# Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

Authors: Tao Long; Yuexin Yu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we first introduce the problem classKpμ,λ,ζ 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 Kpμ,λ,ζ 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.


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## Body

## 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 k,l-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 Section2, the definition of the problem class Kpμ,λ,ζ is given. In Section 3, for the problem class Kpμ,λ,ζ, 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 ClassKpμ,λ,ζ

LetX be a real or complex Banach space with the norm ·, consider the following nonlinear IDEs:(1)y′t=ft,yt,t>0,t≠τk,k=1,2,…,Δy|t=τk=Ikyτk,k=1,2,…,y0+=y0,where Δy|t=τk=yτk+−yτk, yτk+ denotes the right limit of yt at t=τk; Ik:X⟶Xk=1,2,…, are continuous mapping satisfying the following inequality:(2)η1+Ikη1−Ikη2+η2≤γkη1−η2,∀η1,η2∈X,k=1,2,…,where γkk=1,2,…, are real constants with suitable size, and let ζ=γ1,γ2,….In this paper, we also assume thatf:0,+∞×X⟶X is a given continuous mapping and define a nonnegative function (cf., [14]):(3)Hu,v,t,fξ=u−v+∑q=1p−1qξqfq−1t,u−fq−1t,v,where u,v∈X,t≥0,ξ=ξ1,ξ2,…,ξpT∈Rpp∈N+; the functions fqt,x are obtained by the following recurrence formula:(4)f0t,x=ft,x,fq+1t,x=∂fqt,x∂t+∂fqt,x∂xft,x.For the convenience of illustration, we abbreviateHu,v,t,fξ as Hξ. In addition, we define the following symbols:(5)R−n≔α1,α2,…,αnT∈Rn|αi≤0,i=1,2,…,n,α≤β≔∀α=α1,α2,…,αnT,β=β1,β2,…,βnT∈Rn|αi≤βi,i=1,2,…,n.Definition 1.
Suppose thatμ,λ∈R−p. The problem class consisting of problem (1) is called the problem class Kpμ,λ,ζ if the following inequality holds:(6)Hα≤1−μTα−λ1−μTβ−λHβ,∀α,β∈Rp,λ≤α≤β.Remark 1.
Whenζ=0, 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 functiony:0,+∞⟶X is said to be a solution of system (1) if the following conditions are satisfied:(i)
Fort>0,t≠τk,k=1,2,…, the function yt is differentiable and y′t=ft,yt(ii)
The functionyt is left continuous in the interval 0,+∞ and yτk+=yτk+Ikyτk(iii)
limt⟶0+yt=y0=y0+


