Lipschitz stability criteria for functional differential systems of fractionalorderIvanka Stamova and Gani Stamov Citation: J. Math. Phys. 54, 043502 (2013); doi: 10.1063/1.4798234 View online: http://dx.doi.org/10.1063/1.4798234 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i4 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

JOURNAL OF MATHEMATICAL PHYSICS 54, 043502 (2013)

Lipschitz stability criteria for functional differential systemsof fractional order

Ivanka Stamova1,a) and Gani Stamov2

1Department of Mathematics, The University of Texas at San Antonio, One UTSA Circle,San Antonio, Texas 78249, USA2Department of Mathematical Physics, Technical University–Sofia, 8800 Sliven, Bulgaria

(Received 8 December 2012; accepted 11 March 2013; published online 1 April 2013)

In this paper, a class of fractional functional differential equations is investigated.Using differential inequalities and Lyapunov-like functions, Lipschitz stability,uniform Lipschitz stability, and global uniform Lipschitz stability criteria are proved.Since the problem of Lipschitz stability of dynamic systems is relevant in variouscontexts, including many inverse and control problem, our results can be appliedin the qualitative investigations of many practical problems of diverse interest.C© 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4798234]

I. INTRODUCTION

There has been a growing interest in the theory of functional dynamical systems in the pastdecades because of their applications to almost every domain of applied sciences, see, e.g., cf.Refs. 7, 10, and 23–28. Since the stability properties are of practical importance, the framework ofthe more than 50 years old stability theory studies for such systems has not lost their attraction andhas been extended widely.

On the other hand, the use of fractional calculus and fractional differential equations to dealwith engineering and physical problems has become increasingly popular in recent decades. It iswidely applied in material and quantum mechanics, signal processing and systems identification,anomalous diffusion, wave propagation, etc., see cf. Refs. 4, 8, 9, 12, 14, 17–20, and 22. Functionaldifferential equations of fractional order also have been of great interest recently (see Refs. 3 and11). However, the work on the stability theory of such equations is relatively sparse, see cf. Ref. 6.

One type of stability, very useful in real world problems, deals with the so called Lipschitzstability. For nonlinear integer-order dynamic systems, this notion was introduced in Ref. 5. Forlinear systems, the notions of uniform Lipschitz stability and that of uniform stability are equivalent.However, for nonlinear systems, the two notions are quite distinct, see cf. Refs. 5 and 15. In fact,uniform Lipschitz stability lies between uniform stability on one side and the notions of asymptoticstability in variation and uniform stability in variation on the other side. Furthermore, uniformLipschitz stability neither implies asymptotic stability nor is it implied by it. However, to the best ofour knowledge, there has not been any work so far considering the Lipschitz stability of fractionalfunctional differential equations, which is very important in theories and applications and also is avery challenging problem.

The methods of Lyapunov-like functions and differential inequalities are standard techniques instudying stability, asymptotic properties, existence of periodic, and almost periodic solutions, andhave been proposed as techniques in the investigation of persistence, see cf. Ref. 7. The Razumikhintechnique is an important method which is utilized in studying the stability of functional differentialsystems, see, e.g., cf. Refs. 10, and 23–28.

In this paper, motivated by the above discussion, a class of fractional functional differentialequations is investigated, which provides mathematical models for real-world problems in which the

a)Author to whom correspondence should be addressed. Electronic mail: ivanka.stamova@utsa.edu

0022-2488/2013/54(4)/043502/11/$30.00 C©2013 American Institute of Physics54, 043502-1

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-2 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

fractional rate of change depends on the influence of their hereditary effects. Using the comparisonprinciple proved in Ref. 16, Lipschitz stability criteria are obtained, extending the existing theory.Lyapunov-like functions and Razumikhin technique are also used to derive uniform and globaluniform Lipschitz criteria for the fractional functional differential system. Our results can be appliedin the investigation of Lipschitz stability properties of many practical problems of diverse interest.Indeed, the problem of Lipschitz stability of dynamic systems is relevant in various contexts,including many inverse and control problems, see, e.g., cf. Refs. 1, 2, 13, and 29.

II. PRELIMINARIES

Let Rn be the n-dimensional Euclidean space with norm | . |, and let R+ = [0,∞). For a givenτ > 0 we denote by C the space of continuous functions mapping [−τ , 0] into Rn , and for ϕ ∈ C,define the norm

|ϕ|0 = max−τ≤s≤0

|ϕ(s)|.

Suppose that x ∈ C([t0 − τ,∞),Rn), t0 ∈ R+. For any t ≥ t0, we denote by xt a translation ofthe restriction of x to the interval [t − τ , t], i.e., xt is an element of C defined by xt(s) = x(t + s),− τ ≤ s ≤ 0.

