Research Article Stability, Boundedness, and Lagrange Stability of Fractional Differential Equations with Initial Time Difference

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Research Article

Stability, Boundedness, and Lagrange Stability of Fractional

Differential Equations with Initial Time Difference

Muhammed Çiçek, Co

G

kun Yakar, and Bülent O

L

ur

Department of Mathematics, Gebze Institute of Technology, Gebze, Kocaeli 141-41400, Turkey

Correspondence should be addressed to Muhammed C¸ ic¸ek; [email protected] Received 31 August 2013; Accepted 25 November 2013; Published 12 February 2014

Academic Editors: A. Atangana, A. M. A. El-Sayed, A. Kılıc¸man, S. C. O. Noutchie, and A. Secer

Copyright © 2014 Muhammed C¸ ic¸ek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Differential inequalities, comparison results, and sufficient conditions on initial time difference stability, boundedness, and Lagrange stability for fractional differential systems have been evaluated.

1. Introduction

The problem of stability of solutions is one of the major pro-blems in the theory of differential equations. Lyapunov fun-ction and the Lyapunov direct method allow us to obtain suf-ficient conditions for the stability of a system without

explic-itly solving the differential equations [1,2]. The method

gene-ralizes the idea which shows that the system is stable if there are some Lyapunov function candidates for the system.

Only a few decades ago, it was realized that fractional cal-culus provides an attractive tool for modelling the real world problems. The differentiation and integration of arbitrary orders have found applications in diverse fields of science and engineering like viscoelasticity, electrochemistry, diffu-sion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals [3–5]. Recently, some attention has been drawn on stability analysis of fractional differential equations (FDE) [6–9].

In practical situations, it is possible to have not only a change in initial position but also in initial time because of all kinds of disturbed factors. When we do consider such a deviation in initial time, it causes measuring the difference between any two different solutions starting with different initial times. From this point of view, several studies have been made on this problem to explore the stability and boun-dedness, criteria for differential systems relative to initial time difference (ITD) by using variation of parameters and differential inequalities technique [10–13]. In this paper, the stability and boundedness criteria for FDE relative to initial

time difference have been investigated by using comparison

method. InSection 2the differences between classical notion

of stability and the notion of initial time difference (ITD) stability have been discussed and compared by giving basic

definitions. In Section 3 new differential inequalities and

comparison results relative to initial time difference are obtained. Then, we investigate ITD stability, boundedness

and Lagrange stability by using results obtained inSection 3.

Lastly the conclusions are given inSection 4.

2. Fractional Calculus

Fractional calculus generalizes the derivative and the integral

of a function to a noninteger order [3,4,14]. Although there

are several definitions of fractional derivatives and fractional

integrals, only the relative ones are given below. Let 𝑓 :

[𝑎, 𝑏] → Rbe a function.

Definition 1. The fractional integral (or the

Riemann-Liou-ville (RL) type integral) of order𝑞 > 0is defined as

𝐼_𝑎𝑞𝑓 (𝑡) = _{Γ (𝑞)}1 ∫𝑡

𝑎(𝑡 − 𝑠)

𝑞−1_{𝑓 (𝑠) 𝑑𝑠.} ₍₁₎

Definition 2. The RL fractional derivatives of order𝑞 ∈ [𝑛 −

1, 𝑛)of𝑓(𝑡)are defined as 𝑎𝐷𝑞𝑡𝑓 (𝑡) = 𝑑 𝑛 𝑑𝑡𝑛 𝑎𝐼𝑡𝑛−𝑞𝑓 (𝑡) = 1 Γ (𝑛 − 𝑞) 𝑑𝑛 𝑑𝑡𝑛 ∫ 𝑡 𝑎(𝑡 − 𝑠) 𝑛−𝑞−1_{𝑓 (𝑠) 𝑑𝑠.} (2)

Volume 2014, Article ID 939027, 7 pages http://dx.doi.org/10.1155/2014/939027

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Definition 3. The Caputo fractional derivative of order𝑞 ∈ (𝑛 − 1, 𝑛]of𝑓(𝑡)is defined as 𝑐 𝑎𝐷𝑞𝑡𝑓 (𝑡) = 𝐼𝑎𝑛−𝑞[𝑑 𝑛 𝑑𝑡𝑛𝑓 (𝑡)] =_{Γ (𝑛 − 𝑞)}1 ∫𝑡 𝑎(𝑡 − 𝑠) 𝑛−𝑞−1_𝑓(𝑛)_{(𝑠) 𝑑𝑠.} (3)

