Existence of solutions for impulsive fractional integrodifferential

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Abstract. In this paper, by using some fixed point theorems and the measure of noncompactness, we discuss the existence of solutions for a boundary value ...
Li and Gou Advances in Difference Equations (2017) 2017:218 DOI 10.1186/s13662-017-1244-9

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Existence of solutions for impulsive fractional integrodifferential equations with mixed boundary conditions Baolin Li* and Haide Gou *

Correspondence: [email protected] College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

Abstract In this paper, by using some fixed point theorems and the measure of noncompactness, we discuss the existence of solutions for a boundary value problem of impulsive integrodifferential equations of fractional order α ∈ (1, 2]. Our results improve and generalize some known results in (Zhou and Chu in Commun. Nonlinear Sci. Numer. Simul. 17:1142-1148, 2012; Bai et al. in Bound. Value Probl. 2016:63, 2016). Finally, an example is given to illustrate that our result is valuable. MSC: 34A37; 34B15; 34A08 Keywords: impulsive fractional integrodifferential equations; boundary value problem; measure of noncompactness

1 Introduction Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [–]. The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, population dynamics, ecology, chemical, biological systems, and engineering background. Therefore, it has been an object of intensive investigation in recent years, some basic results on impulsive differential equations have been obtained and applications to different areas have been considered by many authors, see [–]. However, the concept of solutions for impulsive fractional differential equations [–] has been argued extensively, while the concept presented in Refs. [–, ] could be controversial and deserves a further argument and mending. In [, , ], Wang et al. and Shu 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 [, , ]. Furthermore, the theory of boundary value problems for nonlinear impulsive fractional differential equations is still in the initial stages and many aspects of this theory need to be explored, we refer the readers to [, , – ]. © 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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In [], Zhou discussed the existence of solutions for a nonlinear multi-point boundary value problem of integro-differential equations of fractional order as follows: ⎧ ⎨c Dα+ u(t) = f (t, u(t), (Hu)(t), (Ku)(t)),

t ∈ [, ], α ∈ (, ],



⎩a u() – b u () = d u(ξ ),

a u() + b u () = d u(ξ ),

where c Dα+ denotes the fractional Caputo derivative and 



t

(Hu)(s) =

g(t, s)u(s) ds,

(Ku)(s) =

t

h(t, s)u(s) ds





with respect to strong topology. In [], Bai studied the existence of solutions for an impulsive fractional differential equation with nonlocal conditions in a Banach space E ⎧ c α  ⎪ ⎪ ⎨ D+ u(t) = f (t, u(t)), t ∈ J , u(tk ) = Ik (u(tk )), u (tk ) = Ik∗ (u(tk )), ⎪ ⎪ ⎩ u() + u () = , u() + u () = ,

k = , , . . . , p,

by using the contraction mapping principle and Krasnoselskii’s fixed point theorem. Dong et al. [] investigated the boundary value problem for p-Laplacian fractional differential equations ⎧ ⎨Dα (φ (Dα u(t))) = f (t, u(t)),  < t < , p ⎩u() = u() = Dα u() = Dα u() = , where  < α ≤  is a real number, Dα is the conformable fractional derivative, φp (s) = |s|p– s, p > , φp– = φq , p + q = , f : [, ] × [, +∞) → [, +∞) is continuous. By the approximation method and fixed point theorems on cone, some existence and multiplicity results of positive solutions are obtained. Bai et al. [] investigated the boundary value problem of fractional differential equations ⎧ ⎨Dα+ u(t) = f (t, u(t)), t ∈ (, h),  ⎩t –α u(t)|t= = u , where f ∈ C([, h] × R, R), Dα+ u(t) is the standard Riemann-Liouville fractional derivative,  < α < . The existence of the blow-up solution, that is to say, u ∈ C(, h] and limt→+ u(t) = ∞, is obtained by the use of the lower and upper solution method. In this paper, motivated by the above references, we investigate the existence of solutions to the following impulsive fractional integro-differential equations with mixed boundary conditions: ⎧ c α ⎪ ⎪ ⎨ D+ u(t) = f (t, u(t), (Tu)(t), (Su)(t)), ⎪ ⎪ ⎩

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

au() + bu () = γ (u),

t ∈ J ,

u (tk ) = Ik∗ (u(tk )), 

k = , , . . . , p,

au() + bu () = γ (u),

(.)

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where f ∈ C(J × E × E × E, E), Ik , Ik∗ ∈ C(E, E), J = [, ], J  = J \ {t , t , . . . , tp }, c Dα+ is the Caputo fractional derivative of order α ∈ (, ], {tk } satisfy  = t < t < t < · · · < tp < tp+ = , p ∈ N , u(tk ) = u(tk+ ) – u(tk– ), u (tk ) = u (tk+ ) – u (tk– ), u(tk+ ) and u(tk– ) represent the right and left limits of u(t) at t = tk , respectively. T and S are two linear operators defined by  (Tu)(t) = 

t

  k(t, s)p s, u(s) ds,

 (Su)(t) =

a

  h(t, s)p s, u(s) ds,



where k ∈ C(D, R+ ), h ∈ C(D , R+ ), D = {(t, s) ∈ R :  ≤ s ≤ t ≤ a}, D = {(t, s) ∈ R :  ≤ t, s ≤ a} and pi ∈ C(J × E, E), γi : J → E (i = , ) is to be specified later. The paper is organized as follows. In Section  we recall some basic known results. In Section  we discuss the existence theorem of solutions for problem (.). In Section , we provide an example to illustrate our result.

