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Oscillation of RungeKutta methods for advanced impulsive differential equations with piecewise constant arguments
Advances in Difference Equations volume 2017, Article number: 32 (2017)
Abstract
The purpose of this paper is to study oscillation of RungeKutta methods for linear advanced impulsive differential equations with piecewise constant arguments. We obtain conditions of oscillation and nonoscillation for RungeKutta methods. Moreover, we prove that the oscillation of the exact solution is preserved by the θmethods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution. We give some numerical examples to confirm the theoretical results.
Introduction
In the past three decades, the theory of differential equations with piecewise constant arguments has been intensively studied. In 1984, Cooke and Wiener [1] studied differential equations without impulses and noted that such equations were related to impulsive and difference equations. Later, oscillation of discontinuous solutions of differential equations with piecewise constant arguments has been proposed by Wiener as an open problem [2], p.380. In recent years, the Euler method for impulsive and stochastic delay differential equations has been studied in [3] and [4], and impulsive delay difference equations have been studied in [5]. Especially, oscillation of advanced impulsive differential equations with piecewise constant arguments has been studied in [6]. Furthermore, in [7], asymptotical stability of RungeKutta methods for the advanced linear impulsive differential equation with piecewise constant arguments was studied. In the present paper, we study oscillation of RungeKutta methods for the following equation:
where a, b, c, d, and \(x_{0}\) are real constants, \(\Delta x(k)=x(k)x(k^{})\), and \([\cdot]\) denotes the greatest integer function.
The rest of the paper is organized as follows. In Section 2, the results about oscillation of the exact solutions of (1.1) in [6] are introduced. In Section 3, the conditions of oscillation and nonoscillation for RungeKutta methods are obtained. In Section 4, the conditions of oscillation and nonoscillation for θmethods are obtained. Moreover, it is proved that the oscillation of the exact solution is preserved by the θmethods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution with the order of accuracy 1 (\(\theta\neq\frac{1}{2}\)) and 2 (\(\theta=\frac{1}{2}\)). In the last section, two simple numerical examples are given to confirm the theoretical results.
Preliminaries
Definition 2.1
A function \(x(t)\) defined on \([0,\infty)\) is said to be a solution of (1.1) if it satisfies the following conditions:

(1)
\(x:[0,\infty)\rightarrow\mathbb{R}\) is continuous for \(t\in [0,+\infty)\) with the possible exception of the points \([t]\in[ 0, \infty)\),

(2)
\(x(t)\) is right continuous and has lefthand limit at the points \([t]\in[ 0, \infty)\),

(3)
\(x(t)\) is differentiable and satisfies \(x'(t)+ax(t)+bx([t])+cx([t+1])=0\) for any \(t\in\mathbb{R}^{+}\) with the possible exception of the points \([t]\in[ 0, \infty)\) where onesided derivatives exist,

