Abstract
In this paper, we first introduce the problem class with respect to the initial value problems of nonlinear impulsive differential equations in Banach spaces. The stability and asymptotic stability results of the analytic solution of the problem class are obtained. Then, the numerical stability and asymptotic stability conditions of multistage one-step multiderivative methods are also given. Two numerical experiments are given to confirm the theoretical results in the end.
1. Introduction
Impulsive differential equations (IDEs) are widely applied in population control, infectious disease control, pest control, and control systems (see [1–4] and so on). Recently, the theory of impulsive differential equations has been widely studied by authors in [5–7] and so on. However, it is difficult to obtain the analytic solution of IDEs. Hence, taking numerical methods is a good choice. In [8], Wen et al. obtained the stability and asymptotic stability of the analytic solution and implicit Euler method for nonlinear IDEs in Banach spaces. The exponential stability of the numerical and analytical solutions of nonlinear IDEs is discussed by using a new technique in [9]. Zhang and Song studied the asymptotic stability of the analytic solution and Runge–Kutta methods for a class of piecewise continuous linear impulsive differential equations in [10]. The stability and asymptotic stability conditions of -algebraically stable of Runge–Kutta methods for stiff impulsive differential equations have been studied by [11]. In [12], Mei proposed a new algorithm based on the reproducing kernel method and least square method for solving nonlinear IDEs. Zhang gave the asymptotic stability of the analytic solution of nonlinear impulsive differential equations under Lipschitz condition and studied the asymptotic stability of Runge–Kutta methods by using Pade approximation theory in [13].
Because of good stability and high accuracy, the multistage one-step multiderivative methods are widely used to solve ordinary differential equations (see [14–18]). In the present paper, we apply multistage one-step multiderivative methods to impulsive differential equations and analyze the stability and asymptotic stability of the methods in Banach spaces.
The rest of this paper is organized as follows. In Section 2, the definition of the problem class is given. In Section 3, for the problem class , we get the conditions of stability and asymptotic stability of the analytic solution. In Section 4, we obtain the stability and asymptotic stability of the multistage one-step multiderivative methods. In Section 5, we give relevant numerical experiments to verify the theoretical results. Finally, some conclusions are drafted in Section 6.
2. Description of the Problem Class
Let X be a real or complex Banach space with the norm , consider the following nonlinear IDEs:where , denotes the right limit of at ; are continuous mapping satisfying the following inequality:where are real constants with suitable size, and let .
In this paper, we also assume that is a given continuous mapping and define a nonnegative function (cf., [14]):where ; the functions are obtained by the following recurrence formula:
For the convenience of illustration, we abbreviate as . In addition, we define the following symbols:
Definition 1. Suppose that . The problem class consisting of problem (1) is called the problem class if the following inequality holds:
Remark 1. When , the initial value problem (1) degenerates to the initial value problem of ordinary differential equations. It can be seen that Definition 1 is a generalization of Definition 2.1 in [18].
Definition 2 (cf., [19]). The function is said to be a solution of system (1) if the following conditions are satisfied:(i)For the function is differentiable and (ii)The function is left continuous in the interval and (iii)
3. Stability Analysis with respect to the Problem Class
In this section, we focus on the stability analysis of the analytical solution of the problem class . First, we introduce the perturbed problem of (1):
Lemma 1 (cf., [18]). If the initial value problem (1) belongs to the problem class , thenwhere represent the first component of and the solution of the perturbed problem (7), respectively.
Theorem 1. If the initial value problem (1) belongs to the problem class , then for , we have
Inequality (9) characterizes the stability of the initial value problem (1).
Proof. Since , then by Lemma 1, for , we haveOn the other hand, from (2), we getBy induction, when , we haveThis completes the proof of Theorem 1.
It is easy to get the following corollary from Theorem 1.
Corollary 1. If the initial value problem (1) belongs to the problem class , there is a constant such thatthen
The above relation characterizes the asymptotic stability of the initial value problem (1).
4. Stability and Asymptotic Stability of Multistage One-Step Multiderivative Methods
The multistage one-step multiderivative methods for solving ordinary differential equationscan be expressed as follows (cf., [18]):
Applying method (16) for solving problem (1), we havewhere the parameters and are real constants, , is a given step-size in , , and , and are approximations to the exact solution , and , respectively. In the present paper, we always assume that .
