Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative

Applied Mathematics Letters 101 (2020) 106072

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Applied Mathematics Letters www.elsevier.com/locate/aml

Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative✩ Tianwei Zhang a ,∗, Lianglin Xiong b a b

City College, Kunming University of Science and Technology, Kunming 650051, China School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, China

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Article history: Received 18 July 2019 Received in revised form 27 September 2019 Accepted 27 September 2019 Available online xxxx

abstract On the basis of some crucial properties for one-parameter and two-parameter Mittag-Leffler functions, the existence, uniqueness and global exponential stability of periodic solution are discussed for a class of semilinear impulsive fractional functional differential equations with piecewise Caputo derivative. Some better result is achieved and it improves and extends some existing research finding. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Piecewise Caputo fractional derivative Impulse Matrix Mittag-Leffler function Exponential stability

1. Introduction In recent literatures, many scholars devoted to the study of asymptotically periodic solutions for fractional differential equations, because conventional fractional differential equations cannot generate nonconstant periodic solutions [1,2]. Therefore, this article proceeds to this problem and concentrates on periodic dynamics for semilinear impulsive piecewise fractional functional differential equations (IPFFDEs) as follows: ⎧c α ⎨ Dtk ,t X(t) = AX(t) + F (t, Xt ), t ∈ (tk , tk+1 ), t > 0, (1.1) X(t+ k ∈ N, k ) = X(tk ) + ∆k (X(tk )), ⎩ X(s) = ϕ(s), s ∈ [−δ, 0], where δ > 0, 0 < α < 1, c Dtαk ,t is the Caputo fractional derivative with lower limit at tk ; X = (X1 , X2 , . . . , Xn )T ∈ C([−δ, +∞), Rn ), A = (aij )n×n ∈ Rn×n is nonsingular real matrix; F = (F1 , F2 , . . . , Fn )T ∈ Rn and ∆k = (∆1k , ∆2k , . . . , ∆nk )T ∈ Rn are real continuous functions defined on [0, +∞) × Rn and Rn , respectively, k ∈ N = {0, 1, 2, . . .}; {tk } is a set of pulse points; Xt : [−δ, 0] → Rn is defined by ✩ This work is supported by National Nature Science Foundation of China under Grant No. 11461082. ∗ Corresponding author. E-mail address: [email protected] (T. Zhang).

https://doi.org/10.1016/j.aml.2019.106072 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

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T. Zhang and L. Xiong / Applied Mathematics Letters 101 (2020) 106072

Xt (θ) = X(t + θ) for θ ∈ [−δ, 0], ϕ ∈ C([−δ, 0], Rn ). Suppose that there are two constants N ∈ N and T > 0 such that tk+N +1 = tk + T and ∆k+N +1 = ∆k for any k ∈ N. Without loss of generality, assume that t0 = 0 is the first pulse point. The main contribution of this paper is to give a new technique to investigate the existence, uniqueness and global exponential stability of T -periodic solution for IPFFDEs (1.1). The current research finding improves and extends the corresponding result in literature [3], and the details can be viewed in Remark 3.2. To authors’ knowledge, the problems solved in this paper are novel and can stimulate the studies of many other impulsive fractional-order systems. The organization of the rest is as follows. In Section 2, a Volterra integral expression for the solution of IPFFDEs (1.1) and some useful properties of Mittag-Leffler functions are established. In Section 3, a decision theorem is derived for the existence and uniqueness of globally exponentially stable T -periodic solution for IPFFDEs (1.1). An illustrative example and its numerical simulations are presented in Section 4. 2. Preliminaries Let Rn be the n-dimensional real vector space. P C(J, Rn ) denotes the space including piecewise continuous functions from J = [0, +∞) to Rn with points of discontinuity of the first kind tk , at which it is left continuous; C n ([t0 , ∞), Rn ) denotes the space consisting of n-order continuously differentiable functions. Definition 2.1 ([4]). The α-order Caputo fractional derivative for f ∈ C n ([t0 , ∞), Rn ) is defined by ∫ t f (n) (s) 1 c α ds (0 < n − 1 < α < n, n ∈ N), Dt0 ,t f (t) = Γ (α − n) t0 (t − s)α−n+1 where Γ (·) is Euler’s Gamma function. Lemma 2.1. Let g(t) =c Dtαk ,t f (t) for t ∈ (tk , tk+1 ], α ∈ (n − 1, n), k, n ∈ N. If f ∈ C n (J, Rn ) ∩ P C n (J, Rn ) is nonconstant T -periodic, then g(t + T ) = g(t) for t ∈ (tk , tk+1 ], k ∈ N. In other words, the piecewise Caputo derivative c Dtαk ,t f (t) (t ∈ (tk , tk+1 ]) can generate nonconstant periodic signal. Proof . By the periodicity of {tk }, if t ∈ (tk , tk+1 ), then t + T ∈ (tk + T, tk+1 + T ) = (tk+N +1 , tk+N +2 ), k ∈ N. Then ∫ t+T 1 f (n) (s) g(t + T ) = c Dtαk+N +1 ,t+T f (t + T ) = ds Γ (α − n) tk+N +1 (t − s)α−n+1 ∫ t 1 f (n) (s) = ds = g(t), Γ (α − n) tk (t − s)α−n+1 where t ∈ (tk , tk+1 ], k ∈ N. Then g is nonconstant periodic function. This completes the proof.