(4)
\(x(n)\) satisfies \(\Delta x(n)=d x(n)\) for \(n\in\mathbb{N}\) and \(x(0)=x_{0}\).
Theorem 2.2
When \(a \neq0\), Eq. (1.1) has on \(t\in[n,n+1)\), \(n=0,1,2,\ldots \) , a unique solution \(x(t)\) defined by
where \(\{t\}=t[t]\), \(m_{0}(t)=\mathrm{e}^{at}+\frac{b}{a}(\mathrm {e}^{at}1)\), and \(m_{1}(t)=\frac{c}{a} (\mathrm{e}^{at}1)\).
Definition 2.3
See [6]
The solution \(x(t)\) of (1.1) is said to be oscillatory if there exist two realvalued sequences \((t_{n})_{n\geq0}, (t'_{n})_{n\geq 0} \subseteq[0,\infty)\) such that \(t_{n}\rightarrow\infty\), \(t'_{n}\rightarrow\infty\) as \(n\rightarrow\infty\) and \(x(t_{n})\leq0\leq x(t'_{n})\) for \(n\geq N\) with N sufficiently large. Otherwise, the solution is called nonoscillatory.
Remark 2.4
When \(1d>0\), Definition 2.3 is equivalent to the following one: The solution \(x(t)\) of (1.1) is said to be oscillatory if \(x(t)\) has arbitrarily large zeros, that is, for every \(T>0\), there exists a point \(\hat{t}>T\) such that \(x(\hat{t})=0\).
Theorem 2.5
See [6]
Let \(a\neq0\), \(c>0\), and \(1d>0\). Then, all solutions of (1.1) are oscillatory if and only if
RungeKutta methods
Oscillation of RungeKutta methods
Consider the RungeKutta methods for (1.1):
where \(h=\frac{1}{m}\), \(m\geq1\), m is an integer, and v is referred to as the number of stages. The weights \(b_{i}\), the abscissaes \(c_{i}=\sum_{j=1}^{v}a_{ij}\), and the matrix \(A=[a_{ij}]_{i,j=1}^{v}\) will be denoted by \((A,b,c)\). Define
where \(n=km+l\), and \(x_{n}\) is an approximation of the solution \(x(nh)\) of (1.1), \(n=0, 1,\ldots \) .
For the abovementioned RungeKutta methods, in the following, we always assume that there exist \(\delta_{1}<0\) and \(\delta_{2}>0\) such that \(0< R(z)<1\) for \(\delta_{1}< z<0\) and \(R(z)>1\) for \(0< z<\delta_{2}\), which means that
where \(z=ah\), \(R(z)=1+zb^{T}(IzA)^{1}e\), and \(e=(1, 1, \ldots, 1)^{T}\) is a vector of dimension v.
Definition 3.1
A nontrivial solution \(x_{n}\) of (3.1)(3.2) is said to be oscillatory if there exists a sequence \(n_{k}\) such that \(n_{k}\rightarrow \infty\) as \(k\rightarrow\infty\) and \(x_{n_{k}}x_{n_{k+1}}\leq0\); otherwise, it is called nonoscillatory. We say that the RungeKutta method (3.1)(3.2) for (1.1) is oscillatory if all the nontrivial solutions of (3.1)(3.2) are oscillatory; we say that the RungeKutta method (3.1)(3.2) for (1.1) is nonoscillatory if all the nontrivial solutions of (3.1)(3.2) are nonoscillatory.
Definition 3.2
We say that the RungeKutta method preserves oscillations of (1.1) if (1.1) oscillates and there is \(h_{0}\) such that (3.1)(3.2) oscillates for \(h< h_{0}\).
Theorem 3.3
When \(a\neq0\), \(c>0\), and \(1d>0\), then the RungeKutta method (3.1)(3.2) for (1.1) is oscillatory if \(Iz A\) is invertible and
for \(\delta_{1}< z<\delta_{2}\).
Proof
Assume that \(Iz A\) is invertible. For \(k=0,1,\ldots \) and \(l=0,1,\ldots ,m\), we can obtain that
which implies
For \(\delta_{1}< z<\delta_{2}\), equation (3.4) implies \(R(z)^{m}+\frac{b}{a}(R(z)^{m}1)\leq0\) and \(\frac{c}{a}(R(z)^{m}1)> 0\). So \(a\neq0\), \(c>0\), \(1d>0\), and (3.5) imply \(x_{k,0}x_{k+1,0}\leq0\), that is, \(x_{km}x_{(k+1)m}\leq0\). Hence, (3.1)(3.2) is oscillatory. □
Lemma 3.4
The \((j,k)\)Padé approximation to \(\mathrm{e}^{z}\) is given by
where
with error
It is the unique rational approximation to \(\mathrm{e}^{z}\) of order \(j+k\) such that the degrees of numerator and denominator are j and k, respectively.
Lemma 3.5
Let \(R(z)\) be the \((j,k)\)Padé approximation to \(\mathrm{e}^{z}\). Then, for \(z>0\) (\(a<0\), \(z=ah\)),