2 Preliminaries In this section, we introduce notations, definitions and preliminary results which will be used throughout this paper. Let E be a real Banach space and denote by  the family of all functions ψ : R+ → R+ satisfying the following conditions: () ψ is nondecreasing; ∞ n n () n= ψ < ∞ for all t > , where ψ is the nth iterate of ψ. For each ψ ∈ , the following assertions hold: () limn→∞ ψ n (t) =  for all t > ; () ψ(t) < t for all t > ; () ψ() = . Furthermore, we write B(x, r) to denote the closed ball centered at x with radius r and X, ConvX to denote the closure and closed convex hull of X, respectively. Moreover, let mE indicate the family of all nonempty bounded subsets of E and nE indicate the family of all relatively compact sets. Let J = (, t ], J = (t , t ], . . . , Jp– = (tp– , tp ], Jp = (tp , ] and C(J, E) denote the Banach space of all continuous E-valued functions on the interval J, PC(J, E) = {u : J → E|u ∈ C(J  , E), u(tk+ ), u(tk– ) exist and u(tk– ) = u(tk ),  ≤ k ≤ p}. Obviously, PC(J, E) is a Banach space with the norm u = supt∈J u(t)E . We use the following definition of the measure of noncompactness given in []. Definition . A mapping μ : mE → R+ is said to be a measure of noncompactness in E if it satisfies the following conditions: () The family ker μ = {X ∈ mE : μ(X) = } is nonempty and ker μ ⊂ nE . () X ⊂ Y ⇒ μ(X) ≤ μ(Y ). () μ(X) = μ(X). () μ(ConvX) = μ(X). () μ(λX + ( – λ)Y ) ≤ λμ(X) + ( – λ)μ(Y ) for λ ∈ [, ]. () If (Xn ) is a sequence of closed sets from mE such that Xn+ ⊂ Xn (n = , , . . .) and if

limn→∞ μ(Xn ) = , then the intersection set X∞ = ∞ n= Xn is nonempty. Lemma . ([]) Let ϕ : R+ → R+ be a nondecreasing and upper semicontinuous function. Then the following two conditions are equivalent:

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(i) limn→∞ ϕ n (t) =  for any t > ; (ii) ϕ(t) < t for any t > . For any nonempty bounded subset, X ∈ mE . For x ∈ X; T >  and ε > , let

ω(x, ε) = sup x(t) – x(s) : t, s ∈ J, |t – s| ≤ ε ,

ω(X, ε) = sup wT (x, ε) : x ∈ X ,

ω (X) = lim ω(X, ε), X(t) = x(t) : x ∈ X , ε→



diam X(t) = sup x(t) – y(t) : x, y ∈ X , and μ(X) = ω (X) + lim sup diam X(t).

(.)

t→∞

In [], Banaś has shown that the function μ is a measure of noncompactness in the spaces PC(J, E). For completeness, we recall the definition of the Caputo derivative of fractional order. Definition . The fractional integral of order α of a function f : [, ∞) → R is defined as Iα+ f (t) =



(α)



t

(t – s)α– f (s) ds,

t > , α > ,



provided the right-hand side is point-wise defined on (, ∞), where (·) is the gamma function. Definition . The Riemann-Liouville derivative of order α with the lower limit zero for a function f : [, ∞) → R can be written as Dα+ f (t) =



dn 

(n – α) dt n



t

f (s) ds, (t – s)α+–n

t > , n –  < α < n.

Definition . The Caputo fractional derivative of order α for a function f : [, ∞) → R can be written as  c

Dα+ f (t) = Dα+

f (t) –

n– k  t k=

k!

 f

(k)

() ,

t > , n –  < α < n,

where n = [α] +  and [α] denotes the integer part of α. Remark . In the case f (t) ∈ C n [, ∞), then c

Dα+ f (t) =



(n – α)

 

t n–α n (t – s)n–α– f (n) (s) ds = I+ f (t),

t > , n –  < α < n.

