Abstract and Applied Analysis

Abstract and Applied Analysis / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 762857 | 16 pages | https://doi.org/10.1155/2010/762857

Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

Academic Editor: Dumitru Bǎleanu
Received16 Sep 2009
Revised29 Nov 2009
Accepted07 Jan 2010
Published16 Feb 2010

Abstract

The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.

1. Introduction

We have investigated that the stability of perturbed solution with respect to unperturbed solution with initial time difference of the nonlinear differential equations of fractional-order. The differential operators are taken in the Riemann-Liouville and Caputo’s sense and the initial conditions are specified according to Caputo’s suggestion [1], thus allowing for interpretation in a physically meaningful way [2].

Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of differential systems [36] along with a comparison result [2, 7] that allows the prediction of behavior of a differential system when the behavior of the null solution of a comparison system is known. However, there has been difficulty with this approach when trying to apply it to unperturbed fractional differential systems [2, 6, 8] and associated perturbed fractional differential systems with an initial time difference. The difficulty arises because there is a significant difference between initial time difference (ITD) stability [7, 916] and the classical notion of stability for fractional differential systems [2, 6]. The classical notions of stability [26, 8, 17] are with respect to the null solution, but ITD stability [7, 916] is with respect to the unperturbed fractional-order differential system where the perturbed fractional-order differential system and the unperturbed fractional-order differential system differ both in initial position and initial time [7, 916].

In this paper, we have dissipated this complexity and have a new comparison result which again gives the null solution a central role in the comparison fractional-order differential system. This result creates many paths for continuing research by direct application and generalization [13, 15, 16].

In Section 2, we present basic definitions and necessary rudimentary material. In Section 3, we discuss and compare the differences between the classical notion of stability and the recent notion of initial time difference (ITD) stability by means of fractional-order differential systems. In Section 4, we have a comparison result in which the stability properties of the null solution of the comparison system imply the corresponding stability properties of the unperturbed fractional-order differential system. In Section 5, we have an other comparison result in which the stability properties of the null solution of the comparison system imply the corresponding (ITD) stability properties of the perturbed fractional-order differential system with respect to the unperturbed fractional-order differential system.

2. Preliminaries

In this section we give relation among the fractional-order derivatives: Caputo, Reimann-Liouville and Grünwald-Letnikov fractional-order derivatives and necessary definition of initial value problems of fractional-order differential equations with these sense.

2.1. Fractional-Order Derivatives: Caputo, Reimann-Liouville and Grünwald-Letnikov

Caputo’s and Reimann-Liouville’s definitions of fractional derivatives, are namely,

respectively, order of and where denotes the Gamma function.

The most important advantage for fractional-order differential equations with Caputo’s derivative is the initial conditions that are the same form as that of ordinary differential equations with integer derivatives. Another difference is that the Caputo derivative for a constant is zero, while the Riemann-Liouville fractional derivative for a constant is not zero but equals to By using (2.1), therefore,

In particular, if we obtain

Hence, we can see that Caputo’s derivative is defined for functions for which Riemann-Liouville fractional-order derivative exists.

Let us write that Grünwald-Letnikov's notion of fractional-order derivative in a convenient form

where If we know that is continuous and exist and integrable, then Riemann-Liouville and Grünwald-Letnikov fractional-order derivatives are connected by the relation

Using (2.3)implies that we have the following relations among the Caputo, Riemann-Liouville and Grünwald-Letnikov fractional derivatives

The foregoing equivalent expressions are very useful in the study of qualitative properties of solutions of fractional differential equations.

2.2. Existence of Euler Solution

We consider the initial value problem of the fractional-order differential equation with Reimann-Liouville's derivative

where is any function from . Let

be a partition of .

Consider the interval and observe that the right hand side of the initial value problem of fractional-order differential equation with Reimann-Liouville’s derivative

on is constant.

Therefore, the initial value problem has a unique solution of (2.10) of the fractional-order differential equation with Reimann-Liouville’s derivative given by

Define the node and iterate next by considering on the initial value problem

The next node is and we proceed this way till an arc has been defined on all Let us employ the notation to emphasize the role played by the particular partition in determining which is the Euler curved arc corresponding to the partition The diameter of the partition is given by

Definition 2.1. An Euler solution is any curved arc which is the uniform limit of Euler curved arcs corresponding to some sequence such that , which means the convergence of the diameter as

Now, we can give the following result on existence of an Euler solution of the initial value problem of fractional-order differential equation with Reimann-Liouville’s derivative for (2.8).

Theorem 2.2. Assume that
(i) where is nondecreasing in (ii)The maximal solution of the fractional-order scalar differential equation with Reimann-Liouville's derivative exists on
Then   (a)there exists at least one Euler solution to the initial value problem (2.8), which satisfies a Hölder condition;(b)any Euler solution of (2.8) satisfies the relation where and

For proof of Theorem 2.2, please see in [6].

If in (2.8) is assumed to be continuous, then , an Euler solution, is actually a solution of the initial value problem (2.8).

Theorem 2.3. Under the assumptions of Theorem 2.2 and if we suppose that , then is a solution of initial value problem (2.8).

For proof Theorem 2.3, please see in [6].

2.3. Fractional-Order Differential Equations with Caputo’s Derivative

Consider the initial value problems of the fractional-order differential equations with Caputo's derivative

where and exist, and the perturbed system of initial value problem of the fractional-order differential equation with Caputo’s derivative of (2.17)

where exists, and ; satisfy a local Lipschitz condition on the set and for . In particular, we have a special case of (2.18) and is said to be perturbation term.

Corollary 2.4. Let , and a function such that for any If , then satisfies a.e. the initial value problems of the fractional-order differential equations with Reimann-Liouville's derivative (2.19) if, and only if, satisfies a.e. the Volterra fractional-order integral equation (2.20).

For proof of Corollary 2.4, please see in [2].

We assume that we have sufficient conditions to the existence and uniqueness of solutions through and . If and is the solution of

where is the Reimann-Liouville fractional-order derivative of as in (2.2) then it also satisfies the Volterra fractional-order integral equation

that is, every solution of (2.20) is also a solution of (2.19); for detail please see [2].

We will only give the basic existence and uniqueness result with the Lipschitz condition by using contraction mapping theorem and a weighted norm with Mittag-Leffler function in [6].

Theorem 2.5. Assume that(i) and bounded by on , where (ii) where the inequalities are componentwise. Then there exists a unique solution on for the initial value problem of the fractional-order differential equation with Caputo's derivative of (2.16), where

For proof of Theorem 2.5, please see in [6].

2.4. Stability Criteria with ITD and Lyapunov-Like Function

Before we can establish our comparison theorem and Lyapunov stability criteria for initial time difference, we need to introduce the following definitions of ITD stability and Lyapunov-like functions.

Definition 2.6. The solution of the initial value problems of fractional-order differential equation with Caputo’s derivative of (2.18) through is said to be initial time difference stable with respect to the solution where is any solution of the initial value problems of fractional-order differential equation with Caputo's derivative of (2.16) for and if and only if given any there exist and such that If are independent of , then the solution of the initial value problems of fractional-order differential equation with Caputo’s derivative of (2.18) is initial time difference uniformly stable with respect to the fractional solution . If the solution of initial value problems of fractional-order differential equation with Caputo’s derivative of of the fractional system (2.18) through is initial time difference stable and there exist and such that for all and with and for , then it is said to be initial time difference asymptotically stable with respect to the fractional solution . It is initial time difference uniformly asymptotically stable with respect to the fractional solution if and are independent of .

Definition 2.7. A function is said to belong to the class if , and is strictly monotone increasing in .

Definition 2.8. For any Lyapunov-like function we define the fractional-order Dini derivatives in Caputo’s sense and as follows for .

Definition 2.9. For a real-valued function , we define the generalized fractional-order derivatives (Dini-like derivatives) in Caputo's sense and as follows for

3. Comparing Fractional Stability with Fractional (ITD) Stability

3.1. Fractional Classical Notion of Stability

Let and be any solutions of the initial value problems of fractional-order differential equations with Caputo’s derivative of (2.16) and of (3.1), respectively,

where

Assume that so that is a null solution of fractional-order differential equation with Caputo’s derivative of (3.1) through . Now, we can state the well-known definitions concerning the stability of the null solution.

Definition 3.1. The null solution of fractional-order differential equation with Caputo’s derivative of (3.1) is said to be stable if and only if for each and for all , there exists a positive function that is continuous in for each such that If is independent of , then the null solution of initial value problems of fractional-order differential equation with Caputo’s derivative of (3.1) is said to be uniformly stable.

Definition 3.2. The solution of initial value problems of fractional-order differential equation with Caputo’s derivative of (3.1) through is said to be stable with respect to the solution of fractional-order differential equation with Caputo’s derivative of (3.1) for if and only if given any there exists a positive function that is continuous in for each such that If is independent of , then the solution of the fractional-order differential equation with Caputo’s derivative of (3.1) is uniformly stable with respect to the solution of (2.16).
We remark that for the purpose of studying the classical stability of a given solution of the initial value problem of fractional-order differential equation with Caputo's derivative of (3.1), it is convenient to make a change of variable. Let and be the unique solutions of the fractional-order differential equations with Caputo’s derivative (2.16) and (3.1), respectively, and set for . Then
It is easy to observe that is a solution of the transformed initial value problems of the fractional differential equation with Caputo’s derivative if which implies . Since and is the null solution, the solution of initial value problems of the fractional-order differential equation with Caputo’s derivative of (2.16) corresponds to the identically null solution of where . Hence, we can always assume, without any loss of generality, that is the null solution of the given fractional-order differential equation with Caputo’s derivative of (3.1) and we can limit our study of stability to that of the null solution [2, 6, 8]. However, it is impossible to do the same transformation for fractional (ITD) stability which we deal with it.

3.2. New Notion of Fractional (ITD) Stability

Let be a fractional solution of (2.17) and where is any solution of initial value problems of the fractional-order differential equations with Caputo’s derivative of the system (2.16) for . Let us make a transformation similar to that in (3.5). Set for . Then

One can observe that even if is not zero and is not the null solution of the initial value problems of transformed fractional-order differential equation with Caputo’s derivative and the solution. does not correspond to the identically zero solution of . Therefore, we cannot use stability properties of the fractional-order differential equation with Caputo's derivative of null solution in order to find fractional (ITD) stability properties using this approach.

4. A Fractional Comparison Result

In our earlier work and in the work of others [46], the differences between the classical notion of fractional stability and fractional ITD stability did not allow the use of the behavior of the null solution in our fractional ITD stability analysis. The main result presented in this section resolves those difficulties with a new approach that allows the use of the fractional stability of the null solution of the comparison system to predict the Caputo’s fractional stability properties of the solution of fractional-order differential equation with Caputo’s derivative of (2.18) with respect to where is any solution of the fractional-order differential equation with Caputo’s derivative of (2.16).

Let and The function space is denoted by as follows:

The Riemann-Liouville fractional derivative is defined by

Now, we will prove the following comparison result.

Theorem 4.1. Assume that is locally Hölder continuous, and Let be the maximal solution of the initial value problem of fractional-order scalar differential equation with Riemann-Liouville's derivative existing on such that where Then we have

Proof. In view of the definition of the maximal solution of the fractional-order differential equation with Riemann-Liouville’s derivative of (4.4) it is enough to prove that where is any solution of the initial value problem of fractional-order scalar differential equation with Riemann-Liouville's derivative Now it follows from (4.7) that Then by applying the comparison result [2, heorem 2.1 in page 23], we get (4.6) and since uniformly on each compact set Hence, The proof is complete.

Theorem 4.2. Assume that and where and Then the stability properties of the trivial solution of the comparison initial value problem of fractional-order differential equation with Caputo's derivative imply the corresponding stability results of the solution of the initial value problem of the fractional-order differential equation with Caputo's derivative of (2.17), respectively.

Proof. Let of (4.10) be stable. Then given and there exists for a with the property that We claim that the trivial solution of (2.17) is stable for these and If this was false, then there would exists a solution of (2.17) and such that For we set and choose Then we have which shows that where as Now, by using the fractional-order Dini derivatives in Caputo’s sense in Definition 2.8 as in (2.23), we have for and
This yields by comparison Theorem 4.1, the estimate
where is the maximal solution of (4.4). At we therefore arrive at the following contradiction Therefore, this do justify our claim.Hence the trivial solution of initial value problem of the fractional-order differential equation with Caputo’s derivative of (2.17) is stable.
Next suppose that is asymptotically stable. Since this implies by definition of the stability of , the stability of the trivial solution of (4.10) is the foregoing argument. This means the inequality (4.15) holds for all and hence, it is clear, by hypothesis, that if then The proof is therefore complete.

5. An Initial Time Difference Fractional Comparison Result

In this section, we have an other comparison result in which the stability properties of the null solution of the comparison system imply the corresponding initial time difference stability properties of the perturbed fractional-order differential system in Caputo’s sense with respect to the unperturbed fractional-order differential system in Caputo’s sense.

Theorem 5.1. Let , and let where and is closed ball with center at and radius . Assume that is the maximal solution of initial value problem of fractional-order differential equation with Caputo’s derivative through . where is any solution of fractional-order differential equation of (2.16) for and , and is the solution of fractional-order differential equation (2.18) with Caputo's derivatives. Then

Proof. Let for . Then by using the fractional-order Dini derivatives in Caputo's sense in Definition 2.8 as in (2.23), we obtain Therefore, and by using a standard result in [6], we get .

Now, we can formulate the comparison results via Lyapunov-like functions.

Theorem 5.2. Let and be locally Lipschitzian in . Assume that the generalized fractional-order derivatives (Dini-like derivatives) in Caputo’s sense satisfies with , where . Let be the maximal solution of the fractional-order differential equation , for . If where is any solution of the system (2.16) for , and and is any solution of (2.18) for such that , then for .

Proof. Let where is any solution of the system fractional-order differential equation (2.16) for , and and is any solution of fractional-order differential equation (2.18) for such that holds. Define for so that . Then for small enough we get Since is locally Lipschitzian in and fractional-order Dini derivatives in Caputo’s sense where is the Lipschitz constant, and and are error terms where Since is the generalized fractional-order derivatives (Dini-like derivatives) in Caputo’s sense, we have

By using Theorem 5.1, this implies

where is the maximal solution of for .

Now, we present the main comparison result that yields knowledge of initial time difference fractional stability properties if we know the stability properties of the null solution of the fractional comparison system.

Theorem 5.3. Assume that
(i) let be locally Lipschitzian in , positive definite and decrescent where the fractional-order Dini derivatives in Caputo’s sense
satisfies for and where and the generalized fractional-order (Dini-like) derivatives in Caputo's sense
(ii) Let be the maximal solution of the fractional-order differential equation with Caputo’s derivative
Then the stability properties of the null solution of the fractional-order differential system with Caputo’s derivative (5.12) with imply the corresponding stability properties of any solution of fractional-order differential system with Caputo’s derivative (2.18) with respect to where is any solution of fractional-order differential system with Caputo’s derivative of (2.16).

Proof. Let where is any solution of the fractional-order differential system with Caputo's derivative of (2.16) for , and and is any solution of fractional-order differential system with Caputo's derivative of (2.18) for such that holds. If we define for so that , then Theorems 5.1 and 5.2 imply that where the fractional-order Dini derivatives in Caputo’s sense and the generalized fractional-order (Dini-like) derivatives in Caputo's sense have been used. Thus, where is the maximal solution of the fractional comparison system of (5.12) with Caputo's derivative. Let the null solution of the fractional comparison system of (5.12) with Caputo's derivative be stable. Given any , since is positive definite and by (i) we have Hence Since exists, we have and . Therefore, the solution of fractional-order differential equation of (2.18) with Caputo's derivative is stable with respect to , where is any solution of the fractional-order differential equation of (2.16) with Caputo's derivative. If the null solution of the fractional comparison system of (5.12) with Caputo's derivative is asymptotically stable, then That implies since . Hence, the solution of fractional differential equation (2.18) is asymptotically stable with respect to where is any solution of the system (2.16). Since is decrescent and by (i) we have and the choice of is independent of .
Thus, uniform stability and uniform asymptotic stability of the fractional comparison system of (5.12) with Caputo's derivative imply the corresponding uniform stability and uniform asymptotic stability of the fractional solution of (2.18) in Caputo’s sense with respect to where is any solution of the fractional system of (2.16) in Caputo's sense. Hence, is any solution of the fractional system of (2.16) in Caputo’s sense that is uniformly stable and uniformly asymptotically stable.

Acknowledgments

This work has been supported by The Scientific and Technological Research Council of Turkey and Department of Mathematics Yeditepe University.

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Copyright © 2010 Coşkun Yakar. 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.


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