Mahmudov and Unul Advances in Difference Equations (2017) 2017:15 DOI 10.1186/s13662-016-1063-4

RESEARCH

Open Access

On existence of BVP’s for impulsive fractional differential equations NI Mahmudov and S Unul* *

Correspondence: [email protected] Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey

Abstract In this research, the existence of the solutions for an impulsive fractional differential equation of order q with mixed boundary conditions is studied by using some well-known fixed point theorems. At last, an example is presented to illustrate our results.

1 Introduction The boundary value problems of fractional differential equations have attracted the attention of many authors. Fractional differential equations are used in mathematical modelling, engineering, biology, chemistry, and many other fields of science; see the references. However, the impulsive fractional differential equations has become a new topic, therefore more researchers interest focused on the field of impulsive problems for fractional differential equations; see [–] and the references therein. Tian and Bai, [] used the Banach fixed point theorem and Schauder’s fixed point theorem to obtain the existence of the solutions of the problem which is given as follows:   Dα+ u(t) = f t, u(t) ,   u(t)t=tk = Ik u(t) , k = , , . . . , m,   u (t)t=tk = I¯k u(t) , k = , , . . . , m, c

u() + u () = , u() + u (ξ ) = . The existence and uniqueness of the solutions for an anti-periodic BVP of nonlinear impulsive differential equations of order α ∈ (, ] were obtained, in  [], given in the following:    Dα+ u(t) = f t, u(t) ,  < α ≤    u(tk ) = Qk u(tk ) , k = , , . . . , p,   u (tk ) = Ik u(tk ) , k = , , . . . , p, c

© 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.

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

  u (tk ) = Ik∗ u(tk ) ,

Page 2 of 16

k = , , . . . , p,

u() = –u(), u () = –u (), u () = –u (), with the Caputo fractional derivative c Dα+ , f ∈ C(J × R, R) and Qk , Ik , Ik∗ ∈ C(R × R),  = t < t < · · · < tk < · · · < tp < tp+ = . In , Cao and Chen, [], studied the following problem to give some existence results and a continuous version of Filippov’s theorem of a fractional differential inclusion:    Dα+ u(t) ∈ f t, u(t) , a.e. t ∈ J   u(t)t=tk = Ik u(t) , k = , , . . . , m,     β D+ u(t)t=tk = I¯k u(t) , k = , , . . . , m, C

β

u() + D+ u() = A, β

u() + D+ u(ζ ) = B. Here, C Dα+ is the Caputo fractional derivative and multi-valued map with compact values F : J × R → P(R) where P(R) is the family of all nonempty subsets of R,  < α ≤  and  < β < α –  with real numbers A, B. In , the contraction mapping principle, Krasnoselskii’s theorem, Schaefer’s theorem, and the Leray-Schauder alternative were used, in [], to find the existence of the solutions of the following problem: C

  q D+ u(t) = f t, u(t) ,

u(tk ) = yk , u (tk ) = y¯ k ,

k = , . . . , m

u() = u ,

u () = u¯  ;

yk , y¯ k , u , u¯  ∈ R.

By using fixed point theorems, the existence and uniqueness solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order were studied in , [], which is given as   q D+ u(t) = f t, u(t) ,   u(tk ) = Ik u(tk ) , C

u() + u () = ,

t ∈ J

  u (tk ) = Jk u(tk ) , u() + u () = ,

q

where q ∈ (, ) and C D+ is the Caputo derivative of order q. Motivated by the above mentioned work, we focus on the existence of solutions of fractional differential equation: C

  q D+ u(t) = f t, u(t) ,

t ∈ J ,

()

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 3 of 16

with boundary conditions;       u(tk ) = Ik u(tk ) = u tk+ – u tk– ,       u (tk ) = Jk u(tk ) = u tk+ – u tk– ;

k = , . . . , p

u() + μ u () = σ , u () + μ u() = σ ,

()

where C D+ is the Caputo derivative of order q ∈ (, ), J = [, ], J  = J\{t , t , . . . , tp },  = t < t < · · · < tp < tp+ = , u(tk ) = u(tk+ ) – u(tk– ) and u (tk ) = u (tk+ ) – u (tk– ). Here, respectively, the right and the left limits of u(t) at t = tk+ are represented by u(tk+ ) and u(tk– ). q

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

Definition  The Riemann-Liouville fractional integral of order α >  for a function f : [, +∞) → R is defined as α f (t) = I+

 (α)



t

(t – s)α– f (s) ds, 

provided that the right hand side of the integral is pointwise defined on (, +∞) and  is the gamma function. Definition  The Caputo derivative of order α >  for a function f : [, +∞) → R is written as Dα+ f (t) =

 (n – α)



t

(t – s)n–α– f (n) (s) ds, 

where n = [α] + , [α] is the integral part of α. Lemma  Let α > . Then the differential equation Dα+ f (t) =  has solutions f (t) = k + k t + k t  + · · · + kn– t n– and α I+ Dα+ f (t) = f (t) + k + k t + k t  + · · · + kn– t n– ,

where ki ∈ R and i = , , . . . , n = [α] + .

