (, q – ), furthermore, there exist positive constants L , L , M , M such that |Ik (x) – Ik (y)| ≤ L |x – y|, |Jk (x) – Jk (y)| ≤ L |x – y|, |Ik (x)| ≤ M , |Jk (x)| ≤ M , x, y ∈ R, k = , , . . . , p, with p(L + L ) < . Then problem (.) has at least one solution on J. Proof Choose  r ≥ μ



L σ (J)

 (q)( q–σ )–σ –σ

+





+ p(M + M )

– –σ (q – )( q–σ ) –σ

and denote  Br = u ∈ PC(J, R) | uPC ≤ r . Define the operators P and Q on Br as  (Pu)(t) = (q) +



t

(t – s) 

–t (q – )



q–







–t f s, u(s) ds + (q) 



( – s)q– f s, u(s) ds, 



 

  ( – s)q– f s, u(s) ds

Bai et al. Boundary Value Problems (2016) 2016:63

(Qu)(t) = ( – t)

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p p       Jk u(tk ) ( – tk ) + ( – t) Ik u(tk ) k=

k= p p       Jk u(tk ) – Ik u(tk ) .

– (t – tk )

k=j+

k=j+

For any u, v ∈ Br and t ∈ J, using the condition that |f (t, u)| ≤ μ(t) and the Hölder inequality, 

t

 

(t – s)q– f s, u(s) ds



 ≤



–σ 

q–

t

(t – s) –σ ds 

σ

 σ

ds

μ(s)

 σ

ds

μ(s)

 σ

μ(s)

≤



μ



μ



L σ (J) , ( q–σ )–σ –σ

t

 

( – s)q– f s, u(s) ds



 ≤



t

t

–σ 

q–

( – s) –σ ds 

t

σ ≤



L σ (J) , ( q–σ )–σ –σ

t

 

( – s)q– f s, u(s) ds



 ≤

t

( – s)

q– –σ

–σ  ds



t

σ ds



≤

μ



L σ (J) q–σ – –σ ( –σ )

.

Therefore, Pu + QvPC ≤

μ

μ



L σ (J) (q)( q–σ )–σ –σ

+



L σ (J) – –σ (q – )( q–σ ) –σ

+ pM + pM + pM + pM     + p(M + M ). = μ σ + – –σ L (J) (q)( q–σ )–σ (q – )( q–σ ) –σ –σ Thus Pu + Qv ∈ Br . It is obvious that Q is a contraction mapping (the proof is just similar to Theorem .). On the other hand, the continuity of f implies that the operator P is continuous. Also, P is uniformly bounded on Br since PuPC ≤

μ

μ



L σ (J) (q)( q–σ )–σ –σ

+



L σ (J) – –σ (q – )( q–σ ) –σ

≤ r.

Now we prove the quasi-equicontinuity of the operator P. Let  = J × Br , fmax = sup(t,u)∈ |f (t, u)|. For any tk < τ < τ ≤ tk+ , we have



(Pu)(τ ) – (Pu)(τ )

 τ 

      – τ  =

(τ – s)q– f s, u(s) ds + ( – s)q– f s, u(s) ds (q)  (q) 

Bai et al. Boundary Value Problems (2016) 2016:63

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   τ      – τ  ( – s)q– f s, u(s) ds – (τ – s)q– f s, u(s) ds (q – )  (q) 

       

 – τ  – τ q– q– – ( – s) f s, u(s) ds – ( – s) f s, u(s) ds

(q)  (q – ) 

  τ

 fmax

τ  q– q– q–

(τ ds + – s) – (τ – s) (τ – s) ds ≤   



(q)  τ



 

(τ – τ )fmax 

(τ – τ )fmax 

+

( – s)q– ds

+

( – s)q– ds

(q) (q – )     q q (τ – τ )q + τ – τ + τ – τ τ – τ + , ≤ fmax (q + ) (q) +

which tends to zero as τ → τ . This shows that P is quasi-equicontinuous on the interval (tk , tk+ ]. It is obvious that P is compact by Lemma ., so P is relatively compact on Br . Thus all the assumptions of Lemma . are satisfied and problem (.) has at least one solution on J. 

4 Example Example . Consider the following impulsive fractional boundary value problem: ⎧   sin u(t)  C  ⎪ ⎪ ⎪ D+ u(t) = (t+) +u (t) , t ∈ [, ], t =  , ⎨   |u( )| |u( )| u (  ) = +|u(  )| u(  ) = +|u(  )| , ⎪   ⎪ ⎪ ⎩u() + u () = , u() + u () = .

(.)

Obviously, L = /, L = /, L = /, M = /, M = /, p = , √ (q + ) (q + )  π = , L < , ( + q)  ( + q)       L + + p(L + L ) = √ + < . (q + ) (q)  π  Thus, all the assumptions in Theorem . are satisfied. Hence, the impulsive fractional boundary value problem (.) has a unique solution on [, ]. Example . Consider the following impulsive fractional boundary value problem: ⎧  |u(t)| et C  ⎪ ⎪ D u(t) = (t+) t ∈ [, ], t =  ,  +|u(t)| , ⎪ ⎨ +  +|u( )| |u(  )| u (  ) = +|u( )| u(  ) = +|u( )| , ⎪   ⎪ ⎪ ⎩u() + u () = , u() + u () = . Set f (t, u) =

|u| et , (t + )  + |u|

Obviously,



f (t, u) ≤

et . (t + )

(t, u) ∈ [, ] × [, ∞).

(.)

Bai et al. Boundary Value Problems (2016) 2016:63

Page 11 of 11

Set L = L = ,

M = ,

M =  and

μ(t) =

  et ∈ L [, ], R .  (t + )

Thus, all the assumptions in Theorem . are satisfied. Hence, the impulsive fractional boundary value problem (.) has at least one solution on [, ].

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 College of Mathematics and System Science, Shandong University of Science and Technology, Qianwangang Road, Qingdao, 266590, P.R. China. 2 School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, P.R. China. Acknowledgements The authors express their sincere thanks to the anonymous reviews for their valuable suggestions and corrections for improving the quality of the paper. This work is supported by NSFC (11571207, 61503064), the Taishan Scholar project. Received: 26 October 2015 Accepted: 8 March 2016 References 1. Ahmad, B, Sivasundaram, S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3, 251-258 (2009) 2. Bainov, D, Simeonov, P: Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics (1993) 3. Benchohra, M, Seba, D: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2009, 8 (2009) 4. Feckan, M, Zhou, Y, Wang, JR: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050-3060 (2012) 5. Guo, TL, Jiang, W: Impulsive fractional functional differential equations. Comput. Math. Appl. 64, 3414-3424 (2012) 6. Jankowski, T: Initial value problems for neutral fractional differential equations involving a Riemann-Liouville derivative. Appl. Math. Comput. 219, 7772-7776 (2013) 7. Liang, SH, Zhang, JH: Existence and uniqueness of positive solutions to m-point boundary value problem for nonlinear fractional differential equations. J. Appl. Math. Comput. 38, 225-241 (2012) 8. Lin, X, Jiang, D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501-514 (2006) 9. Shen, J, Wang, W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 69, 4055-4062 (2008) 10. Wang, GT, Agarwal, RP, Cabada, A: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25, 1019-1024 (2012) 11. Wang, GT, Ahmad, B, Zhang, LH, Nieto, JJ: Comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 19, 401-403 (2014) 12. Wang, GT, Zhang, LH, Song, GX: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. Nonlinear Anal. TMA 74, 974-982 (2011) 13. Wang, JR, Zhou, Y, Feckan, M: On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl. 64, 3008-3020 (2012)