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That is to say, Definition . is just the usual Caputo fractional derivative. In this paper, we consider an impulsive problem, so Definition . is appropriate. Moreover, we need the following known results. Lemma . Let α > , then the differential equation c

Dα+ u(t) = 

has the solution u(t) = c + c t + c t  + · · · + cn t n– , ci ∈ R, i = , , . . . , n, n = [α] + . In view of Lemma ., we have the following. Lemma . Let α > , then   Iα+ c Dα+ u(t) = u(t) + c + c t + c t  + · · · + cn t n– for some ci ∈ R, i = , , . . . , n, n = [α] + . Definition . ([]) Let (E, d) be a metric space with w-distance p and f : E → E be a given mapping. We say that f is a (γ , ψ, p)-contractive mapping if there exist two functions γ : E × E → [, ∞) and ψ ∈  such that   γ (x, y)p(fx, fy) ≤ ψ p(x, y) for all x, y ∈ E. In the following, we will show some fixed point theorems of Darbo type proved by Aghajani et al. and that (γ , ψ, p) is a contractive mapping, which plays a key role in the proof of our main results. Lemma . ([]) Let  be a nonempty, bounded, closed and convex subset of a Banach space E, and let T :  →  be a continuous operator satisfying the inequality   μ(TX) ≤ φ μ(X)

(.)

for any nonempty subset X of , where μ is an arbitrary measure of noncompactness and φ : R+ → R+ is a nondecreasing function such that limn→∞ ϕ n (t) =  for each t ≥ . Then T has at least one fixed point in the set . Lemma . ([]) Let p be a w-distance on a complete metric space (E, d), and let f : E → E be a (γ , ψ, p)-contractive mapping. Suppose that the following conditions hold: (i) f is a γ -admissible mapping; (ii) there exists a point x ∈ E such that γ (x , fx ) ≥ ; (iii) either f is continuous or, for any sequence {xn } in E, if γ (xn , xn+ ) ≥  for all n ∈ N and xn → x ∈ E as n → ∞, then γ (xn , x) ≥  for all n ∈ N. Then there exists a point u ∈ E such that fu = u. Moreover, if γ (u, u) ≥ , then p(u, u) = .

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3 Main results In this section, we establish the existence theorems of solutions for problem (.). For convenience, we give some notations. For B ⊂ PC(J, E), let B(t) = {u(t) : u ∈ B} and denote BR = {u ∈ B : u ≤ R}. Now, following [, , ], let us introduce the definition of a solution of problem (.). Definition . A function u ∈ PC(J, E) is said to be a solution of problem (.) if u(t) = uk (t) for t ∈ (tk , tk+ ) and uk ∈ C([, tk+ ], E) satisfies the equation c Dα+ u(t) = f (t, u(t), (Tu)(t), (Su)(t)) a.e. on (, tk+ ), and the conditions u(tk ) = Ik (u(tk )), u (tk ) = Ik∗ (u(tk )), k = , , . . . , p, and au() + bu () = γ (u), au() + bu () = γ (u) hold. By using a similar technique as in [], Section , we obtain the following lemma. Lemma . Let ρ ∈ C(J, E) and α ∈ (, ], a function u given by ⎧ t    α– ⎪ (t – s)α– ρ(s) ds + ( ab – t)[ (α) ρ(s) ds ⎪  ( – s)

(α)  ⎪ ⎪  ⎪  b  α– ⎪ ⎪ + a (α–)  ( – s) ρ(s) ds] + a [(a( – t) + b)γ (u) + (at – b)γ (u)], ⎪ ⎪ ⎪ ⎪ ⎪ t ∈ [, t ]; ⎪ ⎪ ⎪ t  ⎪ ⎪  b  α– α– ⎪ ⎪ ⎪ (α)  (t – s) ρ(s) ds + ( a – t)[ (α)  ( – s) ρ(s) ds ⎪  ⎪  p ⎪ b α– ⎪ + a (α–) ρ(s) ds] + ( ab +  – t) j= Ij∗ (u(tj ))( ab – tj ) ⎪  ( – s) ⎪ ⎪ ⎪ ⎨ + ( b +  – t) p I (u(t )) – (t – t ) p I ∗ (u(t )) – p I (u(t )) j j j j j= j j=k+ j j=k+ j a u(t) = (.) ⎪ +  [(a( – t) + b)γ (u) + (at – b)γ (u)], ⎪  ⎪ a ⎪ ⎪ ⎪ ⎪ t ∈ (tk , tk+ ], k = , , . . . , p – ; ⎪ ⎪ ⎪ t  ⎪ ⎪   α– ⎪ (t – s)α– ρ(s) ds + ( ab – t)[ (α) ρ(s) ds ⎪  ( – s)

(α)  ⎪ ⎪  ⎪  p ⎪ b b α– ⎪ + a (α–)  ( – s) ρ(s) ds] + ( a +  – t) j= Ij∗ (u(tj ))( ab – tj ) ⎪ ⎪ ⎪ p ⎪ ⎪ ⎪ + ( ab +  – t) j= Ij (u(tj )) + a [(a( – t) + b)γ (u) + (at – b)γ (u)], ⎪ ⎪ ⎪ ⎩ t ∈ (tp , tp+ ] is a unique solution of the following impulsive problem: ⎧ c α ⎪ ⎪ ⎨ D+ u(t) = ρ(t), ⎪ ⎪ ⎩

t ∈ J ,

u(tk ) = Ik (u(tk )), au() + bu () = γ (u),

u (tk ) = Ik∗ (u(tk )),

k = , , . . . , p,

(.)

au() + bu () = γ (u).