To simplify the notations we will use𝐼𝑞,𝐷𝑞, and𝑐𝐷𝑞instead

of𝐼_𝑎𝑞, _𝑎𝐷𝑞_𝑡, and 𝑐_𝑎𝐷𝑞_𝑡, respectively. There are some relation

between these fractional integral and derivatives. For details

please see [3,4,14]. Property 1. (i)𝐼𝛼(𝐼𝛽𝑓(𝑡)) = 𝐼𝛼+𝛽𝑓(𝑡). (ii)L[𝐷𝑞𝑓(𝑡)] = 𝑠𝑞𝑋(𝑠) − ∑𝑛−1_𝑘=0𝑠𝑘[𝐷𝑞−𝑘−1𝑓(𝑡)]_𝑡=0, where 𝐷𝑞_represents 0𝐷𝑞𝑡. (iii)L[𝑐𝐷𝑞𝑓(𝑡)] = 𝑠𝑞𝑋(𝑠)−∑_𝑘=0𝑛−1𝑠𝑞−𝑘−1𝑓(𝑘)(0), where𝑐𝐷𝑞 represents𝑐₀𝐷𝑞_𝑡. (iv)𝑐𝐷𝑞𝐼𝑞𝑓(𝑡) = 𝑓(𝑡)and𝐷𝑞𝐼𝑞𝑓(𝑡) = 𝑓(𝑡). (v)𝐼𝑞𝐷𝑞𝑓(𝑡) = 𝑓(𝑡) − ∑𝑛−1_𝑘=1[𝐷𝑞−𝑘𝑓(𝑡)]_𝑡=𝑎((𝑡 − 𝑎)𝑞−𝑘/Γ(𝑘 − 𝑞 + 1))and 𝐼𝑞 𝑐𝐷𝑞𝑓(𝑡) = 𝑓(𝑡) − ∑𝑛−1_𝑘=0((𝑡 − 𝑎)𝑘/𝑘!) 𝑓(𝑘)(𝑎). (vi) 𝑐𝐷𝑞𝑓(𝑡) = 𝐷𝑞(𝑓(𝑡) − ∑𝑛−1_𝑘=0((𝑡 − 𝑎)𝑘/𝑘!)𝑓(𝑘)(𝑎)).

(vii)𝐷𝑞𝐶 = 𝐶(𝑡 − 𝑎)−𝑞/Γ(1 − 𝑞)and 𝑐𝐷𝑞𝐶 = 0, where𝐶is

arbitrary constant.

After giving definition and properties of fractional inte-gral and derivatives, we consider fractional order IVP with (RL) and Caputo derivative, respectively;

𝐷𝑞𝑥 (𝑡) = 𝑓 (𝑡, 𝑥) ,

𝑥 (𝑡₀) = 𝑥0= Γ (𝑞) 𝑥 (𝑡) (𝑡 − 𝑡0)1−𝑞|𝑡=𝑡0,

𝑐_𝐷𝑞_{𝑥 (𝑡) = 𝑓 (𝑡, 𝑥)}

𝑥 (𝑡₀) = 𝑥₀,

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where0 < 𝑞 < 1,𝑓 ∈ 𝐶[𝐽 ×R,R]and𝐽 = [𝑡₀, 𝑇]. Then IVP’s

are equivalent to the following Volterra fractional integral equations: 𝑥 (𝑡) = 𝑥0(𝑡 − 𝑡0) 𝑞−1 Γ (𝑞) + 1 Γ (𝑞)∫ 𝑡 𝑡0 (𝑡 − 𝑠)𝑞−1𝑓 (𝑠, 𝑥 (𝑠)) 𝑑𝑠, (5) 𝑥 (𝑡) = 𝑥0+_{Γ (𝑞)}1 𝑡 ∫𝑡 𝑡0 (𝑡 − 𝑠)𝑞−1𝑓 (𝑠, 𝑥 (𝑠)) 𝑑𝑠, (6) respectively. In [15] authors develop a relation between the solutions of Caputo fractional differential equations and those of (RL) fractional differential equations.

3. Stability versus Initial Time

Difference Stability

Consider the IVP for the system of nonlinear fractional differential equation

𝑐_𝐷𝑞_{𝑥 (𝑡) = 𝑓 (𝑡, 𝑥) ,}

𝑥 (𝑡₀) = 𝑥₀, (7)

where𝑡₀∈R₊,𝑓 ∈ 𝐶[R₊×R𝑛,R𝑛]and0 < 𝑞 < 1. Suppose

that the function𝑓is smooth enough to guarantee existence,

uniqueness, and continuous dependence of solutions of IVP

(7). Denote by𝑥(𝑡) = 𝑥(𝑡, 𝑡₀, 𝑥₀)the solution of (7). Consider

also the initial value problem at a different initial data; that

is, let𝜏₀ ∈ R₊ and𝑦(𝑡) = 𝑦(𝑡, 𝜏₀, 𝑦₀)be the solution of the

system (7). Assume that𝑥(𝑡) = 𝑥(𝑡, 𝑡₀, 𝑥₀)is the solution on

which we shall study stability and boundedness criteria with

respect to it. Set𝜂 = 𝜏₀− 𝑡₀> 0and denote𝑆(𝜌) = {𝑥 ∈R𝑛 :

‖𝑥‖ < 𝜌}.

Before giving our comparison theorem, stability, bound-edness criteria, and Lagrange stability for FDE we need to introduce the following definitions.

Definition 4. The solution𝑥(𝑡) = 𝑥(𝑡, 𝑡₀, 𝑥₀)of (7) is said to be

(S1) stable with ITD, if given𝜖 > 0and𝑡₀∈R₊; there exist

𝛿 = 𝛿(𝜖, 𝑡₀) > 0and ̃𝛿 = ̃𝛿(𝜖, 𝑡₀) > 0such that

󵄩󵄩󵄩󵄩𝑦0− 𝑥0󵄩󵄩󵄩󵄩 < 𝛿,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿imply󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂,𝜏₀, 𝑦₀) − 𝑥 (𝑡, 𝑡₀, 𝑥₀)󵄩󵄩󵄩󵄩 < 𝜖 for𝑡 ≥ 𝑡₀;

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(S2) uniformly stable with ITD, if(𝑆₁)holds with𝛿and ̃𝛿

independent of𝑡₀.

