The Scientific World Journal

Volume 2014 (2014), Article ID 939027, 7 pages

http://dx.doi.org/10.1155/2014/939027

Research Article

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

Muhammed Çiçek, Coşkun Yakar, and Bülent Oğur

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

Received 31 August 2013; Accepted 25 November 2013; Published 12 February 2014

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

Copyright © 2014 Muhammed Çiçek 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.

Abstract.

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 problems in the theory of differential equations. Lyapunov function and the Lyapunov direct method allow us to obtain sufficient conditions for the stability of a system without explicitly solving the differential equations [1, 2]. The method generalizes 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 calculus 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, diffusion 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 boundedness, 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. In Section 2 the 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 in Section 3. Lastly the conclusions are given in Section 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 be a function.

Definition 1. The fractional integral (or the Riemann-Liouville (RL) type integral) of order is defined as

Definition 2. The RL fractional derivatives of order of are defined as

Definition 3. The Caputo fractional derivative of order of is defined as

To simplify the notations we will use , , and instead of , , and , respectively. There are some relation between these fractional integral and derivatives. For details please see [3, 4, 14].

Property 1.

(i).(ii), where represents .(iii), where represents .(iv) and .(v) and .(vi).(vii) and , where is arbitrary constant.

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

where , and . Then IVP's are equivalent to the following Volterra fractional integral equations:

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

where , and . Suppose that the function is smooth enough to guarantee existence, uniqueness, and continuous dependence of solutions of IVP (7). Denote by the solution of (7). Consider also the initial value problem at a different initial data; that is, let and be the solution of the system (7). Assume that is the solution on which we shall study stability and boundedness criteria with respect to it. Set and denote .

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

Definition 4. The solution of (7) is said to be

(S1)stable with ITD, if given and ; there exist and such that

(S2)uniformly stable with ITD, if holds with and independent of .

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

(B1)equibounded with ITD, if given and ; there exist and such that

(B2)uniformly bounded with ITD, if holds with and independent of ;(A1)attractive in the large with ITD, if for every and each there exists and such that

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

Definition 6. A function is said to belong to the class such that

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

for , where is the solution of (7) and , where is the solution of (7) and .

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). Let be a solution of (7). Study the stability of . Consider the solution of (7). Set the IVP as

where . The function is a solution of the IVP (13).

The IVP (13) has a zero solution and the study of stability properties of the nonzero solution 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 . Consider the solution of (7) with different initial data as . Set the IVP as

where . Then is a solution of (14). The system (14) has no zero solution since . Therefore, in this case the study of stability with ITD of 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]). is said to be continuous; that is, , if and only if the Caputo derivative of exists and satisfies

Lemma 9. Let . Suppose that for any , one has and for ; then it follows that

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

Since we have . Therefore, we obtain

Finally using the lemma for R-L derivative from [6] and (18) implies that

Theorem 10. Let , and

Then implies

Proof. Suppose that relation (21) is false. Then since and , are continuous, there exists a with and for . Set . Then and for . Hence, the hypothesis of Lemma 9 holds and we conclude that , which means that . Consider

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

Theorem 11. Let , and

Assume satisfies the Lipschitz condition

Then, implies

Proof. We set , where is the solution of linear fractional differential equation , . Then from , we get . In order to get (25), we need to apply Theorem 10 to and . Using and Lipschitz condition we have

Applying now Theorem 10 to and , we get for every and consequently making , we get the desired estimate (25).

4.1.2. Comparison Results

The most commonly used technique 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 that and

where . Let be the maximal solution of the IVP

existing on such that . Then one has

Proof. In view of the definition of the maximal solution , it is enough to prove that , , where is any solution of the IVP

Using Lemma 9, we get . And from uniformly on each compact set , we get the desired estimate (29).

Theorem 13. Assume that

(i), and

(ii)the maximal solution of the IVP

exists for ;(iii) is nondecreasing in for each and .

Then (a) , and (b) , .

Proof. (a) It is well known that if is any solution of

sufficiently small then on every compact set .

Setting , we have . Thus we get . On the other hand, using (iii)

Then we get ,   and hence it follows that ,  .

(b) We set so that and

Then we get , . The conclusion follows taking the limit as . The proof of theorem is complete.

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

Theorem 14. Assume that

(i), is locally Lipschitzian in , and

(ii)the maximal solution of the fractional scalar differential equation

exists for ;(iii) is nondecreasing in for each .

Then implies

Proof. Define so that

Let so that

where . Since is locally Lipschitzian in and is the Lipschitz constant and as , we have

By using Theorem 13 we obtain that

4.2. Stability and Boundedness Criteria

A comparison principle 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 transformation 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

Theorem 15. Assume that

(i), is locally Lipschitzian in , and

(ii)the maximal solution of (37) exists for ;(iii)there exists such that for ;(iv) is nondecreasing in for each and ;

Then the stability properties of the null solution of (37) imply the corresponding initial time difference stability properties of the solution .

Proof. Assume that the null solution of (37) is equistable. Let be given. Then by definition of equistability given , , there exist a such that

where is any solution of the (37). Choose as . Obviously . Then given and , and such that

Let and choose . Then we claim that

If it is not true, from (45), there exist a and a solution with and such that

Moreover since , by (iii) we have

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

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

which proves that the solution of (7) is equistable with ITD.

4.2.2. Boundedness Criteria

In this section, Lagrange stability, which includes boundedness criteria, of fractional dynamic systems are discussed by employing comparison method.

Theorem 16. Let the assumption of Theorem 15 hold. 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). Let be given, and let . In view of (iii) . Assume that (37) is Lagrange stable. It follows that (37) is bounded. Then, given and there exists a such that

Moreover, as ; we can choose a verifying the relation

Then from (45), given and , and such that

Let . Then we claim that

If this were false, there would exist a and a solution of (7) such that . Consequently, the relations (49), (51), (52), and (iii) lead to the contradiction

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

To prove attractivity, we let , , be given and . In view of (iii), . Since (37) is attractive in the large with ITD, given , and there exist a such that

We have (49) from the proof of Theorem (45). Now, suppose that there exists a sequence , as , and a solution of (7) such that

The relations (iii), (49), (56), and (57) lead to the contradiction,

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

5. Conclusion

Firstly, differential inequalities and a new comparison principle for fractional differential equations relative to initial time difference have been developed and then stability, boundedness 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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