(i)
\(R(z)<\mathrm{e}^{z}\) for all \(z>0\) if and only if k is even,

(ii)
\(R(z)>\mathrm{e}^{z}\) for \(0< z<\eta\) if and only if k is odd;
and, for \(z<0\) (\(a>0\), \(z=ah\)),

(i)
\(R(z)> \mathrm{e}^{z}\) for all \(z<0\) if and only if j is even,

(ii)
\(R(z)<\mathrm{e}^{z}\) for \(\varsigma< z<0\) if and only if j is odd,
where η is a real zero of \(Q_{k}(z)\), and ς is a real zero of \(P_{j}(z)\).
Theorem 3.6
If \(a\neq0\), \(c>0\), \(1d>0\), and \(b\geq\frac{a}{\mathrm{e}^{a}1}\), then the RungeKutta method (3.1)(3.2) for (1.1) is oscillatory if any one of the following conditions holds:

(1)
\(a<0\), and k is odd for \(h=\frac{1}{m}<\min\{\frac{\eta }{a},\frac{\delta_{2}}{a}\}\);

(2)
\(a>0\), and j is odd for \(h=\frac{1}{m}<\min\{\frac{\zeta }{a},\frac{\delta_{1}}{a}\}\).
Proof
When \(a<0\) and k is odd, by Lemma 3.5 we obtain that
which implies that
which also implies that
Hence, (3.1)(3.2) is oscillatory by Theorem 3.3. □
Nonoscillation of RungeKutta methods
Lemma 3.7
If \(a\neq0\), \(c>0\), \(1d>0\), and \(b<\frac{a}{\mathrm{e}^{a}1}\), then the RungeKutta method (3.1) satisfies \(b<\frac{a R(z)^{l}}{1R(z)^{l}}\), that is, \(R(z)^{l}+\frac{b}{a} (R(z)^{l}1)>0\), \(l=1,2,\ldots, m\), if any one of the following conditions holds:

(1)
\(a<0\), and k is even for \(h=\frac{1}{m}<\frac{\delta_{2}}{a}\);

(2)
\(a>0\), and j is even for \(h=\frac{1}{m}<\frac{\delta_{1}}{a}\).
Proof
For brevity, we prove only part (1) of the lemma. When \(a<0\) and k is even, by Lemma 3.5 we obtain that
which implies that, for \(l=1,2,\ldots,m1\) and \(h=\frac{1}{m}\),
which also implies that
Hence, we obtain that
□
Lemma 3.8
If \(c>0\), \(x+\frac{b}{a}(x1)>0\), and \(f(x)=\frac{\frac{c}{a} (x1)}{x+\frac{b}{a} (x1)}\), then

(1)
\(f(x)\) is decreasing if \(a<0\);

(2)
\(f(x)\) is increasing if \(a>0\).
Proof
It follows from \(c>0\), \(x+\frac{b}{a}(x1)>0\), and
that \(f'(x)>0\) if \(a<0\) and \(f'(x)<0\) if \(a>0\). Hence,

(1)
\(f(x)\) is decreasing if \(a<0\),

(2)
\(f(x)\) is increasing if \(a>0\).
□
Theorem 3.9
If \(a\neq0\), \(c>0\), \(1d>0\), and \(b<\frac{a}{\mathrm{e}^{a}1}\), then the RungeKutta method (3.1)(3.2) for (1.1) is nonoscillatory if any one of the following conditions holds:

(1)
\(a<0\), and k is even for \(h=\frac{1}{m}<\frac{\delta_{2}}{a}\);

(2)
\(a>0\), and j is even for \(h=\frac{1}{m}<\frac{\delta_{1}}{a}\).
Proof
For brevity, we prove only part (1) of the lemma. Without loss of generality, assume that \(x_{0}>0\). Obviously, the conditions of Lemma 3.7 are fulfilled, and hence \(R(z)^{m}+\frac{b}{a}(R(z)^{m}1)>0\). It follows from (3.3) that \(\frac{c}{a} (R(z)^{m}1)>0\). Therefore, we can obtain that
which implies
which also implies
Consequently, we have
By Lemma 3.8 we obtain that, for \(l=1,2,\ldots,m\), \(k=0,1,2,\ldots \) ,
which means that
Hence, the RungeKutta method (3.1)(3.2) is nonoscillatory. □
Piecewise linear interpolation of θmethods
Oscillation of θmethods
Consider the following θmethods for (1.1):
where \(h=\frac{1}{m}\) with integer \(m\geq1\). Define
which is an approximation of the solution \(x(nh)\) of (1.1), \(n=0, 1,\ldots \) .
Lemma 4.1
For all \(m>a\), we have

