Razumikhintype theorems for impulsive differential equations with piecewise constant argument of generalized type
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Abstract
In this paper, we focus on developing Razumikhin technique for stability analysis of impulsive differential equations with piecewise constant argument. Based on the Lyapunov–Razumikhin method and impulsive control theory, we obtain some Razumikhintype theorems on uniform stability, uniform asymptotic stability, and global exponential stability, which are rarely reported in the literature. The significance and novelty of the results lie in that the stability criteria admit the existence of piecewise constant argument and impulses, which may be either slight at infinity or persistently large. Examples are given to illustrate the effectiveness and advantage of the theoretical results.
Keywords
Impulsive differential equations Piecewise constant argument of generalized type Lyapunov–Razumikhin method Asymptotic stability Exponential stabilityAbbreviations
 IDE
impulsive differential equation
 DEPCA
differential equation with piecewise constant argument
 DDE
delay differential equation
 FDE
functional differential equation
 IFDE
impulsive functional differential equation
MSC
34K20 34A371 Introduction
Qualitative theories of impulsive differential equations (IDEs) have been investigated by many researchers in the past three decades due to their potential applications in many fields such as biology, engineering, economics, physics, and so on. Among these theories, the stability problem is of great importance. By now a large number of results on stability problem for various IDE have been obtained by some classical methods and techniques; see [1, 2, 3, 4, 5, 6, 7, 8, 9] and the references therein.
In the 1980s, differential equations with piecewise constant argument (DEPCA) that contain deviation of arguments were initially proposed for investigation by Cooke, Wiener, Busenberg, and Shah [10, 11, 12]. Later, many interesting results have been obtained and applied efficiently to approximation of solutions and various models in biology, electronics, and mechanics [13, 14, 15, 16, 17]. Such equations represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations. Akhmet [18, 19, 20] generalized the concept of DEPCA by considering arbitrary piecewise constant functions as arguments; the proposed approach overcomes the limitations in the previously used method of study, namely reduction to discrete equations. Afterward, the results of the theory have been further developed [21, 22] and applied for qualitative analysis and control problem of real models, for example, in neural network models with or without impulsive perturbations [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], which have great significance in solving engineering and electronic problems.
Razumikhin technique was originally proposed by Razumikhin [34, 35] for delay differential equations (DDE) and was generalized by other researchers to functional differential equations (FDE) and impulsive functional differential equations (IFDE) [36, 37, 38, 39, 40, 41, 42, 43, 44]. The idea of Razumikhin technique is to build a relationship between history and current states using Lyapunov functions, so it is usually called the Lyapunov–Razumikhin method. This method avoids the construction of complicated Lyapunov functionals and provides a technically efficient way to study stability problems for delayed systems or impulsive delayed systems. Considering that DEPCA is a delayedtype system, Akhmet et al. [41] investigated the stability of DEPCA and established some Razumikhintype theorems on uniform stability and asymptotical stability and applied the results to a logistic equation, whereas impulsive perturbations were not taken into consideration. To the best of our knowledge, there have been few results on stability analysis obtained by the Lyapunov–Razumikhin method for impulsive DEPCA.
Motivated by this discussion, in this paper, we develop the Lyapunov–Razumikhin method for stability of impulsive differential equations with piecewise constant argument and establish some Razumikhintype theorems on uniform stability, uniform asymptotic stability, and global exponential stability, which are rarely reported in the literature. To overcome the difficulties created by piecewise constant argument and impulses, which may be persistently large [44], as we will see, more complicated and interesting analysis is demanded, in which a (persistent) impulsive control plays an important role to achieve stability. This paper is organized as follows. In Sect. 2, we introduce some basic notations, lemmas, and definitions. In Sect. 3, we present the main theoretical results. In Sect. 4, we give some practical examples to illustrate the effectiveness and novelty of our results. Finally, we conclude the paper in Sect. 5.
2 Preliminaries
Let \(\mathbb{R}\) be the set of real numbers, \(\mathbb{R}_{+}\) the set of positive real numbers, \(\mathbb{Z}_{+}\) the set of nonnegative integers, and \(\mathbb{R}^{n}\) the ndimensional real space equipped with the Euclidean norm \(\cdot \). Fix a realvalued sequence \(\{\theta_{k}\}\) such that \(0=\theta_{0}<\theta_{1}<\cdots <\theta _{k}<\cdots \) with \(\theta_{k}\rightarrow \infty \) as \(k\rightarrow \infty \).

