Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system Hasib Khan a, Aziz Khan b, Fahd Jarad c,∗, Anwar Shah d a

Department of Mathematics, Shaheed Benazir Bhutto University Sheringal, Dir Upper 18000, Khybar Pakhtunkhwa, Pakistan Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia c Department of Mathematics, Çankaya University, Etimesgut Ankara 06790, Turkey d Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khybar Pakhtunkhwa, Pakistan b

a r t i c l e

i n f o

Article history: Received 23 August 2019 Revised 24 September 2019 Accepted 8 October 2019 Available online xxx Keywords: ABC-fractional order impulsive system Existence and uniqueness of solution Data dependence

a b s t r a c t The study of existence of solution ensures the essential conditions required for a solution. Keeping the importance of the study, we initiate the existence, uniqueness and data dependence of solutions an Atangana-Baleanu-Caputo (ABC)-fractional order differential impulsive system. For this purpose, the suggested ABC-fractional order differential impulsive system is transferred into equivalent fixed point problem via integral operator. The operator is then analyzed for boundedness, continuity and equicontinuity. Then Arzela-Ascolli theorem ensures the relatively compactness of the operator and the Schauder’s fixed point theorem and Banach’s fixed point theorem are utilized for the existence and uniqueness of solution. Data dependence and expressive application are also provided.

1. Introduction Fractional order modeling is considered as a generalization of the integer order modeling and has attracted the consideration of analysts of numerous discipline of science and engineering. Due to the large number of applications, experts of theory and numeric techniques have developed a lot of new results. For the readers, we suggest the monographs of several scientists [1–3]. The customary fractional operators involve singularities in their kernels. Such singularities are reported to make some obstacles to the scientists searching for the best in order to model real world phenomena. For the purpose of overcoming such obstacle, some researchers proposed new fractional operators embodying no singular kernels. The first ever work discussing such kind of fractional operators appeared recently. In this work, the Caputo–Fabrizio fractional derivatives were discussed [4]. Then many works appeared to help in developing the theory of the fractional calculus in the setting of the Caputo–Fabrizio fractional derivatives. We refer the readers to the work done in [5,6] and the references mentioned there. The most well known fractional derivative containing no singularities appeared in the work done by Atangana and Baleanu [7]. It was called Atangana–Baleanu (AB)derivative after the name of the authors of the aforementioned work. In the same work the Caputo version (ABC) of such derivative was also presented. This ∗

Corresponding author. E-mail address: [email protected] (F. Jarad).

© 2019 Published by Elsevier Ltd.

derivative was showed to be more applicable and more effective in a huge number of works. To complete the work done, many authors contributed in advancing the theory of the fractional calculus in the frame of AB or ABC derivatives. We recall for example the works done in [8–12] One of the most important issues in the theoretical aspect of the fractional differential equation FDEs is the existence and uniqueness of solution (EUS). This area ensures to the experts the existence of solution with the assumption of necessary conditions. Many discussed the EUS to FDEs in the sense of different derivatives both singular and nonsingular [11–19] The study of the impulsive fractional differential systems have been attracted the attention of scientists in science and engineering due to the wideness of its applications. The study of EUS for impulsive-FDEs is under development. The impulsive-FDEs can be studied for applications in nonlinear oscillations of earthquakes, fluid dynamics, seepage flow in porous media, hereditary problems like biological and many more dynamical aspects subjected to abrupt changes. For detail, the readers may get help from [20– 22]. Recently, Ge and Xin [23] considered a hybrid impulsive FDE in the Caputo’s sense for the study of existence of solution and presented an illustrative application of their results. Dabas and Chauhan [24] discussed EUS of mild solutions for an impulsive FDE involving Caputo’s fractional derivative. They also considered data dependence results and prescribed the applications of the results. Aimene et al. [25] analyzed controllability of an impulsive AB-FDE

https://doi.org/10.1016/j.chaos.2019.109477 0960-0779/© 2019 Published by Elsevier Ltd.

Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477

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and illustrated the application of their results by an expressive example. Sousa et al. [26] discussed the EUS and stabilities for an impulsive FDE in the ψ -Helfer sense and described application of their work. Guo and Jiang [27] studied the EUS for fractional impulsive system in the Caputo’s sense. Mophou [28] studied EUS of mild solution for a FDE with impulses in Caputo’s sense. Kumar and Malik [29] produced existence and stability results for an impulsive system with integral boundary conditions and time scale. Jaradat et al. [30] described an impulsive system of semi-linear initial value problem of fractional order for the EUS and presented applications. Jarad et al. [11] considered fractional differential equation (DE) in Atangana-Baleanu-Caputo fractional derivative sense (ABCfractional derivative) for the EUS of the type:

ABC 0

D ϑ x(t ) = W0 (t , x(t )),

where ϑ ∈ (0, 1), ABC D ϑ is ABC-fractional order differential operator 0 ϑ x (t ), W (t, x (t )) ∈ C[a, b]. (DO), ABC D 0 0 To the best of our knowledge in the subject, no one considered EUS and data dependence of ABC-FDE with impulsive conditions. Therefore, to overwhelm the gape in the study, we consider the following impulsive ABC-FDE of the kind: 0D

ϑ x (t ) = −W (t , x (t )), t ∈ J c = J − {t , t , . . . , t }, J = [0, T ], m 0 1 2

x(tk ) = x(tk+ ) − x(tk− ) = Ik∗ x(t¯k ), x|[−τ ∗ ,0] = ℘ + G ∗ (t ), (1.2) m

m

where ϑ ∈ (0, 1], = i=1 λi xi (t ), xi i=1 λi < 1, for i = 1, 2, . . . , m, 0 = t0 < t1 < t2 < . . . < tm < tm+1 = T , Ik∗ : Rn → Rn are G ∗ (t )

AB

ϑ

0 Iτ ∗

∈ PC1 ,

continuous functions for k = 1, 2, . . . , m, ABC 0 D ϑ is a Atangana– Baleanu fractional derivative in Caputo’s sense. The Lebesgue measurable function W0 (t, x(t )) : J × PC 1 → Rn is continuous on PC1 with PC 1 = PC 1 ([−τ ∗ , 0], Rn ) is the space of piecewise continuous functions ℘ : [−τ ∗ , 0] → Rn . The xt (s ) = x(t + s ) for −τ ∗ ≤ s ≤ 0, x(tk+ ) = limδ →0+ x(tk + δ ) while x(tk− ) = limδ →0− x(tk + δ ). We remark that the system (1.1) is just a special case of the system (1.2) under consideration. Throughout this paper, we consider the Banach space PC 1 ([−τ ∗ , T ], Rn ) of all the piecewise continuous functions

ψ (τ ∗ ) =

1−ϑ ψ (τ ∗ ) B ∗ (ϑ ) +

ϑ B ∗ (ϑ ) (ϑ )

 τ∗ 0

ψ (s )(τ ∗ − s )ϑ −1 ds.

(1.4)

Lemma 1.3 [8], [10]. The following Newton–Leibniz formula holds for the ABC-fractional order DO ABC Dθϑ and integral operator (IO) AB 0 Iθϑ , 0 is



AB ϑ ABC ϑ 0 Iτ ∗ 0 Dτ ∗

 ( θ ) = ψ ( τ ∗ ) − ψ ( 0 ).

(1.5)

Proposition 1.1 [8], [10]. For ψ (t) defined on [c, d] and ϑ ∈ (n, n + 1], for some n ∈ N0 , we have

(1.1)

x ( 0 ) = x0 ,

ABC

Definition 1.2 [8,10]. The AB-fractional-integral of a function ∈ H∗ (a, b), b > a, 0 < ϑ < 1 is

• (A)

ABR ϑ AB ϑ c D c I ψ (t )

= ψ (t ).

• (B)

AB ϑ ABR ϑ c I c D ψ (t )

= ψ (t ) −

n−1 ψ (k ) (c ) (t − c )k . k=0 k!

Theorem 1.4 (Ho¨ lder’s inequality). Let a, θ ≥ 1 and for

u ∈ La (J),

v ∈ Lθ (J), we have

1 a

+ θ1 = 1. Then, uvL1 ( J ) ≤ uLa ( J ) vLθ ( J ) .

Partition of the paper: In this paper, we have considered a study of EUS and data dependence of an ABC-fractional DE with impulses which is divided in four sections. The literature of the topic is illustrated in this section. In the next section, the ABCfractional DE with impulses (1.2) is transformed into an integral form with the help of AB-fractional calculus. In the third section, EUS are studied with the use of classical fixed point approach. Data dependence theorem is proved in the fourth section. Finally, an expressive example is given to illustrate the application of results. 2. Integral form This section is reserved for the integral form of the ABCfractional order impulsive DE (1.2) Theorem 2.1. For ϑ ∈ (0, 1] and W0 (t, x(t )) ∈ C[0, 1] such that W0 (0, x(0 )) = 0 , x(t) is a solution of ABC

