Abstract
A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.
1. Introduction
In this paper, we study the stability of nonlinear impulsive differential equations: where , , is a positive constant, is a continuous function on , and exists. Assume that .
Definition 1 (see [1]). is said to be the solution of (1), if(1),(2) is differentiable and for , ,(3) is left continuous in and , .
Impulsive differential equations are widely used in actual modeling such as epidemic, optimal control and population dynamics (see [2, 3]). Up to now, extensive work has been done in the area of qualitative theories of IDDEs (see [4–6]). The numerical properties of impulsive differential equations attracted attentions of scholars since Ran et al.’s work in [7], which showed that the explicit Euler method is stable for impulsive differential equations, while the implicit Euler method is not. Some excellent works have been done in existing literature. In [8], the convergence of Euler method for linear IDDEs is studied. Zhang et al. studied the numerical stability of Runge-Kutta methods to IDDEs in [9, 10] and proved that backward Euler method and 2-stage Lobatto IIIC method can preserve the stability of nonlinear IDDEs with fixed impulses. Unfortunately, the numerical scheme may not converge when the impulses are variable. It is necessary to find a convergent stable method for IDDEs with variable pulses.
The rest of this paper is organized as follows. In Section 2, we establish a theorem that enables us to transform the stability of IDDEs to corresponding delay differential equations without impulsive perturbations. In Sections 3 and 4, stability of exact solutions is studied. In Section 5, a convergent numerical scheme is constructed. In Section 6, the stability of numerical solutions is researched. In Section 7, numerical experiments are given to confirm the conclusions.
2. Preliminaries
Recently, a new technique has been introduced in the stability analysis of exact and numerical solutions to impulsive differential equations by constructing equivalent equations (see [9–12]). But the equivalent equations may not work when the impulses are variable. In this section, we will propose an improved equivalent equation.
Hypothesis 1. Assume that the function satisfies the following:(1) is continuously differentiable on , .(2), .(3), .
Remark 2. It is easy to check thatsatisfy Hypothesis 1.
Consider the equationwhere has been defined in Hypothesis 1. Here is the left derivative of .
Corollary 3. In the case that , , the transformation function (2) can be reduced to . Then (4) is the same as the equivalent equations proposed in [9, 13].
Theorem 4. Assume that Hypothesis 1 holds.(1)If is the solution of (1), then is the solution of (4).(2)If is the solution of (4), then is the solution of (1).
Proof. (1) It is obvious that is continuous on each interval . On the other hand, for any , which implies that is continuous in . It is easy to verify that is the solution of (4).
(2) On each interval ,On the other hand, at each impulsive point ,It follows from Definition 1 that is the solution of (1).
3. Stability Analysis of Exact Solutions
Let be an inner product on and is the corresponding norm. We assume that there exist real constants and such that the function in (1) satisfies
To study the stability of (1), we also consider the equation with differential initial value: Based on the transformationwe can get an equivalent equation to (9) as follows:
Definition 5. The zero solution of (1) is said to be stable, if there exists a constant such that
Definition 6. The zero solution of (1) is said to be asymptotically stable, if
Lemma 7. Assume thathold. Then(1)if ,(2)if ,
Proof. Note thatTherefore,It follows thatOn the other hand, when and when . The proof is complete.
By Theorem 4 and Lemma 7, we can obtain the stability of (1).
Theorem 8. Assume that is bounded. Then(1)if , then zero solution of (1) is stable,(2)if , then zero solution of (1) is asymptotically stable.
To investigate the stability of (1) by Theorem 8, we need to establish a suitable function . Different stability conditions may be given under different function . In the next section, some specific stability conditions are given.
4. Some Specific Stability Conditions
In this section, some specific stability conditions are given based on the choice of proposed in (2) and (3).
Case 1 (take as in (2)). Taking as in (2), we can obtain the following conclusions by Theorem 8.
Theorem 9. Assume thatThen the zero solution of (1) is asymptotically stable if .
Remark 10. The zero solution of (1) is asymptotically stable if , when , . The zero solution of (1) is asymptotically stable if , when , . Therefore, in the case that impulses are fixed or absent, the conclusions in Theorem 8 can be reduced to the same as in [9, 14].
Corollary 11. Assume that Then the zero solution of (1) is asymptotically stable if .
