Rend. Circ. Mat. Palermo DOI 10.1007/s12215-016-0230-5

Stability in delay nonlinear fractional differential equations Hamid Boulares1,2 · Abdelouaheb Ardjouni3,4 · Yamina Laskri1

Received: 18 November 2015 / Accepted: 5 January 2016 © Springer-Verlag Italia 2016

Abstract In this paper, we give sufficient conditions to guarantee the asymptotic stability of the zero solution to a kind of delay nonlinear fractional differential equations of order α (1 < α < 2). By using the Krasnoselskii’s fixed point theorem in a weighted Banach space, we establish new results on the asymptotic stability of the zero solution provided that g (t, 0) = f (t, 0, 0), which include and improve some related results in the literature. Keywords

Delay fractional differential equations · Fixed point theory · Stability

Mathematics Subject Classification

34K20 · 34K30 · 34K40

1 Introduction Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control sys-

B

Abdelouaheb Ardjouni [email protected] Hamid Boulares [email protected] Yamina Laskri [email protected]

1

Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, 23000 Annaba, Algeria

2

Advanced Control Laboratory (LABCAV), Guelma University, 24000 Guelma, Algeria

3

Department of Mathematics and Informatics, Faculty of Sciences and Technology, Univ Souk Ahras, P.O. Box 1553, 41000 Souk Ahras, Algeria

4

Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, Univ Annaba, P.O. Box 12, 23000 Annaba, Algeria

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tems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1–13,15] and the references therein. Recently, Agarwal et al. [2] discussed the existence of solutions for the neutral fractional differential equation with bounded delay C α D (x(t) − g(t, xt )) = f (t, xt ) , t  t0 , xt0 = φ, where C D α is the standard Caputo’s fractional derivative of order 0 < α < 1. By employing the Krasnoselskii’s fixed point theorem, the authors obtained existence results. The delay fractional differential equation ⎧ α ⎨ d x(t) = f (t, x(t), x(t − τ )) , t ∈ [0, T ] , α ⎩ xdt(t) = φ (t) , t ∈ [−τ, 0] , 0 < α < 1, dα has been investigated in [1], where α denotes Riemann–Liouville fractional derivative of dt order 0 < α < 1. By using the Krasnoselskii’s fixed point theorem, the existence of solutions has been established. In [5], Ge and Kou investigated the asymptotic stability of the zero solution of the following nonlinear fractional differential equation C α D0+ x(t) = f (t, x(t)) , t  0, x(0) = x0 , x  (0) = x1 , α is the standard Caputo’s fractional derivative of order 1 < α < 2. By employing where C D0+ the Krasnoselskii’s fixed point theorem in a weighted Banach space, the authors obtained stability results. In this paper, we are interested in the analysis of qualitative theory of the problems of the asymptotic stability of the zero solution to delay fractional differential equations. Inspired and motivated by the works mentioned above and the papers [1–13,15] and the references therein, we concentrate on the asymptotic stability of the zero solution for the nonlinear fractional differential equation with variable delay C α α−1 D0+ x(t) = f (t, x(t), x(t − τ (t))) + C D0+ g(t, x(t − τ (t))), t  0, (1.1)  x(t) = φ(t), t ∈ [m 0 , 0] , x (0) = x1 ,

where 1 < α < 2, R+ = [0, +∞), τ : R+ → R+ is continuous with t − τ (t) → ∞ as t → ∞, m 0 = inf t 0 {t − τ (t)}, x1 ∈ R, g : R+ × R → R and f : R+ × R × R → R are α is the standard Caputo fractional continuous functions and g(t, 0) = f (t, 0, 0) = 0, C D0+ derivative and we denote the solution of (1.1) by x (t, φ, x1 ). To show the asymptotic stability of the zero solution, we transform (1.1) into an integral equation and then use Krasnoselskii’s fixed point theorem. The obtained integral equation is the sum of two mappings, one is a contraction and the other is compact. This paper is organized as follows. In Sect. 2, we introduce some notations and lemmas, and state some preliminaries results needed in later sections. Also, we present the inversion of (1.1) and the Krasnoselskii’s fixed point theorem. For details on Krasnoselskii’s theorem we refer the reader to [14]. In Sect. 3, we give and prove our main results on stability.