0D

ϑ x (t ) = −W (t , x (t )), t ∈ J c = J − {t , t , . . . , t }, J = [0, T ], m 0 1 2

x(tk ) = x(tk+ ) − x(tk− ) = Ik∗ x(t¯k ), x|[−τ ∗ ,0] = ℘ + G ∗ ,

(2.1)

provided that

⎧ ℘ + G∗, ⎪  ⎪ ⎪℘ (0 ) + G ∗ (0 ) − B1∗−(ϑϑ) W0 (t , x(t )) − B∗ (ϑϑ) (ϑ ) 0t (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ , ⎪ ⎪ t ⎪ ⎪I ∗ (x(t − )) − 1∗−ϑ W0 (t , x(t )) − ∗ ϑ (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ ⎪ 1 B (ϑ ) B (ϑ ) (ϑ ) 0 ⎨ 1 ∗ x(t ) = +℘ (0 ) + G (0 ), .. ⎪ ⎪ ⎪ . ⎪ ⎪ t ⎪℘ (0 ) + G ∗ (0 ) − 1−ϑ W (t , x(t )) − ϑ ⎪ (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ 0 ⎪ B ∗ (ϑ ) B ∗ (ϑ ) (ϑ ) 0 ⎩ k ∗ − + r=1 Ir (x(tr )), with the norm x∞ = supt∈[−τ ∗ ,T ] x(t ), where . is a complete norm in Rn , where PC 1 ([−τ ∗ , T ] )(μ ) = {x ∈ PC 1 ([−τ ∗ , T ], Rn ) : x∞ ≤ μ for all μ > 0. Definition 1.1 [7]. The ABC-fractional derivative in Caputo’s sense of order ϑ ∈ [0, 1) for ψ ∈ H∗ (a, b), where b > a, is

B ∗ (ϑ ) ABC ϑ ∗ 0 Dτ ∗ ψ ( τ ) = 1−μ where

B ∗ (ϑ )

 τ∗ 0

ψ







−ϑ (τ ∗ − s )ϑ ( s )Eϑ ds, 1−ϑ

satisfies the property B ∗ (0 ) = B ∗ (1 ) = 1.

(1.3)

for t ∈ [−τ ∗ , 0], for t ∈ [0, t1 ], for t ∈ [t1 , t2 ],

(2.2)

for t ∈ [tm , T ].

Proof. For t ∈ P0 , the solution x(t) of the problem (2.1), for t ∈ [0, t1 ], we have

x(t ) = −AB I0ϑ [W0 (t, xt )] + c0



= ℘ (0 ) + G ∗ (0 ) − +

ϑ (ϑ )B∗ (ϑ )



1−ϑ W0 (t, xt ) B ∗ (ϑ ) t

0

 (t − η∗ )ϑ −1 W0 (η∗ , xη∗ )dη∗ ,

(2.3)

Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477

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and

x(



t1−

) = ℘ (0 ) + G (0 ) − ∗

1−ϑ ϑ W0 (t1− , x(t1− )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )



t1 0

3

 ∗ ϑ −1

(t1 − η )

W0 ( η , xη ∗ )d η ∗

∗

.

(2.4)

For t ∈ (t1 , t2 ], x(tk ) = Ik∗ x(t¯k ) and with the help of (2.3), we have

 t1 1−ϑ 1−ϑ ϑ W0 (t,1 , x(t1 )) + ∗ (t1 − η∗ )ϑ −1 W0 (η∗ , xη∗ )dη∗ − ∗ W0 (t, x(t )) ∗ B (ϑ ) B (ϑ ) (ϑ ) 0 B (ϑ )  t ϑ (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗

x(t ) = x(t1+ ) + −

B ∗ (ϑ ) (ϑ )

= x(

t1−

)+

I1∗

0

(x( )) + t1−

ϑ − ∗ B (ϑ ) (ϑ )

 0

t

ϑ B ∗ (ϑ ) (ϑ ) ∗ ϑ −1

(t − η )

= ℘ (0 ) + G ∗ (0 ) + I1∗ (x(t1− )) − For t ∈ (t2 , t3 ], we have

x(t ) = x(t2+ ) +

−

ϑ B ∗ (ϑ ) (ϑ )



t2 0

t1 0

1−ϑ 1−ϑ W0 (t,1 , x(t1 )) − ∗ W0 (t, x(t )) B ∗ (ϑ ) B (ϑ )

(t1 − η∗ )ϑ −1 W0 (η∗ , xη∗ )dη∗ +

W0 (η∗ , x(η∗ ))dη∗

1−ϑ ϑ W0 (t , x(t )) − ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

(t2 − η∗ )ϑ −1 W0 (η∗ , xη∗ )dη∗ +

1−ϑ ϑ W0 (t, x(t ))− ∗ B ∗ (ϑ ) B (ϑ ) (ϑ ) +





t 0



t 0

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ .

1−ϑ W0 (t,1 , x(t1 )) B ∗ (ϑ )

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ =x(t2− ) + I1∗ (x(t2− ))+

1−ϑ 1−ϑ ϑ W0 (t,1 , x(t1 )) − ∗ W0 (t, x(t )) − ∗ B ∗ (ϑ ) B (ϑ ) B (ϑ ) (ϑ )

= ℘ (0 ) + G ∗ (0 ) + I1∗ (x(t1− )) + I2∗ (x(t2− )) −

(2.5)



t 0

(2.6)

ϑ B ∗ (ϑ ) (ϑ )



t1 0

(t2 − η∗ )ϑ −1 W0 (η∗ , xη∗ )dη∗

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗

1−ϑ ϑ W0 (t , x(t )) − ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )



t 0

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ .

(2.7)

Continuing this process of iteration we reach to the case when t ∈ (tm , T], and ultimately

x(t ) = ℘ (0 ) + G ∗ (0 ) +

m

Ik∗ (x(tk− )) −

k=1

1−ϑ ϑ W0 (t , x(t )) − ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

This accomplishes the required proof.



t 0

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ .

(2.8)



3. Theorems for EUS We assume Banach space Lθ (J, R ) for a measurable function f : J → R with the norm:

 

 f Lθ =

θ j |W0 (t )| dt 



1

θ

,



infμ(J¯) supt ∈J−J¯|W0 (t )| ,

1 ≤ θ < ∞,

(3.1)

θ = ∞,

where μ(J¯) is the Lebesgue measure on J¯ and  f θL < ∞.

⎧ ℘ + G∗, ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪℘ (0 ) + G ∗ (0 ) − B1∗−(ϑϑ) W0 (t , x(t )) − B∗ (ϑϑ) (ϑ ) 0 (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ , ⎪ ⎪ ⎪ t ⎪ ⎪ I1∗ (x(t1− )) − B1∗−(ϑϑ) W0 (t , x(t )) − B∗ (ϑϑ) (ϑ ) 0 (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ ⎪ ⎪ ⎨ T x(t ) = +℘ (0 ) + G ∗ (0 ), ⎪ ⎪ ⎪ ⎪.. ⎪ ⎪ ⎪ ⎪. t ⎪ 1 −ϑ ϑ ∗ ∗ ϑ −1 ∗ ∗ ∗ ⎪ ⎪ ⎪℘ (0 ) + G (0 ) − B∗ (ϑ ) W0 (t , x(t )) − B∗ (ϑ ) (ϑ ) 0 (t − η ) W0 (η , x(η ))dη ⎪ ⎩ k ∗ + r=1 Ir (x(tr− )),

for t ∈ [−τ ∗ , 0], for t ∈ [0, t1 ],

for t ∈ [t1 , t2 ],

(3.2)

for t ∈ [tm , T ].

where μ(J¯) is the Lebesgue measure on J¯ and  f θL < ∞. For the proofs of our main results, we need the following assumptions: 1

• (A1 ) Let for a θ ∈ (0, h) and a function m(t ) ∈ L θ (J ) with W0 (t , x(t )) ≤ m(t ) for t ∈ J and x(t) ∈ PC1 . 1

• (A2 ) Let for a δ ∈ (0, h) and a function μ(t ) ∈ L δ (J ) with W0 (t , x(t )) − W0 (t, W0 (t, )) ≤ μ(t )x − y for t ∈ J and x(t ), W0 (t , ) ∈ PC 1 . 1 • (A3 ) Let the impulses Ik∗ ∈ C (Rn , Rn ) be bounded and for a ν ∗ > 0, we have μ(t ) ∈ L δ (J ) with Ik∗ x(tk− ) − Ik∗ (W0 (t,− )) ≤ ν ∗ (t )x − k 1 ∗ y∞ for each x, y ∈ PC ([−τ , T ] )(ρ ), k=1,2,...,m. • (A4 ) Assume that for some positive constant L > 0, W0 (t, x(t )) ≤ L for t ∈ J and x(t) ∈ PC1 (ρ ). Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477

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Theorem 3.1. If the condition (A ) holds, then the AB-fractional order impulsive system (1.2) has a solution provided that the following inequality is satisfied:

℘ (0 ) + G ∗ (0 ) + mN ∗ + where M∗ =

T 0

1

(m(s )) θ

θ

1 −ϑ B ∗ (ϑ )

W0 (t , x(t )) +

B∗

ϑ (ϑ ) (ϑ )



1 −θ ϑ −θ

1 − θ

t1ϑ −θ M∗

≤ 1,

ρ

(3.3)

, N ∗ = max{Ik∗ (x(tk− )) : x ≤ rho}, k = 1, . . . , m.