Case 2 (take as in (3)). Taking as in (3), we can obtain the following conclusion by Theorem 8.
Corollary 12. If there exists a constant , the following hold:The zero solution of (1) is stable when and is asymptotically stable when .
Note that it is difficult to find sequences satisfying conditions (22) and (23) simultaneously. It is necessary to find other stability conditions in this choice of .
Theorem 13. Assume that the following holds:Then the zero solution of (1) is stable when and is asymptotically stable when .
Proof. Note that while we take as in (3), inequality (17) becomesHence,By Theorem 4 and inequality (17), for , we haveNote that if and if . The proof is complete.
5. Numerical Process
In this section, we establish a convergent numerical process of methods for (1).
5.1. Methods for Ordinary Differential Equations
The application of one-stage method in case of an ordinary differential equation, yieldswhere denotes the step size and is an approximation of .
5.2. Methods for IDDEs
Let be a given step size with integer . The application of one-stage methods, in case of (4) on interval , yieldswhere denotes an approximation to for . Based on Theorem 4, the solution of (1) can be approximated by (30) andIn the same way, we can define the numerical solutions of (3).
Remark 14. If we take as in (2), numerical processes (30) and (31) can be reduced towhen .
Remark 15. If we take as in (3), numerical processes (30) and (31) can be reduced to
The convergence of numerical processes (30) and (31) is described in the following theorem.
Theorem 16. Assume that is continuously differentiable on , , and is bounded. Then numerical processes (30) and (31) are of order when and of order when .
Proof. On each interval , (4) reduces to a DDE. Therefore for and for . On the other hand, . The conclusion follows because is bounded.
6. Stability Analysis of Numerical Solutions
In this section, we study the stability property of numerical solutions. As usual, we expect that the numerical solution can reproduce the property of the exact solutions. The stability of the numerical solutions can be defined according to Theorem 4.
Definition 17. A numerical method is said to be stable for (1), if there exists a constant , such that the numerical solutions and of (1) and (4) satisfyfor every step size under the constraint , where is a positive integer.
Definition 18. A numerical method is said to be asymptotically stable for (1), if the numerical solutions and of (1) and (4) satisfy (35) andfor every step size under the constraint , where is a positive integer.
Denote
It follows from (30) that
Lemma 19. Assume that (8) holds. Thenholds if . Furthermore, if , inequalityholds.
Proof. By difference equation (38), it is obvious thatThe proof is complete.
Using a similar technique to the one in [15, 16], we can obtain the numerical stability of methods to (1).
Theorem 20. Assume that is bounded and . Then the numerical methods (30) and (31) are asymptotically stable for (1) with
Proof. It follows from (38) thatTherefore,Note thatHence,In view of inequality (39), we haveHence,By induction, we haveThe conclusion follows because (42) implies thatOn the other hand,Therefore, the conclusion holds because is bounded.
Corollary 21. The implicit Euler method can preserve the asymptotic stability of (1) with .
Corollary 22. Assume that is bounded and . Then the numerical solution (33) is asymptotically stable for (1) satisfying (42). Furthermore, ifthe numerical solution (33) is asymptotically stable for (1) satisfying
It is necessary to find other stability criteria of numerical process (34), since the criteria in Theorem 20 are too tough for numerical scheme (34).
Theorem 23. Assume that andThen the numerical solution calculated by numerical process (34) is asymptotically stable for (1) satisfying
Proof. Note that (54) implies that . Hence, by Lemma 19, whereBy inequality (46), we haveTherefore,Then, by induction, we haveTherefore,The conclusion follows, since (55) implies that
Corollary 24. The implicit Euler method can preserve the asymptotic stability of (1) with (54).
Corollary 25. Ifthen (54) holds when . Then the conclusion in this paper coincides with the numerical stability of delay differential equations in [14].
7. Numerical Experiments
In this section, we will do some numerical experiments to illustrate the conclusion.
Consider the following equations:We can choose , . Then (64) and (65) can be transformed asWe chose and calculated the numerical solutions by numerical process (34) in case of implicit Euler method. Figure 1 shows the relationship of the solutions between (64) and (66). It can be seen from Figure 2 that the difference of solutions to (64) and (65) tends to zero as , which coincides with the conclusion in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 11505090) and Shandong Provincial Natural Science Foundation (no. ZR2017BA026 and no. BS2015SF009).