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 4 of 16

Lemma  ([]) The set F ⊂ PC([, ], Rn ) is relatively compact if and only if F is bounded, that is, x ≤ C for each x ∈ F and some C > , and/or F is quasi-equicontinuous in [, ]. That is to say, for any ε >  there exists δ >  such that if x ∈ F, k ∈ N ; τ , τ ∈ (tk– , tk ] and |τ – τ | < δ, we have |x(τ ) – x(τ )| < ε. Lemma  ([]) Let M be a closed, convex, and nonempty subset of Banach space X, and the operators A and B be such that (i) Ax + By ∈ M whenever x, y ∈ M; (ii) A is compact and continuous; (iii) B is contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Lemma  For q ∈ (, ), and the continuous function f : J → R, we have the following impulsive fractional boundary value problem:   q D+ u(t) = f t, u(t) ,       u(tk ) = Ik u(tk ) = u tk+ – u tk– ,       u (tk ) = Jk u(tk ) = u tk+ – u tk– ;

C

k = , . . . , p,

u() + μ u () = σ , u () + μ u() = σ , has a unique solution, and Green’s function is given by ⎧  (–s)q– t (t–s)q– ⎪ ⎪ ⎪  (q) f (s) ds – μ ω (t)  (q) f (s) ds ⎪ ⎪  q– ⎪ ⎪ – μ ω (t)  (–s) f (s) ds + σ ω (t) + σ ω (t), t ∈ [, t ], ⎪ ⎪ (q–) ⎪ ⎪  q– q– t ⎪ (t–s) ⎪ f (s) ds – μ ω (t)  (–s) f (s) ds ⎪  (q) (q) ⎪ ⎪ ⎪ q–  ⎪ ⎪ – μ ω (t)  (–s) f (s) ds + σ ω (t) + σ ω (t) ⎪ (q–) ⎪ ⎪

p

p ⎪ ⎪ ⎪ ⎪ ⎨ – ω (t) j= Jj (u(tj ))tj – ω (t) j= Jj (u(tj )) p u(t) = + ω (t) j= Ij (u(tj )) ⎪ ⎪

p

p ⎪ ⎪ + j=k+ Jj (u(tj ))(tj – t) – j=k+ Ij (u(tj )); t ∈ [tk , tk+ ] ⎪ ⎪ ⎪ ⎪  (–s)q– t (t–s)q– ⎪ ⎪ ⎪  (q) f (s) ds – μ ω (t)  (q) f (s) ds ⎪ ⎪  ⎪ q– ⎪ ⎪ – μ ω (t)  (–s) f (s) ds + σ ω (t) + σ ω (t) ⎪ (q–) ⎪ ⎪

p

p ⎪ ⎪ ⎪ – ω (t) j= Jj (u(tj ))tj – ω (t) j= Jj (u(tj )) ⎪ ⎪ ⎪ ⎪ ⎩ + ω (t) p Ij (u(tj )), t ∈ [tp , tp+ ], j=

where  + μ – μ t  + μ – μ μ

ω (t) =

and ω (t) =

t – μ .  + μ – μ μ

q

Proof A general solution C D+ u(t) = f (t, u(t)), on (tk , tk+ ], k = , . . . , p,  u(t) = 

t

(t – s)q– f (s) ds + ak + bk t, (q)

for t ∈ (tk , tk+ ],

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

where t = , tp+ =  and taking the derivative, 



t

(t – s)q– f (s) ds + bk , (q – )

u (t) = 

for t ∈ (tk , tk+ ].