Let ρ be a given constant, and let Cρ = {ϕ ∈ C : |ϕ|0 < ρ}. Consider the following system offractional functional differential equations

c Dq x(t) = f (t, xt ), (2.1)

where f : [t0,∞) × C → Rn and cDq is the Caputo’s fractional derivative of order q, 0 < q < 1.

Definition 2.1 (Ref. 22): For any t ≥ t0, the Caputo’s fractional derivative of order q, 0 < q< 1, with the lower limit t0 for a function l ∈ C1[[t0, b],Rn], b > t0, is defined as

c Dql(t) = 1

�(1 − q)

∫ t

t0

l ′(s)

(t − s)qds.

Here and in what follows � denotes the Gamma function.Let ϕ0 ∈ C. Denote by x(t) = x(t; t0, ϕ0) the solution of system (2.1), satisfying the initial

condition

xt0 = ϕ0. (2.2)

We also suppose that the function f is smooth enough on [t0,∞) × C to guarantee existence,uniqueness, and continuability of the solution x(t) = x(t; t0, ϕ0) of the initial value problem (IVP)(2.1), (2.2) on the interval [t0 − τ , ∞) for each ϕ0 ∈ C and t ≥ t0. The existence and uniquenesscriteria are given in Ref. 16 and we have not included them here.

Then, the IVP (2.1), (2.2) is equivalent to the following Volterra fractional integral with memory,see cf. Ref. 16:

xt0 = ϕ0,

x(t) = ϕ0(0) + 1

�(q)

∫ t

t0

(t − s)q−1 f (s, xs)ds, t0 ≤ t < ∞, (2.3)

that is, every solution of (2.3) is also a solution of (2.1) and vice versa.We shall introduce the following Lipschitz stability properties of the zero solution of (2.1).

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-3 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

Definition 2.2: The zero solution of system (2.1) is said to be:

(a) Lipschitz stable, if

(∀t0 ∈ R+)(∃M > 0)(∃δ = δ(t0) > 0)

(∀ϕ0 ∈ C : |ϕ0|0 < δ)(∀t ≥ t0) : |x(t ; t0, ϕ0)| ≤ M |ϕ0|0;

(b) uniformly Lipschitz stable, if the number δ in (a) is independent of t0 ∈ R+;(c) globally uniformly Lipschitz stable, if

(∃M > 0)(∀ϕ0 ∈ C : |ϕ0|0 < ∞)(∀t ≥ t0) : |x(t ; t0, ϕ0)| ≤ M |ϕ0|0;

(d) uniformly stable, if

(∀ε > 0)(∃δ = δ(ε) > 0)(∀t0 ∈ R+)

(∀ϕ0 ∈ C : |ϕ0|0 < δ)(∀t ≥ t0) : |x(t ; t0, ϕ0)| ≤ ε.

Definition 2.3: A function V : [t0,∞) × Rn → R+ belongs to class. C0, if V is continuous in[t0,∞) × Rn and locally Lipschitz continuous with respect to its second argument.

For a function V ∈ C0 we define the following fractional order derivative.

Definition 2.4: Given a function V ∈ C0. For t ∈ [t0, ∞) and ϕ ∈ C the fractional derivative ofV in Caputo’s sense of order q with respect to system (2.1) is defined by

c Dq V (t, ϕ(0)) = limh→0+

sup1

hq[V (t, ϕ(0)) − V (t − h, ϕ(0) − hq f (t, ϕ))].

Together with system (2.1) we consider the following scalar fractional differential equation

c Dqu = g(t, u), (2.4)

where g : [t0,∞) × R+ → R+.Let u0 ∈ R+. We denote by η(t) = η(t; t0, u0) the maximal solution of Eq. (2.4), which satisfies

the initial condition

η(t0; t0, u0) = u0. (2.5)

We shall consider such solutions u(t) of Eq. (2.4) for which u(t) ≥ 0. That is why the followingdefinitions for Lipschitz stability properties of the zero solution of this equation will be used.

Definition 2.5: The zero solution of Eq. (2.4) is said to be:

(a) Lipschitz stable, if

(∀t0 ∈ R+)(∃M > 0)(∃δ = δ(t0) > 0)

(∀u0 ∈ R+ : u0 < δ)(∀t ≥ t0) : η(t ; t0, u0) ≤ Mu0;

(b) uniformly Lipschitz stable, if the number δ in (a) is independent of t0 ∈ R+;(c) globally uniformly Lipschitz stable, if

(∃M > 0)(∀u0 ∈ R+ : u0 < ∞)(∀t ≥ t0) : η(t ; t0, u0) ≤ Mu0;

(d) uniformly stable, if

(∀ε > 0)(∃δ = δ(ε) > 0)(∀t0 ∈ R+)

(∀u0 ∈ R+ : u0 < δ)(∀t ≥ t0) : η(t ; t0, u0) ≤ ε.