□

α Remark 2.1. In [2], the authors proved that FDEs with c D0,t f cannot generate nonconstant periodic c α solution because D0,t f is not nonconstant periodic under the condition that f is nonconstant periodic function. This paper considers IFDEs with piecewise Caputo derivative c Dtαk ,t f for t ∈ (tk , tk+1 ], k ∈ N, which could yield nonconstant periodic solution. Thus, it is worth studying the periodic dynamics for IPFFDEs (1.1). α Remark 2.2. Regarding fractional-order derivative c D0,t f (t) for a function f (t), Guo et al. [5] described that the history of f (t) closing to initial point t = 0 can be ignored for large t. This is the short memory principle, i.e., it only considers the behavior of f (t) in the recent past. Therefore, it is rational to discuss the dynamics of piecewise fractional-order dynamic systems.

T. Zhang and L. Xiong / Applied Mathematics Letters 101 (2020) 106072

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Together with (1.1), the following linear impulsive piecewise fractional differential equations is considered: ⎧c α ⎨ Dtk ,t X(t) = AX(t) + h(t), t ∈ (tk , tk+1 ), (2.1) X(t+ k ) = X(tk ) + ∆k , ⎩ X(0) = ϕ, where h ∈ C(J, Rn ), ϕ is a constant, ∆k ∈ Rn is constant vector, k ∈ N. By a simple calculation, the solution of system (2.1) can be expressed by ∫ tk ∑ X(t) = G(t, mT )X(mT ) + G(t, tk ) q(tk , s)h(s)ds tk−1

k:tk−1 ,tk ∈[mT,t)

∫

t

q(t, s)h(s)ds +

+ ρ(t)

∑

G(t, tk )∆k ,

t ∈ (mT, mT + T ], m ∈ N,

(2.2)

k:tk ∈[mT,t)

where t ∈ (mT, mT + T ], m ∈ N, ρ(t) = max{tk : tk ≤ t}, G(t, s) = Eα [−A(t − s)α ] as tk−1 ≤ s ≤ t ≤ tk , G(t, s) = Eα [A(t − tk )α ]Eα [A(tk − tk−1 )α ] · · · Eα [A(tp+1 − tp )α ]Eα [A(tp − s)α ], as tp−1 < s ≤ tp < tk < t ≤ tk+1 , q(t, s) = (t − s)α−1 Eα,α [A(t − s)α ], s, t > 0; Eα (·) and Eα,α (·) are the one-parameter and two-parameter Mittag-Leffler functions defined as (1.8.1) and (1.8.17) in [4], respectively. In terms of (2.2), the definition of the solution for IPFFDEs (1.1) is listed as follows. X ∈ P C(J, Rn ) is called the solution of IPFFDEs (1.1) in case it satisfies ∫ tk ∑ X(t) = G(t, mT )X(mT ) + G(t, tk ) q(tk , s)F (s, Xs )ds

Definition 2.2.

tk−1

k:tk−1 ,tk ∈[mT,t)

∫

t

q(t, s)F (s, Xs )ds +

+ ρ(t)

Lemma 2.2 ([4]).