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



(α)



t

(t – s)α– ρ(s) ds + ak + bk t,

for t ∈ (tk , tk+ ],



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



(α – )



t

(t – s)α– ρ(s) ds + bk , 

for t ∈ (tk , tk+ ].

(.)

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We have u () = b ,

u() = a , u() =



(α)

u () =





( – s)α– ρ(s) ds + ap + bp , 



(α – )





( – s)α– ρ(s) ds + bp . 

So, applying the boundary conditions (.), we have aa + bb = γ (u),     b a ( – s)α– ds + ( – s)α– ds + aap + abp + bbp = γ (u).

(α) 

(α – ) 

(.) (.)

Furthermore, in view of u (tk ) = u (tk+ ) – u (tk– ) = Ik∗ (u(tk )), we have   bk = bk– + Ik∗ u(tk ) , bk = bp –

p 

  Ij∗ u(tj )

(.) (k = , , . . . , p – ).

(.)

j=k+

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

(.)

which by (.) implies that     ak = ak– – Ik∗ u(tk ) tk + Ik u(tk ) .

(.)

Thus

ak = ap +

p  j=k+

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

(.)

j=k+

Combining (.), (.), (.) with (.) yields

ap =

  (a + b)γ (u) – bγ (u) b + ( – s)α– ρ(s) ds a a (α)        p   b b b ( – s)α– ρ(s) ds – Ij∗ u(tj ) tj – + +  a (α – )  a a j=  +

 b   b + Ij u(tj ) , a j=

(.)

Li and Gou Advances in Difference Equations (2017) 2017:218

bp =

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   γ (u) – γ (u) – ( – s)α– ρ(s) ds a

(α)      p    b b – ( – s)α– ρ(s) ds + Ij∗ u(tj ) tj – a (α – )  a j= –

p    Ij u(tj ) .

(.)

j=

Furthermore, by (.), (.), (.) and (.), we have p 

ak = ap +

p      Ij∗ u(tj ) tj + Ij u(tj )

j=k+

j=k+

  (a + b)γ (u) – bγ (u) b  + ( – s)α– ρ(s) ds = a a (α)        p   b b  b α– ∗ +  ( – s) ρ(s) ds – Ij u(tj ) tj – + a (α – )  a a j=  p b   ∗   b + Ij u(tj ) + Ij u(tj ) tj + a j= 

j=k+

   Ij u(tj ) p

+

(k = , , , . . . , p – ),

(.)

j=k+

bk = bp –

p 

  Ij∗ u(tj )

j=k+

=

   γ (u) – γ (u) – ( – s)α– ρ(s) ds a

(α)      p     b b – ( – s)α– ρ(s) ds + Ij∗ u(tj ) tj – a (α – )  a j= –

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

(.)

j=k+

Hence, for k = , , , . . . , p – , (.) and (.) imply        b b –t ( – s)α– ρ(s) ds + ( – s)α– ρ(s) ds a

(α)  a (α – )        p p    b  b b ∗ +–t – tj + +–t Ij u(tj ) Ij u(tj ) + a a a j= j=

 a k + bk t =

– (t – tj )

p  j=k+

+

p      Ij∗ u(tj ) – Ij u(tj ) j=k+

    a( – t) + b γ (u) + (at – b)γ (u) .  a

(.)

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For k = p, (.) and (.) imply       b  b –t ( – s)α– ρ(s) ds + ( – s)α– ρ(s) ds a

(α)  a (α – )        p p    b  b b Ij∗ u(tj ) Ij u(tj ) +–t – tj + +–t + a a a j= j=

 a k + bk t =

+

    a( – t) + b γ (u) + (at – b)γ (u) .  a

(.)

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 (.). This completes the proof.  To prove our main results, we state the following basic assumptions of this paper. (H) f : R+ × E × E × E → E is continuous and there exists a nondecreasing and upper semicontinuous function ϕ ∈ ; furthermore, there exist nondecreasing continuous functions  ,  : R → R with  () =  () =  such that      f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t)        ≤ ϕ u – v +  Tu – Tv +  Su – Sv . (H) The function defined by |f (t, , , )| is bounded on J, i.e.,

M = sup f (t, , , ) : t ∈ J < ∞. (H) γi (i = , ): E → E are continuous and compact mappings and there exists a nondecreasing and upper semicontinuous function ϕi ∈ ; furthermore, there exist constants Ni such that γi (u) – γi (v) ≤ ϕi (u – v) and γi (u) ≤ Ni (i = , ) for any u, v ∈ BR . (H) The functions Ik , Ik∗ : E → E are continuous, and there exists a nondecreasing and upper semicontinuous function ϕ , ϕ ∈ ; furthermore, there exist constants μ >  and ρ >  such that     Ik (u) – Ik (v) ≤ ϕ u – v

  and Ik (u) ≤ μ,

for all u, v ∈ BR , k = , , . . . , p,    ∗   I (u) – I ∗ (v) ≤ ϕ u – v and I ∗ (u) ≤ ρ, k k k for all u, v ∈ BR , k = , , . . . , p. (H) There exists a positive solution r of the inequality 