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-4 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

In the proof of the main results we shall use the following lemma:

Lemma 2.1 (Ref. 16): Let m ∈ C[[t0 − τ,∞),R] and satisfy the inequality

c Dqm(t) ≤ g(t, |mt |0), t > t0,

where g ∈ C[[t0,∞) × R+,R+]. Assume that the maximal solution η(t) of the IVP (2.4), (2.5)exists on [t0, ∞). Then, if |mt0 |0 ≤ u0, we have

m(t) ≤ η(t), t ∈ [t0,∞).

III. MAIN RESULTS

Following Dannan and Elaydi, see cf. Ref. 5, we shall first show the equivalence of variousnotions of stability in linear differential systems of fractional order without delays.

The Mittag-Leffler function is an important function that finds widespread use in the worldof fractional calculus. Just as the exponential naturally arises out of the solution to integer orderdifferential equations, the Mittag-Leffler function plays an important role in the solution of non-integer order differential equations. The standard definition of the Mittag-Leffler function (see cf.Ref. 22) is given as

Eα(z) =∞∑

k=0

zk

�(αk + 1),

where α > 0. It is also common to represent the Mittag-Leffler function with two parameters, α andβ, such that

Eα,β(z) =∞∑

k=0

zk

�(αk + β),

where α > 0 and β > 0. For β = 1, we have Eα(z) = Eα, 1(z). Also, E1,1(z) = ez.We shall consider the linear system

c Dq x(t) = Ax, t ≥ t0, (3.1)

where A is an n × n matrix.Let x0 ∈ Rn and x(t) = x(t; t0, x0) be the solution of (3.1) with the initial condition

x(t0) = x0. (3.2)

We shall note that if A is a sectorial operator, then the unique solution of the IVP (3.1), (3.2) is(see cf. Refs. 4 and 22)

x(t) = Tq (t − t0)x0, (3.3)

where

Tq (t) = Eq,1(Atq )

is the solution operator, generated by A.

Theorem 3.1: For systems (3.1), the following assertions are equivalent:

(i) The zero solution of (3.1) is uniformly Lipschitz stable in variations.(ii) The zero solution of (3.1) is globally uniformly Lipschitz stable.(iii) The zero solution of (3.1) is uniformly Lipschitz stable.(iv) The zero solution of (3.1) is uniformly stable.(v) The zero solution of (3.1) is globally uniformly Lipschitz stable in variation.

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-5 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

Proof: (i)⇒(ii) Let the zero solution of (3.1) be uniformly Lipschitz stable in variations. Thenthere exist constants M > 0 and δ > 0 such that

|Tq (t − t0)| ≤ M for t ≥ t0, |x0| < δ.

Since the solution operator Tq(t) of (3.1) does not depend on x0, then (3.3) holds for any x0 ∈ Rn .Consequently,

|x(t ; t0, x0)| = |Tq (t − t0)x0| ≤ |Tq (t − t0)||x0| ≤ M |x0|,for t ≥ t0 and x0 ∈ Rn , i.e., (ii) holds.

(ii)⇒(iii) This follows immediately from Definitions 2.2(b) and 2.2(c).(iii)⇒(iv) Let (iii) holds. Then there exist constants M > 0 and δ1 > 0 such that |x(t; t0, x0)|

≤ M|x0| whenever |x0| < δ1 and t ≥ t0.Let ε > 0 be given and let δ = δ(ε) = min (δ1, ε/M). Then for |x0| < δ and t ≥ t0, we have |x(t;

t0, x0)| ≤ M|x0| ≤ Mδ < ε. It follows that the zero solution of (3.1) is uniformly stable.(iv)⇒(v) Let the zero solution of (3.1) be uniformly stable. Then there exists a constant

M > 0 such that |Tq(t − t0)| ≤ M, where Tq(t) is the solution operator, generated by A (seecf. Ref. 4). Consequently, (v) is obtained.

(v)⇒(i) This follows from the corresponding definitions.

Remark: As in the integer-order cases (see cf. Refs. 5 and 15), we proved that for linearfractional-order systems, the notions of uniform Lipschitz stability and that of uniform stability areequivalent. But, it is easy to see, that for functional differential systems of fractional order the twonotions are different, even for the linear systems

c Dq x(t) = Ax(t) + Bxt , t ≥ t0,

whose solutions are given by

x(t) = Tq (t − t0)x0 +∫ t

t0

BSq (t − θ )xθdθ,

where Sq(t) = tq − 1Eq, q(Atq) is the q − resolvent family, and B is an n × n matrix.The efficient applications of fractional functional equations require the finding of criteria for

various types of Lipschitz stability of such systems, and the investigation of the relations betweenthese notions. This is an important problem from a theoretical point of view as well as applications,and is also a challenging problem.