\(\Omega_{1}=\{\varphi (s)\in C(\mathbb{R}_{+},\mathbb{R}_{+}), \mbox{strictly increasing}, \varphi (0)=0, 0<\varphi (s)<s, s>0\}\)

\(\Omega_{2}=\{\varphi (s)\in C(\mathbb{R}_{+},\mathbb{R}_{+}), \mbox{strictly increasing}, \varphi (0)=0\}\)

\(\Omega_{3}=\{\varphi (s)\in C(\mathbb{R}_{+},\mathbb{R}_{+}), \mbox{strictly increasing}, \varphi (0)=0, \varphi (s)>s, s>0\}\)
 (\(A_{1}\))
 \(f(t,x,y):\mathbb{R}_{+}\times \mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is piecewise continuous with respect to t and is rightcontinuous at the possible discontinuous points \(\theta_{k}\), \(k\in \mathbb{Z}_{+}\{0\}\); \(f(t,0,0)=0\) for all \(t\geq 0\), and f satisfies the Lipschitz conditionfor all \(t\in \mathbb{R}_{+}\) and \(x_{1}, x_{2}, y_{1}, y_{2} \in \mathbb{R}^{n}\), where \(l>0\) is a constant;$$ \bigl\vert f(t,x_{1},y_{1})f(t,x_{2},y_{2}) \bigr\vert \leq l\bigl(\vert x_{1}x_{2}\vert +\vert y_{1}y_{2}\vert \bigr) $$
 (\(A_{2}\))

\(I_{k}(t,x):\mathbb{R}_{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is continuous with respect to t and \(x, I_{k}(\theta _{k},0)=0\), and \(x_{1}\neq x_{2}\) implies \(x_{1}+I_{k}(\theta_{k}, x _{1})\neq x_{2}+I_{k}(\theta_{k}, x_{2})\) for all \(k\in \mathbb{Z} _{+}\{0\}\).
 (\(A_{3}\))

there exists a positive constant θ such that \(\theta_{k+1}\theta_{k}\leq \theta \) for all \(k\in \mathbb{Z}_{+}\);
 (\(A_{4}\))

\(l\theta [1+(1+l\theta)e^{l\theta }]<1\);
 (\(A_{5}\))