Proof. By the definition of PC 1 ([−τ ∗ , T ] )(ρ ), we have that PC 1 ([−τ ∗ , T ] )(ρ ) is closed, convex and bounded subset of PC 1 ([−τ ∗ , T ], Rn ). For the existence of solution, we follow the Schauder fixed point approach. For this, we divide the proof in steps below. Step 1. In this step we are interested to show that T : PC 1 ([−τ ∗ , T ] )(ρ ) → PC 1 ([−τ ∗ , T ] )(ρ ). For this, we need to use the Ho¨ lder’s inequality. For t ∈ [0, t1 ], (3.2) proceed to

T x(t ) = ℘ (0 ) + G ∗ (0 ) −

1−ϑ ϑ W0 (t , x(t )) − ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )



0

t

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ 

≤ ℘ (0 ) + G ∗ (0 ) + 

1−ϑ ϑ W0 (t , x(t )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

≤ ℘ (0 ) + G ∗ (0 ) + 

1−ϑ ϑ W0 (t , x(t )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

×



t 0

(W0 (η∗ , x(η∗ ))) θ dη∗ 1

≤ ℘ (0 ) + G ∗ (0 ) + 



t 0

 t  0

θ

1−ϑ ϑ W0 (t , x(t )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ (t − η∗ )ϑ −1

1−θ  ϑ −θ



1 1−θ

1 −θ ϑ −θ t1 M ∗

1−θ

≤ ρ.

(3.4)

Now, for the t ∈ (tk , tk+1 ], we have

T x(t ) = ℘ (0 ) + G ∗ (0 ) +

k

Ir∗ (x(tr− )) −

r=1

≤ ℘ (0 ) + G ∗ (0 ) + 

k

1−ϑ ϑ W0 (t , x(t )) − ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

Ir∗ (x(tr− ))

r=1

≤ ℘ (0 ) + G (0 ) + mN ∗ + ∗



t

0

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗ 

1−ϑ ϑ W0 (t , x(t )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

1−ϑ ϑ W0 (t , x(t )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )



1−θ  ϑ −θ

t 0

(t − η∗ )ϑ −1 W0 (η∗ , x(η∗ ))dη∗

1 −θ ϑ −θ t1 M ∗

≤ ρ.

(3.5)

With the help of (3.4) and (3.5), we have T x ≤ ρ . This completes the first step. Step 2. In this step, we are interested to prove T a continuous operator. For this, consider a convergent sequence xn → x on the PC 1 ([−τ ∗ , T ] )(ρ ). Since, W0 (t, x(t )) and the impulses Ik∗ x(tk− ) are continuous operators on PC 1 ([−τ ∗ , T ] )(ρ ) for k = 1, 2, . . . , m. Then for t ∈ [0, t1 ], we have

T x n − T x  ≤ =

1−ϑ ϑ W0 (t, xn (t )) − W0 (t, x(t )) + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )



0

t

(t − η∗ )ϑ −1 W0 (η∗ , xn (η∗ )) − W0 (η∗ , x(η∗ ))dη∗

1−ϑ tϑ W0 (t, xn (t )) − W0 (t, x(t )) + ∗ W0 (η∗ , xn (η∗ )) − W0 (η∗ , x(η∗ )) → 0. ∗ B (ϑ ) B (ϑ ) (ϑ )

(3.6)

Similarly, for t ∈ (t1 , t2 ], we have

T xn − T x ≤ I (xn (t¯1 )) − I (x(t¯1 )) + + =

B∗

ϑ (ϑ ) (ϑ )

ν ∗ xn − x +



t 0

1−ϑ W0 (t, xn (t )) − W0 (t, x(t )) B ∗ (ϑ )

(t − η∗ )ϑ −1 W0 (η∗ , xn (η∗ )) − W0 (η∗ , x(η∗ ))dη∗

1−ϑ tϑ W0 (t, xn (t )) − W0 (t, x(t )) + ∗ W0 (η∗ , xn (η∗ )) − W0 (η∗ , x(η∗ )) → 0. B ∗ (ϑ ) B (ϑ ) (ϑ )

(3.7)

Ultimately, for t ∈ (tk , tk+1 ], we have

T x n − T x  ≤

m

I (xn (t¯k )) − I (x(t¯k )) +

k=1

+

ϑ B ∗ (ϑ ) (ϑ )



= mν ∗ xn − x +

t 0

1−ϑ W0 (t, xn (t )) − W0 (t, x(t )) B ∗ (ϑ )

(t − η∗ )ϑ −1 W0 (η∗ , xn (η∗ )) − W0 (η∗ , x(η∗ ))dη∗

1−ϑ tϑ W0 (t, xn (t )) − W0 (t, x(t )) + ∗ W0 (η∗ , xn (η∗ )) − W0 (η∗ , x(η∗ )) → 0. B ∗ (ϑ ) B (ϑ ) (ϑ )