We shall investigate the Lipschitz stability of the zero solution of system (2.1). That is why thefollowing conditions will be used:

A1. f(t, 0) = 0, t ∈ [t0, ∞).A2. g(t, 0) = 0, t ∈ [t0, ∞).

Theorem 3.2: Let the condition A1 holds, and let the zero solution of (2.1) is uniformly Lipschitzstable. Then the zero solution of (2.1) is uniformly stable.

Proof: Let the zero solution of (2.1) is uniformly Lipschitz stable. Then there exist constants M> 0 and δ1 > 0 such that |x(t; t0, ϕ0)| ≤ M|ϕ0|0 whenever |ϕ0|0 < δ1, ϕ0 ∈ C and t ≥ t0.

Let ε > 0 be given and let δ = δ(ε) = min (δ1, ε/M). Then for ϕ0 ∈ C , |ϕ0|0 < δ, and t ≥ t0, theinequalities |x(t; t0, ϕ0)| ≤ M|ϕ0|0 ≤ Mδ < ε are valid. Hence, the zero solution of (2.1) is uniformlystable.

Theorem 3.3: Assume that conditions A1 and A2 hold, f ∈ C[[t0,∞) × Cρ,Rn] and for (t, ϕ)∈ [t0, ∞) × Cρ ,

| f (t, ϕ)| ≤ g(t, |ϕ|0), (3.4)

where g ∈ C[[t0,∞) × R+,R+]. Assume that the maximal solution η(t) of the IVP (2.4), (2.5)exists on [t0, ∞).

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-6 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

Then the Lipschitz stability properties of the zero solution of (2.4) imply the correspondingLipschitz stability properties of the zero solution of system (2.1).

Proof: We shall first prove Lipschitz stability of the zero solution of (2.1). Suppose that the zerosolution of (2.4) is Lipschitz stable. Then, there exist M > 0 and δ = δ(t0) > 0, (M > 1, Mδ < ρ),such that

0 ≤ u0 < δ implies η(t ; t0, u0) ≤ Mu0, t ≥ t0 (3.5)

for some given t0 ∈ R+, where the maximal solution η(t; t0, u0) of (2.4) is defined in the interval[t0, ∞), and

η(t ; t0, u0) ≤ Mu0 < Mδ < ρ, t ≥ t0.

Setting m(t) = |x(t; t0, ϕ0)| and |mt0 |0 = |ϕ0|0 = u0, we get by Lemma 2.1,

|x(t ; t0, ϕ0)| ≤ η(t ; t0, |ϕ0|0) for t ≥ t0. (3.6)

Let ϕ0 ∈ C be such that

|ϕ0|0 < δ. (3.7)

Then

|ϕ0|0 < δ < Mδ < ρ.

From (3.5) and (3.7), it follows

η(t ; t0, |ϕ0|0) ≤ M |ϕ0|0,which due to (3.6) implies

|x(t ; t0, ϕ0)| ≤ η(t ; t0, |ϕ0|0) ≤ M |ϕ0|0, t ≥ t0.

Hence, |x(t; t0, ϕ0)| ≤ M|ϕ0|0, t ≥ t0 for the given t0 ∈ R+, which proves the Lipschitz stabilityof the zero solution of (2.1).

Suppose now, that the zero solution of (2.4) is uniformly Lipschitz stable. Therefore, there existconstants M > 0 and δ > 0 (M > 1, Mδ < ρ) such that 0 ≤ u0 < δ implies

η(t ; t0, u0) ≤ Mu0, t ≥ t0 (3.8)

for every t0 ∈ R+.We claim that ϕ0 ∈ C, |ϕ0|0 < δ implies |x(t; t0, ϕ0)| ≤ M|ϕ0|0, t ≥ t0 for every t0 ∈ R+. If the

claim is not true, there exist t0 ∈ R+, a corresponding solution x(t; t0, ϕ0) of (2.1) with |ϕ0|0 < δ,and t* > t0, t* < ∞, such that

|x(t∗)| > M |ϕ0|0 and |x(t ; t0, ϕ0)| ≤ M |ϕ0|0, t ∈ [t0, t∗).