Definition 5. The system (7) is said to be

(B1) equibounded with ITD, if given𝛼 > 0and𝑡₀ ∈ R₊;

there exist̃𝛿 = ̃𝛿(𝛼, 𝑡₀) > 0and𝛽 = 𝛽(𝛼, 𝑡₀) > 0such

that

󵄩󵄩󵄩󵄩𝑦0− 𝑥0󵄩󵄩󵄩󵄩 ≤ 𝛼,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿imply 󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂,𝜏₀, 𝑦₀) − 𝑥 (𝑡, 𝑡₀, 𝑥₀)󵄩󵄩󵄩󵄩 < 𝛽 𝑡 ≥ 𝑡₀; (9)

(B2) uniformly bounded with ITD, if(𝐵₁)holds with̃𝛿and

𝛽independent of𝑡₀;

(A1) attractive in the large with ITD, if for every𝜖 > 0

and each𝛼 > 0there exists̃𝛿 = ̃𝛿(𝜖, 𝛼, 𝑡₀)and𝑇 =

𝑇(𝜖, 𝛼, 𝑡₀)such that

󵄩󵄩󵄩󵄩𝑦0− 𝑥0󵄩󵄩󵄩󵄩 ≤ 𝛼,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿imply󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂,𝜏₀, 𝑦₀) − 𝑥 (𝑡, 𝑡₀, 𝑥₀)󵄩󵄩󵄩󵄩 < 𝜖 for 𝑡 ≥ 𝑡₀+ 𝑇;

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(L1) Lagrange stable if (B1) and (A1) hold.

Definition 6. A function𝑎is said to belong to the classK such that

K= [𝑎 ∈ 𝐶 [R₊,R₊] : 𝑎 (0) = 0and𝑎is strictly increasing] . (11)

Definition 7. We define the generalized derivative with respect to the system (7) as follows:

𝑐_𝐷𝑞 +𝑉 (𝑡, ̃𝑦 − 𝑥) = lim ℎ → 0+sup 1 ℎ𝑞 [𝑉 (𝑡, ̃𝑦 − 𝑥) − 𝑉 (𝑡 − ℎ, ̃𝑦 − 𝑥 − ℎ𝑞 × (𝑓 (𝑡 + 𝜂, ̃𝑦) − 𝑓 (𝑡, 𝑥)))] (12)

for(𝑡, ̃𝑦 − 𝑥) ∈R₊×R𝑛, where𝑥 = 𝑥(𝑡) = 𝑥(𝑡, 𝑡₀, 𝑥₀)is the

solution of (7) and ̃𝑦 = ̃𝑦(𝑡, 𝑡₀, 𝑦₀) = 𝑦(𝑡 + 𝜂, 𝜏₀, 𝑦₀), where

𝑦(𝑡, 𝜏₀, 𝑦₀)is the solution of (7) and𝜂 = 𝜏₀− 𝑡₀> 0.

The stability with ITD gives us an opportunity to compare solutions of FDE where both initial time and position are different. In the case of differential equation, stability with

ITD is studied in [10–13, 16]. We will give a brief overview

of both concepts of stability.

Case 1(classical notion of stability). Let𝑥(𝑡) = 𝑥(𝑡, 𝑡₀, 𝑥₀)

be a solution of (7). Study the stability of𝑥(𝑡). Consider the

solution𝑌(𝑡) = 𝑌(𝑡, 𝑡₀, 𝑦₀)of (7). Set the IVP as

𝑐_𝐷𝑞_{𝑧 = ̃}_{𝑓 (𝑡, 𝑧) ,}

𝑧 (𝑡₀) = 𝑦₀− 𝑥₀, (13)

where 𝑓(𝑡, 𝑧) = 𝑓(𝑡, 𝑧 + 𝑥(𝑡)) − 𝑓(𝑡, 𝑥(𝑡))̃ . The function

𝑧(𝑡; 𝑡₀, 𝑦₀, 𝑥₀) = 𝑌(𝑡) − 𝑥(𝑡)is a solution of the IVP (13). The IVP (13) has a zero solution and the study of stability

properties of the nonzero solution𝑥(𝑡)of (7) is reduced to

the stability of the zero solution of transformed system (13).

Case 2 (stability with ITD). Study the stability with initial

time difference of 𝑥(𝑡). Consider the solution of (7) with

different initial data as𝑦(𝑡) = 𝑦(𝑡, 𝜏₀, 𝑦₀). Set the IVP as

𝑐_𝐷𝑞_{𝑧 = ̃}_{𝑓 (𝑡, 𝑧) ,}

𝑧 (𝑡₀) = 𝑦₀− 𝑥₀, (14)

where𝑓(𝑡, 𝑧; 𝜂) = 𝑓(𝑡 + 𝜂, 𝑧 + 𝑥(𝑡)) − 𝑓(𝑡, 𝑥(𝑡))̃ . Then𝑧(𝑡; 𝑡₀,

𝜏₀, 𝑦₀, 𝑥₀) = 𝑦(𝑡 + 𝜂) − 𝑥(𝑡)is a solution of (14). The system

(14) has no zero solution since 𝑓(𝑡, 0) = 𝑓(𝑡 + 𝜂, 𝑥(𝑡)) −̃

𝑓(𝑡, 𝑥(𝑡)) ̸= 0. Therefore, in this case the study of stability with

ITD of𝑥(𝑡)could not be reduced to the study of stability of the

zero solution of an appropriate fractional differential system.