## 3. Stability Analysis with respect to the Problem ClassKpμ,λ,ζ

In this section, we focus on the stability analysis of the analytical solution of the problem classKpμ,λ,ζ. First, we introduce the perturbed problem of (1):(7)z′t=ft,zt,t>0,t≠τk,k=1,2,…,Δz|t=τk=Ikzt,k=1,2,…,z0+=z0.Lemma 1 (cf., [18]).
If the initial value problem (1) belongs to the problem class Kpμ,λ,ζ, then(8)yt2−zt2≤eμ1/1+μTλt2−t1yt1−zt1,∀τk−1<t1≤t2≤τk,k=1,2,…,where μ1,zt represent the first component of μ and the solution of the perturbed problem (7), respectively.Theorem 1.
If the initial value problem (1) belongs to the problem class Kpμ,λ,ζ, then for ∀t∈τk−1,τkk=1,2,…, we have(9)yt−zt≤eμ1/1+μTλt−τk−1∏i=1k−1γieμ1/1+μTλτi−τi−1y0−z0.Inequality (9) characterizes the stability of the initial value problem (1).Proof.
Sincey0=y0+andz0=z0+, then by Lemma 1, for ∀t∈0,τ1, we have(10)yt−zt≤eμ1/1+μTλt−0+y0−z0.
On the other hand, from (2), we get(11)yτk+−zτk+=yτk+Ikyτk−zτk−Ikzτk≤γkyτk−zτk≤γkeμ1/1+μTλτk−τk−1yτk−1+−zτk−1+,k=1,2,….
By induction, whent∈τk−1,τkk=1,2,…, we have(12)yt−zt≤eμ1/1+μTλt−τk−1γk−1eμ1/1+μTλτk−1−τk−2yτk−2+−zτk−2+≤eμ1/1+μTλt−τk−1∏i=1k−1γieμ1/1+μTλτi−τi−1y0−z0.
This completes the proof of Theorem1.
It is easy to get the following corollary from Theorem1.Corollary 1.
If the initial value problem (1) belongs to the problem class Kpμ,λ,ζ, there is a constant 0<δ<1 such that(13)γieμ1/1+μTλτi−τi−1≤δ,i=1,2,…,then(14)limt⟶+∞yt−zt=0.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 equations(15)y′t=ft,yt,t>0,y0=y0,can be expressed as follows (cf., [18]):(16)Yi=yl+∑q=1p−1q−1hq∑j=1iaijqfq−1tl+cjh,Yj,i=1,2,…,s,yl+1=yl+∑q=1p−1q−1hq∑j=1sbjqfq−1tl+cjh,Yj.Applying method (16) for solving problem (1), we have(17)Yik,l=yk,l−1+∑q=1p−1q−1hkq∑j=1iaijqfq−1tk,l−1+cjhk,Yjk,l,i=1,2,…,s,yk,l=yk,l−1+∑q=1p−1q−1hkq∑j=1sbjqfq−1tk,l−1+cjhk,Yjk,l,l=1,2,…,m,yk+1,0=yk,m+Ik+1yk,m,k=1,2,…,n,y0,0=y0,where the parameters aijq and bjq are real constants, 0≤cj≤1, hk=τk+1−τk/mm≥1 is a given step-size in τk,τk+1, tk,l−1=τk+l−1hk, and yk,l,yk+1,0, and Yjk,l are approximations to the exact solution ytk,l,ytk+1,0+, and ytk,l−1+cjhk, respectively. In the present paper, we always assume that aiiq≥0i=1,…,s,q=1,…,p.Remark 2.
If there exists at least onei such that aiiq>0, then method (17) is implicit; otherwise, it is explicit; if p=1, method (17) is the explicit or diagonally implicit Runge–Kutta methods.
Similarly, applying the same methods (16) to the perturbed problem (7) yields(18)Zik,l=zk,l−1+∑q=1p−1q−1hkq∑j=1iaijqfq−1tk,l−1+cjhk,Zjk,l,i=1,2,…,s,zk,l=zk,l−1+∑q=1p−1q−1hkq∑j=1sbjqfq−1tk,l−1+cjhk,Zjk,l,l=1,2,…,m,zk+1,0=zk,m+Ik+1zk,m,k=1,2,…,n,z0,0=z0.
For the convenience of the following analysis, we introduce the following symbols:(19)Wik,l=Yik,l−Zik,l,i=1,2,…,s,Qj,k,lq=hkqfq−1tk,l−1+cjhk,Yjk,l−fq−1tk,l−1+cjhk,Zjk,l,wk,l=yk,l−zk,l,as+1,jq=bjq,Qs+1,k,lq=as+1,s+1q=0,j=1,2,…,i,q=1,2,…,p.
Then, a combination of (17) and (18) leads directly to(20)Wik,l=wk,l−1+∑q=1p−1q−1∑j=1iaijqQj,k,lq,i=1,2,…,s,wk,l=wk,l−1+∑q=1p−1q−1∑j=1sbjqQj,k,lq,l=1,2,…,m,k=1,2,…,n,wk+1,0=φk+1yk,m−φk+1zk,m,where φk+1yk,m=yk,m+Ik+1yk,m.
Now, we give a set ofs×s lower triangular matrix Dq=dijqq=1,2,…,p and construct a set of s×s lower triangular matrix Gq=gijqq=1,2,…,p by matrix group Dq and method (17), where(21)gijq=0,∑q˜=1pdijq˜=0,∑q¯=1p∑k=jidikq¯akjq−ai+1,jq=0,+∞,∑q˜=1pdijq˜=0,∑q¯=1p∑k=jidikq¯akjq−ai+1,jq≠0,∑q¯=1p∑k=jidikq¯akjq−ai+1,jq∑q˜=1pdijq˜,∑q˜=1pdijq˜≠0,j=1,2,…,i,i=1,2,…,s.Definition 3 (cf., [18]).