     (b + a) b + ϕ u +  (D ) +  (D ) + M a (α + ) a (α) +

   a+b p a(b + a)μ + a + ab + b ρ +  (N + N ) ≤ r,  a a

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where D is a positive constant defined by the equality    t   D = sup k(t, s)p s, u(s) ds : t, s ∈ J, u ∈ BR ,    a    D = sup h(t, s)p s, u(s) ds : t, s ∈ J, u ∈ BR . 

Moreover,  t       k(t, s) p s, u(s) – p s, v(s)  ds =  lim t→∞ 

and

 lim

a→∞ 

a

     h(t, s) p s, u(s) – p s, v(s)  ds = .

Theorem . Let E be a Banach space, suppose that conditions (H)-(H) are satisfied. Then problem (.) has at least one solution in the space PC(J, E). Proof First we consider the operator Q : PC(J, E) → PC(J, E) defined by (Qu)(t) :=

 t    (t – s)α– f s, u(s), (Tu)(s), (Su)(s) ds

(α)        b  + –t ( – s)α– f s, u(s), (Tu)(s), (Su)(s) ds a

(α)       b ( – s)α– f s, u(s), (Tu)(s), (Su)(s) ds + a (α – )        p p    b  b b ∗ +–t – tj + +–t Ij u(tj ) Ij u(tj ) + a a a j= j= – (t – tj )

p 

p      Ij∗ u(tj ) – Ij u(tj )

j=k+

j=k+

    +  a( – t) + b γ (u) + (at – b)γ (u) , a

t ∈ (tk , tk+ ], k = , , . . . , p – .

It is easy to see that the fixed points of Q are the solutions of nonlocal problem (.). Set Br = {u ∈ B : u(t) ≤ r , t ∈ J}, then Br is a closed ball in PC(J, E) with center θ and radius r . For ∀u ∈ Br , by means of (H), (H) and the triangle inequality, we get          f s, u(s), (Tu)(s), (Su)(s)  ≤ f s, u(s), (Tu)(s), (Su)(s) – f (s, , , ) + f (s, , , )       ≤ ϕ u +  Tu +  Su + M   ≤ ϕ u +  (D ) +  (D ) + M . (.) First, we notice that the continuity of Q(u)(t) for any u ∈ PC(J, E) is obvious, and by (.), we have  t      (Qu)(t) ≤  (t – s)α– f s, u(s), (Tu)(s), (Su)(s)  ds

(α)     b     + – t ( – s)α– f s, u(s), (Tu)(s), (Su)(s)  ds a

(α) 

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      b ( – s)α– f s, u(s), (Tu)(s), (Su)(s)  ds a (α – )    p  ∗ p   b   I u(tj )  · b – tj + b +  – t Ij u(tj )  + +  – t j a a a j= j=

+

– |t – tj |

p p   ∗      I u(tj )  – Ij u(tj )  j j=k+

j=k+

      a( – t) + b · γ (u) + |at – b| · γ (u) a  t     (t – s)α– ds ≤ ϕ u +  (D ) +  (D ) + M

(α)      b b  ( – s)α– ( – s)α– + ds + ds a 

(α) a  (α – )     p p  ∗     b b I u(tj )  + b +  Ij u(tj )  + + j a a a j= j= +

p p   ∗        Ij u(tj )  + a + b (N + N ) + Ij u(tj ) + a



j=k+

j=k+

     (b + a) b ≤ + ϕ u +  (D ) +  (D ) + M a (α + ) a (α) +

   a+b p a(b + a)μ + a + ab + b ρ +  (N + N ),  a a

and D , D are given by (H). Thus   (Qu)(t) ≤



+

     b (b + a) +  ϕ u +  (D ) +  (D ) + M a (α + ) a (α)    a+b p a(b + a)μ + a + ab + b ρ +  (N + N ) ≤ r.  a a

Now Q is well defined, we have Q(Br ) ⊂ Br , where r is a constant appearing in assumption (H). We shall show that Q is continuous from Br into Br . To show this, take u, v ∈ Br and ε >  arbitrarily such that u – v < ε, for t ∈ J, we have   Qu(t) – Qv(t)  t       ≤ (t – s)α– f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t)  ds

(α)    b     ( – s)α–  f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t)  ds + – t a

(α)     α–      b ( – s)   + f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t) ds a  (α – )  p  ∗ b  ∗  b   Ij u(tj ) – Ij v(tj ) · – tj + +  – tk a a j=

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 p   ∗ p   b       Ij u(tj ) – Ij v(tj )  – |t – tj | + I u(tj ) – I ∗ v(tj )  + +  – tk j j a j= j=k+