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2 Preliminaries We introduce some necessary definitions, lemmas and theorems which will be used in this paper. For more details, see [7,8,13,14]. Definition 2.1 [7,13] The fractional integral of order α > 0 of a function x : R+ → R is given by  t 1 α I0+ x(t) = (t − s)α−1 x(s)ds, (α) 0 provided the right side is pointwise defined on R+ . Definition 2.2 [7,13] The Caputo fractional derivative of order α > 0 of a function x : R+ → R is given by  t 1 n−α (n) C α D0+ x(t) = I0+ x (t) = (t − s)n−α−1 x (n) (s)ds, (n − α) 0 where n = [α] + 1, provided the right side is pointwise defined on R+ . Lemma 2.3 [7,13] Let (α) > 0. Suppose x ∈ C n−1 [0, +∞) and x (n) exists almost everywhere on any bounded interval of R+ . Then   n−1 x (k) (0) k αC α t . D0+ x (t) = x(t) − I0+ k! k=0

αC α  In particular, when 0 < (α) < 1, I0+ D0+ x (t) = x(t) − x(0). Remark 2.4 From Definitions 2.1, 2.2 and Lemma 2.3, it is easy to see that α I α x(t) = x(t) holds for all t ∈ R+ . (1) Let (α) > 0. If x is continuous on R+ , then D0+ 0+ (2) The Caputo derivative of a constant is equal to zero.

The following Banach space plays a fundamental role in our discussion. Let h : [m 0 , +∞) → [1, +∞) be a strictly increasing continuous function with h(m 0 ) = 1, h(t) → ∞ as t → ∞, h(s)h(t − s)  h(t) for all m 0  s  t  ∞. Let

E = x ∈ C ([m 0 , +∞)) : sup |x(t)| / h(t) < ∞ . t m 0

Then E is a Banach space equipped with the norm x = supt m 0 of this Banach space, see [8]. Moreover, let

|x(t)| h(t) . For more properties

ϕt = max {|ϕ(s)| : m 0  s  t} , for any t  m 0 , any given ϕ ∈ C ([m 0 , +∞)) and let (ε) = {x ∈ E : x (t)  ε for t ∈ [m 0 , +∞) and x (t) = φ (t) if t ∈ [m 0 , 0]} for any ε > 0. Lemma 2.5 [5] Let r ∈ C ([m 0 , +∞)). Then x ∈ C ([m 0 , +∞)) is a solution of the Cauchy type problem C α D0+ x(t) = r (t), t ∈ R+ , 1 < α < 2, (2.1) x(t) = φ(t), t ∈ [m 0 , 0] , x  (0) = x1 , if and only if x is a solution of the Cauchy type problem   α−1 x (t) = I0+ r (t) + x1 , t ∈ R+ , x(t) = φ(t), t ∈ [m 0 , 0] .

(2.2)

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Lemma 2.6 Let k ∈ R. Then x ∈ C ([m 0 , +∞)) is a solution of (1.1) if and only if 1 − e−kt x(t) = φ(0)e−kt + (x1 − g(0, φ(−τ (0)))) k  t e−k(t−s) (kx(s) + g(s, x(s − τ (s)))) ds + 0  t t 1 + e−k(t−s) (s − u)α−2 ds f (u, x(u), x(u − τ (u)))du. (α − 1) 0 u

(2.3)

Proof Let x ∈ C ([m 0 , +∞)) be a solution of (1.1). From Lemma 2.5, we have   α−1 α−1 f (t, x(t), x(t − τ (t))) + C D0+ x  (t) = I0+ g(t, x(t − τ (t))) + x1 , t ∈ R+ , x(t) = φ(t), t ∈ [m 0 , 0] . Then

⎧  ⎨ x (t) =

t 1 (α−1) 0 (t

− s)α−2 f (s, x(s), x(s − τ (s)))ds +g(t, x(t − τ (t))) − g(0, φ(−τ (0))) + x1 , t ∈ R+ , ⎩ x(t) = φ(t), t ∈ [m 0 , 0] .

