Boundary value problems for impulsive multi-order Hadamard ... - Core

Yukunthorn et al. Boundary Value Problems (2015) 2015:148 DOI 10.1186/s13661-015-0414-5

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Boundary value problems for impulsive multi-order Hadamard fractional differential equations Weera Yukunthorn1 , Suthep Suantai2 , Sotiris K Ntouyas3,4 and Jessada Tariboon1* *

Correspondence: [email protected] 1 Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand Full list of author information is available at the end of the article

Abstract In this paper, we study the existence and uniqueness of solutions for impulsive multi-orders Caputo-Hadamard fractional differential equations equipped with boundary and integral conditions. The Banach, Schaefer, and Rothe fixed point theorems and degree theory are used to establish our main results. Examples illustrating the main results are presented. MSC: 34A08; 34B10; 34A37 Keywords: impulsive differential equations; Hadamard fractional calculus; fixed point theorems; existence

1 Introduction During the last years, fractional calculus has gained considerable importance due to the applications in almost all applied sciences. It was pointed out that fractional derivatives and integrals are more convenient for describing real materials, and some physical problems were treated by using derivatives of non-integer orders. For details, and some recent results on the subject we refer to [–] and the references cited therein. It has been noticed that most of the work on the topic is based on Riemann-Liouville and Caputo type fractional differential equations. Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in  [], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains a logarithmic function of arbitrary exponent. Details and properties of the Hadamard fractional derivative and integral can be found in [, –]. However, this calculus with Hadamard derivatives is still studied less than that of Riemann-Liouville. On the other hand, integer order impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and the social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see for instance [–]. Recently in [], Wang et al. studied existence and uniqueness results for the following impulsive multipoint fractional integral boundary value problem involving multi-order © 2015 Yukunthorn 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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fractional derivatives and a deviating argument: ⎧ c αk ⎪ D + u(t) = f (t, u(t), u(θ (t))), ⎪ ⎪ ⎨ tk

 < αk ≤ ,

u (tk ) = Ik∗ (u(tk )), k = , , . . . , p, u(tk ) = Ik (u(tk )), ⎪ ⎪ ⎪ ⎩u() = p λk J β+k u(ηk ), u () = , tk < ηk < tk+ , 

k=

(.)

tk

α

β

where c Dt+k is the Caputo fractional derivative of order αk , Jt+k is Riemann-Liouville frack k tional integral of order βk > , f ∈ C(J × R × R, R), Ik , Ik∗ ∈ C(R, R), θ ∈ C(J, J), J = [, T] (T > ),  = t < t < · · · < tk < · · · < tp < tp+ = T, u(tk ) = u(tk+ ) – u(tk– ), and u (tk ) = u (tk+ ) – u (tk– ) where u(tk+ ), u (tk+ ) and u(tk– ), u (tk– ) denote the right and left hand limits of u(t) and u (t) at t = tk (k = , , . . . , p). In , Wang et al. [] established the existence of solutions for a class of nonlinear impulsive Hadamard fractional differential equations with initial condition of the form ⎧ α ⎪ ⎪ ⎨H D+ u(t) = f (t, u(t)), ⎪ ⎪ ⎩

α ∈ (, ), t ∈ (, e] \ {t , t , . . . , tm },

+ –α – u(ti ) = H J–α + u(ti ) – H J+ u(ti ) = pi , –α + H J+ u( ) = u ,

pi ∈ R, i = , , . . . , m,

(.)

u ∈ R,

where H Dα+ is the left-side Hadamard fractional derivative of order α with the lower limit  and H J–α + denotes left-side Hadamard fractional integral of order  – α. The existence results were obtained by using the Banach contraction principle and Schauder’s fixed point theorem on the weight spaces of piecewise continuous functions. The Hadamard and Riemann-Liouville fractional derivatives have one similar property, which is the fact that the derivative of a constant is not equal to zero. It is caused by the definitions of them containing the usual derivative outside the integrals. In , Jarad et al. [] presented the modifications of the Hadamard fractional derivative into a more suitable one having physically interpretable initial conditions similar to the Caputo sense. In , Gambo et al. [] proved the fundamental theorem of fractional calculus, some interesting results and also semigroup properties of Caputo-Hadamard operators. In this paper we are concerned with the existence of solutions for boundary value problems of impulsive Hadamard fractional differential equations of the form ⎧ p k C ⎪ ⎪ Dtk x(t) = f (t, x(t)), ⎨ ⎪ ⎪ ⎩

t ∈ Jk ⊂ [t , T], t = tk ,

x(tk ) = ϕk (x(tk )), k = , , . . . , m,  qi αx(t ) + βx(T) = m i= γi Jti x(ti+ ), p