4. Main Results

4.1. Differential Inequalities and Comparison Results 4.1.1. Differential Inequalities

Definition 8(see [15]). 𝑚is said to be𝐶𝑞continuous; that is,

𝑚 ∈ 𝐶𝑞_([𝑡

0, 𝑇],R), if and only if the Caputo derivative of

𝑐_𝐷𝑞_𝑚(𝑡)_{exists and satisfies}

𝑐_𝐷𝑞_{𝑚 (𝑡) =} 1

Γ (1 − 𝑞)∫

𝑡 𝑡0

(𝑡 − 𝑠)−𝑞𝑚󸀠_{(𝑠) 𝑑𝑠.} ₍₁₅₎

Lemma 9. Let𝑚 ∈ 𝐶𝑞([𝑡₀, 𝑇],R). Suppose that for any𝑡₁ ∈

(𝑡₀, 𝑇], one has𝑚(𝑡₁) = 0and𝑚(𝑡) < 0for𝑡₀ ≤ 𝑡 < 𝑡₁; then it follows that

𝑐_𝐷𝑞_{𝑚 (𝑡}

1) > 0. (16)

Proof. From the relation between Riemann-Liouville and Caputo fractional derivatives we write

𝑐_𝐷𝑞_{𝑚 (𝑡) = 𝐷}𝑞_{[𝑚 (𝑡) − 𝑚 (𝑡} 0)] = 𝐷𝑞𝑚 (𝑡) −𝑚 (𝑡0) (𝑡 − 𝑡0) −𝑞 Γ (1 − 𝑞) . (17)

Since𝑚(𝑡₀) < 0we have𝐷𝑞𝑚(𝑡₀) < 0. Therefore, we obtain

𝑐_𝐷𝑞_{𝑚 (𝑡) > 𝐷}𝑞_{𝑚 (𝑡) .} ₍₁₈₎

Finally using the lemma for R-L derivative from [6] and (18) implies that 𝑐_𝐷𝑞_{𝑚 (𝑡} 1) > 𝐷𝑞𝑚 (𝑡1) ≥ 0. (19) Theorem 10. LetV, 𝑤 ∈ 𝐶𝑞([𝑡₀, 𝑇],R),𝑓 ∈ 𝐶([𝑡₀, 𝑇] ×R,R) and (𝑖) 𝑐𝐷𝑞V(𝑡) ≤ 𝑓 (𝑡,V(𝑡)) , (𝑖𝑖) 𝑐𝐷𝑞𝑤 (𝑡) > 𝑓 (𝑡, 𝑤 (𝑡)) , 𝑡0≤ 𝑡 ≤ 𝑇. (20) ThenV(𝑡₀) < 𝑤(𝑡₀)implies V(𝑡) < 𝑤 (𝑡) , 𝑡₀≤ 𝑡 ≤ 𝑇. (21)

Proof. Suppose that relation (21) is false. Then sinceV(𝑡₀) <

𝑤(𝑡₀)andV(𝑡),𝑤(𝑡)are continuous, there exists a𝑡₁∈ (𝑡₀, 𝑇]

withV(𝑡₁) = 𝑤(𝑡₁)andV(𝑡) < 𝑤(𝑡)for𝑡₀ ≤ 𝑡 < 𝑡₁. Set𝑚(𝑡) = V(𝑡) − 𝑤(𝑡). Then𝑚(𝑡₁) = 0and𝑚(𝑡) < 0for𝑡₁ ∈ [𝑡₀, 𝑡₁).

Hence, the hypothesis ofLemma 9 holds and we conclude

that𝑐𝐷𝑞𝑚(𝑡₁) > 0, which means that𝑐𝐷𝑞V(𝑡₁) > 𝑐𝐷𝑞𝑤(𝑡₁). Consider

𝑓 (𝑡₁,V(𝑡₁)) ≥ 𝑐𝐷𝑞V(𝑡₁) > 𝑐𝐷𝑞𝑤 (𝑡₁) > 𝑓 (𝑡₁, 𝑤 (𝑡₁))

(22) which is a contradiction. Thus the conclusion of the theorem holds and the proof is complete.

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Theorem 11. LetV, 𝑤 ∈ 𝐶𝑞([𝑡₀, 𝑇],R),𝑓 ∈ 𝐶([𝑡₀, 𝑇] ×R,R) and

(𝑖) 𝑐𝐷𝑞V(𝑡) ≤ 𝑓 (𝑡,V(𝑡)) ,

(𝑖𝑖) 𝑐𝐷𝑞𝑤 (𝑡) ≥ 𝑓 (𝑡, 𝑤 (𝑡)) , 𝑡0≤ 𝑡 ≤ 𝑇.

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Assume𝑓satisfies the Lipschitz condition

𝑓 (𝑡, 𝑥) − 𝑓 (𝑡, 𝑦) ≤ 𝐿 (𝑥 − 𝑦) , 𝑥 ≥ 𝑦, 𝐿 > 0. (24)

Then,V(𝑡₀) ≤ 𝑤(𝑡₀)implies

V(𝑡) ≤ 𝑤 (𝑡) , 𝑡0≤ 𝑡 ≤ 𝑇. (25)

Proof. We set𝑤_𝜖(𝑡) = 𝑤(𝑡) + 𝜖𝜆(𝑡), where𝜆(𝑡) = 𝜆(𝑡₀)

𝐸_𝑞(2𝐿(𝑡 − 𝑡₀)𝑞₎_{is the solution of linear fractional differential}

equation 𝑐𝐷𝑞𝜆(𝑡) = 2𝐿𝜆(𝑡),𝜆(𝑡₀) = 𝜆₀. Then from𝑤_𝜖(𝑡₀) =

𝑤(𝑡₀) + 𝜖𝜆(𝑡₀), we get𝑤_𝜖(𝑡₀) > 𝑤(𝑡₀) ≥V(𝑡₀). In order to get

(25), we need to applyTheorem 10toV(𝑡)and𝑤_𝜖(𝑡). Using(𝑖𝑖)

and Lipschitz condition we have

𝑐_𝐷𝑞_𝑤 𝜖(𝑡) = 𝑐𝐷𝑞𝑤 (𝑡) + 𝑐𝐷𝑞𝜆 (𝑡) ≥ 𝑓 (𝑡, 𝑤 (𝑡)) − 𝑓 (𝑡, 𝑤𝜖(𝑡)) + 𝑓 (𝑡, 𝑤𝜖(𝑡)) + 𝜖2𝐿𝜆 (𝑡) , 𝑐_𝐷𝑞_𝑤 𝜖(𝑡) > 𝑓 (𝑡, 𝑤𝜖(𝑡)) . (26)

Applying nowTheorem 10toV(𝑡)and𝑤_𝜖(𝑡), we getV(𝑡) <

𝑤_𝜖(𝑡)for every𝜖 > 0and consequently making𝜖 → 0, we

get the desired estimate (25).