Since

M |ϕ0|0 < Mδ < ρ,

and the function x(t) is continuous on [t0, t*], there exists t0, t0 < t0 ≤ t*, such that

M |ϕ0|0 < |x(t0)| < ρ, |x(t)| < ρ, t0 < t ≤ t0. (3.9)

Define for t ∈ [t0 − τ , t0] the notation m(t) = |x(t; t0, ϕ0)|. Using assumption (3.4), it is easyto obtain for t ∈ [t0, t0], the inequality

c Dqm(t) ≤ g(t, |mt |0).

Choosing |mt0 |0 = |ϕ0|0 = u0, we obtain by Lemma 2.1,

|x(t ; t0, ϕ0)| ≤ η(t ; t0, u0), t ∈ [t0, t0]. (3.10)

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-7 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

From (3.9), (3.10), and (3.8) we get

M |ϕ0|0 < |x(t0)| ≤ η(t0) ≤ Mu0 = M |ϕ0|0.The contradiction obtained shows that

|x(t ; t0, ϕ0)| ≤ M |ϕ0|0for |ϕ0|0 < δ and t ≥ t0. It, therefore, follows that the zero solution of system (2.1) is uniformlyLipschitz stable.

We can now formulate the basic comparison Lemma 2.1 in terms of Lyapunov functions.

Lemma 3.1: Assume that:

1. The function g : [t0,∞) × R+ → R+ is continuous in [t0,∞) × R+, and is nondecreasing inu for each t ≥ t0.

2. The maximal solution η(t) of the IVP (2.4), (2.5) exists on [t0, ∞).3. The function V ∈ C0 is such that for t ≥ t0, ϕ ∈ C the inequality

c Dq V (t, ϕ(0)) ≤ g(t, V (t, ϕ(0)))

is valid whenever V (t + s, ϕ(s)) ≤ V (t, ϕ(0)) for − τ ≤ s ≤ 0.

Then max−τ≤s≤0V (t0 + s, ϕ0(s)) ≤ u0 imply

V (t, x(t ; t0, ϕ0)) ≤ η(t ; t0, u0), t ∈ [t0,∞).

Proof: Let x(t) = x(t; t0, ϕ0) be the solution of IVP (2.1), (2.2) such that max−τ≤s≤0

V (t0 + s, ϕ0(s))

≤ u0. Introduce the notations m(t) = V (t, ϕ), t ≥ t0. From condition 3 of Lemma 3.1, and thedefinition of the fractional derivative of the function V ∈ C0, we get the differential inequalities

c Dqm(t) ≤ g(t, V (t, ϕ(0))) ≤ g(t, |mt |0), t ≥ t0,

whenever V (t + s, ϕ(s)) ≤ V (t, ϕ(0)) for − τ ≤ s ≤ 0.Since

max−τ≤s≤0

V (t0 + s, ϕ0(s)) = |mt0 |0 ≤ u0,

then by Lemma 2.1, we are led to the inequality

V (t, x(t ; t0, ϕ0)) ≤ η(t ; t0, u0) for t ∈ [t0,∞).

In the next, we shall apply Lyapunov-like functions from the class C0 and Razumikhin technique,and we shall obtain sufficient conditions for uniform Lipschitz stability (global uniform Lipschitzstability) of the zero solution of (2.1).

Introduce the following condition:A3. V (t, 0) = 0, t ≥ t0.

Theorem 3.4: Assume that:

1. Condition A3 and conditions of Lemma 3.1 hold.2. There exist positive constants A and B, such that

A|x | ≤ V (t, x) ≤ B|x |, (t, x) ∈ [t0,∞) × Sρ, (3.11)

where Sρ = {x ∈ Rn : |x | < ρ}.3. The zero solution of Eq. (2.4) is uniformly Lipschitz stable (globally uniformly Lipschitz

stable).

Then, the zero solution of system (2.1) is uniformly Lipschitz stable (globally uniformly Lips-chitz stable).

Proof: Let the zero solution of (2.4) is uniformly Lipschitz stable. Therefore, there exist constantsM > 0 and δ1 > 0 such that 0 ≤ u0 < δ1 implies (3.8) for every t0 ∈ R+.

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-8 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

Choose M1 > 1 and δ2 > 0 such that

M1 >B

A, M1 >

M B

A, M1δ2 < ρ. (3.12)

Let δ = min{ δ1B , δ1, δ2}, and ϕ0 ∈ C be such that inequality (3.7) holds. Then, using (3.11), we

get

|ϕ0|0 <B

Aδ < M1δ2 < ρ,

i.e., ϕ0 ∈ Cρ .We shall prove that if inequality (3.7) is satisfied, then

|x(t ; t0, ϕ0)| ≤ M |ϕ0|0, t ≥ t0, (3.13)

for every t0 ∈ R+, where x(t; t0, ϕ0) is the solution of IVP (2.1), (2.2) with the chosen above initialfunction ϕ0 ∈ Cρ .