If the elements of the matrix groupDq=dijqq=1,2,…,p satisfying(22)gijq≤ajjq,j=1,2,…,i,i=1,2,…,s,q=1,2,…,p,then it is said that the matrix group Dqq=1,2,…,p and method (17) is Gp−consistent.Definition 4 (cf., [18]).
If a set of nonnegatives×s lower triangular matrix Dqq=1,2,…,p is Gp−consistent, and Dq satisfies max1≤q≤pDq∞≤1/p, then method (17) is said to be Gp−stable with respect to Dqq=1,2,…,p.Theorem 2.
Suppose that method (17) is Gp−stable. Then, the numerical solutions yk,l and zk,l, obtained by using method (17) to problems (1) and (7) which belong to the problem class Kpμ,λ,ζ, respectively, satisfy the stability inequality(23)yk,l−zk,l≤ϕlhk∏i=1kϕmhi−1γiy0−z0,l=1,2,…,m,∀hk∈0,H\+∞,where(24)H=min1≤q≤pmin1≤i≤smin1≤j≤ihijq,hijq=λqgijq1/q,gijq<0,+∞,gijq≥0.ϕhk is a positive, continuous, and nonincreasing function in the interval 0,H, and it can be calculated by the following recurrence formula:(25)ϕ1hk=1,ϕi+1hk=1−∑q=1p∑j=1idijq+∑q=1p∑j=1idijqχijhkϕjhk,ϕhk=ϕs+1hk,χijhk=1−μTgijhk−λ1−μTajjhk−λ,i=1,2,…,s,j=1,2,…,i,where gijhk=hkgij1,…,hkpgijpT∈Rpi,j=1,…,s,ajjhk=hkajj1,…,hkpajjpT∈Rpj=1,2,…,i,i=1,2,…,s.Proof.
For∀hk∈0,H\+∞, from (20), we have(26)Wik,l−∑q=1p−1q−1aiiqQi,k,lq=wk,l−1+∑q=1p−1q−1∑j=1i−1aijqQj,k,lq,where i=1,2,…,s+1.
Next, we shall prove the following inequality by mathematical induction:(27)Wik,l−∑q=1p−1q−1aiiqQi,k,lq≤ϕihkwk,l−1,i=1,2,…,s+1.
Wheni=1, by (26), we can easily get(28)W1k,l−∑q=1p−1q−1a11qQ1,k,lq≤ϕ1hkwk,l−1.
If there existsi=k˜, such that the following inequality holds:(29)Wk˜k,l−∑q=1p−1q−1ak˜k˜qQk˜,k,lq≤ϕk˜hkwk,l−1,then for i=k˜+1, by (26), we have(30)Wk˜+1k,l−∑q=1p−1q−1ak˜+1,k˜+1qQk˜+1,k,lq=wk,l−1+∑q=1p−1q−1∑i=1k˜ak˜+1,iqQi,k,lq=1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑q=1p−1q−1∑i=1k˜ak˜+1,iqQi,k,lq+∑q=1p∑i=1k˜dk˜iqwk,l−1.
From (20), we can get(31)wk,l−1=Wik,l−∑q=1p−1q−1∑j=1iaijqQj,k,lq.
Substituting (31) into (30), we have(32)Wk˜+1k,l−∑q=1p−1q−1ak˜+1,k˜+1qQk˜+1,k,lq=1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑q=1p−1q−1∑i=1k˜ak˜+1,iqQi,k,lq+∑q=1p∑i=1k˜dk˜iqWik,l−∑q¯=1p−1q¯−1∑j=1iaijq¯Qj,k,lq¯=1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑q=1p∑i=1k˜−1q−1ak˜+1,iqQi,k,lq+dk˜iqWik,l−∑q=1p∑q¯=1p∑j=1k˜∑i=jk˜−1q¯−1dk˜iqaijq¯Qj,k,lq¯=1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑q˜=1p∑i=1k˜dk˜iq˜Wik,l+∑q=1p∑i=1k˜∑q¯=1p∑j=ik˜−1qdk˜jq¯ajiq+−1q−1ak˜+1,iqQi,k,lq=1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑i=1k˜∑q˜=1pdk˜iq˜Wik,l+∑q=1p−1q∑q¯=1p∑j=ik˜dk˜jq¯ajiq−ak˜+1,iqQi,k,lq∑q˜=1pdk˜iq˜=1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑i=1k˜∑q˜=1pdk˜iq˜Wik,l+∑q=1p−1qgk˜iqQi,k,lq,where gk˜iq can be calculated by (21). Then, a combination of (22) and (6) leads directly to(33)Hjgijhk≤χijhkHjajjhk,j=1,2,…,i;i=1,2,…,s.
By (29), we obtain(34)Hjajjhk=Yjk,l−Zjk,l+∑q=1p−1qhkqajjqfq−1tk,l−1+cjhk,Yjk,l−fq−1tk,l−1+cjhk,Zjk,l=Wjk,l+∑q=1p−1qajjqQj,k,lq≤ϕjhkwk,l−1.
From equations (33) and (34) and through the induction method, we can obtain(35)Wk˜+1k,l−∑q=1p−1q−1ak˜+1,k˜+1qQk˜+1,k,lq≤1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑i=1k˜∑q˜=1pdk˜iq˜Higk˜ihk≤1−∑q=1p∑i=1k˜dk˜iqwk,l−1+∑i=1k˜∑q˜=1pdk˜iq˜χk˜ihkHiaiihk≤1−∑q=1p∑i=1k˜dk˜iq+∑i=1k˜∑q˜=1pdk˜iq˜χk˜ihkϕihkwk,l−1=ϕk˜+1hkwk,l−1.
For (26), let i=s+1, then we have(36)wk,l=Ws+1k,l−∑q=1p−1q−1as+1,s+1qQs+1,k,lq≤ϕhkwk,l−1,which implies that(37)yk,l−zk,l≤ϕhkyk,l−1−zk,l−1.
