SURVEY PAPER A SURVEY OF LYAPUNOV

SURVEY PAPER A SURVEY OF LYAPUNOV FUNCTIONS, STABILITY AND IMPULSIVE CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Ravi Agarwal 1 , Snezhana Hristova 2 , Donal O’Regan

3

Abstract We present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results. MSC 2010 : Primary 34A34; Secondary 34A08, 34D20 Key Words and Phrases: stability, Caputo derivative, Lyapunov functions, impulses, fractional differential equations 1. Introduction Fractional calculus has attracted much attention in the literature since it plays an important role in many fields of science and engineering. For c 2016 Diogenes Co., Sofia pp. 290–318 , DOI: 10.1515/fca-2016-0017

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A SURVEY OF LYAPUNOV FUNCTIONS, STABILITY . . .

291

example the behavior of many systems, such as physical phenomena having memory and genetic characteristics, can be adequately modeled by fractional differential systems (see, for example, [9], [29]). The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. One of the main difficulties on the application of a Lyapunov function to fractional order differential equations is the appropriate definition of its derivative. In this paper we give a brief overview of the literature on derivatives of Lyapunov functions among Caputo fractional differential equations. In real life there are many processes and phenomena that are characterized by rapid changes in their state. Fractional differential equations were extended to impulsive fractional differential equations (see for example [2]). Impulsive fractional differential equations is an important area of study. There are two different concepts of solutions of Caputo fractional equations with fixed points of impulses. In this paper both approaches are presented and discussed. We choose a particular one to study stability properties of the solutions of impulsive Caputo fractional differential equations. Piecewise continuous Lyapunov functions are applied. An appropriate generalization of the Caputo fractional Dini derivative of Lyapunov functions is given, and some of its advantages are illustrated by examples. Sufficient conditions for stability and uniform stability of the zero solution are presented. 2. Notes on Fractional Calculus Fractional calculus generalizes the derivative and the integral of a function to a non-integer order [25, 27, 35, 37] and there are several definitions of fractional derivatives and fractional integrals. In engineering, the fractional order q is often less than 1, so we restrict our attention to q ∈ (0, 1). 1. The Riemann–Liouville (RL) fractional derivative of order q ∈ (0, 1) of m(t) is given by (see, for example, Section 1.4.1.1 [16], or [35]) t d 1 RL q (t − s)−q m(s)ds, t ≥ t0 , t0 D m(t) = Γ (1 − q) dt t0

where Γ(.) denotes the Gamma function. 2. The Caputo fractional derivative of order q ∈ (0, 1) is defined by (see, for example, Section 1.4.1.3 [16]) t 1 c q (t − s)−q m (s)ds, t ≥ t0 . (2.1) t0 D m(t) = Γ (1 − q) t0

The properties of the Caputo derivative are quite similar to those of ordinary derivatives. Also, the initial conditions of fractional differential


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R. Agarwal, S. Hristova, D. O’Regan

equations with the Caputo derivative has a clear physical meaning and as a result the Caputo derivative is usually used in real applications. 3. The Gr¨ unwald−Letnikov fractional derivative is given by (see, for example, Section 1.4.1.2 [16]) t−t0

[ h ] 1 GL q (−1)r (qCr)m(t − rh), t ≥ t0 , t0 D m(t) = lim q h→0 h r=0 and the Gr¨ unwald−Letnikov fractional Dini derivative by t−t0

[ h ] 1 GL q (−1)r (qCr)m(t − rh), t0 D+ m(t) = lim sup q h→0+ h

t ≥ t0 ,

(2.2)

r=0

0 and [ t−t where qCr = q(q−1)...(q−r+1) r! h ] denotes the integer part of the fract−t0 tion h . q We note the relation ct0 D q m(t) =RL t0 D [m(t) − m(t0 )].

Proposition 2.1. (Theorem 2.25, [18]) Let m ∈ C 1 [t0 , b]. Then, for t ∈ (t0 , b] GL q t0 D m(t)

q =RL t0 D m(t).

Also, according to Lemma 3.4 ([18]) we have (t − t0 )−q q c RL q . D m(t) = D m(t) − m(t ) 0 t0 t t0 Γ(1 − q) From the relation between the Caputo fractional derivative and the Gr¨ unwald−Letnikov fractional derivative, using (2.2) we define the Caputo fractional Dini derivative as q c t0 D+ m(t)

i.e. q c t0 D+ m(t)

= lim sup h→0+

=

GL q t0 D+ [m(t)

− m(t0 )],

(2.3)

1 m(t) − m(t0 ) hq [

t−t0

] h (−1)r+1 (qCr) m(t − rh) − m(t0 ) . −

(2.4)

r=1

Definition 2.1. ([17]) We say that m ∈ C q ([t0 , T ], Rn ), if m(t) is differentiable (i.e. m (t) exists), the Caputo derivative ct0 D q m(t) exists and satisfies (2.1) for t ∈ [t0 , T ]. Remark 2.1. Definition 2.1 could be extended to any interval I ⊂ R+ .


A SURVEY OF LYAPUNOV FUNCTIONS, STABILITY . . . Remark 2.2. If m ∈ C q ([t0 , T ], Rn ) then

c D q m(t) t0 +

=

293

c D q m(t). t0

([8]) Let x ∈ C q ([t0 , ∞), Rn ). Then for any t ≥ t0 the c q T D (t)x(t) ≤ 2 xT (t) cτ0 D q x(t) x τ0

Lemma 2.1. inequality holds.

Remark 2.3. In the scalar case we have equality. 3. Impulses in Fractional Differential Equations Consider the initial value problem (IVP) for the system of fractional differential equations (FrDE) with a Caputo derivative for 0 < q < 1, n

c q τ0 D x

= f (t, x) for t ≥ τ0

with x(τ0 ) = x0 ,

× Rn , Rn ],

(3.1)

n

and (τ0 , x0 ) ∈ R+ × R is an arbitrary where x ∈ R , f ∈ C[R+ initial data. We suppose that the function f (t, x) is smooth enough on R+ × Rn , such that for any initial data (τ0 , x0 ) ∈ R+ × Rn the IVP for FrDE (3.1) has a solution x(t) = x(t; τ0 , x0 ) ∈ C q ([τ0 , ∞), Rn ). Some sufficient conditions for global existence of solutions of (3.1) are given in [11], [27]. The IVP for FrDE (3.1) is equivalent to the following integral equation t 1 (t − s)q−1 f (s, x(s))ds for t ≥ τ0 . x(t) = x0 + Γ(q) τ0 Here and in what follows in the paper we will assume the points ti , i = 1, 2, . . . are fixed such that t1 < t2 < . . . and limk→∞ tk = ∞. Let τ ∈ R+ and define the set Ωτ = {k : tk > τ }. Let the initial time t0 be given. Without loss of generality, in the paper we will assume 0 ≤ t0 < t1 . The idea of impulses in differential equations was originated in the 1960’s by Millman and Mishkis ([33], [34]) and for recent developments see for example the monographs [22], [26], [38] and the cited references therein. Let us briefly recall the idea of impulses in differential equations. Consider the IVP for a system of differential equations with fixed moments of impulses (IDE) x = f (t, x) for t ≥ t0 , t = ti , x(ti + 0) = Φi (x(ti ))

for i = 1, 2, . . . ,

(3.2)

x(t0 ) = x0 , where x, x0 ∈ R , f : R+ × Rn → Rn , t0 ∈ R+ , Φi : Rn → Rn , i = 1, 2, 3, . . . . n