(.)

where C Dtkk is the Hadamard fractional derivative of Caputo type of order  < pk ≤  on intervals Jk := (tk , tk+ ], k = , , . . . , m, with J = [t , t ],  < t < t < t < · · · < tk < · · · < tm < tm+ = T are the impulse points, J := [t , T], f : J × R → R is a continuous function, ϕk ∈ q C(R, R), Jti i is the Hadamard fractional integral of order qi > , i = , , . . . , m. The jump conditions are defined by x(tk ) = x(tk+ ) – x(tk ), x(tk+ ) = limε→+ x(tk + ε), k = , , , . . . , m.

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The paper is organized as follows: Section  contains some preliminary notations, definitions and lemmas that we need in the sequel. In Section  we present the main results for the problem (.), where existence and uniqueness results are proved by using Banach and Rothe fixed point theorems, Leray-Schauder alternative and degree theory. Examples illustrating the obtained results are also presented.

2 Preliminaries In this section, we introduce some notations and definitions of Hadamard fractional calculus (see []) and present preliminary results needed in our proofs later. Definition . For an at least n-times differentiable function g : [a, b] → R, a, b > , the Caputo type Hadamard derivative of fractional order α is defined as C

Daα g(t) =

 (n – α)

 t ds t n–α– n δ g(s) , log s s a

n –  < α < n, n = [α] + ,

where δ = t dtd , t ∈ [a, b], and [α] denotes the integer part of the real number α and log(·) = loge (·). Definition . The Hadamard fractional integral of order α is defined as

Jaα g(t) =

 (α)

 t t α– ds log g(s) , s s a

α > ,

provided the integral exists on [a, b]. Lemma . [] Let x ∈ ACδn [a, b] or Cδn [a, b] and α ∈ C, where Xδn [a, b] = {g : [a, b] → C : δ n– g(t) ∈ X[a, b]}. Then we have

C  Jaα Daα x(t) = x(t) –

n– k δ x(a) k=

k!

 k t log . a

The key tools for proving of our results are based on the following fixed point theorems. ¯ → E is a completely continuous operator. If one of Theorem . [] Suppose that A : the following conditions is satisfied: (i) (Altman) Ax – x  ≥ Ax  – x  , for all x ∈ ∂ , (ii) (Rothe) Ax ≤ x , for all x ∈ ∂ , (iii) (Petryshyn) Ax ≤ Ax – x , for all x ∈ ∂ , then deg(I – A, , θ ) = , and hence A has at least one fixed point in . ¯ → E is completely continuous operator. If Theorem . [] Suppose that A : Ax = λx,

∀x ∈ ∂ , λ ≥ ,

¯ then deg(I – A, , θ ) =  and A has at least one fixed point in .

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Theorem . [] Let E be a Banach space. Assume that T : E → E is a completely continuous operator and the set V = {u ∈ E : u = λTu,  < λ < } is bounded. Then T has a fixed point in E.  γi (log(ti+ /ti ))qi Lemma . Assume that  = α + β – m = . Then the solution of the probi= (qi +) lem (.) is equivalent to the following integral equation: k–

pi

   p

x(t) = Jtk k f t, x(t) + Jti f ti+ , x(ti+ ) + ϕi+ x(ti+ )

+

 

–β

i= m

q +pi

γi Jti i



 p f ti+ , x(ti+ ) – β Jtmm f T, x(T)

i=

m–  

pi

Jti f ti+ , x(ti+ ) + ϕi+ x(ti+ ) i=

+

m  i=

γi (log(ti+ /ti ))qi (qi + )

 i–



  pj

Jtj f tj+ , x(tj+ ) + ϕj+ x(tj+ )

.