+

p          Ij u(tj ) – Ij v(tj )  +  a( – t) + b γ (u) – γ (v)  a

j=k+

  + (at – b)γ (u) – γ (v)     b (b + a) +  ϕ u – v ≤ a (α + ) a (α)    t      +  k(t, s) p s, u(s) – p s, v(s) ds 

  + 



a

        p h(t, s) p s, u(s) – p s, v(s) ds +  a(b + a)ϕ u – v a

    b + a      + a + ab + b ϕ u – v +  ϕ u – v + ϕ u – v a         b (b + a) +  ϕ u – v +  δ(t, u, v) +  γ (t, u, v) ≤ a (α + ) a (α)       p +  a(b + a)ϕ u – v + a + ab + b ϕ u – v a    b + a  +  ϕ u – v + ϕ u – v , a

(.)

where  t      δ(t, u, v) = k(t, s) p s, u(s) – p s, v(s) ds ,   a      h(t, s) p s, u(s) – p s, v(s) ds . γ (t, u, v) = 

Furthermore, considering conditions (H) and (H), there exists a >  such that for t ≥ a we have    δ(t, u, v) < ε,

   γ (t, u, v) < ε.

(.)

Then, from (.) and (.) for t ≥ a, we have   (Qu)(t) – (Qv)(t)    b+a  b p (b + a) +  +  (a(b + a) + a + ab + b +  ϕmax (ε), ≤ a (α + ) a (α) a a where ϕmax (ε) := max{ϕ(ε), ϕi (ε), i = , , , }. Now we assume that t ∈ J, then by using the continuity of p , p on J × J × [–r, r] and condition (H), we can obtain    δ(t, u, v) → ,

   γ (t, u, v) → ,

as ε → . Thus, we proved that Q : Br → Br is a continuous operator.

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Now we show that for any nonempty set X ⊂ Br , μ(QX) ≤ φ(μ(X)). First, we demonstrate that the operator Q : Br → Br is equicontinuous. In view of (.), we define fmax = sup(t,u)∈J×Br f (s, u, Tu, Su). For any u ∈ Br and tk– ≤ τ < τ ≤ tk ∈ J with |τ – τ | ≤ ε,  we get that   (Qu)(τ ) – (Qu)(τ )   τ (τ – s)α–     f s, u(s), (Tu)(s), (Su)(s) ds =  τ

(α)   τ   (τ – s)α– – (τ – s)α–  f s, u(s), (Tu)(s), (Su)(s) ds +

(α)     ( – s)α–  + (τ – τ ) f s, u(s), (Tu)(s), (Su)(s) ds

(α)     α–   ( – s) b f s, u(s), (Tu)(s), (Su)(s) ds + a  (α – )   p p p        b   + (τ – τ ) – tj + (τ – τ ) Ij∗ u(tj ) Ij u(tj ) + (τ – τ ) Ij∗ u(tj ) a j= j= j=k+   (τ – τ )   + γ (u) – γ (u)   a    b   + |τ – τ | + (τ – τ )α ≤ fmax

(α + ) a (α)

(α + )    b |τ – τ | +  ρ + μ |τ – τ | + (μ + ρ). +p a a In conclusion , ω(Qu, ε) →  as ε → , which implies that Q(Br ) is equicontinuous. Linking these statements with the above estimate and formula (.), we deduce the following inequality:  ω (QX) ≤

   b (b + a) +  ϕ ω (X) . a (α + ) a (α)

Also, for fixed t ∈ J and u, v ∈ Br , from (.) we obtain   (Qu)(t) – (Qv)(t)      b + a   p  ≤  a(b + a) + a + ab + b +  ϕmax u – v a a         b (b + a) +  ϕmax u – v +  δ(t, u, v) +  γ (t, u, v) , + a (α + ) a (α) where ϕmax (u – v) := max{ϕ(u – v), ϕi (u – v), i = , , , }. By using condition (.) and t → ∞, we deduce that  lim sup diam(QX)(t) ≤

t→∞

  b + a p a(b + a) + a + ab + b +  a a     b (b + a) +  lim sup diam ϕmax X(t) . + a (α + ) a (α) t→∞

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Consequently, by considering μ defined by (.), we have      b + a   (b + a) p b  μ(QX) ≤  a(b + a) + a + ab + b +  + +  ϕ μ(X) , a a a (α + ) a (α) (b+a) b + a (α+) + a (α) ]ϕ(s) ∈ . Combining where φ(s) = [ ap (a(b + a) + (a + ab + b )) + b+a a the above estimate with all properties of the operator Q, by Lemma ., we complete the proof. 