∑

G(t, tk )∆k (X(tk )),

t ∈ (mT, mT + T ], m ∈ N.

(2.3)

k:tk ∈[mT,t)

] d[ α z Eα,α+1 (λz α ) = z α−1 Eα,α (λz α ), where α, λ, z ∈ R. dz

Lemma 2.3 ([6]). Assume that α ∈ (0, 1) and λ ∈ R, then α (1) Eα (λtα 1 ) ≤ Eα (λt2 ) for λ < 0, t1 ≥ t2 ≥ 0. 1 , Eα (λtα ) > 0 and Eα,α (λtα ) > 0, ∀t > 0. (2) Eα (0) = 1, Eα,α (0) = Γ (α)

Lemma 2.4. If λ > 0 and α ∈ (0, 1), then [ ] d α α (1) t Eα,α+1 (−λt ) = tα−1 Eα,α [−λtα ] ≥ 0, ∀t > 0. dt 1 1 (2) lim tα Eα,α+1 (−λtα ) = and tα Eα,α+1 (−λtα ) ∈ [0, ), ∀t > 0. t→∞ λ λ [ ] d α α Proof . By Lemmas 2.2–2.3, t Eα,α+1 (−λt ) = tα−1 Eα,α [−λtα ] ≥ 0, ∀t > 0. By (1.8.28) in [4], it gets dt [ ] N 1 ∑ 1 tα tα α α t Eα,α+1 (−λt ) = − +O with t → ∞, λ Γ (α + 1 − αk) (−λtα )k (−λtα )N +1 k=2

which derives limt→∞ t Eα,α+1 (−λtα ) = λ1 . Together with (1) and 0α Eα,α+1 (−λ0α ) = 0, then tα Eα,α+1 (−λtα ) ∈ [0, λ1 ) for t ≥ 0. This completes the proof. □ α

T. Zhang and L. Xiong / Applied Mathematics Letters 101 (2020) 106072

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3. Main result For X = (X1 , X2 , . . . , Xn )T and A = (aij )n×n , define ∥X∥1 = max1≤i≤n |Xi | and ∥A∥0 = ∑n max1≤i≤n j=1 |aij |. Let P CT (Rn ) = P CT ([−δ, +∞), Rn ) denote the space consisting of all Rn -valued T -periodic piecewise continuous functions with initial condition ϕ(s), s ∈ [−δ, 0]. P CT (Rn ) is a Banach space with ∥φ∥∞ = supt≥0 ∥φ(t)∥1 = supt≥0 max1≤i≤n |φi (t)| for φ ∈ P CT (Rn ). Let a = max1≤k≤N +1 {tk+1 − tk } and b = min1≤k≤N +1 {tk+1 − tk }. Theorem 3.1. IPFFDEs (1.1) possess a unique globally exponentially stable T -periodic solution in case the following conditions are valid. (P1 ) F (t, X) ∈ C(R × Rn , Rn ) is T -periodic in its first argument. Further, F and ∆k satisfy the Lipschitz condition, i.e., there exist positive constants L and K such that ∥F (t, X) − F (t, Y )∥1 ≤ L∥X − Y ∥1 , ∥∆k (X) − ∆k (Y )∥1 ≤ K∥X − Y ∥1 , ∀X, Y ∈ Rn , t ≥ 0, k ∈ N. (P2 ) A is a diagonalization matrix, i.e., there exist diagonal matrix Λ ( = diag{λ )1 , λ2 , . . . , λn } and invertible matrix P such that P −1 AP = Λ, and

+1 (1−µN +1 EN (−λbα )) α (1−µEα (−λbα ))