(i)
for \(a>0\),
$$\begin{aligned}& \biggl(1+\frac{z}{1z\theta}\biggr)^{m}\geq\mathrm{e}^{a} \quad \textit{if and only if}\quad \varphi(1)\leq\theta\leq1, \\& \biggl(1+\frac{z}{1z\theta}\biggr)^{m}\leq\mathrm{e}^{a} \quad \textit{if and only if}\quad 0\leq\theta\leq\frac{1}{2}; \end{aligned}$$ 
(ii)
for \(a<0\),
$$\begin{aligned}& \biggl(1+\frac{z}{1z\theta}\biggr)^{m}\geq\mathrm{e}^{a} \quad \textit{if and only if}\quad \frac{1}{2} \leq\theta\leq1, \\& \biggl(1+\frac{z}{1z\theta}\biggr)^{m}\leq\mathrm{e}^{a} \quad \textit{if and only if}\quad 0\leq\theta\leq\varphi(1), \end{aligned}$$where \(\varphi(x)=\frac{1}{x} \frac{1}{\mathrm{e}^{x}1}\).
Applying Lemma 4.1, we can obtain the following two results.
Theorem 4.2
If \(a\neq0\), \(c>0\), \(1d>0\), and \(b\geq\frac{a}{\mathrm{e}^{a}1}\), th the en θmethod (4.1)(4.2) preserves the oscillation of (1.1) if any of the following conditions is satisfied:

(1)
\(\frac{1}{2} \leq\theta\leq1\) and \(a<0\) for \(h=\frac{1}{m}\), \(m>a\);

(2)
\(0 \leq\theta\leq\frac{1}{2}\) and \(a>0\) for \(h=\frac{1}{m}\), \(m>a\).
Theorem 4.3
If \(a\neq0\), \(c>0\), \(1d>0\), and \(b< \frac{a}{\mathrm{e}^{a}1}\), then the θmethod (4.1)(4.2) preserves the nonoscillation of (1.1) if any of the following conditions is satisfied:

(1)
\(0 \leq\theta\leq\varphi(1)\) and \(a<0\) for \(h=\frac{1}{m}\), \(m>a\);

(2)
\(\varphi(1) \leq\theta\leq1\) and \(a>0\) for \(h=\frac{1}{m}\), \(m>a\).
Piecewise linear interpolation of θmethods
For convenience, define the functions \(y_{k}(t)\) on the closed intervals \([k,k+1]\), \(k=0,1,2,\ldots \) , as follows:
where \(x(t)\) is the exact solution of (1.1). Obviously,
Theorem 4.4
Let \(a\neq0\), \(c>0\), \(1d>0\), and \(b>\frac{a}{\mathrm{e}^{a}1}\). For any integer k, we have

(1)
\(x(k)x(k+1)<0\);