294


Note that between two consecutive points of impulses the solution is determined by the differential equation and the initial condition is changeable and determined by the impulsive function at the point of impulse. The IVP for ODE is equivalent to the following integral equations (see, for example [26]) ⎧ t ⎪ ⎪ ⎪ f (s, x(s))ds for t ∈ [t0 , t1 ] x0 + ⎪ ⎪ ⎨ t0 (3.3) x(t) = Φk (x(tk − 0)) t ⎪ ⎪ ⎪ ⎪ ⎪ f (s, x(s))ds for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . ., ⎩ + tk

and equivalently, t ⎧ ⎪ ⎪ f (s, x(s))ds for t ∈ [t0 , t1 ] ⎪x0 + ⎪ ⎪ t0 ⎪ ⎪ t k ti ⎪ ⎨ f (s, x(s))ds + f (s, x(s))ds x(t) = x0 + tk ti−1 ⎪ i=1 ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ Ii (x(ti − 0)) for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . . + ⎩ i=1

or

t ⎧ ⎪ ⎪ f (s, x(s))ds for t ∈ [t0 , t1 ] x0 + ⎪ ⎪ ⎪ t0 ⎪ ⎪ t ⎨ + f (s, x(s))ds x 0 x(t) = t0 ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ Ii (x(ti − 0)) for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . ., ⎩ +

(3.4)

(3.5)

i=1

where Ik (x) = Φk (x) − x, k = 1, 2, . . . . Consider the IVP for the system of impulsive fractional differential equations (IFrDE) with a Caputo derivative for 0 < q < 1, c q t0 D x

= f (t, x) for t ≥ t0 , t = ti ,

x(ti + 0) = Φi (x(ti ))

for i ∈ Ωt0 ,

(3.6)

x(t0 ) = x0 , n

where x, x0 ∈ R , f : R+ × Rn → Rn , t0 ∈ R+ , Φi : Rn → Rn , i = 1, 2, 3, . . . . Remark 3.1. In the literature the second equation in (3.6), the so called impulsive condition is given also in the equivalent form Δx(ti ) = Ii (x(ti )) for i ∈ Ωt0 , where Δx(ti ) = x(ti + 0) − x(ti − 0) and Ii (x) =



295

Φi (x) − x, i ∈ Ωt0 , gives the amount of the jump of the solution at the point ti . We now look at the concept of a solution to Caputo fractional differential equations with impulses. There are mainly two viewpoints: (V1) Using the classical Caputo derivative and working in each subinterval, determined by the impulses (see, for example, [1], [2], [10], [12], [13]). This approach is based on the idea that on each interval between two consecutive impulses (tk , tk+1 ) the solution is determined by the differential equation of fractional order. Since the Caputo fractional derivative depends significantly on the initial point (which is different for the ordinary derivative) it leads to a change of the equation on each interval (tk , tk+1 ). This approach neglects the lower limit of the Caputo fractional derivative at t0 and moves it to each impulsive time tk . Then, see (3.1) and (3.3) for the ordinary case q = 1, in the fractional case the IVP for IFrDE (3.6) is equivalent to the following integral equation ⎧ t 1 ⎪ ⎪ ⎪ + (t − s)q−1 f (s, x(s))ds for t ∈ [t0 , t1 ] x 0 ⎪ ⎪ Γ(q) ⎨ t0 t 1 (3.7) x(t) = (x(t − 0)) + (t − s)q−1 f (s, x(s))ds Φ ⎪ k k ⎪ Γ(q) ⎪ tk ⎪ ⎪ ⎩ for t ∈ (t , t ], k = 1, 2, 3, . . . , k

k+1

or equivalently ⎧ t 1 ⎪ ⎪ ⎪ (t − s)q−1 f (s, x(s))ds for t ∈ [t0 , t1 ] x0 + ⎪ ⎪ Γ(q) ⎪ t0 ⎪ ⎪ k ti ⎪ ⎪ 1 ⎪x + ⎪ (ti − s)q−1 f (s, x(s))ds ⎪ ⎨ 0 Γ(q) t i−1 i=1 x(t) = t 1 ⎪ ⎪ (t − s)q−1 f (s, x(s))ds + ⎪ ⎪ ⎪ Γ(q) ⎪ tk ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ Ii (x(ti − 0)) for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . . ⎪ ⎩ +

(3.8)

i=1

where Ik (x) = Φk (x) − x, k = 1, 2, . . . . Note that if for some natural k, Φk (x) ≡ x then there will be no impulse at the point tk but formula (3.8) is not the correct equation for the solution. In [52] the authors using approach (V1) considered the case when Ik (x(tk − 0)) → 0.


296

R. Agarwal, S. Hristova, D. O’Regan Using approach (V1), the solution x(t; t0 , x0 ) of (3.6) is given by: x(t; t0 , x0 ) ⎧ X0 (t; t0 , x0 ) for t ∈ [t0 , t1 ] ⎪ ⎪ ⎪ ⎨X (t; t , Φ (X (t ; t , x ))) for t ∈ (t , t ] 1 1 1 0 1 0 0 1 2 = ⎪X2 (t; t2 , Φ2 (X1 (t2 ; t1 , Φ2 (X0 (t1 ; t0 , x0 ))) for t ∈ (t2 , t3 ] ⎪ ⎪ ⎩ .........

(3.9)

where - X0 (t; t0 , x0 ) is the solution of IVP for FrDE (3.1) with τ0 = t0 , - X1 (t; t1 , Φ1 (X0 (t1 ; t0 , x0 ))) is the solution of IVP for FrDE (3.1) with τ0 = t1 , x0 = Φ1 (X0 (t1 ; t0 , x0 )), - X2 (t; t2 , Φ2 (X1 (t2 ; t1 , Φ1 (X0 (t1 ; t0 , x0 ))) is the solution of Eq. (3.1) with τ0 = t2 and x0 = Φ2 (X1 (t2 ; t1 , Φ1 (X0 (t1 ; t0 , x0 )), and so on. (V2) Keeping the lower limit t0 of the Caputo derivative for all t ≥ t0 but considering different initial conditions on each interval (tk , tk+1 ) (see, for example, [20], [21], [43], [44], [45]). This approach is based on the fact that the restriction of the fractional derivative ct0 D q x(t) on any interval (tk , tk+1 ), k = 1, 2, . . . does not change. Then the fractional equation is kept on each interval between two consecutive impulses with only the initial condition changed. Similarly (see (3.4), (q = 1)) in the fractional case problem (3.6) is equivalent to the following integral equation (see formula (10) [20]) t ⎧ 1 ⎪ ⎪ (t − s)q−1 f (s, x(s))ds for t ∈ [t0 , t1 ] x0 + ⎪ ⎪ Γ(q) ⎪ t ⎪ 0t ⎪ ⎨ 1 (t − s)q−1 f (s, x(s))ds x(t) = x0 + Γ(q) t ⎪ 0 ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ Ii (x(ti − 0)) for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . . ⎩ +

(3.10)

i=1

Note that in approach (V2) the application of induction w.r.t. to the intervals of continuity can lead to some problems. Approach (V2) is based on the significant dependence on the Caputo fractional derivative on the initial time point. For example, in the proof of Theorem 1 in [48] there are problems when one applies directly the results of Theorem 2 in [47].