We use the boundary conditions u() + μ u () = σ and u () + μ u() = σ to get 



( – s)q– f (s) ds + μ bp = σ (q – )

t

(t – s)q– f (s) ds + μ ap + μ bp = σ , (q)

a + μ 

and  b + μ 

where u() = a , u () = b ,  t ( – s)q– f (s) ds + ap + bp , u() = (q)   t ( – s)q–  u () = f (s) ds + bp .  (q – ) That is,   u (tk ) = Jk u(tk )     = u tk+ – u tk– = bk – bk– ,   bk = bk– + Jk u(tk ) ,   bk+ = bk + Jk+ u(tk + ) , bp = bk– +

p    Jj u(tj ) , j=k

bk = bp –

p    Jj u(tj ) , j=k+

and       u(tk ) = Ik u(tk ) = u tk+ – u tk– ,   ak + bk tk = ak– + bk– tk + Ik u(tk ) . Since bk = bk– + Jk (u(tk )), we have      ak + bk– + Jk u(tk ) tk = ak– + bk– tk + Ik u(tk ) ,     ak + bk– tk + Jk u(tk ) tk = ak– + bk– tk + Ik u(tk ) ,

Page 5 of 16

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 6 of 16

    ak + Jk u(tk ) tk = ak– + Ik u(tk ) ,     ak = ak– – Jk u(tk ) tk + Ik u(tk ) , ak = ap +

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

j=k+

Then 



a + μ bp + μ 

( – s)q– f (s) ds = σ (q – )

()

and 



b + μ ap + μ bp + μ 

( – s)q– f (s) ds = σ . (q)

()

Also we get   bk = bk– + Jk u(tk ) , bk = bp –

p    Jj u(tj ) ,

()

j=k+

and     ak = ak– – Jk u(tk ) tk + Ik u(tk ) , ak = ap +

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

()

j=k+

By combining (), (), (), and () 



( – s)q– f (s) ds = σ , (q – )



      ( – s)q– Jj u(tj ) tj – Ij u(tj ) = σ , f (s) ds + (q – ) j= j=

a + μ bp + μ 

 ap + μ bp + μ 

p

p

and 







( – s)q– f (s) ds = σ , (q)      p    ( – s)q– bp – f (s) ds = σ , Jk u(tj ) + μ ap + μ bp + μ (q)  j=

b + μ ap + μ bp + μ

μ ap + ( + μ )bp + μ 

   ( – s)q– f (s) ds – Jj u(tj ) = σ . (q) j= p

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 7 of 16

Then 



ap = σ – μ bp – μ 

ap =

σ ( + μ ) – bp – μ μ

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



 

p

p     ( – s)q– f (s) ds + Jj u(tj ) . (q) μ j=

Also we have 



σ – μ bp – μ 

=

      ( – s)q– f (s) ds – Jj u(tj ) tj + Ij u(tj ) (q – ) j= j=

σ ( + μ ) – bp – μ μ

p

 



p

p  ( – s)q–    f (s) ds + Jj u(tj ) . (q) μ j=

Therefore ap and bp are found as follows:    μ  σ + σ  + μ – μ μ  + μ – μ μ    ( – s)q– μ f (s) ds –  + μ – μ μ (q)     ( – s)q– μ μ f (s) ds +  + μ – μ μ  (q – )     p p     μ  + Jj u(tj ) tj + Jj u(tj )  + μ – μ μ j=  + μ – μ μ j=

 bp = –



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

()

and  ap =

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

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

()

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 8 of 16

By (), (), (), and () since

bk = bp –

p    Jj u(tj ) , j=k+

p p       ak = ap + Jj u(tj ) tj – Ij u(tj ) , j=k+

j=k+

are known. By (),    μ  σ + σ  + μ – μ μ  + μ – μ μ       ( – s)q– ( – s)q– μ μ μ – f (s) ds + f (s) ds  + μ – μ μ (q)  + μ – μ μ   (q – )     p p      μ Jj u(tj ) tj + Jj u(tj ) +  + μ – μ μ j=  + μ – μ μ j=

 bk = –

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

()

j=k+

with the help of (),  ak =

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

j=k+

for k = , , . . . , p – . By using () and (), we get     + μ – μ t t – μ σ + σ  + μ – μ μ  + μ – μ μ    –μ (t – μ ) ( – s)q– f (s) ds +  + μ – μ μ  (q)    ( – s)q– –μ ( + μ – μ t) + f (s) ds  + μ – μ μ  (q – )  p    + μ ( – t)   Jj u(tj ) tj –  + μ – μ μ j=

 a k + bk t =

j=k+

()

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15



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



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

+

+

Page 9 of 16

p p       Jj u(tj ) (tj – t) – Ij u(tj ) .