µ λL

+ K

+

µ λL

< 1, where λ =

−1

− max1≤i≤n λi > 0 and µ = ∥P ∥0 ∥P ∥0 . (P3 ) There exist positive constants c and ω such that ∥Eα (Atα )∥0 ≤ e−ωt , ∥Eα,α (Atα )∥0 ≤ ce−ωt for t ∈ [0, a], ] [ α ] 2(1−α) [ 2−α ξµcL K α 1 , γ = 2−α and ωγ 1−e−ωb + 1 + 1−e−ωb < 1, where ξ = γ γ 2aα2 . Proof . According to (2.3), Φφ(t) is defined on P CT (Rn ) as follows: ∫ tk ∑ Φφ(t) = G(t, mT )ϕ(0) + G(t, tk ) q(tk , s)F (s, φs )ds tk−1

k:tk−1 ,tk ∈[mT,t)

∫

t

∑

q(t, s)F (s, φs )ds +

+ ρ(t)

G(t, tk )∆k (φ(tk )),

t ∈ (mT, mT + T ], k, m ∈ N. (3.1)

k:tk ∈[mT,t)

In (3.1), φ(mT ) = φ(0) = ϕ(0) is utilized, m ∈ N. To start with, Φ : P CT (Rn ) → P CT (Rn ) should be demonstrated. For t ∈ (mT, mT + T ], then t + T ∈ (mT + T, mT + 2T ], m ∈ N. By using (3.1), it follows from (P1 ) that ∫ tk+N +1 ∑ q(tk+N +1 , s)F (s, φs )ds Φφ(t + T ) = G(t + T, mT + T )ϕ(0)ϕ(0) + G(t + T, tN +k+1 ) tk+N

k:tk−1 ,tk ∈[mT,t)

∫

t

∑

q(t + T, s + T )F (s + T, φs+T )ds +

+ ρ(t)

= G(t, mT )ϕ(0) +

∫

∑

∫

t

q(t, s)F (s, φs )ds + ρ(t)

tk

G(t, tk )

q(tk , s)F (s, φs )ds tk−1

k:tk−1 ,tk ∈[mT,t)

+

G(t + T, tN +k+1 )∆N +k+1 (φ(tk+N +1 ))

k:tk ∈[mT,t)

∑

G(t, tk )∆k (φ(tk )) = Φφ(t),

(3.2)

k:tk ∈[mT,t)

where t ∈ (mT, mT + T ], k, m ∈ N. Further, it is easy to obtain the piecewise continuity of Φφ according to the piecewise continuity of φ. For reducing tedious narratives, it should be ignored. By (3.2), Φφ ∈ P CT (Rn ). In view of assumption (P2 ) and by the computation theory of matrix functions, it gains ∥Eα (Atα )∥0 ≤ ∥P ∥0 ∥P −1 ∥0 max Eα (λi tα ), 1≤i≤n

∥Eα,α (Atα )∥0 ≤ ∥P ∥0 ∥P −1 ∥0 max Eα,α (λi tα ), 1≤i≤n

∀t ≥ 0. (3.3)

T. Zhang and L. Xiong / Applied Mathematics Letters 101 (2020) 106072

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The next step is to show that Φ is a contraction mapping. For φ, ψ ∈ P CT (Rn ), it derives from (3.1), (3.3), Lemmas 2.2–2.4 that [ ∥Φφ(t) − Φψ(t)∥∞ ≤ sup µL t≥0

∑

  G(t, tk ) max 0

1≤i≤n

k:tk−1 ,tk ∈[mT,t)

∫

tk

(tk − s)α−1 Eα,α [λi (tk − s)α ]ds

tk−1

t

∑

(t − s)α−1 Eα,α [λi (t − s)α ]ds + K

+ µL max

1≤i≤n

∫

ρ(t)

]   G(t, tk ) ∥φ − ψ∥∞ 0

k:tk ∈[mT,t)