297

Following approach (V1) many authors discussed existence and uniqueness with q ∈ (0, 1) ([52]), with q ∈ (1, 2] ([1]), for an anti-periodic boundary value problem with q ∈ (2, 3] ([41]), nonlocal boundary value problem ([50]), and existence in Banach spaces ([46]). Following approach (V2) many authors discussed existence of decay IFrDE ([24]), existence and the multiplicity of solutions for IFrDE ([14], [36]), and existence and uniqueness of mild solutions for impulsive fractional integro-differential equations ([49]). Remark 3.2. If f (t, x) ≡ 0 both formulas (3.8) and (3.10) coincide and both approaches (V1) and (V2) are equivalent. Example 3.1. Consider the IVP for the scalar impulsive Caputo fractional differential equation c q t0 D u

= 0, for t ≥ t0 , t = ti ,

u(ti + 0) = Φi (u(ti − 0)) for i = 1, 2, . . . ,

(3.11)

u(t0 ) = u0 , If Φi (u) = ai + u, i = 1, 2, . . . , where ai = 0, i = 1, 2, 3, . . . are constants, then applying (V1) and (3.9) or (V2) and (3.10) we obtain the solution u(t; t0 , u0 ) = u0 + ki=1 ai for t ∈ (tk , tk+1 ], k = 0, 1, 2, . . . (note the sum is 0 if k = 0). If Φi (u) = ai u, i = 1, 2, . . . , where ai = 1, i = 1, 2, 3, . . . are constants, then applying (V1) and (3.9) or (V2) and (3.10) we obtain the solution

u(t; t0 , u0 ) = u0 ki=1 ai for t ∈ (tk , tk+1 ], k = 0, 1, 2, . . . (note the product is 1 if k = 0). In the general case, since we have t tk f (s, x(s))ds + f (s, x(s))ds = tk−1

tk

t

f (s, x(s))ds, tk−1

the formulas (3.3) and (3.4) are equivalent. However since t tk q−1 (tk−1 − s) f (s, x(s))ds + (tk − s)q−1 f (s, x(s))ds tk−1

=

tk

t tk−1

(tk−1 − s)q−1 f (s, x(s))ds,

formulas (3.8) and (3.10) differ, and so in the general case both approaches (V1) and (V2) differ.


298

R. Agarwal, S. Hristova, D. O’Regan Example 3.2. Consider the IVP for the scalar IDE x = Ax, for t ≥ t0 , t = ti , x(ti + 0) = Φi (x(ti − 0)) for i = 1, 2, 3, . . . ,

(3.12)

x(t0 ) = x0 , where x ∈ R, A is a given real constant. Consider the initial value problem for the scalar IFrDE with a Caputo derivative for 0 < q < 1, c q t0 D x = Ax, for t ≥ t0 , t = ti , x(ti + 0) = Φi (x(ti − 0)) for i = 1, 2, . . . ,

(3.13)

x(t0 ) = x0 . Case 1. Let Φi (x) = ai + x, where ai = 0 , i = 1, 2, . . . . The solution of IVP for the impulsive differential equation (3.12) is k ai eA(t−ti ) for t ∈ (tk , tk+1 ], k = 0, 1, 2, . . . (3.14) x(t) = x0 eA(t−t0 ) + i=1

Applying (V1) and (3.9), we obtain the solution of (3.13), namely k Eq (A(ti − ti−1 )q ) x(t; t0 , x0 ) = x0 i=1

+

k

ai

i=1

for

k

Eq (A(tj − tj−1 )q ) Eq (A(t − tk )q )

(3.15)

j=i+1

t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . . ,

where the Mittag-Leffler function (with one parameter) is defined by the power series ∞ zk . Eq (z) = Γ(qk + 1) k=0

Applying (V2) and (3.10), we get the solution of (3.13), namely x(t; t0 , x0 ) = x0 Eq (A(t − t0 )q ) +

k

ak for t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . .

(3.16)

i=1

Therefore, by applying (V2) the amount ak of jump at point tk reflects by the shift ak units of the solution on the whole interval of continuity (tk , tk+1 ] in comparison to the solution on the previous interval (tk−1 , tk ]. In this case it looks like (3.15) is closer to (3.14) for the ordinary case (q = 1).



299

Case 2. Let Φi (x) = ai x, where ai = 1, i = 1, 2, . . . are constants. The solution of IVP for the ordinary differential equation (3.12) is x(t) = x0 eA(t−t0 )

k

ai

for t ∈ (tk , tk+1 ], k = 0, 1, 2, . . .

(3.17)

i=1

Applying (V1) and (3.9) we obtain the solution of (3.13), namely k ai Eq (A(ti − ti−1 )q ) Eq (A(t − tk )q ) x(t; t0 , x0 ) = x0 (3.18) i=1 t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . . . t Eq (λsq ) λ q Applying (V2) and (3.10), using Γ(q) 0 (t−s)q ds = Eq (λt ) − 1, we get the solution of (3.13), namely x(t; t0 , x0 ) = x0 Eq (A(t − t0 )q ) for

+

k

q

Eq (A(ti − t0 ) )(ai − 1)

i=1

k

aj ,

(3.19)

j=i+1

for t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . . It looks like (3.18) is closer to the ordinary case (q = 1). The concept of the FrDE with impulses is rather problematic. In [20], the authors pointed out that the formula, based on (V1) of solutions for IFrDE in [1], [3], [10] is incorrect and gave a new formula using approach (V2). In [42], [44], the authors established a general framework to find solutions for impulsive fractional boundary value problems and obtained some sufficient conditions for the existence of solutions to impulsive fractional differential equations based on (V1). In [42] the authors discussed (V1) criticized the viewpoint (V2) in [20], [44], [45]. Next, in [21] the authors considered the counterexample in [20] and provided further explanations about (V2). We give a simple example to illustrate the difference between both approaches (V1) and (V2). Example 3.3. Let us consider the initial value problem for the scalar impulsive differential equations (q = 1), x (t) = t, for t ≥ 0, t = ti , x(ti + 0) = x(ti − 0) + ai for

i = 1, 2, . . . , ,

x(t0 ) = x0 ,


(3.20)