+

j=k+

j=k+

Thus 

t

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

u(t) =

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

+

j=

      Jj u(tj ) (tj – t) – Ij u(tj ) , p

p

j=k+

j=k+

where ω (t) =

 + μ – μ t  + μ – μ μ

and

ω (t) =

t – μ .  + μ – μ μ



2.1 Existence and uniqueness results In this section, we state and prove existence and uniqueness results of the fractional BVP ()-() by using the Banach fixed point theorem. We use the following notations throughout this paper: ω (t) =

 + μ – μ t ,  + μ – μ μ

ω (t) =

t – μ  + μ – μ μ

and ω (t) ≤ ω :=

 + |μ | , | + μ – μ μ |

ω (t) ≤ ω :=

 + |μ | . | + μ – μ μ |

By using the following conditions, we state and prove our first result. (A) The function f : [, ] × R → R is jointly continuous. (A) There exist positive constants L , L , L , M , M such that   f (t, x) – f (t, y) ≤ L |x – y|, t ∈ [, ], x, y ∈ R;     Ik (x) – Ik (y) ≤ L |x – y|, Jk (x) – Jk (y) ≤ L |x – y|,     Jk (x) ≤ M . Ik (x) ≤ M ,

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 10 of 16

Also it is clear that       f (t, x) ≤ f (t, x) – f (t, ) + f (t, ) ≤ L |x| + M, where supt∈[,] |f (t, )| = M. Theorem  Assume (A)-(A) holds. If  L

 + |μ |ω |μ |ω + (q + ) (q)

 + (ω + )p(L + L ) + ω pL < ,

()

then our boundary value problem ()-() has a unique solution on [, ]. Proof By using () r can be chosen as follows:  r > –

 L L   + |μ |ω – |μ |ω (q + ) (q)

– 

M + ω |σ | + ω |σ | (q + )

M M + |μ |ω (q + ) (q)

+ |μ |ω

 + (ω + )p(M + M ) + ω pM . Define an operator T : PC([, ], R) → PC([, ], R) to transform ()-() into the fixed point problem  (T u)(t) =

 (t – s)q–  f s, u(s) ds + ω (t)σ + ω (t)σ (q)     ( – s)q–  f s, u(s) ds – μ ω (t) (q)     ( – s)q–  f s, u(s) ds – μ ω (t)  (q – ) t

– ω (t)

p p       Jj u(tj ) tj – ω (t) Jj u(tj ) j=

+ ω (t)

j=

p    Ij u(tj ) j=

p       Jj u(tj ) (tj – t) – Ij u(tj ) , p

+

j=k+

j=k+

where tk < t < tk+ , k = , . . . , p. Then   Tu(t) ≤



     (t – s)q–   f s, u(s)  ds + ω (t)|σ | + ω (t)|σ | (q)      ( – s)q–    f s, u(s)  ds + |μ |ω (t) (q)  t

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

  + |μ |ω (t)







Page 11 of 16

 ( – s)q–   f s, u(s)  ds (q – )

p           Jj u(tj )  + ω (t) Jj u(tj )  + ω (t) p

j=

j=

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

j=k+

j=k+

and then   T u(t)  t  t    (t – s)q–   (t – s)q–  f s, u(s) – f (s, ) ds + f (s, ) ds ≤ (q) (q)       + ω (t)|σ | + ω (t)|σ |           ( – s)q–   ( – s)q–      f s, u(s) – f (s, ) ds + f (s, ) ds + |μ | ω (t) (q) (q)             ( – s)q–   ( – s)q–  + |μ |ω (t) f s, u(s) – f (s, ) ds + f (s, ) ds  (q – )  (q – ) p p p                  Ij u(tj )          Jj u(tj ) + ω (t) Jj u(tj ) + ω (t) + ω (t) j=

+

j=

j=

       Jj u(tj )  + Ij u(tj ) . p

p

j=k+

j=k+

Thus   T u(t) ≤

M L r + + ω |σ | + ω |σ | (q + ) (q + )   M L r + |μ |ω + (q + ) (q + )   M L r + + |μ |ω (q) (q) + (ω + )p(M + M ) + ω pM < r.