(2.4)

Rewrite (2.4) as ⎧  t 1 α−2 f (s, x(s), x(s − τ (s)))ds ⎨ x (t) + kx (t) = kx (t) + (α−1) 0 (t − s) +g(t, x(t − τ (t))) − g(0, φ(−τ (0))) + x1 , t ∈ R+ , ⎩ x(t) = φ(t), t ∈ [m 0 , 0] . By the variation of constants formula, we obtain (2.3). Since each step is reversible, the converse follows easily. This completes the proof.

Definition 2.7 The trivial solution x = 0 of (1.1) is said to be (i) stable in Banach space E, if for every ε > 0, there exists a δ = δ (ε) > 0 such that |φ(t)| + |x1 |  δ implies that the solution x(t) = x(t, φ, x1 ) exists for all t  m 0 and satisfies x  ε. (ii) asymptotically stable, f it is stable in Banach space E and there exists a number σ > 0 such that |φ(t)| + |x1 |  σ implies limt→∞ x(t) = 0. Lastly in this section, we state Krasnoselskii’s fixed point theorem which enables us to prove the asymptotic stability of the zero solution to (1.1). For its proof we refer the reader to [14]. Theorem 2.8 (Krasnoselskii [14]) Let be a non-empty closed convex subset of a Banach space (S, .). Suppose that A and B map into S such that (i) Ax + By ∈ for all x, y ∈ , (ii) A is continuous and A is contained in a compact set of S, (iii) B is a contraction with constant l < 1. Then there is a x ∈ with Ax + Bx = x. In order to prove (ii), the following modified compactness criterion is needed. Theorem 2.9 [8] Let M be a subset of the Banach space E. Then M is relatively compact in E if the following conditions are satisfied

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(i) {x(t)/ h(t) : x ∈ M} is uniformly bounded, (ii) {x(t)/ h(t) : x ∈ M} is equicontinuous on any compact interval of R+ , (iii) {x(t)/ h(t) : x ∈ M} is equiconvergent at infinity i.e. for any given ε > 0, there exists a T0 > 0 such that for all x ∈ M and t1 , t2 > T0 , if holds |x(t2 )/ h(t2 ) − x(t1 )/ h(t1 )| < ε.

3 Main results Before stating and proving the main results, we introduce the following hypotheses. (h1) g and f are continuous functions and g(t, 0) = f (t, 0, 0) = 0. g is also supposed to be locally Lipschitz continuous in x. That is, there is a L g > 0 so that if |x| , |y| ≤ l then |g (t, x) − g (t, y)| ≤ L g x − y .

(3.1)

(h2) There exists a constant β1 ∈ (0, 1) such that   Lg β1 1 + < 1, |k| and e−kt / h(t) ∈ BC ([m 0 , +∞)) ∩ L 1 ([m 0 , +∞)) , |k|

(3.2) 

t

e−kt / h(t)ds  β1 < 1.

(3.3)

0

(h3) There exists constants η > 0, β2 ∈ (0, 1 − β1 ) and a continuous function f˜ : [0, ∞) × (0, η] × (0, η] → R+ such that | f (t, υ1 h(t), υ2 h(t − τ (t)))|  f˜(t, |υ1 | , |υ2 |), h(t) holds for all t  0, 0 < |υ1 | , |υ2 |  η and  t K (t − u) ˜ sup f (u, r1 , r2 )du  β2 < 1 − β1 , t 0 0 h(t − u)

(3.4)

(3.5)

holds for every 0 < r1 , r2  η, where f˜(t, r1 , r2 ) is nondecreasing in r1 and r2 for fixed t, f˜(t, r1 , r2 ) ∈ L 1 ([0, +∞)) in t for fixed r1 and r2 , and  1  t −k(t−s) e (s − u)α−2 ds, t − u  0, (3.6) K (t − u) = (α−1) u 0, t − u < 0. Theorem 3.1 Suppose that (h1)–(h3) hold. Then the trivial solution x = 0 of (1.1) is stable in Banach space E. Proof For any given ε > 0, we first prove the existence of δ > 0 such that |φ (t)| + |x1 | < δ implies x  ε. In fact, according to (3.3), there exists a constant M1 > 0 such that e−kt  M1 . h(t)

(3.7)