(3.8)

With the help of (3.6)-(3.8), we have the operator T : PC 1 ([−τ ∗ , T ] ) → PC 1 ([−τ ∗ , T ] ) is continuous. Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477

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5

Step 3. This section is reserved for the proof of equicontinuity of the operator T : PC 1 ([−τ ∗ , T ] ) → PC 1 ([−τ ∗ , T ] ). For this, consider

T x(t2 ) − T x(t1 ) ≤ +

ϑ

1−ϑ W0 (t1 , x(t2 )) − W0 (t1 , x(t1 )) B ∗ (ϑ )  t2   t2  (t2 − η∗ )ϑ −1 − (t2 − η∗ )ϑ −1 W0 (η∗ , xn (η∗ ))dη∗

B ∗ (ϑ ) (ϑ )

0

(3.9)

0

(t2ϑ − t1ϑ ) 1−ϑ  W ( t , x ( t )) − W ( t , x ( t ))  + W0 (η∗ , xn (η∗ )) 0 2 2 0 1 1 B ∗ (ϑ ) B ∗ (ϑ ) (ϑ ) → 0. =

With the help of (3.9), we have T x(t2 ) − T x(t1 ) → 0 as t1 → t2 . Similarly, we have for t ∈ (tk , tk+1 ]. This implies that the operator T is an equicontinuous operator. Thus, with the use of Arzela-Ascolli theorem, T is a relatively compact which implies T is completely continuous. Thus, by Step 1. to 3., and Schauder fixed point theorem, the operator T has a fixed point and consequently the AB-fractional order impulsive system (1.2) has a solution.  Corollary 3.2. Let the assumption (A4 ) hold true. Then, the AB-fractional order impulsive system (1.2) has a solution provided that the following is satisfied

℘ (0 ) + G ∗ (0 ) + mN ∗ + ρ where M∗ =

T 0

1

(m(s )) θ

θ

1 −ϑ L B ∗ (ϑ )

+

ϑ

B ∗ (ϑ ) (ϑ )

TϑL

≤ 1,

, N ∗ = max{Ik∗ (x(tk− )) : x ≤ ρ}, k = 1, . . . , m.

Proof. The proof is similar to the proof of Theorem 3.1. Therefore, we omit the proof.



Theorem 3.3. If the assumptions (A1 ) − (A3 ), the ABC-fractional order impulsive system (1.2) has a unique solution provided that the following inequality is satisfied

mν +

1−ϑ 1 μ+ ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

where ϑ = maxt∈J (

t

0 (μ (t ))

1−δ  ϑ −δ

1 −δ

T ϑ −δ ϑ < 1 ,

(3.10)

1

δ )δ .

Proof. For the uniqueness of solution we assume the contrary path that is let there exist two solutions say x1 and x2 . Then we consider three cases Case 1. For t ∈ [−τ ∗ , 0], we have T x1 − T x2  → 0. This means that the x1 = x2 . Case 2. Then for t ∈ [0, t1 ], we have

 t 1−ϑ ϑ  W ( t, x ( t )) − W ( t, x ( t ))  + (t − η∗ )ϑ −1 W0 (η∗ , x1 (η∗ )) − W0 (η∗ , x2 (η∗ ))dη∗ 0 1 0 2 B ∗ (ϑ ) B ∗ (ϑ ) (ϑ ) 0  t 1−ϑ 1−ϑ tϑ ≤ ∗ μx1 − x2  + ∗ (t − η∗ )ϑ −1 μ(t )x1 − x2 dt ≤ ∗ μx1 − x2  B (ϑ ) B (ϑ ) (ϑ ) 0 B (ϑ )  t    t  1 tϑ 1 + ∗ (t − η∗ )ϑ −1 1−δ dη∗ 1−δ (μ(t )) δ dt δ x1 − x2  B (ϑ ) (ϑ ) 0 0 1−δ  1−ϑ 1 1 −δ ϑ −δ ≤ ∗ μx1 − x2  + ∗ T ϑx1 − x2  B (ϑ ) B (ϑ ) (ϑ ) ϑ − δ  1−ϑ   1 1 − δ 1 −δ ϑ −δ ≤ μ+ ∗ T ϑ x1 − x2 . B ∗ (ϑ ) B (ϑ ) (ϑ ) ϑ − δ

T x 1 − T x 2  ≤

(3.11)