(.)

j=

Proof By Lemma ., the solution of (.) on interval J can be written as  p

x(t) = Jt f t, x(t) + x , where x ∈ R. For t ∈ J , by using Lemma . and the impulse condition x(t ) = ϕ (x(t )), we obtain   p

x(t) = Jt  f t, x(t) + x t+

   p

p

= Jt  f t, x(t) + Jt f t , x(t ) + ϕ x(t ) + x . Again, for t ∈ J , we have   p

x(t) = Jt f t, x(t) + x t+



     p

p

p

= Jt f t, x(t) + Jt  f t , x(t ) + ϕ x(t ) + Jt f t , x(t ) + ϕ x(t ) + x . Repeating the above process, for t ∈ J, we obtain k–

pi

   p

Jti f ti+ , x(ti+ ) + ϕi+ x(ti+ ) + x . x(t) = Jtk k f t, x(t) + i=

Applying the boundary condition of (.), it follows that

 p αx(t ) + βx(T) = (α + β)x + β Jtmm f T, x(T) +β

m–

pi

  Jti f ti+ , x(ti+ ) + ϕi+ x(ti+ ) i=

(.)

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and m

q

γi Jti i x(ti+ )

i=

=

m

m

 γi (log(ti+ /ti ))qi f ti+ , x(ti+ ) + x (qi + ) i= i=   m  i–

j–

  γi (log(ti+ /ti ))qi + Jtj f tj+ , x(tj+ ) + ϕj+ x(tj+ ) , (qi + ) i= j= p +qi

γi Jti i

which leads to

m 

  q +p

p x = γi Jti i i f ti+ , x(ti+ ) – β Jtmm f T, x(T)  i= –β

m–



  p

Jti i f ti+ , x(ti+ ) + ϕi+ x(ti+ )

i=

  m  i–  

j–

γi (log(ti+ /ti ))qi . Jtj f tj+ , x(tj+ ) + ϕj+ x(tj+ ) + (qi + ) i= j= 

Replacing the constant x into (.), we obtain (.), as desired.

3 Main results Let PC(J, R) = {x : J → R; x(t) is continuous everywhere except for some tk at which x(tk+ ) and x(tk– ) exist and x(tk– ) = x(tk ), k = , , . . . , m}. Obviously, PC(J, R) is a Banach space with the norm x = sup{|x(t)|; t ∈ J}. A function x ∈ PC is called a solution of the problem (.) if it satisfies (.). In this section, we investigate the existence and uniqueness of solutions for the problem (.) via a variety of fixed point theorems by defining an operator K : PC → PC as k–

pi

   p

Kx(t) = Jtk k f t, x(t) + Jti f ti+ , x(ti+ ) + ϕi+ x(ti+ )

+

 

–β

i= m

q +pi

γi Jti i

  p

f ti+ , x(ti+ ) – β Jtmm f T, x(T)

i=

m–



  p

Jti i f ti+ , x(ti+ ) + ϕi+ x(ti+ )

i=

+

m  i=

γi (log(ti+ /ti )qi ) (qi + )

 i–



  pj

Jtj f tj+ , x(tj+ ) + ϕj+ x(tj+ )

.

j=

Clearly, the boundary value problem (.) becomes a fixed point problem x = Kx.

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For convenience, we set the notations of constants, thus (log(t/tk ))pk (log(ti+ /ti ))pi + (pk + ) (pi + ) i=  m m  |γi |(log(ti+ /ti ))qi +pi (log(ti+ /ti ))pi + + |β| || i= (qi + pi + ) (pi + ) i= m–

 =

  m i–  (log(T/tm ))pm (log(tj+ /tj ))pj |γi |(log(ti+ /ti ))qi + , + |β| (pm + ) (pj + ) (qi + ) i= j=

 m  (log(T/tm ))pm (log(t/tk ))pk |γi |(log(ti+ /ti ))qi + |β|m + + |β| .  = k + (pk + )  (qi + ) (pm + ) i= Theorem . Assume that f : J × R → R and ϕk : R → R, k = , , . . . , m, are continuous functions which satisfy the following conditions: (H ) |f (t, x) – f (t, y)| ≤ L |x – y|, ∀t ∈ J, L > , x, y ∈ R; (H ) |ϕk (u) – ϕk (v)| ≤ L |u – v|, L > , for all u, v ∈ R, ∀k = , , . . . , m. If L  + L  <  then the problem (.) has a unique solution on J. Proof We define a closed ball Br by Br = {x ∈ PC; x ≤ r} where r ≥ (M  + M  )( – L  – L  )– , where M = supt∈J |f (t, )| and M = max{|ϕi ()|, i = , , . . . , m}. We will show that K : Br → Br . For any x ∈ Br , we have k–  