Suppose that (3.13) is not true. Then, there exist a solution x(t; t0, ϕ0) of (2.1) for which |ϕ0|0< δ, and t* > t0, t* < ∞, such that

|x(t∗)| > M |ϕ0|0 and |x(t ; t0, ϕ0)| ≤ M |ϕ0|0, t ∈ [t0, t∗).

Then, due to the choice of δ, and the fact that the function x(t) is continuous on [t0, t*], we canfind t0, t0 < t0 ≤ t*, such that

M1|ϕ0|0 < |x(t0)| < ρ, |x(t)| < ρ, t0 < t ≤ t0. (3.14)

Set max−τ≤s≤0

V (t0 + s, ϕ0(s)) = u0. From the choice of δ, we have

V (t0 + s, ϕ0(s)) ≤ B|ϕ0|0 < Bδ ≤ δ1, s ∈ [−τ, 0],

i.e., u0 < δ1.Since for t ∈ [t0, t0] all the conditions of Lemma 3.1 are met, then

V (t, x(t ; t0, ϕ0)) ≤ η(t ; t0, u0), t ∈ [t0, t0]. (3.15)

From (3.14), (3.11), (3.15), and (3.12) there follow the inequalities

M1|ϕ0|0 < |x(t0)| ≤ 1

AV (t0, x(t0; t0, ϕ0)) ≤ 1

Aη(t0; t0, u0) ≤ M

Au0

= M

Amax

−τ≤s≤0V (t0 + s, ϕ0(s)) ≤ M B

A|ϕ0|0 < M1|ϕ0|0.

The obtained contradiction proves the validity of inequality (3.13) and the uniform Lipschitzstability of the zero solution of (2.1). The proof of the global uniform Lipschitz stability is analogous.

We obtain the following partial case of Theorem 3.4.

Theorem 3.5: Assume that conditions A1 and A2 hold, and the function g(t, u) is nondecreasingin u for each t ≥ t0. If in Theorem 3.3 condition (3.4) is replaced by the condition

|ϕ(0)| ≤ |ϕ(0) − hq f (t, ϕ)| + hq g(t, |ϕ(0)|) + ε(hq ), t ≥ t0, (3.16)

where h > 0 is sufficiently small, ε(hq )hq → 0 as h → 0+ , and ϕ ∈ Cρ is such that |ϕ(t + s)|

≤ |ϕ(t)|, s ∈ [−τ , 0], then the uniform Lipschitz stability (global uniform Lipschitz stability) of thezero solution of (2.4) imply the uniform Lipschitz stability (global uniform Lipschitz stability) ofthe zero solution of system (2.1).

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-9 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

Proof: Let V (t, x(t)) = |x(t)|. Note that V ∈ C0.According to (3.16), for ϕ ∈ Cρ such that |ϕ(t + s)| ≤ |ϕ(t)|, s ∈ [−τ , 0], we have

c Dq V (t, ϕ(0))

= limh→0+

sup1

hq[V (t, ϕ(0)) − V (t − h, ϕ(0) − hq f (t, ϕ))]

= limh→0+

sup1

hq[|ϕ(0)| − |ϕ(0) − hq f (t, ϕ)|]

≤ g(t, |ϕ(0)|) + limh→0+

supε(hq )

hq

= g(t, |ϕ(0)|) = g(t, V (t, ϕ(0))).

Later the proof of Theorem 3.5 is completed as the proof of Theorem 3.3.

Theorem 3.6: If in Theorem 3.5 condition (3.16) is replaced by the condition

[ϕ(0), f (t, ϕ)]+ ≤ g(t, |ϕ(0)|)

where ϕ ∈ Cρ is such that |ϕ(t + s)| ≤ |ϕ(t)|, s ∈ [−τ , 0], and [x, y]+ = limh→0+

sup1

hq[|x | − |x

− hq y|], x, y ∈ Rn , then the uniform Lipschitz stability (global uniform Lipschitz stability) of thezero solution of (2.4) imply the uniform Lipschitz stability (global uniform Lipschitz stability) ofthe zero solution of system (2.1).

The proof of Theorem 3.6 is analogous to the proof of Theorem 3.5.

Theorem 3.7: Assume that condition A1 holds, f ∈ C[[t0,∞) × Cρ,Rn] and for (t, ϕ) ∈ [t0,∞) × Cρ ,

| f (t, ϕ)| ≤ m(t)p(|ϕ(t)|),where p(u) is non-decreasing positive submultiplicative continuous function on (0, ∞), and

G−1(

G(1) + p(|ϕ0|0)

|ϕ0|01

�(q)

∫ t

θ

(t − s)q−1m(s) ds)

≤ M, 0 < M = const,

for all θ ≥ t0 ≥ 0, and G(u) =∫ u

u0

ds

p(s), u ≥ u0 > 0.