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   L 1−β1 1+ |k|g −β2 |k|

Let 0 < δ  M |k|+(1+M ) 1+L ε. Consider the non-empty closed convex subset  (ε) ⊆ E, g) 1 1 ( for t  0, we denote two mapping A and B on  (ε) as follows:  t t 1 e−k(t−s) (s − u)α−2 ds f (u, x(u), x(u − τ (u)))du (α − 1) 0 u  t K (t − u) f (u, x(u), x(u − τ (u)))du, =

Ax(t) =

(3.8)

0

and 1 − e−kt Bx(t) = φ(0)e−kt + (x1 − g(0, φ(−τ (0)))) k  t e−k(t−s) (kx(s) + g(s, x(s − τ (s)))) ds. +

(3.9)

0

Obviously, for x ∈  (ε), both Ax and Bx are continuous functions on [m 0 , +∞). Furthermore, for x ∈  (ε), by (3.3)–(3.5) for any t  0, we have 

K (t − u) | f (u, x(u), x(u − τ (u)))| du h(t − u) h(u)    t |x(u)| |x(u − τ (u))| K (t − u) ˜  , f u, du h(u) h(u − τ (u)) 0 h(t − u)  β2 x  β2 ε < ∞,

|Ax(t)|  h(t)

t

0

(3.10)

and  |Bx(t)|  e−kt 1 − e−kt = φ(0) + (x1 − g(0, φ(−τ (0)))) h(t) h(t) kh(t)   t −k(t−s)  e  + (kx(s) + g(s, x(s − τ (s)))) ds   h(t) 0 1 + M1 (|x1 | + |g(0, φ(−τ (0)))|) |k|    ∞ −ku Lg e x + |k| du 1 + |k| h(u) 0    Lg 1 + M1

|x1 | + L g |φ(−τ (0))| + β1 1 +  M1 |φ(0)| + ε < ∞. |k| |k| (3.11)  M1 |φ(0)| +

Then A (ε) ⊆ E and B (ε) ⊆ E. Next, we shall use Theorem 2.8 to prove there exists at least one fixed point of the operator A + B in  (ε). Here, we divide the proof into three steps. Step 1. We prove that Ax + By ∈  (ε) for all x, y ∈  (ε).

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For any x, y ∈  (ε), from (3.10) and (3.11), we obtain that sup t 0

  |Ax (t) + By (t)| e−kt 1 − e−kt = sup φ(0) + (x1 − g(0, φ(−τ (0)))) h(t) h(t) kh(t) t 0  t −k(t−s) e + (ky(s) + g(s, y(s − τ (s)))) ds h(t) 0   t  K (t − u) + f (u, x(u), x(u − τ (u)))du  h(t) 0  1 + M1

|x1 | + L g δ  M1 |φ(0)| + |k|    ∞ −ku Lg e y + β2 x + |k| du 1 + |k| h(u) 0 

  M1 |k| + (1 + M1 ) 1 + L g Lg  δ + β1 1 + ε + β2 ε  ε, |k| |k|

which implies Ax + By ∈  (ε) for all x, y ∈  (ε). Step 2. It is easy to see that A is continuous. Now we only prove that A (ε) is a relatively compact in E. In fact, from (3.10), we get that {x(t)/ h(t) : x ∈  (ε)} is uniformly bounded in E. Moreover, a classical theorem states the fact that the convolution of an L 1 -function with a function tending to zero, does also tend to zero. Then we conclude that for t − u  0, we have  t −k(t−s) K (t − u) 1 e (s − u)α−2 ds  lim t→∞ h(t − u) t→∞ (α − 1) u h(t − u) h(s − u)  t −k(t−u−s) α−2 s 1 e = lim ds = 0, t→∞ (α − 1) 0 h(t − u − s) h(s)

0  lim

(3.12)

α−2

due to the fact limt→∞ th(t) = 0. Together with the continuity of K and h, we get that there exists a constant M2 > 0 such that    K (t − u)     h(t − u)   M2 ,

(3.13)

and for any T0 ∈ R+ , the function K (t − u)h(u)/ h(t) is uniformly continuous on {(t, u) : 0  u  t  T0 }. For any t1 , t2 ∈ [0, T0 ], t1 < t2 , we have     Ax(t2 ) Ax(t1 )   t2 K (t2 − u)  − f (u, x(u), x(u − τ (u)))du =  h(t ) h(t1 )   0 h(t2 ) 2   t1  K (t1 − u) − f (u, x(u), x(u − τ (u)))du  h(t1 ) 0   t1   K (t2 − u) K (t1 − u)   | f (u, x(u), x(u − τ (u)))| du  −  h(t ) h(t1 )  2 0  t2 K (t2 − u) ˜ + f (u, ε, ε) du t1 h(t2 − u)