(2)
\(x(t)\) has at most one zero at \([k,k+1]\).
Proof
(1) If \(a\neq0\), \(c>0\), \(1d>0\), and \(b>\frac{a}{\mathrm {e}^{a}1}\), then
By (2.2), that is,
we have
(2) Let \(t=k+\alpha\), \(\alpha\in[0,1]\). For convenience, define \(A_{k}=x(k)\). It follows from Theorem 2.2 that
Suppose that \(x_{k}(t)\) has two zeros \(k+\alpha_{1}\), \(k+\alpha_{2}\), and \(\alpha_{1}\neq\alpha_{2}\), \(\alpha_{1}, \alpha_{2}\in[0,1]\). Then we have
It follows from
that \(A_{k}=A_{k+1}=0\), which implies \(y_{k}(t)\equiv0\). Hence, \(y_{k}(t)\) has at most one zero at \([k,k+1]\), which implies that \(x(t)\) has at most one zero at \([k,k+1]\). □
Let \(\bar{x}_{k}(t)\), \(t\in[k,k+1]\), be the linear interpolation of \((x_{k,l})_{l=0,1,\ldots,m}\) given by
where \(\xi\in[0,1]\). Define
which is a piecewise continuous numerical solution of (1.1). The following theorems give the properties of \(\bar{x}(t)\), \(t\geq0\).
Theorem 4.5
Under the conditions of Theorem 4.4, the piecewise linear interpolation function \(\bar{x}(t)\) defined by (4.4)(4.5) satisfies

(1)
\(\bar{x}(t)x(t)=O(h)\) (\(\theta\neq\frac{1}{2}\)), \(\bar {x}(t)x(t)=O(h^{2})\) (\(\theta=\frac{1}{2}\));

(2)
\(\bar{x}(t)\) has at most one zero in \([k,k+1]\) for any integer k.
Proof
(1) Obviously, by mathematical induction we can prove that, for any nonnegative integer k,
which implies
(2) Suppose that \(\bar{x}_{k}(t)\) has two zeros \(t_{1}=t_{k,l_{1}}+\eta_{1}h\), \(t_{2}=t_{k,l_{2}}+\eta_{2}h\). Then by (4.4) we have
where
Hence,
In the following, we will prove that \(M\neq0\).
• If \(l_{1}=l_{2}\), \(\eta_{1}\neq\eta_{2}\), then
• Else, if \(l_{1}\neq l_{2}\), \(\eta_{1}=\eta_{2}\), then
• Else, if \(l_{1}\neq l_{2}\), \(\eta_{1}\neq\eta_{2}\), then, without loss of generality, let \(l_{2}>l_{1}\) and \(M=0\). Then
and these contradictions lead to \(M\neq0\).
Consequently, \(x_{k,0}=x_{k+1,0}=0\), which implies \(\bar{x}_{k}(t)\equiv 0\), which is a contradiction to part (1) of the theorem. Hence, \(\bar {x}_{k}(t)\) has at most one zero in \([k,k+1]\), which implies \(\bar{x}(t)\) has at most one zero in \([k,k+1]\) for any integer k. □
Theorem 4.6
Assume that the conditions of Theorem 4.4 hold and \(x(t)=0\). Then there is \(h_{0}>0\) such that, for \(h< h_{0}\), there is a unique t̄ such that

(1)
\(\bar{x}(\bar{t})=0\); moreover, \(t\bar{t}=O(h)\) (\(\theta\neq \frac{1}{2}\)), \(t\bar{t}=O(h^{2})\) (\(\theta=\frac{1}{2}\));

(2)
\(\bar{x}(t)\) intersects the axis of abscissas at t̄, that is, \(\bar{x}(\bar{t})=0\),

(i)
\(\bar{x}(k+(l+1)h)\bar{x}(k+(l1)h)<0\) for \(\bar{t}=k+lh\),

(ii)
\(\bar{x}(k+(l+1)h)\bar{x}(k+lh)<0\) for \(\bar{t}=k+(l+\mu)h\) (\(0<\mu<1\)).