Theorem . Let ξ : E × E → R+ be a given function. Assume that the following conditions hold: (A) There exists ψ ∈  such that      f s, u(s), (Tu)(s), (Su)(s) – f s, v(s), (Tv)(s), (Sv)(s)    a (α) (α + ) ψ u – v , [a (α)(a + b) + b (α + )]      γi (u) – γi (v) ≤ a ψ u – v , (a + b)    –    –    a Ik u t – Ik v tk  ≤ ψ u – v , k p(a(b + a))  ∗   –  ∗   –    a I u t – Ik v t k  ≤ ψ u – v k k p(a + ab + b ) ≤

for all t ∈ J and for all u, v ∈ E with ξ (u, v) ≥ , i = , , k = , , . . . , p. (B) There exists u ∈ PC(J, E) such that ξ (u (t), Qu (t)) ≥  for all t ∈ J, where a mapping Q : PC(J, E) → PC(J, E) is defined by (Qu)(t) :=

 t    (t – s)α– f s, u(s), (Tu)(s), (Su)(s) ds

(α)        b  + –t ( – s)α– f s, u(s), (Tu)(s), (Su)(s) ds a

(α)       b + ( – s)α– f s, u(s), (Tu)(s), (Su)(s) ds a (α – )        p p    b  b b ∗ +–t – tj + +–t Ij u(tj ) Ij u(tj ) + a a a j= j= – (t – tj )

p  j=k+

p      Ij∗ u(tj ) – Ij u(tj ) j=k+

    +  a( – t) + b γ (u) + (at – b)γ (u) , a

t ∈ (tk , tk+ ], k = , , . . . , p – .

(C) For each t ∈ J, and u, v ∈ PC(J, E), ξ (u(t), v(t)) ≥  implies that ξ (Qu(t), Qv(t)) ≥ . (D) For each t ∈ J, if {un } is a sequence in PC(J, E) such that un → u in PC(J, E) and ξ (un (t), un+ (t)) ≥  for all n ∈ N, then   ξ un (t), u(t) ≥  for all n ∈ N. Then problem (.) has at least one solution.

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Proof First of all, let E = PC(J, E). By Lemma ., it is easy to see that u ∈ E is a solution of (.) given by (.), then problem (.) is equivalent to finding u∗ ∈ E which is a fixed point of Q. Now, let u, v ∈ E such that ξ (u(t), v(t)) ≥  for all t ∈ J. By condition (A), we have   Qu(t) – Qv(t)  t       ≤ (t – s)α– f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t)  ds

(α)    b     ( – s)α–  f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t)  ds + – t a

(α)         b ( – s)α–  f t, u(t), (Tu)(t), (Su)(t) – f t, v(t), (Tv)(t), (Sv)(t)  ds + a  (α – )  p  ∗ b  ∗  b   Ij u(tj ) – Ij v(tj ) · – tj + +  – tk a a j= p p     ∗ b       Ij u(tj ) – Ij v(tj )  – |t – tj | + I u(tj ) – I ∗ v(tj )  + +  – tk j j a j= j=k+

+

p          Ij u(tj ) – Ij v(tj )  +  a( – t) + b γ (u) – γ (v)  a

j=k+

  + (at – b)γ (u) – γ (v)   ≤ ψ u – v . This implies that for each u, v ∈ E with ξ (u(t), v(t)) ≥  for all t ∈ J, we obtain that     Qu(t) – Qv(t) ≤ ψ u – v

(.)

for all u, v ∈ E. Now, we define the function γ : E × E → [, ∞) by ⎧ ⎨ if ξ (u(t), v(t)) ≥  for all t ∈ J, γ (u, v) = ⎩ otherwise, and also we define the w-distance p on E by p(u, v) = u – v. From (.), we have   γ (u, v)p(Qu, Qv) ≤ ψ p(u, v) for all u, v ∈ E. This implies that Q is a (γ , ψ, p)-contractive mapping. From condition (B), there exists u ∈ E such that γ (u , Qu ) ≥ . Next, by using condition (C), the following assertions hold for all u, v ∈ E: γ (u, v) ≥ 



  ξ u(t), v(t) ≥    ξ Qu(t), Qv(t) ≥ 



γ (Qu, Qv) ≥ 



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and hence Q is a γ -admissible mapping. Finally, from condition (D) we get that condition (iii) of Lemma . holds. Therefore, by Lemma ., we find x∗ ∈ E such that x∗ = Qx∗ , and so x∗ is a solution of problem (.), which completes the proof. 

4 An example In this section we give an example to illustrate the usefulness of our results. Example . We consider the following impulsive fractional differential equation: ⎧   t se–t cos u(t) t c  ⎪ ⎪ ⎪ D+ u(t) = +t ln( + |u(t)|) +  +| sin u(t)| ds ⎪ ⎪ ⎪ t s (+sin u(t)) ⎪ ⎨ ds, + s| cos u(t)|+e t 

e (+sin u(t))

⎪ |u(  )| |u(  )| ⎪ ⎪ u (  ) = +|u(  )| , u(  ) = +|u(  )| , ⎪ ⎪   ⎪ ⎪ ⎩  η u( ), u() + u () = m i u(  ), u() + u () = m i= i i= η 

(.)