Remark 2. If there exists at least one such that , then method (17) is implicit; otherwise, it is explicit; if , method (17) is the explicit or diagonally implicit Runge–Kutta methods.
Similarly, applying the same methods (16) to the perturbed problem (7) yieldsFor the convenience of the following analysis, we introduce the following symbols:Then, a combination of (17) and (18) leads directly towhere .
Now, we give a set of lower triangular matrix and construct a set of lower triangular matrix by matrix group and method (17), where
Definition 3 (cf., [18]). If the elements of the matrix group satisfyingthen it is said that the matrix group and method (17) is .
Definition 4 (cf., [18]). If a set of nonnegative lower triangular matrix is , and satisfies , then method (17) is said to be with respect to .
Theorem 2. Suppose that method (17) is . Then, the numerical solutions and , obtained by using method (17) to problems (1) and (7) which belong to the problem class , respectively, satisfy the stability inequalitywhere
is a positive, continuous, and nonincreasing function in the interval , and it can be calculated by the following recurrence formula:where .
Proof. For , from (20), we havewhere .
Next, we shall prove the following inequality by mathematical induction:When , by (26), we can easily getIf there exists , such that the following inequality holds:then for , by (26), we haveFrom (20), we can getSubstituting (31) into (30), we havewhere can be calculated by (21). Then, a combination of (22) and (6) leads directly toBy (29), we obtainFrom equations (33) and (34) and through the induction method, we can obtainFor (26), let , then we havewhich implies thatNote thatTherefore, we obtainThis completes the proof of Theorem 2.
From Theorem 2, the following corollary can be obtained.
Corollary 2. If there is a real constant such thatand the conditions in Theorem 2 hold, then
The above formula shows that method (17) is asymptotically stable when solving the problem class .
Remark 3. Most of the literature on the stability analysis of the numerical method for impulsive differential equations is based on the classic Lipschitz or one-sided Lipschitz conditions in the sense of the standard inner product norm (see [9, 13, 20]). If the value of the one-sided Lipschitz constant of the problem is very large (see problem (42)), these classic stability theories will fail. Therefore, in order to overcome the standard inner product norm, we analyze the stability in Banach space. So, the research in this paper has certain value.
5. Numerical Experiments
Example 1. Consider the following nonlinear IDEs:The corresponding perturbed problem isThe Jacobi matrix of the right-side function of problem (42) is . In the standard inner product space, the logarithmic matrix norm is . That is to say, the one-sided Lipschitz constant is a huge positive value, so the classical stability theory fails. However, under the 1 norm, it is easy to know that problem (42) belongs to the problem class , where .
We consider the third-order Obrechkoff method as follows:Takewhich implies that the matrix groupSo, we can verify that method (44) is stable. For , when method (44) is applied to problems (42) and (43), we have the following stability inequalities:where , can be obtained by (25).
According to Corollary 2, when we take , the third-order Obrechkoff method (44) for (42) is asymptotically stable (see Figure 1).
Example 2. Consider the following nonlinear IDEs:The corresponding perturbed problem isThe Jacobi matrix of the right-side function of problem (48) is . Under the 1 norm, the logarithmic matrix norm is . So, it is easy to know that problem (48) belongs to the problem class , where .
We consider the fourth-order explicit method (cf., [17]):Takewhich implies that the matrix groupThus, we can verify that method (50) is . For , when method (50) is applied to problems (48) and (49), we have the following stability inequalities:where can be obtained by (25).
According to Corollary 2, when we take , the fourth-order explicit method (50) for (48) is asymptotically stable (see Figure 2).
6. Conclusion
In this paper, we focus on the stability and asymptotic stability of multistage one-step multiderivative methods for the problem class in Banach spaces and confirm the correctness of the results through two numerical experiments. In the future work, we might analyze the convergence of multistage one-step multiderivative methods for nonlinear impulsive differential equations.
Data Availability
The data used to support the results of this study are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the NSF of China (11 571 291) and Scientific Research Fund of Hunan Provincial Education Department.