For t ∈ [, ], the expression is well defined. The fixed point of the operator T is the solution of our boundary value problem ()-(). To show the existence and uniqueness of the solution, the Banach fixed point theorem is used and then it is shown that T is a contraction and we get   (T x)(t) – (T y)(t)  t    (t – s)q–   f s, x(s) – f s, y(s)  ds ≤ (q) 

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

   + |μ |ω (t)   + |μ |ω (t)

 



 

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

Page 12 of 16



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

       Jj x(tj ) – Jj y(tj )  + ω (t) p

j= p        Jj x(tj ) – Jj y(tj )    + ω (t) j= p        Ij x(tj ) – Ij y(tj )  + ω (t) j=

+

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

j=k+

Thus   (T x)(t) – (T y)(t)    |μ ||ω (t)| |μ ||ω (t)|  + + ≤ L (q + ) (q + ) (q)       + ω (t) +  p(L + L ) + ω (t)pL x – y.

()

T is contraction mapping. By condition (), we have Tx – Ty     + |μ |ω |μ |ω + ≤ L (q + ) (q)

 + (ω + )p(L + L ) + ω pL x – y.

Thus T is a contraction mapping. T has a fixed point, and that is the solution of the BVP by the Banach fixed point theorem.  

Theorem  Assume |f (t, u)| ≤ ρ(t) for (t, u) ∈ J × R where ρ ∈ L σ (J × R) and σ ∈ (, q – ), moreover, there exist positive constants L , L , L , M , M and M such that   f (t, x) – f (t, y) ≤ L |x – y|, t ∈ [, ], x, y ∈ R;     Ik (x) – Ik (y) ≤ L |x – y|, Jk (x) – Jk (y) ≤ L |x – y|,     Ik (x) ≤ M , Jk (x) ≤ M , with (ω + )p(L + L ) + ω pL < . Then our boundary value problem has at least one solution on J.

()

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 13 of 16

Proof Let us choose  r ≥ ρ



|μ |ω ( + |μ |ω ) q–σ –σ + – –σ (q)( –σ ) (q – )( q–σ ) –σ  + (ω + )p(M + M ) + ω pM ,



 Lσ

and Br = {u ∈ PC(J, R) | uPC ≤ r}. The operators S and N on Br are defined as     (t – s)q–  ( – s)q–  f s, u(s) ds – μ ω (t) f s, u(s) ds (q) (q)     q–   ( – s) – μ ω (t) f s, u(s) ds  (q – )

 (Su)(t) =

t

and p p p          Jj u(tj ) tj – ω (t) Jj u(tj ) + ω (t) Ij u(tj ) (Nu)(t) = –ω (t) j=

+

j=

j=

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

j=k+

For any u, v ∈ Br and t ∈ J, by using |f (t, u)| ≤ ρ(t) and the Hölder inequality,  (q)



t

  (t – s)q– f s, u(s)  ds ≤

 (q)





t

(t – s)

  ( – s)q– f s, u(s)  ds ≤



≤

 (q)

–σ  ds

ρ(s)

 σ

σ ds





ρ

t



Lσ (q)( q–σ )–σ –σ



q– –σ



ρ

≤  (q)



,



( – s)

q– –σ

–σ 



ds

ρ(s)



σ

 σ

ds





Lσ (q)( q–σ )–σ –σ

,

and at last  (q – )





  ( – s)q– f s, u(s)  ds ≤



≤

 (q – )



ρ



( – s) 

We get Su + Nv ≤

( + |μ |ω )ρ (q)( q–σ )–σ –σ



Lσ

+

|μ |ω ρ



Lσ – –σ (q – )( q–σ ) –σ

+ (ω + )p(M + M ) + ω pM .

–σ 



ds

ρ(s)





Lσ – –σ (q – )( q–σ ) –σ

q– –σ

.

 σ

σ ds

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 14 of 16

Thus Su + Nv ∈ Br . By (), it is obvious that N is a contraction mapping. Moreover, the continuity of f implies S is continuous. And the operator S is uniformly bounded on Br where Su ≤

( + |μ |ω )ρ (q)( q–σ )–σ –σ



Lσ

+

|μ |ω ρ



Lσ q–σ – –σ (q – )( –σ )

≤ r.

Here the quasi-equicontinuity of the operator S is proved. Let  = J × Br , fsup = sup(t,u)∈ |f (t, u)|. For any tk < t < t < tk+ , we have   (Su)(t ) – (Su)(t )    t  fsup  t  (t – s)q–  q– q– (t ds + – s) – (t – s) ds ≤    (q)   (q) t      ( – s)q–  ds + |μ |(t – t ) (q)       ( – s)q–  + |μ ||μ |(t – t ) ds  (q – )  q q q q q [(t – t )] (t – t ) + t – t + |μ |   ≤ fsup (q + ) (q + ) q q  |μ |(t – t ) + |μ | . (q) It tends to zero as t → t . On the interval (tk , tk+ ], S is quasi-equicontinuous. Also by lemma ( ), S is compact and is relatively compact on Br . Therefore our BVP has at least one solution on J = [, ]. 