Step 3. For t ∈ (tk , tk+1 ]. For this, consider

T x 1 − T x 2  ≤

k

Ik∗ x1 (tk ) − Ik∗ x2 (tk ) +

r=1

+ +

ϑ B ∗ (ϑ ) (ϑ ) tϑ B∗

(ϑ ) (ϑ )



t

0



0

t

1−ϑ W0 (t, x1 (t )) − W0 (t, x2 (t )) B ∗ (ϑ )

(t − η∗ )ϑ −1 W0 (η∗ , x1 (η∗ )) − W0 (η∗ , x2 (η∗ ))dη∗ ≤ mνx1 − x2  + (t − η∗ )ϑ −1 μ(t )x1 − x2 dt ≤ mνx1 − x2  +

1−ϑ μx1 − x2  B ∗ (ϑ )

  t   1−ϑ 1 ∗ ϑ −1 1−1 δ ∗ 1 −δ δ dt δ x − x  ≤ mνx − x  + ( t − η ) d η × ( μ ( t )) μx1 − x2  1 2 1 2 ∗ (ϑ ) B ∗ (ϑ ) (ϑ ) 0 B 0       1 1 − δ 1 −δ ϑ −δ 1−ϑ 1 1 − δ 1 −δ ϑ −δ + ∗ T ϑx1 − x2  ≤ mν + ∗ μ+ ∗ T ϑ x1 − x2 . B (ϑ ) (ϑ ) ϑ − δ B (ϑ ) B (ϑ ) (ϑ ) ϑ − δ +

tϑ

 t 

1−ϑ μx1 − x2  B ∗ (ϑ )

(3.12)

Thus, with the help of Case 1 to 3 and condition (3.10), we have T is a contraction. This with the Banach’s fixed point theorem implies that the fixed point is unique which further implies that the solution of the ABC-fractional order impulsive system has a unique solution. This completes the proof.  Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477

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Corollary 3.4. With the assumption of (A1 ), (A3 ), (A5 ) the ABC-fractional order impulsive system (1.2) has a unique solution provided that the following inequality is satisfied

mν +

1−ϑ 1 μ+ ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

1−δ  ϑ −δ

1 −δ

T ϑ −δ μ < 1 .

(3.13)

Proof. The proof is similar to the proof of Theorem 3.3. Therefore, we omit the proof here.



4. Data dependence In this section, we are interested in the analysis for data dependence of the solution of the ABC-fractional order impulsive system (1.2). Theorem 4.1. Let us assume the condition (A1 ) − (A3 ) and consider x(t) be a solution of the (3.2) and let  x(t ) be a solution of (3.2) sat, for ℘ ∈ PC 1 . If for some  ∗ > 0, ℘ − ℘ 1 < δ ∗ = (1 − c ) ∗ then we have x −  isfying that  x|[−τ ∗ ,0] = ℘ x∞ <  ∗ , where c = mν + B1∗−(ϑϑ ) μ + 1



B ∗ (ϑ ) (ϑ )

1−δ ϑ −δ



1−δ T ϑ −δ ϑ .

Proof. With the help of Theorem 2.1, we have

x(t ) −  x(t ) =

⎧ (0 ) − 1∗−ϑ [W0 (t , x(t )) − W0 (t ,  (0 ) − G ℘ (0 ) + G ∗ (0 ) − ℘ x(t ))] ⎪ B (ϑ ) ⎪ ⎪ ⎪  ⎪ ⎨− B∗ (ϑϑ) (ϑ ) 0t (t − η∗ )ϑ −1 [W0 (t , x(t )) − W0 (t ,  x(t ))]dη∗ ,

for t ∈ [0, t1 ],

(4.1)

k

⎪ ℘ (0 ) + G ∗ (0 ) − B1∗−(ϑϑ) [W0 (t , x(t )) − W0 (t ,  x(t ))] + r=1 [Ir∗ (x(tr− )) − Ir∗ ( x(tr− ))] ⎪ ⎪ ⎪ ⎪  ⎩ t − B∗ (ϑϑ) (ϑ ) 0 (t − η∗ )ϑ −1 [W0 (t , x(t )) − W0 (t ,  x(t ))]dη∗ for t ∈ [tm , T ],

Case 1. Then for t ∈ [0, t1 ], we have

1−ϑ W0 (t , x(t )) − W0 (t ,  x(t )) B ∗ (ϑ )

(0 ) + (0 ) − G x(t ) −  x(t ) ≤ ℘ (0 ) + G ∗ (0 ) − ℘ +

ϑ B ∗ (ϑ ) (ϑ )



tϑ

t

0

(0 ) + (0 ) − G (t − η∗ )ϑ −1 W0 (η∗ , x(η∗ )) − W0 (η∗ ,  x(η∗ ))dη∗ ≤ ℘ (0 ) + G ∗ (0 ) − ℘