 

pi 

   Kx(t) ≤ Jtpk f t, x(t)  + Jti f ti+ , x(ti+ )  + ϕi+ x(ti+ )  k

+

 ||

+ |β|

i= m

q +pi 

|γi |Jti i



 p 

f ti+ , x(ti+ )  + |β|Jtmm f T, x(T) 

i=

m–



   p

Jti i f ti+ , x(ti+ )  + ϕi+ x(ti+ ) 

i=

+

m  i=

|γi |(log(ti+ /ti ))qi (qi + )

 i–

j=

    p 

≤ Jtk k f t, x(t) – f (t, ) + f (t, ) +

k–    

pi 

Jti f ti+ , x(ti+ ) – f (ti+ , ) + f (ti+ , ) i=

   

 + ϕi+ x(ti+ ) – ϕi+ () + ϕi+ ()

m      q +p 

|γi |Jti i i f ti+ , x(ti+ ) – f (ti+ , ) + f (ti+ , ) + || i=     p 

+ |β|Jt m f T, x(T) – f (T, ) + f (T, ) m

+ |β|

m–



i=



   pj 

Jtj f tj+ , x(tj+ )  + ϕj+ x(tj+ ) 

    p 

Jti i f ti+ , x(ti+ ) – f (ti+ , ) + f (ti+ , )

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 + ϕi+ x(ti+ ) – ϕi+ () + ϕi+ ()  m  i– 

pj 

 |γi |(log(ti+ /ti ))qi Jtj f tj+ , x(tj+ ) – f (tj+ , ) + (qi + ) i= j=

      

 + f (tj+ , ) + ϕj+ x(tj+ ) – ϕj+ () + ϕj+ ()

 k–  (log(t/tk ))pk (log(ti+ /ti ))pi (L r + M ) + + (L r + M ) ≤ (L r + M) (pk + ) (pi + ) i=

m (log(ti+ /ti ))qi +pi (log(T/tm ))pm  + |β|(L r + M ) |γi |(L r + M ) + || i= (qi + pi + ) (pm + ) m– 

 (log(ti+ /ti ))pi + |β| (L r + M ) + (L r + M ) (pi + ) i=    m i– (log(tj+ /tj ))pj |γi |(log(ti+ /ti ))qi + (L r + M ) (L r + M ) + (qi + ) (pj + ) i= j= ≤ (L  + L  )r + (M  + M  ) ≤ r. Then KBr ⊆ Br . Next we will show that K is a contraction mapping. For x, y ∈ Br , we get |Kx – Ky| k–

pi 



 

 p 

≤ Jtk k f t, x(t) – f t, y(t)  + Jti f ti+ , x(ti+ ) – f ti+ , y(xi+ )  i=





  + ϕi+ x(ti+ ) – ϕi+ y(ti+ ) 

m 

  q +p 

|γi |Jti i i f ti+ , x(ti+ ) – f ti+ , y(ti+ )  + || i= 

 p 

+ |β|Jtmm f T, x(T) – f T, y(T)  + |β|

m–

pi 





    Jti f ti+ , x(ti+ ) – f ti+ , y(ti+ )  + ϕi+ x(ti+ ) – ϕi+ y(ti+ )  i=

+

m  i=

|γi |(log(ti+ /ti ))qi (qi + )

 i–

pj 



 Jtj f tj+ , x(tj+ ) – f tj+ , y(tj+ ) 

j=





  + ϕj+ x(tj+ ) – ϕj+ y(tj+ ) 



≤ (L  + L  ) x – y . Since (L  + L  ) < , the operator K is contractive. Hence K has a unique fixed point on Br . Therefore the problem (.) has a unique solution on J.  Theorem . Let f and ϕk , k = , , . . . , m, be continuous functions. Assume that there are two positive real numbers N and N such that:

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(H ) |f (t, x)| ≤ N and |ϕk (x)| ≤ N , for t ∈ J, x ∈ R and k = , , . . . , m. Then the problem (.) has at least one solution on J. Proof Define a ball Bω = {x ∈ PC; x < ω}. The proof is divided into  steps. Step . We will show that K is continuous. To prove this, we let {xn } be a sequence in PC such that xn → x as n → ∞. Then we have   Kxn (t) – Kx(t) 

 p 

≤ Jtk k f t, xn (t) – f t, x(t)  +

k–







    p

Jti i f ti+ , xn (ti+ ) – f ti+ , x(ti+ )  + ϕi+ xn (ti+ ) – ϕi+ x(ti+ ) 

i=

m 

  q +p 

|γi |Jti i i f ti+ , xn (ti+ ) – f ti+ , x(ti+ )  + || i= 

 p 

+ |β|Jtmm f T, xn (T) – f T, x(T)  + |β|

m–

pi 



 Jti f ti+ , xn (ti+ ) – f ti+ , x(ti+ )  i=





  + ϕi+ xn (ti+ ) – ϕi+ x(ti+ )   m  i–

pj 



 |γi |(log(ti+ /ti ))qi + Jtj f tj+ , xn (tj+ ) – f tj+ , x(tj+ )  (qi + ) i= j= 



  + ϕj+ xn (tj+ ) – ϕj+ x(tj+ ) 



.

Using the continuity of f and ϕk for k = , , . . . , m, we have |f (t, xn ) – f (t, x)| and |ϕk (xn ) – ϕk (x)| vanish as n → ∞. Therefore Kxn – Kx → , which yields the continuity of the operator K. Step . K maps a bounded set into a bounded set. For each x ∈ B¯ ω , we have k–

pi 

    p 

|Kx| ≤ Jtk k f t, x(t)  + Jti f ti+ , x(ti+ )  + ϕi+ x(ti+ ) 

+

 ||

+ |β|

i= m

q +pi 

|γi |Jti i

  p 

f ti+ , x(ti+ )  + |β|Jtmm f T, x(T) 

i=

m–



   p

Jti i f ti+ , x(ti+ )  + ϕi+ x(ti+ ) 

i=

+

m  i=

|γi |(log(ti+ /ti ))qi (qi + )

 i–

≤  N  +   N  , which yields the boundedness of KB¯ ω .

j=

   pj 

Jtj f tj+ , x(tj+ )  + ϕj+ x(tj+ ) 



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Step . K maps a bounded set into an equicontinuous set. Let τ , τ ∈ (tk , tk+ ), for each k = , , , . . . , m, we have  

 

 Kx(τ ) – Kx(τ ) ≤ Jtpk f τ , x(τ ) – f τ , x(τ ) . k The continuity of x and f implies that Kx(τ ) → Kx(τ ) as τ → τ . Consequently K is completely continuous by applying the Azelá-Ascoli theorem. Let V = {x ∈ Bω ; μKx = x for μ ∈ (, )}. For all x ∈ V , x = μKx, we have |x| ≤ μ|Kx| ≤  N +  N . Hence V is bounded. By Theorem ., the problem (.) has at least one solution on J.  Theorem . Assume that (H ) limx→

f (t,x) x

=  and limx→

ϕk (x) x

=  for k = , , . . . , m.

Then the problem (.) has at least one solution on J. Proof From (H ), choosing  = /( +  ), there exist constants δ , δ ∈ R+ such that   f (t, x) < |x| where |x| < δ

and

  ϕ(x) < |x|

where |x| < δ .

Now, we define an open ball = {u ∈ PC; u < min{δ , δ }}. By Theorem ., the operator ¯ → PC is completely continuous. For any x ∈ ∂ , we have K: k–

pi 

    p 

Jti f ti+ , x(ti+ )  + ϕi+ x(ti+ )  |Kx| ≤ Jtk k f t, x(t)  +

+

 ||

+ |β|

i= m

q +pi 

|γi |Jti i

  p 

f ti+ , x(ti+ )  + |β|Jtmm f T, x(T) 

i=

m–



   p

Jti i f ti+ , x(ti+ )  + ϕi+ x(ti+ ) 

i=

+

m  i=

|γi |(log(ti+ /ti ))qi (qi + )

 i–





  

pj 

Jtj f tj+ , x(tj+ )  + ϕj+ x(tj+ ) 

j=

≤ (  +  ) x = x . It follows from Theorem ., case (ii), that the problem (.) has at least one solution on J.  Theorem . Let f and ϕk for k = , , . . . , m, be continuous functions and satisfy the following inequalities: (H ) |f (t, x)| ≤ a|x| + b, ∀(t, x) ∈ J × R and |ϕk (x)| ≤ c|x| + d, ∀x ∈ R, k = , . . . , m, where constants a, c >  and b, d ≥ . Then the problem (.) has at least one solution on J.