2.2 Examples Example  Consider the following boundary value problem of fractional differential equation: ⎧  u(t)  ⎪ u(t) = (t+)cos (+u D+ ⎪  (t)) , ⎪ ⎪  ⎪ |u(  )| ⎪  ⎪ u(  ) = +|u(  )| , ⎪ ⎪ ⎨  |u(  )|   u ( ) = ,  ⎪ +|u(  )| ⎪ ⎪ ⎪ ⎪ ⎪ u() + u() = , ⎪ ⎪ ⎪ ⎩ u() + u() = . Here t ∈ [, ], let  q= , 

 t= , 

σ = σ = , L = L = L = .,

μ = μ = ,

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 15 of 16

and since . < (.) < . and . < (.) < ., we found ω ≤ 

ω ≤ .

Therefore,  .

 . ( + .) + + (. + .)(. + ) + .(.) < , (.) (.)

.(. + .) + (.)(.) + (.)(.) < , . + . + . < , . < . Thus, by Theorem , the BVP has a unique 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. Received: 2 August 2016 Accepted: 22 December 2016 References 1. Bai, Z, Dong, X, Yin, C: Existence results for impulsive non-linear fractional differential equation with mixed boundary conditions. Bound. Value Probl. (2016) 2. Feckan, M, Zhou, Y, Wang, J: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050-3060 (2012) 3. Shen, J, Wang, W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 69, 4055-4062 (2008) 4. Agarwal, R, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973-1033 (2010) 5. Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4, 134-141 (2010) 6. Cao, J, Chen, H: Some results on impulsive boundary value problem for fractional differential inclusions. Electron. J. Qual. Theory Differ. Equ. 2010 11 (2011) 7. Tian, Y, Bai, Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59, 2601-2609 (2010) 8. Benchohra, M, Seba, D: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I 2009 8 (2009) 9. Bainov, D, Simeonov, P: Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics (1993) 10. Wang, J, Feckan, M, Zhou, Y: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19(2016), 806-831 (2016) 11. Wang, J, Feckan, M, Zhou, Y: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64(2012), 3389-3405 (2012) 12. Wang, J, Feckan, M, Zhou, Y: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395(2012), 258-264 (2012) 13. Wang, J, Zhang, Y: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 39(2015), 85-90 (2015) 14. Wang, J, Ibrahim, AG, Feckan, M: Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Appl. Math. Comput. 257(2015), 103-118 (2015) 15. Yu, X, Debbouche, A, Wang, J: On the iterative learning control of fractional impulsive evolution equations in Banach spaces. Math. Methods Appl. Sci. (2015) 16. Liu, S, Wang, J, Wei, W: Iterative learning control based on a noninstantaneous impulsive fractional-order system. J. Vib. Control 22(2016), 1972-1979 (2016) 17. Wang, G, Ahmad, B, Zhang, L: Impulsive anti-periodic boundary value problem for non-linear differential equations of fractional order. Nonlinear Anal. 74(2011), 792-804 (2010) 18. Wang, J, Li, X, Wei, W: On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2). Commun. Nonlinear Sci. Numer. Simul. 17, 4384-4394 (2012) 19. Wang, J, 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) 20. Diethelm, K: The Analysis of Fractional Differential Equations. Lecture Notes in Math. (2010)

Mahmudov and Unul Advances in Difference Equations (2017) 2017:15

Page 16 of 16

21. Guo, T, Jiang, W: Impulsive fractional functional differential equations. Comput. Math. Appl. 64, 3414-3424 (2012) 22. Kilbas, A, Srivastava, H, Trujillo, J: Theory and Applications of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) 23. Liang, S, Zhang, J: 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) 24. 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) 25. Mahmudov, N, Unul, S: Existence of solutions of α ∈ (2, 3] order fractional three-point boundary value problems with integral conditions. Abstr. Appl. Anal. 2014, Article ID 198632 (2014) 26. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) 27. Wang, G, Ahmad, B, Zhang, L, Nieto, J: Comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 19, 401-403 (2014) 28. Wang, G, Zhang, L, Song, G: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. Nonlinear Anal. TMA 74, 974-982 (2011)