(i)
Proof
(1) Assume that \(x(t)=0\), \(t\in(k,k+1)\). Then it follows from Theorem 4.4 that
Hence, by Theorem 4.5 there is \(h_{0}\) such that, for \(h< h_{0}\),
It is easy to see from Theorem 4.5 that there is a unique \(\bar {t}\in(k,k+1)\) such that \(\bar{x}(\bar{t})=0\).
Assume that \(t\neq\bar{t}\). By Theorem 4.5 we obtain that
On the other hand,
where ξ is in between t and t̄. So \(x'(\xi)\neq0\); otherwise, \(x(\bar{t})=0\), which is contrary to Theorem 4.4. Hence,
(2) • Assume that \(\bar{x}(k+lh)=0\), \(0\leq l< m\). Obviously, the first equation in (4.1) can be rewritten as
where \(R(z)=1+\frac{z}{1z\theta}\), \(z=ah\). Similarly, we can obtain
It follows from (4.6) and (4.7) that
Hence, since \(R(z)\neq0\), by Theorem 4.5(2) we immediately have \(x_{k,l1}x_{k,l+1}<0\), that is, \(\bar{x}(k+(l+1)h)\bar{x}(k+(l1)h)<0\).
• Assume that \(\bar{x}(k+(l+\mu)h)=0\), \(0<\mu<1\), \(0\leq l< m\). It follows from Theorem 4.5(2) and \(\bar{x}(k+(l+\mu)h)=\mu x_{k,l+1}+(1\mu)x_{k,l}=0\) that
□
Numerical experiments
Example 5.1
Consider the following equation:
where \(x_{0}>0\), \(c>0\), and \(1d>0\). Solving this equation, we get
Obviously, the exact solution \(x(t)\) of (1.1) is oscillatory, which can also be proved by Theorem 2.5. By Theorem 3.6(2), the RungeKutta method (3.1)(3.2) for (5.1) is oscillatory since j is odd for \(h=\frac{1}{m}<\min\{{\zeta },{\delta_{1}}\}\) with integer m. All the numerical methods in the last line of Table 1 preserve oscillation of (5.1), and by Theorem 4.2(2), the θmethod (3.1)(3.2) for (5.1) is oscillatory as \(0 \leq\theta\leq\frac{1}{2}\) for \(h=\frac{1}{m}\) with integer \(m>1\).
On the other hand, the RungeKutta method (3.1)(3.2) for (5.1) is nonoscillatory as j is an even positive integer, even though the stepsize h is very small. In fact, all the numerical solutions of the RungeKutta method (3.1)(3.2) for (5.1) are positive as j is even for \(h=\frac{1}{m}<\delta_{1}\) with integer m, which can be proved similarly as in the proof of Theorem 3.9. Hence, all the numerical methods in the last line of Table 2 for (5.1) are nonoscillatory for \(h=\frac{1}{m} <\delta_{1}\) with integer m.
Example 5.2
Consider the following example from [6]:
By Theorem 2.5 we obtain that the exact solution \(x(t)\) of (1.1) is oscillatory. By Theorem 3.6(2) the RungeKutta method (3.1)(3.2) for (5.2) is oscillatory as j is odd for \(h=\frac{1}{m}<\min\{ {\zeta},{\delta_{1}}\} \) with integer m. All the numerical methods in the last line of Table 1 preserve oscillation of (5.2).
By Theorem 4.2(2) the θmethod (3.1)(3.2) for (5.2) is oscillatory as \(0 \leq\theta\leq\frac{1}{2}\) for \(h=\frac{1}{m}\) with integer \(m>1\). Tables 3 and 4 roughly illustrate that the zeros of the piecewise linear interpolation of θmethods converge to the corresponding zeros of the exact solution with the order of accuracy 1 (\(\theta\neq\frac{1}{2}\)) and 2 (\(\theta=\frac{1}{2}\)), which is in agreement with Theorem 4.6.
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Acknowledgements
I would like to thank the referees for their helpful comments and suggestions. This work is supported by the Natural Science Foundations of Hebei Province A2015501130, the Research Project of Higher School Science and Technology in Hebei province ZD2015211,the Fundamental Research Funds for Central Universities N152304007 and the Youth Science Foundations of Heilongjiang Province QC2016001.
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Zhang, GL. Oscillation of RungeKutta methods for advanced impulsive differential equations with piecewise constant arguments. Adv Differ Equ 2017, 32 (2017). https://doi.org/10.1186/s1366201610670
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Keywords
 impulsive differential equations
 piecewise constant arguments
 RungeKutta methods
 oscillation
 Padé approximation