1−ϑ μx1 − x2  B ∗ (ϑ )

(0 ) + 1 − ϑ μx −  (0 ) − G (t − η∗ )ϑ −1 μ(t )x −  xdt ≤ ℘ (0 ) + G ∗ (0 ) − ℘ x (ϑ ) (ϑ ) 0 B ∗ (ϑ )       t t 1 tϑ 1 + ∗ (t − η∗ )ϑ −1 1−δ dη∗ 1−δ × (μ(t )) δ dt δ x −  x B (ϑ ) (ϑ ) 0 0   1 1 − δ 1 −δ ϑ −δ (0 ) + 1 − ϑ μx −  (0 ) − G ≤ ℘ (0 ) + G ∗ (0 ) − ℘ x  + T ϑx −  x B ∗ (ϑ ) B ∗ (ϑ ) (ϑ ) ϑ − δ  1−ϑ   1 1 − δ 1 −δ ϑ −δ (0 ) + (0 ) − G ≤ ℘ (0 ) + G ∗ (0 ) − ℘ μ+ ∗ T ϑ x −  x. B ∗ (ϑ ) B (ϑ ) (ϑ ) ϑ − δ +

t

B∗

(4.2)

Now for t ∈ (tk , tk+1 ], where k = 1, 2, . . . , m, we have (0 ) + (0 ) − G x −  x ≤ ℘ (0 ) + G ∗ (0 ) − ℘

ϑ + ∗ B (ϑ ) (ϑ )

k

Ik∗ x(tk ) − Ik∗ x(tk ) +

r=1



t 0

∗ ϑ −1

(t − η )

1−ϑ W0 (t , x(t )) − W0 (t ,  x(t )) B ∗ (ϑ )

W0 (η∗ , x(η∗ )) − W0 (η∗ ,  x(η∗ ))dη∗

(0 ) + mνx −  (0 ) − G ≤ ℘ (0 ) + G ∗ (0 ) − ℘ x +

1−ϑ tϑ μx −  x + ∗ B ∗ (ϑ ) B (ϑ ) (ϑ )

1−ϑ tϑ (0 ) + mνx −  (0 ) − G ≤ ℘ (0 ) + G ∗ (0 ) − ℘ x + ∗ μx −  x + ∗ B (ϑ ) B (ϑ ) (ϑ )



t

(t − η∗ )ϑ −1 μ(t )x −  xdt  t   t    1 1 (t − η∗ )ϑ −1 1−δ dη∗ 1−δ (μ(t )) δ dt δ x −  x 0

0

0

1−ϑ (0 ) + mνx −  (0 ) + G ∗ (0 ) − G ≤ ℘ (0 ) − ℘ x + ∗ μx −  x B (ϑ ) +

1 B ∗ (ϑ ) (ϑ )





1−δ ϑ −δ



1−δ

(0 ) (0 ) + G ∗ (0 ) − G T ϑ −δ ϑx −  x ≤ ℘ (0 ) − ℘

1−ϑ 1 + mν + ∗ μ+ ∗ B (ϑ ) B (ϑ ) (ϑ )



1−δ ϑ −δ

 1−δ

T ϑ −δ ϑ

 x −  x .

(4.3)

With the help of (4.2) and (4.3), we have

x −  x ≤

℘ (0 ) − ℘(0 ) + G ∗ (0 ) − G(0 ) 1−c

<

(4.4)

 In order to give verification of the existence and data dependence theorems, here we give the following illustrative model. Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477

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Example 4.1. Consider the following ABC-fractional order impulsive system:

⎧ t sin(x(t− π2 )) ABC ϑ ⎪ D x(t ) = , ⎪ 5 ⎨ 0  kπ 1 x 2 = 7 , ⎪ ⎪ ⎩ cos(t ) x(t ) = t +t100 ,

t ∈ [0, 52π ],

k = 1, 2, 3, 4,

m(t ) ∈ L ([0, and

5 2 π ] ),

(4.5)

t ∈ [− π2 , 0].

t sin(x(t− π2 ))  ≤ 15 , 5 we have δ = 14 = θ

With the help of  1 4

t = k2π ,

assume m(t ) = and

M∗

=

℘ (0 )+G ∗ (0 ) + mN ∗ + B1∗−(ϑϑ ) W0 (t, x(t ))+ B∗ (ϑϑ) (ϑ )

1 5



= 1 −θ ϑ −θ

1 5, N ∗,



1 −θ

clearly

ρ = 1.5 t1ϑ −θ M∗

ρ = 0.681086 < 1.

(4.6)

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Please cite this article as: H. Khan, A. Khan and F. Jarad et al., Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109477