Note that(38)yk,0−zk,0≤γkyk−1,m−zk−1,m.
Therefore, we obtain(39)yk,l−zk,l≤ϕlhkyk,0−zk,0≤ϕlhkγkyk−1,m−zk−1,m≤ϕlhk∏i=1kϕmhi−1γiy0−z0.
This completes the proof of Theorem2.
From Theorem2, the following corollary can be obtained.Corollary 2.
If there is a real constant0<ρ<1 such that(40)ϕmhi−1γi≤ρ,i=1,2,…,and the conditions in Theorem 2 hold, then(41)limk⟶+∞yk,l−zk,l=0.The above formula shows that method (17) is asymptotically stable when solving the problem class Kpμ,λ,ζ.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:(42)y′t=−21070−2−107yt+siny2t−siny2t,0<t,t≠k,k=1,2,…,Δyt=0.60.60.40.5yt,t=k,y0+=0.5,0.5T.
The corresponding perturbed problem is(43)z′t=−21070−2−107zt+sinz2t−sinz2t,0<t,t≠k,k=1,2,…,Δzt=0.60.60.40.5yt,t=k,z0+=1,1T.
The Jacobi matrix of the right-side functionfyt of problem (42) is J=∂f/∂y=−2107+cosy2t0−2−107−cosy2t. In the standard inner product space, the logarithmic matrix norm is m=2−1/2×107+cosy2t−2. 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 Kpμ,λ,ζ, where p=2,μ=−2,−4T,λ=−1/2,−1/2T,andζ=1.1,…,1.1T.
We consider the third-order Obrechkoff method as follows:(44)Y1k=yk,l−1,Y2k=yk,l−1+hk13ftk,l−1,Y1k+23ftk,l,Y2k−h26f′tk,l,Y2k,yk,l=yk,l−1+hk13ftk,l−1,Y1k+23ftk,l,Y2k−h26f′tk,l,Y2k.
Take(45)D1=1301414,D2=1301316,which implies that the matrix group(46)G1=−120−13−1415,G2=000−730.
So, we can verify that method (44) is Gp− stable. For ∀hk∈0,H, when method (44) is applied to problems (42) and (43), we have the following stability inequalities:(47)yk,l−zk,l≤2.1×ϕlhk∏i=1kϕmhi−1y0−z0,l=1,2,…,m,where H=min−15/14λ1,−30/7λ2=15/28, ϕhk can be obtained by (25).
According to Corollary2, when we take hk=1/10, the third-order Obrechkoff method (44) for (42) is asymptotically stable (see Figure 1).Figure 1
The error ofyk,l−zk,l1 for (42) and (43).Example 2.
Consider the following nonlinear IDEs:(48)y′t=−3100−13yt+cosy2t−cosy2t,0<t,t≠k,k=1,2,…,Δyt=0.70.50.30.5yt,t=k,y0+=0,0T.
The corresponding perturbed problem is(49)z′t=−3100−13zt+cosz2t−cosz2t,0<t,t≠k,k=1,2,…,Δzt=0.70.50.30.5zt,t=k,z0+=1,1T.
The Jacobi matrix of the right-side functionfyt of problem (48) is J=∂f/∂y=−310−siny2t0−13+siny2t. Under the 1 norm, the logarithmic matrix norm is m=−3. So, it is easy to know that problem (48) belongs to the problem class Kpμ,λ,ζ, where p=2,μ=−3,−9T,λ=−1/2,−1/2T,andζ=1,…,1T.
We consider the fourth-order explicit method (cf., [17]):(50)Y1k=yk,l−1,Y2k=yk,l−1+hk2ftk,l−1,Y1k+hk28f′tk,l−1,Y1k,yk,l=yk,l−1+hkftk,l−1,Y1k+hk216f′tk,l−1,Y1k+13f′tk,l−1+hk2,Y2k.
Take(51)D1=1301316,D2=1301613,which implies that the matrix group(52)G1=−340−320,G2=−3160−524−23.
Thus, we can verify that method (50) is Gp−stable. For ∀hk∈0,H, when method (50) is applied to problems (48) and (49), we have the following stability inequalities:(53)yk,l−zk,l≤2×ϕlhk∏i=1kϕmhi−1y0−z0,l=1,2,…,m,where H=min−4/3λ1,−3/2λ2=3/4,ϕhk can be obtained by (25).
According to Corollary2, when we take hk=1/10, the fourth-order explicit method (50) for (48) is asymptotically stable (see Figure 2).Figure 2
The error ofyk,l−zk,l1 for (48) and (49).


## 6. Conclusion

In this paper, we focus on the stability and asymptotic stability of multistage one-step multiderivative methods for the problem classKpμ,λ,ζ 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.



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