Then the zero solution of (2.1) is globally uniformly Lipschitz stable.

Proof: Since the function x(t) = x(t; t0, ϕ0) satisfies the integral equation (2.3), it follows that:

|x(t ; t0, ϕ0)| ≤ |ϕ0(0)| + 1

�(q)

∫ t

t0

(t − s)q−1| f (s, xs)| ds

≤ |ϕ0|0 + 1

�(q)

∫ t

t0

(t − s)q−1m(s)p(|x(s)|) ds,

from which, we obtain the estimates

|x(t ; t0, ϕ0)||ϕ0|0 ≤ 1 + 1

�(q)

∫ t

t0

(t − s)q−1 m(s)

|ϕ0|0 p(|ϕ0|0 |x(s; t0, ϕ0)|

|ϕ0|0)

ds

≤ 1 + 1

�(q)

∫ t

t0

(t − s)q−1 p(|ϕ0|0)

|ϕ0|0 m(s)p( |x(s; t0, ϕ0)|

|ϕ0|0)

ds, t ≥ t0.

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-10 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

To the last inequality, we apply a generalization of Bihari’s inequality (see cf. Ref. 21), and weare led to the inequality

|x(t ; t0, ϕ0)| ≤ |ϕ0|0G−1(

G(1) + p(|ϕ0|0)

|ϕ0|01

�(q)

∫ t

θ

(t − s)q−1m(s) ds).

Hence, |x(t; t0, ϕ0)| ≤ M|ϕ0|0 for all ϕ0 ∈ C, |ϕ0|0 < ∞, and t ≥ t0, and the conclusion of thetheorem follows.

IV. EXAMPLES

Example 4.1: Let x ∈ Rn and τ = const > 0. Consider the following fractional linear functionaldifferential system:

c Dq x(t) = Ax(t) + Bx(t − τ ), t ∈ R+, (4.1)

where A and B are constant matrices of type (n × n).Let μ(A + B) be the logarithmic norm of Lozinskii (see cf. Refs. 5 and 15)

μ(A + B) = limh→0+

|I + h(A + B)| − 1

h,

where I denotes the identity matrix.We have that

|Ax(t) + Bx(t − τ )| ≤ μ(A + B) maxs∈[t−τ,t]

|x(s)|.

Together with system (4.1) consider the equationc Dqu = μ(A + B)u, (4.2)

where u ∈ R+.If there exists a continuous function α(t) on [t0, ∞) such that

(i) μ(A + B) ≤ α(t) for t ≥ t0 ≥ 0, u0 ≤ δ;

(ii)∫ ∞

θ

α(s)ds < ∞ for θ ≥ t0 ≥ 0,

then the zero solution of Eq. (4.2) is uniformly stable (see cf. Ref. 6). Therefore, according toTheorem 3.1, the zero solution of (4.2) is globally uniformly Lipschitz stable, and by Theorem 3.4we conclude that the zero solution of (4.1) is globally uniformly Lipschitz stable.

Example 4.2: Consider the following fractional single–species model exhibiting the so-calledAllee effect in which the per-capita growth rate is a quadratic function of the density and subject todelays:

c Dq N (t) = N (t)[a + bN (t − τ (t)) − cN 2(t − τ (t))], t ≥ 0, (4.3)

where a, c ∈ (0,∞), b ∈ R, and 0 < τ (t) < τ .Let ϕ0 ∈ C[[−τ, 0],R+]. Denote by N(t) = N(t; 0, ϕ0) the solution of (4.3) satisfying the initial

conditions

N (s) = ϕ0(s) ≥ 0, s ∈ [−τ, 0); N (0) > 0.

Define the function V (t, N ) = |N |. Let there exist functions d ∈ C[R+,R] and k ∈ K,K = {k ∈ C[R+,R+] : k(r ) is strictly increasing and k(0) = 0}, such that

[N (0), N (t)(a + bN (t − τ (t)) − cN 2(t − τ (t)))]+ ≤ d(t)k(N (t)),

whenever |N(s)| ≤ |N(t)|, t − τ ≤ s ≤ t, t ≥ 0, and |N(t)| < ρ, ρ = const > 0.Hence, by Theorem 3.6, if the zero solution of the equation

c Dq N (t) = d(t)k(N (t)), t ≥ 0,

is uniformly Lipschitz stable, then the zero solution of (4.3) is uniformly Lipschitz stable.