4.1.2. Comparison Results. The most commonly used tech-nique in the theory of differential equations is related to the estimation of a function satisfying a differential inequality by the extremal solutions of the related differential equation. The following theorems give such estimate with initial time difference.

Theorem 12. Assume that𝑚 ∈ 𝐶𝑞([𝑡₀, 𝑇],R)and

𝑐_𝐷𝑞_{𝑚 (𝑡) ≤ 𝑔 (𝑡, 𝑚 (𝑡)) , 𝑡}

0≤ 𝑡 ≤ 𝑇, (27)

where𝑔 ∈ 𝐶([𝑡₀, 𝑇] ×R,R). Let𝑟(𝑡)be the maximal solution of the IVP

𝑐_𝐷𝑞_{𝑢 = 𝑔 (𝑡, 𝑢) ,} _{𝑢 (𝑡}

0) = 𝑢0, (28)

existing on[𝑡₀, 𝑇]such that𝑚(𝑡₀) ≤ 𝑢₀. Then one has

𝑚 (𝑡) ≤ 𝑟 (𝑡) , 𝑡0≤ 𝑡 ≤ 𝑇. (29)

Proof. In view of the definition of the maximal solution𝑟(𝑡),

it is enough to prove that𝑚(𝑡) < 𝑢(𝑡, 𝜖),𝑡₀ ≤ 𝑡 ≤ 𝑇, where

𝑢(𝑡, 𝜖)is any solution of the IVP

𝑐_𝐷𝑞_{𝑢 = 𝑔 (𝑡, 𝑢) + 𝜖,} _{with the initial value}_{𝑢 (𝑡}

0) + 𝜖, 𝜖 > 0.

(30)

UsingLemma 9, we get𝑚(𝑡) < 𝑢(𝑡, 𝜖). And from lim_{𝜖 → 0}

𝑢(𝑡, 𝜖) = 𝑟(𝑡)uniformly on each compact set𝑡₀≤ 𝑡 ≤ 𝑇₀< 𝑇, we get the desired estimate (29).

Theorem 13. Assume that

(i)𝑚 ∈ 𝐶𝑞([𝑡₀, 𝑇],R₊),𝑔 ∈ 𝐶([𝑡₀, 𝑇] ×R₊,R)and

𝑐_𝐷𝑞_{𝑚 (𝑡) ≤ 𝑔 (𝑡, 𝑚 (𝑡)) , 𝑚 (𝑡}

0) ≤ 𝑤0; (31)

(ii)the maximal solution𝑟(𝑡) = 𝑟(𝑡, 𝜏₀, 𝑤₀)of the IVP

𝑐_𝐷𝑞_{𝑤 = 𝑔 (𝑡, 𝑤) , 𝑤 (𝜏}

0) = 𝑤0≥ 0 (32)

exists for𝑡 ≥ 𝜏₀;

(iii)𝑔(𝑡, 𝑤)is nondecreasing in𝑡for each𝑤and𝜏₀> 𝑡₀.

Then (a)𝑚(𝑡) ≤ 𝑟(𝑡 + 𝜂),𝑡 ≥ 𝑡₀and (b)𝑚(𝑡 − 𝜂) ≤ 𝑟(𝑡),𝑡 ≥ 𝜏₀. Proof. (a) It is well known that if𝑤(𝑡, 𝜖)is any solution of

𝑐_𝐷𝑞_{𝑤 = 𝑔 (𝑡, 𝑤) + 𝜖, 𝑤 (𝜏}

0) = 𝑤0+ 𝜖 for𝜖 > 0 (33)

sufficiently small then lim_{𝜖 → 0}𝑤(𝑡, 𝜖) = 𝑟(𝑡, 𝜏₀, 𝑤₀)on every

compact set[𝜏₀, 𝜏₀+ 𝑇].

Setting𝑤₀(𝑡, 𝜖) = 𝑤(𝑡 + 𝜂, 𝜖), we have𝑤₀(𝑡₀, 𝜖) = 𝑤(𝑡₀+

𝜂, 𝜖) = 𝑤(𝜏₀, 𝜖) = 𝑤₀+ 𝜖 > 𝑤₀ ≥ 𝑚(𝑡₀). Thus we get𝑚(𝑡₀) <

𝑤₀(𝑡₀, 𝜖). On the other hand, using (iii)

𝑐_𝐷𝑞_𝑤

0(𝑡, 𝜖) = 𝑔 (𝑡 + 𝜂, 𝑤0(𝑡, 𝜖)) + 𝜖

> 𝑔 (𝑡 + 𝜂, 𝑤0(𝑡, 𝜖)) > 𝑔 (𝑡, 𝑤0(𝑡, 𝜖)) , 𝑡 ≥ 𝑡0.

(34)

Then we get𝑚(𝑡) < 𝑤₀(𝑡, 𝜖),𝑡 ≥ 𝑡₀and hence it follows that

𝑚(𝑡) ≤ 𝑟(𝑡 + 𝜂),𝑡 ≥ 𝑡₀. (b) We set𝑚₀(𝑡) = 𝑚(𝑡 − 𝜂)so that𝑚₀(𝜏₀) = 𝑚(𝑡₀) ≤ 𝑤₀< 𝑤₀+ 𝜖and 𝑐_𝐷𝑞_𝑚 0(𝑡) ≤ 𝑔 (𝑡 − 𝜂, 𝑚0(𝑡)) < 𝑔 (𝑡, 𝑚0(𝑡)) , 𝑡 ≥ 𝜏0. (35)

Then we get𝑚₀(𝑡) < 𝑤(𝑡, 𝜖),𝑡 ≥ 𝜏₀. The conclusion follows

taking the limit as𝜖 → 0. The proof of theorem is complete.

In the following theorem, we obtain a comparison result in terms of Lyapunov-like functions with ITD.

(i)𝑉 ∈ 𝐶[R₊×R𝑛,R₊],𝑉(𝑡, 𝑥)is locally Lipschitzian in

𝑥 ∈R𝑛_,_{𝑔 ∈ 𝐶[}_R2 +,R]and 𝑐_𝐷𝑞

+𝑉 (𝑡, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡)) ≤ 𝑔 (𝑡, 𝑉 (𝑡, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡))) ;

(36) (ii)the maximal solution𝑟(𝑡) = 𝑟(𝑡, 𝜏₀, 𝑢₀)of the fractional

scalar differential equation

𝑐_𝐷𝑞_{𝑢 = 𝑔 (𝑡, 𝑢) ,} _{𝑢 (𝜏}

0) = 𝑢0≥ 0, (37)

exists for𝑡 ≥ 𝜏₀;

(5)

Then𝑉(𝑡₀, 𝑦₀− 𝑥₀) ≤ 𝑢₀implies

𝑉 (𝑡, 𝑦 (𝑡 + 𝜂, 𝜏₀, 𝑦₀) − 𝑥 (𝑡, 𝑡₀, 𝑥₀)) ≤ 𝑟 (𝑡 + 𝜂, 𝜏₀, 𝑢₀) 𝑓𝑜𝑟 𝑡 ≥ 𝑡₀. (38)

Proof. Define𝑚(𝑡) = 𝑉(𝑡, 𝑦(𝑡 + 𝜂, 𝜏₀, 𝑦₀) − 𝑥(𝑡, 𝑡₀, 𝑥₀))so that

𝑚 (𝑡0) = 𝑉 (𝑡0, 𝑦0− 𝑥0) ≤ 𝑢0. (39) Let𝑧(𝑡, 𝑡₀, 𝑦₀− 𝑥₀) = 𝑦(𝑡 + 𝜂, 𝜏₀, 𝑦₀) − 𝑥(𝑡, 𝑡₀, 𝑥₀)so that 𝑐_𝐷𝑞_{𝑧 = ̃}_{𝑓 (𝑡, 𝑧) = [𝑓 (𝑡 + 𝜂, 𝑦 (𝑡 + 𝜂)) − 𝑓 (𝑡, 𝑥 (𝑡))] ,} 𝑧 (𝑡₀) = 𝑦₀− 𝑥₀, 𝑚 (𝑡) − 𝑚 (𝑡 − ℎ) ℎ𝑞 = 𝑉 (𝑡, 𝑧 (𝑡)) − 𝑉 (𝑡 − ℎ, 𝑧 (𝑡) − ℎ 𝑞_{[ ̃}_{𝑓 (𝑡, 𝑧)])} ℎ𝑞 +𝑉 (𝑡 − ℎ, 𝑧 (𝑡) − ℎ 𝑞_{[ ̃}_{𝑓 (𝑡, 𝑧)]) − 𝑉 (𝑡 − ℎ, 𝑆 (𝑧, ℎ, 𝑞))} ℎ𝑞 , (40)

where𝑆(𝑧, ℎ, 𝑞) = 𝑧(𝑡) − ℎ𝑞( ̃𝑓(𝑡, 𝑧)) − 𝜖(ℎ𝑞). Since𝑉is locally

Lipschitzian in𝑥 and 𝐿 > 0is the Lipschitz constant and

𝜖(ℎ𝑞)/ℎ𝑞 → 0asℎ → 0, we have 𝑐_𝐷𝑞 +𝑚 (𝑡) ≤ lim ℎ → 0+𝐿󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 𝜖 (ℎ𝑞₎ ℎ𝑞 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩 +lim sup ℎ → 0+ ( 𝑉 (𝑡, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡)) ℎ𝑞 −𝑉 (𝑡 − ℎ, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡)) ℎ𝑞 −ℎ𝑞[𝑓 (𝑡 + 𝜂, 𝑦 (𝑡 + 𝜂)) − 𝑓 (𝑡, 𝑥 (𝑡))] ℎ𝑞 ) ≤ 𝑔 (𝑡, 𝑚 (𝑡)) . (41)

By usingTheorem 13we obtain that

𝑚 (𝑡) = 𝑉 (𝑡, 𝑦 (𝑡 + 𝜂, 𝜏0, 𝑦0) − 𝑥 (𝑡, 𝑡0, 𝑥0)) ≤ 𝑟 (𝑡 + 𝜂, 𝜏0, 𝑢0) ,

for𝑡 ≥ 𝑡₀. (42)

4.2. Stability and Boundedness Criteria. A comparison prin-ciple is obtained by employing the notion of Lyapunov function together with the theory of differential inequalities. In this part, one can see Lyapunov-like function as a trans-formation which reduces the study of stability, boundedness, and Lagrange stability properties relative to ITD of a given complicated system to a relatively simpler scalar equation.

4.2.1. Stability Criteria

(i)𝑉 ∈ 𝐶[R₊×R𝑛,R₊],𝑉(𝑡, 𝑥)is locally Lipschitzian in

𝑥 ∈R𝑛,𝑔 ∈ 𝐶[R2₊,R]and

𝑐_𝐷𝑞

+𝑉 (𝑡, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡)) ≤ 𝑔 (𝑡, 𝑉 (𝑡, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡)))

(43) (ii)the maximal solution𝑟(𝑡) = 𝑟(𝑡, 𝜏₀, 𝑢₀)of (37)exists

for𝑡 ≥ 𝜏₀;

(iii)there exists𝑎, 𝑏 ∈ Ksuch that𝑏(‖𝑥‖) ≤ 𝑉(𝑡, 𝑥) ≤

𝑎(‖𝑥‖)for(𝑡, 𝑥) ∈R₊× 𝑆(𝜌);

(iv)𝑔(𝑡, 𝑢)is nondecreasing in𝑡for each𝑢and𝑔(𝑡, 0) = 0; Then the stability properties of the null solution of (37)imply the corresponding initial time difference stability properties of the solution𝑥(𝑡, 𝑡₀, 𝑥₀).

Proof. Assume that the null solution of (37) is equistable. Let

0 < 𝜖 < 𝜌be given. Then by definition of equistability given

𝑏(𝜖) > 0,𝜏₀∈R₊, there exist a𝛿₁= 𝛿₁(𝜖, 𝜏₀)such that

𝑢 (𝑡) < 𝑏 (𝜖)provided that𝑢₀< 𝛿₁, 𝑡 ≥ 𝜏₀, (44)

where 𝑢(𝑡, 𝜏₀, 𝑢₀)is any solution of the (37). Choose 𝛿₂ =

𝛿₂(𝜖, 𝜏₀) > 0as0 < 𝑎(𝛿₂) < 𝛿₁. Obviously lim_(𝜏₀_,𝑦₀_{) → (𝑡}₀_,𝑥₀₎

‖𝑦(𝑡 + 𝜂, 𝜏₀, 𝑦₀) − 𝑥(𝑡, 𝑡₀, 𝑥₀)‖= 0. Then given 𝜖 > 0and

𝜏₀∈R₊,̃𝛿 = ̃𝛿(𝜖, 𝜏₀) > 0and𝛿₃= 𝛿₃(𝜖, 𝜏₀) > 0such that

󵄩󵄩󵄩󵄩𝑦0− 𝑥0󵄩󵄩󵄩󵄩 < 𝛿3,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿implies󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂) − 𝑥(𝑡)󵄩󵄩󵄩󵄩 < 𝜖 for𝑡₀≤ 𝑡 ≤ 𝜏₀ (45) Let𝑢₀ = 𝑎(‖𝑦₀− 𝑥₀‖)and choose𝛿 = min(𝛿₂, 𝛿₃). Then we claim that

󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂) − 𝑥(𝑡)󵄩󵄩󵄩󵄩 < 𝜖provided that󵄩󵄩󵄩󵄩𝑦₀− 𝑥₀󵄩󵄩󵄩󵄩 < 𝛿,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿 for𝑡 ≥ 𝑡₀.

(46)

If it is not true, from (45), there exist a𝑡₁> 𝜏₀and a solution

𝑦(𝑡, 𝜏₀, 𝑦₀)with‖𝑦₀− 𝑥₀‖ < 𝛿and|𝜂| < ̃𝛿such that

󵄩󵄩󵄩󵄩𝑦(𝑡1+ 𝜂) − 𝑥 (𝑡1)󵄩󵄩󵄩󵄩 ≥ 𝜖,

󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂) − 𝑥(𝑡)󵄩󵄩󵄩󵄩 < 𝜖, for 𝑡₀≤ 𝑡 < 𝑡₁. (47)

Moreover since‖𝑦₀− 𝑥₀‖ < 𝛿, by (iii) we have

𝑉 (𝑡₀, 𝑦₀− 𝑥₀_{) ≤ 𝑎 (󵄩󵄩󵄩󵄩𝑦}₀− 𝑥₀󵄩󵄩󵄩󵄩) < 𝑎(𝛿) < 𝛿₁. (48)

Hence, by (i), (ii), (47) and, Theorem 14, we obtain the

following estimate:

𝑉 (𝑡, 𝑦 (𝑡 + 𝜂) − 𝑥 (𝑡)) ≤ 𝑟 (𝑡 + 𝜂, 𝜏0, 𝑢0) , 𝑡0≤ 𝑡 < 𝑡1.

(49) Consequently, the relations (44), (47), (49), and (iii) lead to the contradiction

𝑏 (𝜖) ≤ 𝑏 (󵄩󵄩󵄩󵄩𝑦 (𝑡1+ 𝜂) − 𝑥 (𝑡1)󵄩󵄩󵄩󵄩)

≤ 𝑉 (𝑡₁, 𝑦 (𝑡₁+ 𝜂) − 𝑥 (𝑡₁)) ≤ 𝑟 (𝑡₁+ 𝜂, 𝜏₀, 𝑢₀) < 𝑏 (𝜖) .

(50)

which proves that the solution𝑥(𝑡, 𝑡₀, 𝑥₀)of (7) is equistable

(6)

4.2.2. Boundedness Criteria. In this section, Lagrange stabil-ity, which includes boundedness criteria, of fractional dyna-mic systems are discussed by employing comparison method.

Theorem 16. Let the assumption ofTheorem 15hold. Then the

Lagrange stability properties of (37)imply the corresponding initial time difference Lagrange stability properties of (7). Proof. We need to prove (B1) and (S3) for (7). Let𝛼 ≥ 0be

given, and let‖𝑦₀− 𝑥₀‖ ≤ 𝛼. In view of (iii)𝑉(𝑡₀, 𝑦₀− 𝑥₀) ≤

𝑎(𝛼) = 𝛼1. Assume that (37) is Lagrange stable. It follows that

(37) is bounded. Then, given𝛼₁≥ 0and𝜏₀ ∈R₊there exists

a𝛽₁= 𝛽₁(𝜏₀, 𝛼)such that

𝑢0≤ 𝛼1 imply𝑟 (𝑡, 𝜏0, 𝑢0) < 𝛽1. (51)

Moreover,𝑏(𝑢) → ∞as𝑢 → ∞; we can choose a𝛽 =

𝛽(𝜏₀, 𝛼)verifying the relation

𝑏 (𝛽) ≥ 𝛽₁. (52)

Then from (45), given𝛽 > 0and𝜏₀ ∈ R₊, ̃𝛿 = ̃𝛿(𝜖, 𝜏₀) > 0

and𝛿₁= 𝛿₁(𝜖, 𝜏₀) > 0such that

󵄩󵄩󵄩󵄩𝑦0− 𝑥0󵄩󵄩󵄩󵄩 < 𝛿1,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿implies󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂) − 𝑥(𝑡)󵄩󵄩󵄩󵄩 < 𝛽 for𝑡₀≤ 𝑡 ≤ 𝜏₀. (53) Let𝑢₀= 𝑉(𝜏₀, 𝑦₀− 𝑥₀). Then we claim that

󵄩󵄩󵄩󵄩𝑦(𝑡 + 𝜂) − 𝑥(𝑡)󵄩󵄩󵄩󵄩 < 𝛽provided that󵄩󵄩󵄩󵄩𝑦₀− 𝑥₀󵄩󵄩󵄩󵄩 < 𝛼,

󵄨󵄨󵄨󵄨𝜂󵄨󵄨󵄨󵄨 < ̃𝛿 for𝑡 ≥ 𝑡₀.

(54)

If this were false, there would exist a𝑡∗ > 𝜏₀and a solution

𝑦(𝑡, 𝜏₀, 𝑦₀) of (7) such that ‖̃𝑥(𝑡∗ + 𝜂) − 𝑥(𝑡∗)‖= 𝛽. Consequently, the relations (49), (51), (52), and (iii) lead to the contradiction

𝑏 (𝛽) ≤ 𝑏 (󵄩󵄩󵄩󵄩𝑦 (𝑡∗+ 𝜂) − 𝑥 (𝑡∗_{)󵄩󵄩󵄩󵄩) ≤ 𝑉 (𝑡}∗, 𝑦 (𝑡∗+ 𝜂) − 𝑥 (𝑡∗)) ≤ 𝑟 (𝑡∗+ 𝜂, 𝜏₀, 𝑢₀) < 𝛽₁≤ 𝑏 (𝛽)

(55) which proves that (B1) holds for (7).

To prove attractivity, we let𝜖 > 0,𝛼 ≥ 0,𝜏₀∈R₊be given

and‖𝑦₀− 𝑥₀‖ ≤ 𝛼. In view of (iii),𝑉(𝑡₀, 𝑦₀− 𝑥₀) ≤ 𝑎(𝛼) = 𝛼₁.

Since (37) is attractive in the large with ITD, given𝛼₁ ≥ 0,

𝑏(𝜖)and𝜏₀∈R₊there exist a𝑇 = 𝑇(𝜏₀, 𝛼, 𝜖)such that

𝑢₀≤ 𝛼₁ implies𝑢 (𝑡, 𝜏₀, 𝑢₀) < 𝑏 (𝜖) , for𝑡 ≥ 𝜏₀+ 𝑇. (56)

We have (49) from the proof of Theorem (45). Now, suppose

that there exists a sequence{𝑡_𝑘} ∈R₊,𝑡_𝑘 → ∞as𝑘 → ∞,

𝑡_𝑘> 𝜏₀+ 𝑇and a solution𝑦(𝑡, 𝜏₀, 𝑦₀)of (7) such that

󵄩󵄩󵄩󵄩𝑦(𝑡𝑘+ 𝜂) − 𝑥 (𝑡𝑘)󵄩󵄩󵄩󵄩 ≥ 𝜖. (57)

The relations (iii), (49), (56), and (57) lead to the contradic-tion,

𝑏 (𝜖) ≤ 𝑏 (󵄩󵄩󵄩󵄩𝑦 (𝑡𝑘+ 𝜂) − 𝑥 (𝑡𝑘)󵄩󵄩󵄩󵄩) ≤ 𝑉 (𝑡𝑘, 𝑦 (𝑡𝑘+ 𝜂) − 𝑥 (𝑡𝑘))

≤ 𝑟 (𝑡_𝑘+ 𝜂, 𝜏₀, 𝑢₀) < 𝑏 (𝜖)

(58)

which proves that the system is attractive in the large with ITD.

5. Conclusion

Firstly, differential inequalities and a new comparison prin-ciple for fractional differential equations relative to initial time difference have been developed and then stability, boun-dedness criteria, and Lagrange stability relative to initial time difference have been proved by employing comparison method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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