300


where x ∈ R, ai are constants. The solution of (3.20) is given by ⎧ t2 ⎪ ⎪ for t ∈ [0, t1 ] + x ⎪ 0 ⎪ 2 ⎪ ⎪ k k−1 ⎨ 1 2 (3.21) x(t) = x0 + k (tj+1 − t2j ) aj + ⎪ 2 ⎪ j=1 j=0 ⎪ ⎪ ⎪ ⎪ ⎩ + 1 (t2 − t2 ) for t ∈ (t , t ]. k k+1 k 2 Now, consider the initial value problem for the scalar impulsive fractional differential equations (IFrDE) with a Caputo derivative for 0 < q < 1, c 0.5 0 D x(t)

= t, for t ≥ 0, t = ti ,

x(ti + 0) = x(ti − 0) + ai for

i = 1, 2, . . . ,

(3.22)

x(0) = x0 . Using approach (V1) and formula (3.3) the solution of (3.22) is ⎧ 4t1.5 ⎪ ⎪ for t ∈ [0, t1 ] x0 + ⎪ ⎪ ⎪ 3Γ(q) ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ + aj x 0 ⎪ ⎪ ⎪ ⎨ j=1 k−1 (3.23) x(t) = 2 ⎪ ⎪ + (2t − t ) t − t j+1 j j+1 j ⎪ ⎪ 3Γ(q) ⎪ ⎪ j=0 ⎪ ⎪ √ ⎪ 2 ⎪ ⎪ (2t − tk ) t − tk + ⎪ ⎪ 3Γ(q) ⎪ ⎪ ⎩ for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . . Using approach (V2) and formula (3.4) we obtain for the solution of (3.22) ⎧ 4t1.5 ⎪ ⎪ x0 + for t ∈ [0, t1 ] ⎪ ⎨ 3Γ(q) k (3.24) x(t) = 4t1.5 ⎪ ⎪ aj for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . . ⎪ ⎩x0 + 3Γ(q) + j=1

In this case it looks like the concept of a solution in (V1) is closer to the ordinary case q = 1. Suppose am = 0 holds for some m ∈ {1, 2, ...}. Then from the second equation in (3.20) and (3.22) it follows that x(tm + 0) = x(tm − 0), i.e. there is no impulse at the point tm . In the case q = 1 formula (3.5) could be rewritten as



301

⎧ t2 ⎪ ⎪ for t ∈ [0, t1 ] x0 + ⎪ ⎪ ⎪ 2 ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ + k aj x ⎨ 0 x(t) =

j=1,j=m

(3.25)

⎪ k−1 ⎪ ⎪ 1 1 2 ⎪ ⎪ (tj+1 − t2j ) + (t2 − t2k ) + ⎪ ⎪ 2 2 ⎪ ⎪ j=0,j=m ⎪ ⎪ ⎩ for t ∈ (tk , tk+1 ], k = 1, 2, 3, . . . ,

i.e. the point tm is ignored. The same is true with formula (3.10), i.e. when (V2) is applied. However applying (3.8) we do not obtain the correct result (see our comment in (V1) after equation (3.8)). To avoid the confusing situation in the application of approach (V1) mentioned above, we will assume: H1. If x = 0, then Φk,j (x) = xj for all j = 1, 2 . . . , n and k = 1, 2, 3, . . . where x ∈ Rn , x = (x1 , x2 , . . . , xn ) and Φk : Rn → Rn , Φk = (Φk,1 , Φk,2 , . . . , Φk,n ). Remark 3.3. Condition (H1) is equivalent to Ik,j (x) = 0 if x = 0 for all k = 1, 2, 3, . . . and j = 1, 2 . . . , n, where Ik = (Ik,1 , Ik,2 , . . . , Ik,n ). Let J ⊂ R+ be a given interval and Δ ⊂ Rn . Let Jimp = {t ∈ J : t = tk , k = 1, 2, . . . } and introduce the following classes of functions: C q (Jimp , Δ) =

∞

C q ((tk , tk+1 ), Δ),

∞

C(Jimp , Δ) =

k=0

C((tk , tk+1 ), Δ),

k=0

P C q (J, Δ) = {u ∈ C q (Jimp , Δ) : u(tk ) = lim u(t) < ∞, t↑tk

u(tk + 0) = lim u(t) < ∞, u (tk ) = lim u (t) < ∞, t↓tk

t↑tk

u (tk + 0) = lim u (t) < ∞ for all k : tk ∈ J}, t↓tk

P C(J, Δ) = {u ∈ C(Jimp , Δ) : u(tk ) = lim u(t) < ∞, t↑tk

u(tk + 0) = lim u(t) < ∞ for all k : tk ∈ J}. t↓tk

Remark 3.4. From the above, any solution of (3.6) is from the class n 0 , b), R ), b ≤ ∞.

P C q ([t


302

R. Agarwal, S. Hristova, D. O’Regan 4. Stability and Lyapunov functions for IFrDE

We will assume the following condition is satisfied: (H2) f (t, 0) ≡ 0 for t ∈ R+ and Φi (0) = 0 for i = 1, 2, 3 . . . . In the definition below we let x(t; t0 , x0 ) ∈ P C q ([t0 , ∞), Rn ) be any solution of (3.6). Definition 4.1. The zero solution of (3.6) is said to be: • stable if for every > 0 and t0 ∈ R+ there exist δ = δ(, t0 ) > 0 such that for any x0 ∈ Rn the inequality ||x0 || < δ implies ||x(t; t0 , x0 )|| < for t ≥ t0 ; • uniformly stable if for every > 0 there exist δ = δ() > 0 such that for t0 ∈ R+ , x0 ∈ Rn with ||x0 || < δ the inequality ||x(t; t0 , x0 )|| < holds for t ≥ t0 . Example 4.1. Consider the IVP for the scalar IFrDE (3.11) with Φi (u) = ai u, i = 1, 2, . . . , where ai = 1, i = 1, 2, 3, . . . are constants |a | ≤ M . According to and there exists a constant M > 0 with ∞

i=1 i Example 3.1 the solution u(t; t0 , u0 ) = u0 ki=1 ai for t ∈ (tk , tk+1 ], k = 0, 1, 2, . . . . Therefore, the zero solution of (3.11) is uniformly stable. Example 4.2. Consider the IVP for the scalar IFrDE (3.13) when A = −1, tk = k, ak = 0.5, k = 1, 2, . . . , t0 = 0. From Example 3.2, applying (V1) and Eq. (3.18) we obtain Eq (−1) k Eq (−(t − k)q ) x(t; 0, x0 ) = x0 (4.1) 2 for t ∈ (k, k + 1], k = 0, 1, 2, 3, . . . . Equality (4.1) and inequality 0 < Eq (−(T − τ )q ) ≤ 1 for T ≥ τ prove |x(t; 0, x0 )| < |x0 |, i.e. the zero solution of the scalar IFrDE (3.13) is stable (see the graphs of the solutions with q = 0.2 for various initial values x0 in Figure 4.1). From Example 3.2, applying (V2) and (3.19) we get the solution k Eq (−iq ) q x(t; 0, x0 ) = x0 Eq (−t ) − 2k−i+1 (4.2) i=1

t ∈ (tk , tk+1 ], k = 0, 1, 2, 3, . . . 1 < 2|x0 | which From (4.2) we obtain |x(t; 0, x0 )| = |x0 | 1+ ki=1 2k−i+1 proves the stability of the zero solution of (3.13) (see the graphs of the solutions with q = 0.2 for various initial values x0 in Figure 4.2). for



Figure 4.1. Solutions of (3.13) using (V1) with q = 0.2 and x0 = 0.5, 1, 0.2.