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Proof Define a unit ball as O = {x ∈ PC; x < }. It is straightforward to show that the operator K : O¯ → PC is completely continuous. Suppose that there is x∗ ∈ ∂ O . Then we choose λ = (a + c) + (b + d) +  such that Kx∗ = λx∗ . By taking the norm in both sides of Kx∗ = λx∗ , we obtain K x∗ ≥ λ x∗ . Then we have K = sup |Kx| x =



k–

pi 

    p 

= sup Jtk k f t, x(t)  + Jti f ti+ , x(ti+ )  + ϕi+ x(ti+ )  x =

+

i=

m    q +p 

p 

|γi |Jti i i f ti+ , x(ti+ )  + |β|Jtmm f T, x(T)  || i=

+ |β|

m–

pi 

   Jti f ti+ , x(ti+ )  + ϕi+ x(ti+ )  i=

+

m  i=

|γi |(log(ti+ /ti ))qi (qi + )

 i–



   pj 

Jtj f tj+ , x(tj+ )  + ϕj+ x(tj+ ) 



j=

≤ (a + c) + (b + d) = λ – , which contradicts K ≥ λ. Hence the assumptions of Theorem . hold. Therefore the problem (.) has at least one solution on J. 

4 Examples In this section, we present four examples to illustrate our results. Example . Consider the boundary value problem for an impulsive multi-order Hadamard fractional differential equation of the form ⎧ ( k+ )  (|x(t)|+) ⎪ t ∈ [, e+ ] \ {tk }, ⎪C Dtkk+ x(t) = (t–t  +) ( |x(t)|+ ),  ⎪ ⎨ sin |x(tk )| ke+ x(tk ) = (–k) , tk = k+ , k = , , . . . , , ⎪  ⎪ ⎪ ( i +i+ )  ⎩  e+ i +i+ –i x() + x( ) = ( – e ) J x(ti+ ). ti i=   

(.)

Here α = /, β = /, m = , pk = (k + )/(k + ), γk =  – e–k , qk = (k  + k + )/(k  + k + ) for k = , , . . . , . From the information, we find that  ≈ .,  ≈ ., and  ≈ .. The functions f and ϕk are given by   – t  (|x| + ) f (t, x) = , (t  + ) |x| + 

ϕk (x) =

sin |x| , ( – k)

which satisfy   f (t, x) – f (t, y) ≤  |x – y| 

and

  ϕk (x) – ϕk (y) ≤  |x – y|, 

∀k = , , . . . , .

Then we get L = / and L = /, which implies L  + L  ≈ . < . Therefore the problem (.) has a unique solution on [, (e + )/] due to Theorem ..

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Example . Consider the boundary value problem for an impulsive multi-order Hadamard fractional differential equation of the form ⎧ k+ –t ) log(|x(t)|+) ⎪ C log( i= (/(i+)!)) ⎪ D x(t) = (–e |x(t)|+ – , t ∈ [ π , π] \ {tk }, ⎪ tk ⎨ k– k k x(tk ) = e–  cos(kx(tk )) + e  sin(kx(tk )), tk =   π, k = , , . . . , , ⎪ ⎪ |i–| ⎪ ⎩–e– π x( π ) + e–π x(π) =  (–)i (i + )J ( i+ ) x(t ). i+

ti

i=



(.)

 k  Here α = –e–π / , β = e–π , m = , pk = log( k+ i= (/(i + )!)), γk = (–) (k + ), qk = |k – |/(k + ) for k = , , . . . , . We find that  ≈ –. = . The functions f (t, x) = ( – k k e–t ) log(|x| + )/(|x| + ) –  and ϕk (x) = e–  cos(kx) + e  sin(kx) are bounded as   f (t, x) ≤ 

and

  √ ϕk (x) ≤ e– + e .