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   K (t2 − u)h(u) K (t1 − u)h(u)  ˜   −  f (u, ε, ε) du  h(t2 ) h(t1 ) 0  t2 + M2 f˜ (u, ε, ε) du → 0, 

t1

t1

as t2 → t1 , which means that {x(t)/ h(t) : x ∈  (ε)} is equicontinuous on any compact interval of R+ . By Theorem 2.9, in order to show that A (ε) is a relatively compact set of E, we only need to prove that {x(t)/ h(t) : x ∈  (ε)} is equiconvergent at infinity. In fact, for any ε1 > 0, there exists a L > 0 such that 

∞

M2 L

ε1 f˜ (u, ε, ε) du  . 3

According to (3.12), we get that K (t − u)  max lim sup t→∞ u∈[0,L] h(t − u)



K (t − L) K (t) lim , lim t→∞ h(t − L) t→∞ h(t)

 = 0.

Thus, there exists T > L such that t1 , t2  T , we have         K (t2 − u)h(u)  K (t2 − u)   K (t1 − u)h(u)   + sup  K (t1 − u)   sup   sup −      h(t2 ) h(t1 ) s∈[0,L] s∈[0,L] h(t2 − u) s∈[0,L] h(t1 − u)  ∞ −1 ε1  f˜ (u, ε, ε) du 3 0 Therefore, for t1 , t2  T ,     Ax(t2 ) Ax(t1 )   t2 K (t2 − u)  f (u, x(u), x(u − τ (u)))du  h(t ) − h(t )  =  h(t2 ) 2 1 0   t1  K (t1 − u) f (u, x(u), x(u − τ (u)))du  − h(t1 ) 0   L  K (t2 − u)h(u) K (t1 − u)h(u)  ˜  −    f (u, ε, ε) du h(t2 ) h(t1 ) 0  t2  t1 K (t2 − u) ˜ K (t1 − u) ˜ + f (u, ε, ε) du + f (u, ε, ε) du h(t − u) 2 L L h(t1 − u)  ∞ ε1  + 2M2 f˜ (u, ε, ε) du  ε1 . 3 L Hence the required conclusion is true. Step 3. we claim that B :  (ε) → E is a contraction mapping.

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In fact, for any x, y ∈  (ε), from (3.1)–(3.3), we obtain that     t e−k(t−u)  Bx(t)  By(t)  = sup  − sup  (kx(u) + g(u, x(u − τ (u)))) du  h(t) h(t) t 0 h(t) t 0  0   t −k(t−u)  e  − (ky(u) + g(u, y(u − τ (u)))) du   h(t) 0  t −k(t−u) |x(u) − y(u)| e  sup |k| du h(u) 0 h(t − u) t 0  t −k(t−u) |g(u, x(u − τ (u))) − g(u, y(u − τ (u)))| e du + sup h(t − u) h(u) t 0 0    t −k(t−u) Lg e x − y  |k| du 1 + |k| 0 h(t − u)   Lg x − y < x − y . β1 1 + |k| By Theorem 2.8, we know that there exists at least one fixed  point of the operator A + B in L

 (ε). Finally, for any ε2 > 0, if 0 < δ1  implies that

1−β1 1+ |k|g −β2 |k|

ε , |k|M1 +(1+M1 )(1+L g ) 2

then |φ (t)| + |x1 |  δ1

  e−kt 1 − e−kt x = sup φ(0) + (x1 − g(0, φ(−τ (0)))) h(t) kh(t) t 0  t −k(t−s) e + (kx(u) + g(u, x(u − τ (u)))) du h(t) 0   t  K (t − u) + f (u, x(u), x(u − τ (u)))du  h(t) 0   −kt 1 − e−kt 

 e |x1 | + L g |φ(−τ (0))|  sup φ(0) + |k| h(t) h(t) t 0    t Lg e−k(t−u) |x(u)| du |x(u)| + + |k| |k| 0 h(t − u)h(u)   t K (t − u) | f (u, x(u), x(u − τ (u)))| + du h(u) 0 h(t − u)    Lg 1 + M1

x + β2 x  M1 δ1 + δ1 + L g δ1 + β1 1 + |k| |k| 

|k| M1 + (1 + M1 ) 1 + L g    δ1  ε2 .  L 1 − β1 1 + |k|g − β2 |k| Thus, we know that trivial solution of (1.1) is stable in Banach space E.