303

Figure 4.2. Solutions of (3.13) using (V2) with q = 0.2 and x0 = 0.5, 1, 0.2.

From Figure 4.1 and Figure 4.2 we see that both approaches (V1) and (V2) give different solutions. With positive initial values the solution given by (V1) is positive (see Fig. 4.1), but the solution given by (V2) can be negative (see Fig. 4.2). Comparing with the ordinary derivative (impulsive differential equations) (see, (3.14) and Figure 6.1) it seems (V1) reflects the ordinary case. In this paper we will use the following sets: K = {a ∈ C[R+ , R+ ] : a is strictly increasing and a(0) = 0}, S(A) = {x ∈ Rn : ||x|| ≤ A},

A > 0.

The stability of fractional order systems is quite a recent topic. In studying stability for nonlinear fractional differential equations, there are several approaches in the literature, one of which is the Lyapunov approach. As is noted in [40] there are several difficulties encountered when one applies the Lyapunov technique to fractional differential equations. We give a brief overview of the application of Lyapunov’s second method to Caputo fractional differential systems. Note one of the main difficulties in applications of Lyapunov functions to fractional differential equations is the appropriate definition of its derivative among the differential equation of fractional order. The results on stability for nonlinear fractional differential equations in the literature via Lyapunov functions could be divided into three main groups: (G1) Continuously differentiable Lyapunov functions. The main condition concerns the Caputo fractional derivative of the Lyapunov q function C 0 Dt V (t, x(t)), where x(t) is a solution of the studied fractional equation (see, for example, the papers [8], [19], [23], [31], [32]). This approach requires the function to be smooth enough (at least continuously differentiable).


304


(G2) Continuous Lyapunov functions and fractional Dini derivative – this derivative is based on the Dini derivative of the Lyapunov function V (t, x) among the ordinary differential equation x = f (t, x) 1 V (t, x) − V (t − h, x − hf (t, x)) . (4.3) D+ V (t, x) = lim sup h→0+ h Formula (4.3) is generalized to the Dini fractional derivative V (t, x) among the Caputo FrDE (3.1) by (see, for example, [27], [28]) 1 q V (t, x) = lim sup q V (t, x) − V (t − h, x − hq f (t, x) (4.4) D+ h→0+ h where 0 < q < 1. The fractional Dini derivative (4.4) is a local operator which differs from the Caputo fractional derivative. (G3) Continuous Lyapunov functions and Caputo fractional Dini derivative – recently (see [4]) defined and is based on the Caputo fractional Dini derivative of a function m(t) given by (2.4) and is given by c D q V (t, x; τ0 , x0 ) (3.1) + t−τ [ h0] 1 (−1)r+1 qCr = lim sup q V (t, x) − V (τ0 , x0 ) − h h→0+ r=1 q for t ∈ (τ0 , T ), × V (t − rh, x − h f (t, x)) − V (τ0 , x0 )

(4.5)

where x, x0 ∈ Δ, and there exists h1 > 0 such that t − h ∈ [τ0 , T ), x − hq f (t, x) ∈ Δ for 0 < h ≤ h1 , Δ ⊂ Rn . Formula (4.5) is equivalent to q c D V (t, x; τ0 , x0 ) (3.1) + t−τ [ h0] 1 r+1 q q (−1) V (t − rh, x − h f (t, x)) (4.6) = lim sup q V (t, x) − r h→0+ h

− V (τ0 , x0 )

)−q

(t − τ0 Γ(1 − q)

r=1

for t ∈ (τ0 , T ).

Note that in [17] the authors defined a derivative of a Lyapunov function and called it the Caputo fractional Dini derivative of V (t, x) (see Definition 3.2 [17]): n 1 c q D+ V (t, x) = lim sup q V (t, x) − V (t − rh, x − hq f (t, x)) h + h→0 r=1 (4.7) − V (t0 , x0 ) ,



305

(we feel (−1)r+1 qCr is missing in the formula). Formula (4.7) is quite different than the Caputo fractional Dini derivative of a function (see (2.3) and (2.4)). Also, in Definition 5.1 [17] the authors define the Caputo fractional Dini derivative of Lyapunov function V (s, y(t, s, x)) by 1 c q D+ V (t, y(t, s, x)) = lim sup q V (t, y(t, s, x)) h→0+ h (4.8) n r+1 q (−1) qCrV (s − rh, y(t, s − rh, x − h F (t, x))) . − r=1

Formula (4.8) is also quite different than the Caputo fractional Dini derivative of a function (2.4). We will use definition (4.5) as the definition of Caputo fractional Dini derivative of a Lyapunov function. Using formula (4.5), the stability and uniform stability of FrDE’s ([4], [5]), strict stability of FrDE’s ([6]), practical stability ([7]) are studied. Example 4.3. Differentiable Lyapunov function V(x). Consider the Lyapunov function which does not depend on t, i.e. V (x) for x ∈ R and let x(t) ∈ C q ([τ0 , T ], R) be any solution of FrDE (3.1). In this case (G1) could be applied and the Caputo fractional derivative cτ0 D q V (x(t)) could be used to study stability properties. Apply formula (4.5) to obtain the Caputo fractional Dini derivative of the Lyapunov function c Dq V (3.1) +

(t, x(t); τ0 , x0 ) 1 = lim sup q V (x(t)) − V (x0 ) h→0+ h [

−

t−τ0 ] h

r+1

(−1)

q qCr V x(t) − h f (t, x(t)) − V (x0 )

r=1

1 = lim sup q h→0+ h t−τ [ h0]

+

0] [ t−τ h

(−1) qCr V (x(t − rh)) − V (x0 ) r

r=0

q (−1) qCr V x(t) − h f (t, x(t)) − V (x(t − rh)) r

r=1


306


t−τ [ h0] 1 q D ) + lim sup (−1)r qCrV (ξ)× V (x(t)) − V (x =GL 0 t0 + q h + h→0 r=0 × x(t) − hq f (t, x(t)) − (x(t − h)) . q GL Using τ0 D+ V (x(t)) − V (x0 ) =cτ0 D q V (x(t)) and

[

t−τ0 ] h

(4.9)

q (−1) qCr V x(t) − h f (t, x(t)) − V (x(t − rh)) r

r=1 t−τ0

[ h ] 1 (−1)r qCr x(t) − hq f (t, x(t)) − (x(t − h)) ≤ L lim sup q h + h→0 r=0

[

= Lf (t, x(t)) lim sup h→0+

+ L lim sup h1−q h→0+

t−τ0 ] h

(−1)r qCr

r=0 t−τ [ h0]