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

043502-11 I. Stamova and G. Stamov J. Math. Phys. 54, 043502 (2013)

V. CONCLUSIONS

We have considered a fractional-order model in which the fractional rate of change depends onthe influence of their hereditary effects. The notion of Lipschitz stability, which becomes one of theimportant stability notions through the technique of utilizing the linear variational system aroundan arbitrary solution, is introduced for the model under consideration. By using Lyapunov-likefunctions, Razumikhin technique, and differential inequalities, Lipschitz stability, uniform Lipschitzstability, and global uniform Lipschitz stability criteria are obtained, extending the existing theory.Example are presented to illustrate our results as well as the utility of the used methods. The techniquecan be applied in the study of different types of fractional-order inverse and control problems inclassical physics, control theory, biological physics, chaos, fluid dynamics, etc.

1 V. Bacchelli and S. Vessella, “Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary,”Inverse Probl. 22, 1627–1658 (2006).

2 M. Bellassoued and M. Yamamoto, “Lipschitz stability in determining density and two Lame coefficients,” J. Math. Anal.Appl. 329, 1240–1259 (2007).

3 M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differentialequations with infinite delay,” J. Math. Anal. Appl. 338, 1340–1350 (2008).

4 A. Chauhan and J. Dabas, “Existence of mild solutions for impulsive fractional-order semilinear evolution equations withnonlocal conditions,” Electron. J. Differ. Equations 2011, 1–10 (2011).

5 F. M. Dannan and S. Elaydi, “Lipschitz stability of nonlinear systems of differential equations,” J. Math. Anal. Appl. 113,562–577 (1986).

6 A. M. A. El-Sayed, F. M. Gaafar, and E. M. A. Hamadalla, “Stability for a non-local non-autonomous system of fractionalorder differential equations with delays,” Electron. J. Differ. Equations 2010, 1–10 (2010).

7 H. I. Freedman and S. Ruan, “Uniform persistence in functional differential equations,” J. Differ. Equations 115, 173–192(1995).

8 B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrodinger equations,” J. Math.Phys. 53, 083702 (2012).

9 X. Guo and M. Xu, “Existence of the global smooth solution to the period boundary value problem of fractional nonlinearSchrodinger equation,” J. Math. Phys. 47, 082104 (2006).

10 J. K. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York, 1977).11 J. Henderson and A. Ouahab, “Fractional functional differential inclusions with finite delay,” Nonlinear Anal. 70, 2091–

2105 (2009).12 R. Hilfer, Application of Fractional Calculus in Physics (World Scientific, Singapore, 2000).13 O. Imanuvilov and M. Yamamoto, “Global Lipschitz stability in an inverse hyperbolic problem by interior observations,”

Inverse Probl. 17, 717–728 (2001).14 A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (North-Holland

Mathematics Studies, Elsevier, 2006), Vol. 204.15 G. K. Kulev and D. D. Bainov, “Lipschitz stability of impulsive systems of differential equations,” Int. J. Theor. Phys. 30,

737–756 (1991).16 V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Anal. Theory, Methods Appl. 69,

3337–3343 (2008).17 V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems (Cambridge Scientific

Publishers, 2009).18 N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135 (2000).19 N. Laskin, “Fractional Schrodinger equations,” Phys. Rev. E 66, 056108 (2002).20 R. Metzler and J. Klafter, “The restaurant at the end of the random walk: Recent developments in the description of

anomalous transport by fractional dynamics,” J. Phys. A 37, R161–R208 (2004).21 B. Pachpatte, “On generalizations of Bihari’s inequality,” Soochow J. Math. 31, 261–271 (2005).22 I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).23 B. S. Razumikhin, Stability of Hereditary Systems (Nauka, Moscow, 1988) (in Russian).24 G. T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations (Springer, Berlin, 2012).25 G. T. Stamov, J. Alzabut, P. Atanasov, and A. Stamov, “Almost periodic solutions for an impulsive delay model of price

fluctuations in commodity markets,” Nonlinear Anal.: Real World Appl. 12, 3170–3176 (2011).26 I. M. Stamova, Stability Analysis of Impulsive Functional Differential Equations (Walter de Gruyter, Berlin/New York,

2009).27 I. M. Stamova and A. Stamov, “Impulsive control on the asymptotic stability of the solutions of a Solow model with

endogenous labor growth,” J. Franklin Inst. 349, 2704–2716 (2012).28 I. M. Stamova, T. Stamov, and N. Simeonova, “Impulsive control on global exponential stability for cellular neural networks

with supremums,” J. Vib. Control 19, 483–490 (2013).29 J. R. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Anal.: Real World

Appl. 12, 262–272 (2011).

Downloaded 30 Sep 2013 to 147.188.128.74. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions