Abstract and Applied AnalysisVolume 2012 (2012), Article ID 258329, 14 pagesdoi:10.1155/2012/258329

Research Article

Stability of Analytic and Numerical Solutions for Differential Equations with Piecewise Continuous Arguments

Minghui Song and M. Z. Liu

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 17 January 2012; Accepted 13 February 2012

Academic Editor: Norio Yoshida

Copyright © 2012 Minghui Song and M. Z. Liu. 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.

In this paper, the asymptotic stability of the analytic and numerical solutions for differential equations with piecewise continuous arguments is investigated by using Lyapunov methods. In particular, the linear equations with variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the -methods are obtained. Some examples are illustrated.

1. Introduction

This paper deals with the stability of both analytic and numerical solutions of the following differential equation: where , is continuous, , and denotes the greatest integer function. This kind of equations has been initiated by Wiener [1, 2], Cooke and Wiener [3], and Shah and Wiener [4]. The general theory and basic results for EPCA have by now been thoroughly investigated in the book of Wiener [5].

It seems to us that the strong interest in differential equation with piecewise constant arguments is motivated by the fact that it describes hybrid dynamical system (a combination of continuous and discrete). These equations have the structure of continuous dynamical systems within intervals of unit length. Continuity of a solution at a point joining any two consecutive intervals implies recurrent relations for the values of the solution at such points. Therefore, they combine the properties of differential equations and difference equations.

There are also some authors who have considered the stability of numerical solutions (see [6–8]). However, all of the above results are based on the linear autonomous equations. In this paper, we will use Lyapunov methods to investigate the analytic and numerical solution of the generalized equation (1.1).

Definition 1.1 (see [5]). A solution of (1.1) on is a function that satisfies the following conditions:(1) is continuous on ,(2)the derivative exists at each point , with the possible exception of the points , where one-sided derivatives exist,(3)Equation (1.1) is satisfied on each interval with integral endpoints.

2. The Stability of the Analytic Solution

Definition 2.1 (see [9–11]). The trivial solution of (1.1) is said to be(1)stable if for any given , there exists a number such that if , then for all ,(2)asymptotically stable if it is stable, and there exists an such that for any given , there exists a number such that if , then for all ,(3)globally asymptotically stable if it is asymptotically stable and ,(4)unstable if stability fails to hold, where is a norm in .

Definition 2.2. Given a continuous function , the derivative of along the solution of (1.1) is defined by for .

It is easy to see that if has continuous partial derivatives with respect to and , then (2.1) can be represented by

Theorem 2.3. Suppose that is continuous, are continuous, strictly increasing functions satisfying . Let constants and exist such that for , Then the trivial solution of (1.1) is asymptotically stable.

Proof. Let be given, then there exist and such that , , and . Let , and Let be a solution of (1.1), then it follows from (2.4) that the function is decreasing with respect to . Making use of (2.3) and (2.4), we obtain successively the inequalities for any integer , so .From (2.3), (2.5), and (2.6), we have for , Hence, for all , which implies that the trivial solution is stable.From (2.3), (2.4), and (2.5), we have for , Hence,

Example 2.4. The trivial solution of the following system: is asymptotically stable.

Proof. Let be a constant such that , , , , , and , then Hence, for , we have Therefore, the trivial solution is asymptotically stable.

In the following, we consider the following equation: where and are continuous.

Theorem 2.5. The trivial solution of (2.12) is asymptotically stable if there exist constants and such that for ,

Proof. We define , Then we have for , Let , then Hence, And we have from (2.13), In view of (2.3), the theorem is proved.

Assume , , then (2.12) and conditions (2.13) reduce to If we choose , then (2.19) is automatically satisfied.

Remark 2.6. (1) The conditions (2.13) are necessary and sufficient for the trivial solution of (2.12) being asymptotically stable (see [1, Theorem  1.45]).(2) The condition (2.20) is necessary and sufficient for the trivial solution of (2.18) being asymptotically stable (see [1, Corollary  1.2]).

Again we consider (2.12). Assume , then for , Therefore, we have the following corollary.

Corollary 2.7. Assume that , , then the trivial solution of (2.12) is asymptotically stable if

3. The Stability of the Discrete System

In this section, we will consider the discrete system with the form where , .

We assume that and (3.1) has a unique solution. The solution is the trivial solution of (3.1). Like (2.1), we can define the stability and asymptotical stability.

Theorem 3.1. Suppose are continuous, are continuous, strictly increasing functions satisfying . Let constants and exist such that for Then the trivial solution of (3.1) is asymptotically stable.

Proof. Firstly, we will prove the stability. Let be given, then there exists a and such that , , and . Let , and Let be a solution of (3.1), then it follows from (3.3) that the function is nonincreasing with respect to . Making use of (3.2) and (3.3), we obtain successively the inequalities so .From (3.2), (3.4), and (3.5) Therefore, for all , , .Nextly, we will prove the asymptotic stability. We have, from (3.2) and (3.4), so The proof is complete.

In the rest of the section, we consider the following scalar system: Let . The following corollary is easy to prove.

Corollary 3.2. If there exists a , such that for , , then the trivial solution of (3.1) is asymptotically stable.

4. The Stability of the Numerical Solution

In this section, we will investigate the numerical asymptotic stability of -methods.

4.1. -Methods

Let be a given stepsize with integer and the gridpoints . The linear -method applied to (1.1) can be represented as follows: and the one-leg -method Here, is a parameter with , specifying the method, denotes an approximation to , and is an approximation to defined by

4.2. Numerical Stability

Applying (4.1) and (4.2) to (2.12), we arrive at the following recurrence relations, respectively: Let , then we define , , as according to Definition 1.1. As a result, (4.4) reduce to In fact, in each interval , (2.12) can be seen as ordinary differential equation. Hence, the -methods are convergent of order 1 if and order 2 if .

Definition 4.1. (1) The numerical methods are called asymptotically stable if there exists an , such that as for any given and any stepsize .(2) The numerical methods are called general asymptotically stable if as for any given and any stepsize.

Theorem 4.2. Assume that , , and there exists a such that , then (1)the linear -method and the one-leg -method are asymptotically stable if ,(2)the one-leg -method is general asymptotically stable if , and the linear -method is general asymptotically stable if and is nonincreasing.

Proof. Denote , and .(1)For any integer , and , we have, from For the linear -method, For the one-leg -method, (2)For the linear -method, if , then we have, from (1),If , then since is nonincreasing, we haveFor one-leg -method, we have the following two cases., then ., then .

5. Numerical Experiments

In this section, we will give some examples to illustrate the conclusions in the paper. We consider the following three problems:

It is easy to verify that the above examples satisfy the conditions of Theorem 4.2. Hence, the solutions of three equations are asymptotically stable according to Corollary 2.7.

In Tables 1 and 2, we list the absolute errors (AEs) and the relative errors (REs) at of the -methods for the first problem. We can see from these tables that the methods preserve their orders of convergence.

Table 1: Linear -methods for problem (5.1).

AE
RE
AE
RE
AE
RE

3

5

10

20

50

100

Ratio
1.9857
1.9857
3.9998
3.9999
2.0141
2.0141

Table 2: One-leg -methods for problem (5.1).

AE
RE
AE
RE
AE
RE

3

5

10

20

50

100

Ratio
1.9857
1.9857
3.9999
4.0000
2.0141
2.0141

In Figures 1, 2, 3, 4, 5, and 6, we draw the numerical solutions of the -methods with . It is easy to see that the numerical solutions are asymptotically stable.

Figure 1: Linear -method with for (5.1).

Figure 2: Linear -method with for (5.1).

Figure 3: One-leg -method with for (5.2).

Figure 4: One-leg -method with for (5.2).

Figure 5: One-leg -method with for (5.3).

Figure 6: Linear -method with for (5.3).

Acknowledgment

This work is supported by the NSF of P.R. China (no. 10671047)

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