N ∑

N ∑ µ µ ≤ L∥φ − ψ∥∞ [µEα (−λbα )]k−1 + L∥φ − ψ∥∞ + K∥φ − ψ∥∞ [µEα (−λbα )]k λ λ k=1 k=0 [ ( ) ] N +1 N +1 α (1 − µ Eα (−λb )) µ µ ≤ L + K + L ∥φ − ψ∥∞ , (3.4) (1 − µEα (−λbα )) λ λ

where t ∈ (mT, mT + T ], m, k ∈ N. By (P2 ), Φ is a contraction mapping. Then Φ has a unique fixed point φ∗ = Φφ∗ ∈ P CT (Rn ) and φ∗ is a unique T -periodic solution for IPFFDEs (1.1). In the last part, it needs to show φ∗ is globally exponentially stable. Assume that ψ ∗ = (ψ1∗ , ψ2∗ , . . . , ψn∗ )T is another solution for IPFFDEs (1.1). Let z(t) = φ∗ (t) − ψ ∗ (t), t ∈ [−δ, +∞). Observing (P3 ), there exist τ ∈ (0, ω) small enough and M > 0 large enough such that [ ] ξµcLeτ δ 1 K 1 + +1 + < 1. γ −(ω−τ )b −(ω−τ )b M (ω − τ ) 1 − e 1−e

(3.5)

Making use of (2.3), it follows from (P3 ) that ∑

∥z(t)∥1 ≤ e−ωt ∥z(0)∥1 + µcL

e−ω(t−tk )

∫

tk

tk−1

k:tk−1 ,tk ∈[0,t) t

∫

∑

(t − s)α−1 e−ω(t−s) ∥zs ∥1 ds + K

+ µcL

(tk − s)α−1 e−ω(tk −s) ∥zs ∥1 ds

ρ(t)

e−ω(t−tk ) ∥z(tk )∥1 ,

∀t ≥ 0.

(3.6)

k:tk ∈[0,t)

Let z0 = maxt∈[−δ,0] ∥z(t)∥1 < +∞ and ω0 = ω − τ . At present, assume that ∥z(t)∥1 ≤ M e−τ t z0 ,

∀t ≥ 0.

(3.7)

If (3.7) is invalid, then there must exist p0 ∈ (0, +∞) such that ∥z(p0 )∥1 > M e−τ p0 z0

and ∥z(t)∥1 ≤ M e−τ t z0 ,

∀t ∈ [−δ, p0 ).

(3.8)

In view of (3.6), it follows from (3.5) that [ ∥z(p0 )∥1 ≤ e−ωp0 z0 + µcLeτ δ

∑

e−ω0 (p0 −tk )

τδ

p0

(p0 − s)

+ µcLe

ρ(p0 )

≤ M e−τ p0 z0

α−1 −ω0 (p0 −s)

{

e

tk

(tk − s)α−1 e−ω0 (tk −s) ds

tk−1

k:tk−1 ,tk ∈[0,p0 )

∫

∫

ds + K

∑

e

−ω0 (p0 −tk )

] M e−τ p0 z0

k:tk ∈[0,p0 )

[ ] } 1 K 1 −ω0 p0 ξµcLeτ δ e + + 1 + < M e−τ p0 z0 . M ω0γ 1 − e−ω0 b 1 − e−ω0 b

(3.9)

T. Zhang and L. Xiong / Applied Mathematics Letters 101 (2020) 106072

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1

Fig. 1. e−2.6340t − E 1 (−2.6340t 3 ) (red line: 3

1

) and e−2.6340t − 0.4E 1 , 1 (−2.6340t 3 ) (red line: 3

). (For interpretation of the references

3

to color in this figure legend, the reader is referred to the web version of this article.)

Here the following fact is employed in the calculation of (3.9): ∫

tk

α−1 −ω0 (tk −s)

(tk − s)

e

[∫

tk

2−α (α−1)× 2(1−α)

ds ≤

(tk − s)

tk−1

] 2(1−α) [∫ 2−α ds

α

2a 2 ≤ α

[ ] 2(1−α) 2−α

e

−ω0 (tk −s)× 2−α α

α ] 2−α ds

tk−1

tk−1

[

tk

α (2 − α)ω0

α ] 2−α

≤

ξ , ω0γ

k ∈ N.