i are given positive constants with where  < η < η < · · · < ,  < η < η < · · · < , and ηi , η m m    j <  . Take J := [, ] and α =  , p = , a = , b = . i= ηi <  and j= η Let  Tu = 

t

se–t cos u(t) ds,  + | sin u(t)| 

f (t, u, Tu, Su) = Ik (u) =

|u(  )|

t  + t  ,

 + |u(  )|   m   γ (u) = , ηi u  i=

 Su = 

t

s| cos u(t)| + es ( + sin u(t)) ds, et ( + sin u(t))

  ln  + u(t) + Fu + Gu, Ik∗ (u) =

|u(  )|

,  + |u(  )|   m   γ (u) = , η i u  i=

then the impulsive fractional differential equation (.) can be transformed into the abstract form of problem (.). We show that all the conditions of Theorem . are satisfied for problem (.). Next, let u, v ∈ PC(J, E), we calculate   f (t, u, Tu, Su) – f (t, v, Tv, Sv)    t     =   + t  ln  + u(t) – ln  + v(t)   + (Tu – Tv) + (Su – Sv)          |u| – |v|   +  Tu – Tv +  Su – Sv ≤  ln  +     + |u|           ln  + |u – v| +  Tu – Tv +  Su – Sv ≤          + |u – v|  +  Tu – Tv +  Su – Sv ≤ ln  

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       |u – v| ≤ ln  + +  Tu – Tv +  Su – Sv          ≤ ln  + |u – v| +  Tu – Tv +  Su – Sv        = ϕ |u – v| +  Tu – Tv +  Su – Sv .

Obviously, the function ϕ(t) =  ln( + t) is nondecreasing on J and ϕ(t) < t for all t > ,  (t) =  (t) = t, hence condition (H) holds. On the other hand, the fact that M = sup{|f (t, , , )|, t ∈ R+ } =  shows condition (H) is valid. Moreover, we have     Ik (u) – Ik (v) =  

   ≤  u – v,      + |u(  )|  + |v(  )|     |u(  )| ∗ |v(  )|    I (u) – I ∗ (v) =  k k   + |u(  )| –  + |v(  )|  ≤  u – v,    m    m            γ (u) – γ (v) =  ηi u ηi v –  ≤ u – v,      i= i=  m     m            γ (u) – γ (v) =  – η i η i u v  ≤ u – v,      i= i=       |u(  )|    ∗   |u(  )|   Ik (u) =  I (u) =   ≤  u,  k   + |u(  )|     + |u(  )|  ≤  u,    m   m                  γ (u) =  γ (u) =  ηi u ≤ u, η i u x,  ≤ u,        |u(  )|



i=

|v(  )|

i=

  k(t, s) p (s, u) – p (s, v) ≤ s , et   h(t, s) p (s, u) – p (s, v) ≤ s , et  t       k(t, s) p s, u(s) – p s, v(s) ds ≤ lim s = lim t =  lim t→∞  t→∞ et t→∞ et and  lim

t→∞ 

t

      h(t, s) p s, u(s) – p s, v(s) ds ≤ lim s = lim t =  t→∞ et t→∞ et

for any u, v ∈ PC(J, E). Also, we have  t  t  t     t s k(t, s)p s, u(s) ds ≤ k(t, s)p s, u(s) ds ≤ ds = , (.) t et    e  t  t  t     t  + et –  s + es h(t, s)p s, u(s) ds ≤ h(t, s)p s, u(s) ds ≤ ds = . (.) et et 





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Using (.), (.), we compute   t    D = sup k(t, s)p s, u(s) ds : t, s ∈ J, u ∈ C(J, E) 

   t : t >  ≤ sup et = e– ,    a   D = sup h(t, s)p s, u(s) ds : t, s ∈ J, u ∈ C(J, E)     t + et –  : t >  ≤ sup et =

 + e –  e

= . Finally, let us consider the first inequality in assumption (H). On the basis of the above calculations, we see that each number r ≥  satisfies the inequality in condition (H), i.e., 

     b (b + a) +  ϕ u +  (D ) +  (D ) + M a (α + ) a (α) +

   a+b p a(b + a)μ + a + ab + b ρ +  (N + N ) ≤ r.  a a

Thus, as the number r , we can take r = . Consequently, all the conditions of Theorem . are satisfied. Thus, problem (.) has at least one solution belonging to the ball Br in the space PC(J, E).

5 Conclusions The aim of this paper is to discuss the existence of solutions for a class of mixed boundary value problems of impulsive integrodifferential equations of fractional order α ∈ (, ]. Our results improve and generalize some known results. Acknowledgements The authors wish to thank the referees for their valuable comments. This work is supported by the National Natural Science Foundation of China (11061031). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 23 March 2017 Accepted: 13 June 2017 References 1. Zhou, WX, Chu, YD: Existence of solutions for fractional differential equations with multi-point boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 17, 1142-1148 (2012) 2. Bai, ZB, Dong, XY, Yin, C: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016, 63 (2016)

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