(−1)r qCrx (η)

r=0

= 0, q V (t, x(t); τ0 , x0 ) =cτ0 D q V (x(t)), i.e. the derivatives we obtain that c(3.1) D+ of the Lyapunov function V (x) given in (G1) and (G3) coincide in this particular case. Example 4.4. Quadratic Lyapunov function V (x) = x2 , x ∈ R. First we consider the case q = 1. From (4.3) we get D+ V (t, x) = 2xf (t, x)

(4.10)

Let x(t) ∈ C q ([τ0 , T ], R) be any solution of FrDE (3.1). Applying (G1) and Lemma 1 we get c q 2 (4.11) τ0 D (x(t)) = 2x(t)f (t, x(t)) Apply (G2) and formula (4.4) to obtain the Dini fractional derivative among the FrDE (3.1), and we get x2 − (x − hq f (t, x))2 hq h→0+ hq f (t, x)(2x − hq f (t, x)) = lim sup hq h→0+ = 2xf (t, x).

q V (t, x) = lim sup D+


(4.12)


307

The derivatives (4.11) and (4.12) coincide with the ordinary case (4.10) for q = 1. Example 4.5. Lyapunov function depending on t. Let us for example, consider V (t, x) = g 2 (t)x2 for g ∈ C 1 (R+ , R) and x ∈ R. If V (t, x) is differentiable function and x(t) ∈ C q ([τ0 , T ], R) is a solution of FrDE(3.1), then to obtain the Caputo fractional derivative of V (t, x), i.e. cτ0 D q V (t, x(t)) =cτ0 D q g2 (t)x(t)) we need the multiplication rule from fractional calculus, so, (G1) could lead to some difficulties in calculations of the derivative. Now, apply (G2) and formula (4.4) to obtain the Dini fractional derivative and we get 2 1 q V (t, x) = lim sup q g2 (t)x2 − g2 (t − h) x − hq f (t, x) D+ h h→0+ g(t) − g(t − h) xh1−q + g(t − h)f (t, x) = lim sup (4.13) h h→0+ q × (g(t) + g(t − h))x − g(t − h)h f (t, x) = 2x g 2 (t)f (t, x),

t ∈ R+ , x ∈ R.

The derivative of the Lyapunov function in the ordinary case (q = 1) applying (4.3) is d 2 g (t) , t ∈ R+ , x ∈ R. (4.14) D+ V (t, x) = 2x g2 (t)f (t, x) + x2 dt Now we look at (4.13) and (4.14). Both differ significantly. In the fractional Dini derivative (4.13) one term is missing. Additionally, the Dini fractional derivative (4.13) is independent of the order of the differential equation q. However the behavior of solutions of fractional differential equations depends significantly on the order q. For example, consider the simple fractional differential equation c0 D q x + x(t) = 1, x(0) = 0 whose solution is given by x(t) = tq Eq,1+q (−tq ). Note limt→∞ x(t) varies for different values of the order q of the fractional differential equation. Let t, τ0 ∈ R+ , x, x0 ∈ R. Now use (4.5) to obtain the Caputo fractional Dini derivative of V , namely c Dq V (3.1) +

(t, x; τ0 , x0 )

[ h ] 1 r 2 2 2 2 (−1) qCr g (t − rh)x − g (τ0 )x0 = lim sup q h r=0 h→0+ t−τ0


308

R. Agarwal, S. Hristova, D. O’Regan [

t−τ0

]

h 1 (−1)r qCr + lim sup q h h→0+ r=1 2 2 q 2 2 × g (t − rh) x − h f (t, x) − g (t − rh)x q GL 2 2 2 2 =τ0 D+ g (t)x − g (τ0 )(x0 ) t−τ [ h0]

− 2xf (t, x) lim sup h→0+

(−1)r qCr g2 (t − rh)

r=1 [

+ (f (t, x))2 lim sup hq h→0+

GL q t0 D+

t−t0

] h

(−1)r qCr g2 (t − rh).

r=1

Now using

(4.15)

t−τ0

[ h ] 1 2 q 2 (−1)r qCr g2 (t − rh) =RL D (t) , g (t) = lim sup q g τ0 h h→0+ r=0

we obtain from (4.15), c Dq V (3.1) +

(t, x; τ0 , x0 )

= 2x g 2 (t)f (t, x) + x2

RL q τ0 D

g2 (t) − (x0 )2

RL q τ0 D

g2 (τ0 ) .

(4.16)

Note the Caputo fractional Dini derivative depends not only on the fractional order q but also on the initial data (τ0 , x0 ) of (3.1) which is similar to the Caputo fractional derivative of a function. If x0 = 0 or τ0 = 0 and g(0) = 0, then (4.16) (similar to the ordinary case q = 1 and formula (4.14)) consists of two terms where the ordinary derivative is replaced by the fractional one. It seems that formula (4.5) defined in (G3) is a natural generalization of the one for ordinary derivatives. We recall the following result: Corollary 4.1. ([4]) Assume the following conditions are satisfied: 1. The function x∗ (t) = x(t; τ0 , x0 ), x∗ ∈ C q ([τ0 , T ], Δ), is a solution of the FrDE (3.1) where Δ ⊂ Rn , 0 ∈ Δ, 0 ≤ τ0 < T . 2. The function V (t, x) ∈ C([τ0 , T ] × Δ, R+ ) is locally Lipschitz w.r.t. its second argument, V (t, 0) = 0 and for any points t ∈ [τ0 , T ] and q V (t, x; τ0 , x0 ) ≤ 0 holds. x ∈ Δ the inequality c(3.1) D+

Then for t ∈ [τ0 , T ] the inequality V (t, x∗ (t)) ≤ V (τ0 , x0 ) holds.



309

Now we will use piecewise continuous Lyapunov functions and we will generalize the Caputo fractional Dini derivative (4.5) to appropriate ones for Caputo fractional equations with impulses. Definition 4.2. Let J ∈ R+ be a given interval, and Δ ⊂ Rn , 0 ∈ Δ be a given set. We will say that the function V (t, x) : J × Δ → R+ , V (t, 0) ≡ 0 belongs to the class Λ(J, Δ) if: 1. The function V (t, x) is continuous on J/{tk ∈ J} × Δ and it is locally Lipschitzian with respect to its second argument; 2. For each tk ∈ J and x ∈ Δ there exist finite limits V (tk − 0, x) = lim V (t, x), t↑tk

and

V (tk + 0, x) = lim V (t, x) t↓tk

and the following equalities are valid V (tk − 0, x) = V (tk , x). We define the generalized Caputo fractional Dini derivative of the function V (t, x) ∈ Λ([t0 , T ), Δ) along trajectories of solutions of IVP for the system IFrDE (3.6) as follows: 1 q c D V (t, x; t0 , x0 ) = lim sup q V (t, x) − V (t0 , x0 ) (3.6) + h→0+ h t−t0 ] h r+1 q (−1) qCr V (t − rh, x − h f (t, x)) − V (t0 , x0 ) −

[

(4.17)

r=1

for t ∈ (t0 , T ) : t = tk , where x, x0 ∈ Δ, and there exists h1 > 0 such that t − h ∈ [t0 , T ), x − hq f (t, x) ∈ Δ for 0 < h ≤ h1 . The following result is true for the generalized Caputo fractional Dini derivative of Lyapunov function defined by (4.17). Lemma 4.1. Assume the following conditions are satisfied: 1. Let condition (H1) holds. 2. The function x∗ (t) = x(t; t0 , x0 ), x∗ ∈ P C q ([t0 , T˜], Δ), is a solution of the IFrDE (3.6) where Δ ⊂ Rn , 0 ∈ Δ, 0 ≤ t0 < t1 ≤ T˜. 3. The function V ∈ Λ([t0 , T˜], Rn ) and (i) for any point t ∈ [t0 , T˜], t = tk , k = 1, 2, . . . , p and x ∈ Δ the inequality c D q V (t, x∗ (t); t0 , x0 ) ≤ 0 (3.6) + holds; (ii) for any point tk ∈ (t0 , T˜) and x ∈ Rn the inequality V (tk + 0, Φk (x∗ (tk ))) ≤ V (tk , x∗ (tk )) holds. Then for t ∈ [t0 , T˜] the inequality V (t, x∗ (t)) ≤ V (t0 , x0 ) holds.


310


P r o o f. We use induction to prove Lemma 4.1. Let t ∈ [t0 , t1 ]. According to Corollary 4.1 with τ0 = t0 and T = t1 the claim of Lemma 4.1 is true on [t0 , t1 ]. Let t ∈ (t1 , t2 ]∩[t0 , T˜]. According to approach (V1) the function x1 (t) ∈ q C ([t1 , t2 ] ∩ [t0 , T˜], Rn ) defined by x1 (t) ≡ x∗ (t), t ∈ (t1 , t2 ] ∩ [t0 , T˜] and x1 (t1 ) ≡ x∗ (t1 + 0) = Φ1 (x∗ (t1 − 0)), is a solution of IVP for FrDE (3.1) for τ0 = t1 and x0 = Φ1 (x∗ (t1 − 0)). Using condition 3 (ii) and the above proved inequality V (t1 , x∗ (t1 )) = V (t1 − 0, x∗ (t1 − 0)) ≤ V (t0 , x0 ) we obtain V (t1 + 0, x1 (t1 )) = V (t1 + 0, x∗ (t1 + 0))

= V (t1 + 0, Φ1 (x∗ (t1 − 0))) = V (t1 + 0, Φ1 (x∗ (t1 )))

(4.18)

≤ V (t1 , x∗ (t1 )) ≤ V (t0 , x0 ). According to Corollary 4.1 for τ0 = t1 and T = min{T˜, t2 } we obtain V (t, x1 (t)) ≤ V (t1 + 0, x1 (t1 )) for t ∈ [t1 , t2 ] ∩ [t0 , T˜]. Therefore, V (t, x∗ (t)) ≤ V (t0 , x0 ) for t ∈ (t1 , t2 ] ∩ [t0 , T˜], i.e. the claim of Lemma 4.1 is true on [t0 , t2 ] ∩ [t0 , T˜]. Continuing this process and an induction argument proves the claim is 2 true on [t0 , T˜]. The result of Lemma 4.1 is also true on the half line (recall [4] that Corollary 1 extends to the half line). Lemma 4.2. Suppose all the conditions of Lemma 4.1 are satisfied with [t0 , T˜] replaced by [t0 , ∞) (here k = 1, 2, ..., p is replaced by k = 1, 2, ...). Then for any t ≥ t0 the inequality V (t, x∗ (t)) ≤ V (t0 , x0 ) holds. 5. Sufficient conditions for stability of IFrDE We obtain sufficient conditions for stability of the zero solution of nonlinear impulsive Caputo fractional differential equations. Theorem 5.1. Let the following conditions be satisfied: 1. Let (H1) and (H2) be satisfied. 2. The functions f ∈ P C(R+ , Rn ), Φk : Rn → Rn , k = 1, 2, . . . , are such that for any (t0 , x0 ) ∈ R+ × Rn the IVP for the system of IFrDE (3.6) has a solution x(t; t0 , x0 ) ∈ P C q ([t0 , ∞), Rn ). 3. There exists a function V ∈ Λ(R+ , Rn ) such that (i) for any t0 ∈ R+ and x0 ∈ Rn the inequality c Dq V (3.6) +

(t, x; t0 , x0 ) ≤ 0 for t ≥ t0 , t = tk , k = 1, 2, . . . , x ∈ Rn holds;



311

(ii) the inequality V (tk + 0, Φk (x)) ≤ V (tk , x) for k = 1, 2, . . . , x ∈ Rn holds; (iii) b(||x||) ≤ V (t, x) for t ∈ R+ , x ∈ Rn , where b ∈ K. Then the zero solution of the system of IFrDE (3.6) is stable. P r o o f. Let > 0 and t0 ∈ R+ be given. Without loss of generality we assume t0 < t1 . Since V (t0 , 0) = 0 there exists δ = δ(t0 , ε) > 0 such that V (t0 , x) < b(ε) for ||x|| < δ. Let x0 ∈ Rn with ||x0 || < δ. Then b−1 (V (t0 , x0 )) < ε. Consider any solution x∗ (t) = x(t; t0 , x0 ) ∈ P C q ([t0 , ∞), Rn ) of the IFrDE (3.6) which exists according to condition 2. According to Lemma 4.2 the inequality V (t, x∗ (t)) ≤ V (t0 , x0 ) holds for t ≥ t0 . Then for any t ≥ t0 from condition 3 (iii) we obtain b(||x∗ (t)||) ≤ V (t, x∗ (t)) ≤ V (t0 , x0 ). Therefore ||x∗ (t)|| ≤ b−1 (V (t0 , x0 )) < ε, so the result follows. 2 Now we present sufficient conditions for stability of the zero solution of the IFrDE in the case when the condition for the Caputo fractional Dini derivative of the Lyapunov function is satisfied only on a ball. Theorem 5.2. Let the following conditions be satisfied: 1. The conditions 1, 2 of Theorem 5.1 are fulfilled. 2. There exists a function V ∈ Λ(R+ , Rn ) such that (i) for any t0 ∈ R+ and x0 ∈ S(λ) the inequality c Dq V (3.6) +

(t, x; t0 , x0 ) ≤ 0 for t ≥ t0 , t = tk , k = 1, 2, . . . , x ∈ S(λ)

holds, where λ > 0 is given; (ii) the inequality V (tk + 0, Φk (x)) ≤ V (tk , x) for k = 1, 2, . . . , x ∈ S(λ) holds; (iii) b(||x||) ≤ V (t, x) ≤ a(||x||) for t ∈ R+ , x ∈ S(λ), where a, b ∈ K. Then the zero solution of the system of IFrDE (3.6) is uniformly stable. P r o o f. Let ∈ (0, λ] and t0 ∈ R+ be given. Let δ1 < min{, b()}. From a ∈ K there exists δ2 = δ2 () > 0 so if s < δ2 then a(s) < δ1 . Let δ = min(, δ2 ). Choose the initial value x0 ∈ Rn such that ||x0 || < δ. Therefore x0 ∈ S(λ). Let x∗ (t) = x(t; t0 , x0 ), t ≥ t0 be a solution of the IVP for IFrDE (3.6). We now prove that ||x∗ (t)|| < ,

t ≥ t0 .


(5.1)

312


Assume inequality (5.1) is not true and let t∗ = inf{t > t0 : ||x∗ (t)|| ≥ }. First we prove ||x∗ (t)|| < for t ∈ [t0 , t∗ ) and ||x∗ (t∗ )|| = . According to Lemma 4.1 for T˜ = t∗ and Δ = S(λ) we get V (t, x∗ (t)) ≤ V (t0 , x0 ) on [t0 , t∗ ]. Then applying condition 2 (iii) we obtain b(ε) = b(||x∗ (t∗ )||) ≤ V (t∗ , x∗ (t∗ )) ≤ V (t0 , x0 ) ≤ a(||x0 ||) < δ1 < b(ε). The contradiction proves (5.1) and therefore, the zero solution of IFrDE (3.6) is uniformly stable. 2 Remark 5.1. In [39] some stability properties for IFrDE are studied based on the approach (V2) and a piecewise generalization of the Dini fractional derivative of Lyapunov functions c D q + V (t, x) given in (G2) by (4.4). However the Dini fractional derivative c D q + V (t, x) for V (t, x) = ||x||, x ∈ R does not coincide with the Caputo fractional derivative 0 c Dq t ||x(t)|| (see Section 5, [39]). Also the approach (V2) requires a different type of inductive proof w.r.t. the intervals.

6. Applications Consider the generalized Caputo population model. Example 6.1. Let the points tk , tk < tk+1 , limk→∞ tk = ∞ be fixed. Consider the scalar impulsive Caputo fractional differential equation c q 0D x

= −a(t)x(1 + x2 ) for t ≥ 0, t = tk , k = 1, 2, . . . ,

x(tk + 0) = ck x(tk ),

k = 1, 2, 3, . . . ,

(6.1)

where x ∈ R, ck ∈ [−1, 0) ∪ (0, 1), k = 1, 2, . . . , are given constants and the 1 . function a(t) = 2tq Γ(1−q) Consider the function V (t, x) = x2 . Then (ck x)2 ≤ x2 , i.e. condition 3 (ii) of Theorem 5.1 is satisfied. Let t > 0, t = tk we get 1 q c 2 2 = −2a(t)x4 ≤ 0. D V (t, x; 0, x ) ≤ x ) + − 2a(t)(1 + x 0 + (6.1) tq Γ(1 − q) Then according to Theorem 5.1, the trivial solution of IFrDE (6.1) is stable.



313

Figure 6.1. Graphs of the solution (3.14) Figure 6.2. Graphs of v(t) for various q. for various x0 .

Example 6.2. Consider the initial value problem for the scalar impulsive fractional differential equation c q 0 D x(t)

=

1 −0.25q − 0.75 tq Γ(1−q) − 2q−2 cos(2t +

qπ 2 ))

1 + cos2 (t) π for t ≥ 0, t = k , 2 π π x(k + 0) = ck x(k ), k = 1, 2, 3, . . . , 2 2 x(0) = x0 ,

x, (6.2)

where ck ∈ [−1, 0) ∪ (0, 1), k = 1, 2, . . . are given constants. Let V (t, x) = x2 . For t ≥ 0, t = k π2 we get t−q q c 2 2 D V (t, x; 0, x ) = 2xf (t, x) + x − x 0 0 (6.2) + Γ(1 − q) t−q ≤ 2xf (t, x) + x2 − x20 Γ(1 − q) q−2 cos(2t + qπ ) −0.25q − 0.75 q 1 1 2 t Γ(1−q) − 2 + = x2 2 1 + cos2 (t) tq Γ(1 − q)

(6.3)

≡ x2 v(t). Note the sign of the function v(t) changes on [0, ∞) (see Figure 6.2). Now we consider the Lyapunov function V (t, x) = (1+cos2 (t))x2 . From Example 4.5 we obtain for t = k π2 , x, x0 ∈ R, c Dq V (6.2) +

(t, x; 0, x0 )

1 + cos (t) − (x0 )2 = 2x(1 + cos (t))f (t, x) + q 2 D (t) . 1 + cos ≤ 2x(1 + cos2 (t))f (t, x) + x2 RL 0 2

q x2 RL 0 D

2


RL q 0 D 2

(6.4)

314

R. Agarwal, S. Hristova, D. O’Regan From (6.4), cos2 (t) = 0.5 + 0.5 cos(2t) and we obtain

qπ 2 ),

c Dq V (6.2) +

RL D q (cos(2t)) 0

= 2q cos(2t +

(t, x; 0, x0 )

qπ 1 q−2 − 2 ) cos(2t + tq Γ(1 − q) 2 qπ 1.5 + 2q−1 cos(2t + ) + x2 q t Γ(1 − q) 2

≤ 2x2 − 0.25q − 0.75

= −0.5q x2 ≤ 0. Let tk = k π2 for k = 1, 2, . . . . Then π π π π V (k , ck x) = (1 + cos2 (k ))(ck )2 x2 ≤ (1 + cos2 (k ))x2 = V (k , x). 2 2 2 2 According to Theorem 5.1 the zero solution of (6.2) is stable. Acknowledgements The research was partially supported by the Fund NPD, Plovdiv University, No. MU15-FMIIT-008. References [1] B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3 (2009), 251–258. [2] R. Agarwal, M. Benchohra, B. Slimani, Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44 (2008), 1–21. [3] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109 (2010), 973–1033. [4] R. Agarwal, D. O’Regan, S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions. Appl. Math. 60, No 6 (2015), 653–676. [5] R. Agarwal, D. O’Regan, S. Hristova, Stability and Caputo fractional Dini derivative of Lyapunov functions for Caputo fractional differential equations. In: Intern. Workshop QUALITDE, December 27–29, 2015, Tbilisi, Georgia, 3–6. [6] R. Agarwal, S. Hristova, D. O’Regan, Lyapunov functions and strict stability of Caputo fractional differential equations. Adv. Diff. Eq. 2015 (2015), Article ID 346 (20 p.).



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Dept. of Mathematics, Texas A&M University-Kingsville Kingsville, TX 78363, USA e-mail: [email protected]

2

Dept. of Applied Mathematics, Plovdiv University Tzar Assen 24, Plovdiv - 4000, BULGARIA Received: July 20, 2015 e-mail: [email protected]

Revised: January 25, 2016

3

School of Mathematics, Statistics and Applied Mathematics National University of Ireland, Galway, IRELAND e-mail: [email protected] Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 290–318, DOI: 10.1515/fca-2016-0017