In the above calculation, H¨ older inequality is utilized. Obviously, (3.9) is contradictive with (3.8). Hence, (3.7) holds and the T -periodic solution φ∗ is globally exponentially stable. This completes the proof. □ Remark 3.1. In order to investigate global exponential stability for IPFFDEs (1.1), Mittag-Leffler functions Eα (·) and Eα,α (·) need to be transformed to exponential function e(·) in a local interval [0, a], see (P3 ). With the help of Matlab numerical computing, it is possible to do this, see Example 4.1. Remark 3.2. In [3], Feˇckan and Wang discussed IPFFDEs (1.1) with zero matrix A and δ = 0. The authors [3] achieved the existence, uniqueness and global asymptotical stability of T -periodic solution under ∏N the assumptions ∥X + ∆k (X) − Y − ∆k (Y )∥1 ≤ Lk ∥X − Y ∥1 and k=0 Lk+1 Eα (L(tk+1 − tk )α ) < 1, ∀X, Y ∈ Rn , k ∈ N. Nevertheless, these assumptions are strict and a great many of F and ∆k are difficult to satisfy them, such as those in system (4.1) of Example 4.1. Thus, the current work improves and extends the result in literature [3]. 4. Illustrative example and numerical simulations Example 4.1. Considering the following IPFFDEs: ⎧ [ ] [ ][ ] [ ] 1 X1 (t) sin(2πt)X2 (t − 1) ⎪ ⎪ c Dt3 ,t X1 (t) = −3 1 + 0.1 , ⎨ k X2 (t) 0.5 −4 X2 (t) cos(X1 (t)) + ⎪ ⎪ X(tk ) = 0.8X(tk ), tk = 0.25k, k ∈ N, T ⎩ X(s) = 0.1, s ∈ [−1, 0], X = (X1 , X2 ) .

t ∈ (tk , tk+1 ), (4.1)

Corresponding to IPFFDEs (1.1) and Theorem 3.1, it utilizes Matlab tool to compute λ = 2.6340, 1 µ = 1.5298 ∗ 1.4546 ≈ 2.2252. Taking c = 0.4 and ω = 2.6340, then E 1 (−2.6340t 3 ) ≤ e−2.6340t , 1

3

E 1 , 1 (−2.6340t 3 ) ≤ 0.4e−2.6340t for t ∈ [0, 0.25]. The corresponding numerical simulations can be observed 3 3 in Fig. 1.

T. Zhang and L. Xiong / Applied Mathematics Letters 101 (2020) 106072

7

Fig. 2. Exponential stability of 1-periodic impulsive solution for system (4.1).

Besides,

N +1 (1−µN +1 Eα (−λbα )) (1−µEα (−λbα ))

(

µ λL

) +K

+ µλ L = 0.8345 < 1,

ξµcL ωγ

[

1 1−e−ωb

] +1 +

K 1−e−ωb

= 0.8670 < 1.

Therefore, all the conditions in Theorem 3.1 hold. By Theorem 3.1, system (4.1) admits a unique globally exponentially stable 1-periodic solution. The corresponding numerical simulations can be observed in Fig. 2. References [1] L.L. Ren, J.R. Wang, M. Feˇ ckan, Asymptotically periodic solutions for Caputo type fractional evolution equations, Fract. Calc. Appl. Anal. 21 (2018) 1294–1312. [2] E. Kaslik, S. Sivasundaram, Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions, Nonlinear Analysis RWA 13 (2012) 1489–1497. [3] M. Feˇ ckan, J.R. Wang, Periodic impulsive fractional differential equations, Adv. Nonlinear Anal. 8 (2019) 482–496. [4] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Boston, USA, 2006. [5] B.L. Guo, X.K. Pu, F.H. Huang, Frational Patial Differential Equations and their Numerical Solutions, Science Press, Beijing, China, 2011. [6] J.R. Wang, M. Feˇ ckan, Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. Spec. Top. 222 (2013) 1857–1874.