Hence the assumption (H ) of Theorem . holds. Therefore the problem (.) has at least one solution on [π/, π]. Example . Consider the boundary value problem for an impulsive multi-order Hadamard fractional differential equation of the form ⎧ ( k+ ) tx(t) ⎪ x(t)–x(t)) ⎪ , t ∈ [  , ] \ {tk }, ⎪C Dtkk +k+ x(t) = e (sint+ ⎨  kx (tk ) x(tk ) = log(|x(t , tk = k+ , k = , , . . . , ,  ⎪ k )|+) ⎪ √  ⎪ ⎩ x(  ) +  x() =  ( i + )J arctan i x(t ). 



i= i +

(.)

i+

ti

√ Here α = , β = /, m = , pk = (k + )/(k  + ), γk = (k  + )/(k  + ), qk = arctan(k) for k = , , . . . , . We find that  ≈ . = . The functions f (t, x) = etx (sin x – x)/(t + ) and ϕk (x) = kx / log(|x| + ) satisfy  sin x f (t, x) ext = lim – = x→ x→ t +  x x lim

and ϕk (x) kx = lim = , x→ x→ log(|x| + ) x lim

∀k = , , . . . , .

Thus the condition (H ) of Theorem . holds. Therefore, we conclude that the problem (.) has at least one solution on [/, ]. Example . Consider the boundary value problem for an impulsive multi-order Hadamard fractional differential equation of the form ⎧ √ t –sin (k+) ⎪ C ⎪ x(t) = e  sin x(t) + tx(t) cos x(t) + , ⎪ ⎨ D tk x(tk ) = kx(tk ) – log(|x(tk )| +  ), tk =  ·  ⎪ ⎪ ⎪ ⎩  x(  ) –  x() =  (–)i J ( k+ k+ ) x(t ). 





i= i+

ti

i+

k– 

t ∈ [  , ] \ {tk },

, k = , , . . . , ,

(.)

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 Here α = /, β = –/, m = , pk =  – sin (k + ), γk = (–)k /(k + ), qk = (k + )/(k + ), for k = , , . . . , . We find that  ≈ . = . The functions f (t, x) = t e  sin x + tx cos x +  and ϕ(x) = kx – log(|x| + (/)) satisfy the inequalities   



f (t, x) ≤ t|x| +  + e t ≤ |x| +  + e and   ϕk (x) ≤ |x|(k + ) +  ≤ |x| +  .   Therefore (H ) holds. According to Theorem ., the problem (.) has at least one solution on [/, ].

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally in this article. They read and approved the final manuscript. Author details 1 Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand. 2 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand. 3 Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece. 4 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. Acknowledgements This paper was supported by the Thailand Research Fund under the project RTA5780007. Received: 21 May 2015 Accepted: 11 August 2015 References 1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006) 2. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) 3. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) 4. Ahmad, B, Ntouyas, SK, Alsaedi, A: An existence result for fractional differential inclusions with nonlinear integral boundary conditions. J. Inequal. Appl. 2013, 296 (2013) 5. Nyamoradi, N, Baleanu, D, Agarwal, RP: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys. 2013, Article ID 823961 (2013) 6. Ntouyas, SK, Tariboon, J, Sudsutad, W: Boundary value problems for Riemann-Liouville fractional differential inclusions with nonlocal Hadamard fractional integral conditions. Meditter. J. Math. (2015). doi:10.1007/s00009-015-0543-1 7. Alsaedi, A, Ntouyas, SK, Agarwal, RP, Ahmad, B: On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Adv. Differ. Equ. 2015, 33 (2015) 8. Yukunthorn, W, Ntouyas, SK, Tariboon, J: Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions. Adv. Differ. Equ. 2014, 315 (2014) 9. Tariboon, J, Ntouyas, SK, Sudsutad, W: Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations. Bound. Value Probl. 2014, 253 (2014) 10. Hadamard, J: Essai sur l’étude des fonctions données par leur développement de Taylor. J. Mat. Pure Appl. Ser. 8, 101-186 (1892) 11. Butzer, PL, Kilbas, AA, Trujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387-400 (2002) 12. Butzer, PL, Kilbas, AA, Trujillo, JJ: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1-27 (2002) 13. Butzer, PL, Kilbas, AA, Trujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1-15 (2002) 14. Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191-1204 (2001) 15. Kilbas, AA, Trujillo, JJ: Hadamard-type integrals as G-transforms. Integral Transforms Spec. Funct. 14, 413-427 (2003) 16. Agarwal, RP, Ahmad, B: Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 18, 535-544 (2011) 17. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) 18. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing, New York (2006)

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