Theorem 3.2 Suppose that all conditions of Theorem 3.1 are satisfied, lim e−kt / h(t) = 0,

t→∞

(3.14)

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and for any r > 0, there exists a function ϕr (t) ∈ L 1 ([0, +∞)), ϕr (t) > 0 such that |u| , |v|  r implies | f (t, u, v)| / h(t)  ϕr (t), a.e. t ∈ [0, +∞) .

(3.15)

Then the trivial solution of (1.1) is asymptotically stable. Proof First, it follows from Theorem 3.1 that the trivial solution of (1.1) is stable in the Banach space E. Next, we shall show that the trivial solution x = 0 of (1.1) is attractive. For any r > 0, defining   ∗ (r ) = x ∈  (r ) , lim x(t)/ h(t) = 0 . t→∞

We only need to prove that Ax + By ∈ ∗ (r ) for any x, y ∈ ∗ (r ), i.e. Ax(t) + By(t) → 0 as t → ∞, h(t) where 1 − e−kt Ax(t) + By(t) = φ(0)e−kt + (x1 − g(0, φ(−τ (0)))) k  t e−k(t−s) (ky(u) + g(u, y(u − τ (u)))) du + 0  t + K (t − u) f (u, x(u), x(u − τ (u)))du. 0

In fact, for x, y ∈ ∗ (r ), based on the fact that used in the proof of Theorem 3.1 (Step 2), it follows from (3.1 )–(3.3) and (3.14) that  t −k(t−u) e (ky(u) + g(u, y(u − τ (u)))) du → 0, h(t − u) h(u) 0 and K (t − u) = h(t − u)

t

e−k(t−s) u h(t−u) (s

− u)α−2 ds

(α − 1)

→ 0,

as t → ∞. Together with the hypothesis ϕr (t) ∈ L 1 ([0, +∞)), we obtain that  t  t K (t − u) | f (u, x(u), x(u − τ (u)))| K (t − u) du  ϕr (u)du → 0, h(t − u) h(u) 0 0 h(t − u) as t → ∞. Thus we get the conclusion.



References 1. Abbas, S.: Existence of solutions to fractional order ordinary and delay differential equations and applications. Electron. J. Differ. Equ. 2011(09), 1–11 (2011) 2. Agarwal, R.P., Zhou, Y., He, Y.: Existence of fractional functional differential equations. Computers Math. Appl. 59, 1095–1100 (2010) 3. Burton, T.A., Zhang, B.: Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems. Nonlinear Anal. 75, 6485–6495 (2012) 4. Chen, F., Nieto, J.J., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. Real Word Appl. 13, 287–298 (2012)

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Stability in delay nonlinear fractional differential equations 5. Ge, F., Kou, C.: Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations. Appl. Math. Comput. 257, 308–316 (2015) 6. Ge, F., Kou, C.: Asymptotic stability of solutions of nonlinear fractional differential equations of order 1 ≤ α ≤ 2. J. Shanghai Normal Univ. 44(3), 284–290 7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) 8. Kou, C., Zhou, H., Yan, Y.: Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal. 74, 5975–5986 (2011) 9. Li, Y., Chen, Y., Podlunby, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009) 10. Li, Y., Chen, Y., Podlunby, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010) 11. Li, C., Zhang, F.: A survey on the stability of fractional differential equations. Eur. Phys. J. Special Topics. 193, 27–47 (2011) 12. Ndoye, I., Zasadzinski, M., Darouach, M., Radhy, N. E.: Observer based control for fractional-order continuous-time systems. In: Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, WeBIn5.3, pp. 1932–1937, December 16–18 (2009) 13. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) 14. Smart, D.R.: Fixed point theorems. Cambridge Uni. Press, Cambridge (1980) 15. Wang, J., Zhou, Y., Feˇckan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)

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