\(3l\theta e^{l\theta }<1\).
Lemma 1
([41])
Definition 1
 (i)
\(x(t)\) is continuous on each \([\theta_{k}, \theta_{k+1})\subseteq [t_{0}, \infty)\) and is rightcontinuous at \(t=\theta_{k}\), \(k\in \mathbb{Z}_{+}\);
 (ii)
the derivative \(x'(t)\) exists for \(t\in [t_{0}, \infty)\) with the possible exception of the points \(\theta_{k}\), \(k\in \mathbb{Z}_{+}\), where the righthand derivatives exist;
 (iii)
system (1) is satisfied by \(x(t)\) on \([t_{0}, \infty)\).
We give the following statement assertion on the existence and uniqueness of solutions of the initial value problem (1).
Theorem 1
Assume that conditions (\(A_{1}\))–(\(A_{5}\)) are fulfilled. Then, for every\((t_{0}, x_{0})\in \mathbb{R}_{+}\times \mathbb{R}^{n}\), there exists a unique solution\(x(t)=x(t, t_{0}, x _{0})\)of (1) on\([t_{0}, \infty)\)such that\(x(t_{0})=x_{0}\).
Proof
Remark 1
From the proof of Theorem 1 we can see that every solution of system (1) exists uniquely and is piecewise continuous on \([t_{0}, \infty)\). Moreover, every solution \(x(t)\) is rightcontinuous at the possible discontinuous points \(\theta_{k}\), and \(x(\theta_{k} ^{})\) exists. In addition, system (1) obviously has the zero solution.
Definition 2
 (i)
V is continuous on \([\theta_{k}, \theta_{k+1})\times \mathbb{R} ^{n}\), \(k\in \mathbb{Z}_{+}\) and \(V(t, 0)\equiv 0\) for all \(t\in \mathbb{R}_{+}\);
 (ii)
V is continuously differentiable on \([\theta_{k}, \theta_{k+1}) \times \mathbb{R}^{n}\), \(k\in \mathbb{Z}_{+}\) and for each \(x\in \mathbb{R}^{n}\), its righthand derivative exists at \(t=\theta_{k}, k \in \mathbb{Z}_{+}\).
Definition 3
3 Main results
In this section, under the same assumptions, we obtain the stability of the zero solution of system (1) based on the Lyapunov–Razumikhin method. Firstly, we present some uniform stability results.
Theorem 2
 (i)
\(u(x)\leq V(t,x)\leq v(x)\), \((t,x)\in [t_{0},\infty)\times \mathbb{R}^{n}\);
 (ii)for all\(t\in [\theta_{k},\theta_{k+1})\), \(k\in \mathbb{Z}_{+}\), and\(x, y\in \mathbb{R}^{n}\), \(V(\beta (t),y)\leq V(t,x)\)implies that$$ D^{+}V(t,x,y)\leq 0; $$
 (iii)
for all\(k\in \mathbb{Z}_{+}\{0\}\)and\(x\in \mathbb{R}^{n}\), we have\(V(\theta_{k},x+I_{k}(\theta_{k},x))\leq (1+b_{k})V(\theta ^{}_{k},x)\), where\(b_{k}\geq 0\)with\(\sum^{\infty }_{k=1} b_{k}< \infty\).
Theorem 3
 (i)
\(u(x)\leq V(t,x)\leq v(x)\), \((t,x)\in [t_{0},\infty)\times \mathbb{R}^{n}\);
 (ii)for all\(t\in [\theta_{k},\theta_{k+1})\), \(k\in \mathbb{Z}_{+}\), and\(x, y\in \mathbb{R}^{n}\), \(V(\beta (t),y)\leq \psi^{1}(V(t,x))\)implies thatwhere\(g:[t_{0},\infty)\rightarrow \mathbb{R}_{+}\)is locally integrable;$$ D^{+}V(t,x,y)\leq g(t)W\bigl(V(t,x)\bigr), $$
 (iii)
for all\(k\in \mathbb{Z}_{+}\{0\}\)and\(x\in \mathbb{R}^{n}\), \(V( \theta_{k},x+I_{k}(\theta_{k},x))\leq \psi (V(\theta^{}_{k},x))\);
 (iv)
for all\(k\in \mathbb{Z}_{+}\), \(\inf_{\mu \in \mathbb{R}_{+}} \int^{\mu }_{\psi (\mu)}\frac{ds}{W(s)}>\int^{\theta_{k+1}}_{\theta _{k}}g(s)\,ds\).
Remark 2
We may observe that the two theorems generalize the existing corresponding results, and their proofs can be formulated by combining the corresponding theorems in [38] and [41], and so we omit them.
Theorem 4
 (i)
\(u(x)\leq V(t,x)\leq v(x)\), \((t,x)\in [t_{0},\infty)\times \mathbb{R}^{n}\);
 (ii)for all\(t\in [\theta_{k},\theta_{k+1})\), \(k\in \mathbb{Z}_{+}\), and\(x, y\in \mathbb{R}^{n}\), \(V(\beta (t),y)< \psi (V(t,x))\)implies thatwhere\(g:[t_{0},\infty)\rightarrow \mathbb{R}_{+}\)is locally integrable;$$ D^{+}V(t,x,y)\leq g(t)W\bigl(V(t,x)\bigr), $$
 (iii)
for all\(k\in \mathbb{Z}_{+}\{0\}\)and\(x\in \mathbb{R}^{n}\), \(V( \theta_{k},x+I_{k}(\theta_{k},x))\leq \psi (V(\theta^{}_{k},x))\);
 (iv)
for all\(k\in \mathbb{Z}_{+}\), \(\sup_{\nu \in \mathbb{R}_{+}} \int^{\psi (\nu)}_{\nu }\frac{ds}{W(s)}<\int^{\theta_{k+1}}_{\theta _{k}}g(s)\,ds\).
Proof
For given \(\varepsilon >0\), we may choose \(\delta >0\) such that \(\psi (v(\delta))< u(\varepsilon)\). For any \(t_{0}\geq 0\) and \(x_{0}<\delta \), we shall show that \(x(t)<\varepsilon\), \(t\geq t _{0}\). To show that this δ is the needed one, we consider two cases where \(t_{0}=\theta_{i}\) for some \(i\in \mathbb{Z}_{+}\) and another one where \(t_{0}\neq \theta_{j}\) for all \(j\in \mathbb{Z}_{+}\).
Now, let \(t_{0}\geq 0\) with \(t_{0}\neq \theta_{i}\) for any \(i\in \mathbb{Z}_{+}\). By the idea in [41] and Lemma 1 we take \(\delta_{1}=\frac{ \delta }{K(l)}\), where δ satisfies \(\psi (v(\delta))< u( \varepsilon)\). Then \(x_{0}<\delta_{1}\) implies \(x(t)<\varepsilon\), \(t\geq t_{0}\). We see that the evaluation of \(\delta_{1}\) is independent of \(t_{0}\). The proof of Theorem 4 is complete. □
Remark 3
It should be noted that Theorem 4 allows for significant increases in V at impulse times, which may be persistently large (\(\psi (s)>s\) for \(s>0\)) and appropriately controlled by the length of impulsive intervals. So, the length of impulsive intervals cannot be too small; in other words, the disturbed impulses cannot happen too frequently, with the same idea as that in [44]; yet the impulse conditions presented in Theorem 4 are more general than that in [44] and can be verified more easily and conveniently.
Now, we present uniform asymptotical stability and exponential stability results.
Theorem 5
 (i)
\(u(x)\leq V(t,x)\leq v(x)\), \((t,x)\in [t_{0},\infty)\times \mathbb{R}^{n}\);
 (ii)
for all\(k\in \mathbb{Z}_{+}\{0\}\)and\(x\in \mathbb{R}^{n}\), \(V( \theta_{k},x+I_{k}(\theta_{k},x))\leq (1+b_{k})V(\theta^{}_{k},x)\), where\(b_{k}\geq 0\)with\(\bar{M}=\sum^{\infty }_{k=1} b_{k}<\infty\).
 (iii)for all\(t\in [\theta_{k},\theta_{k+1})\), \(k\in \mathbb{Z}_{+}\)and\(x, y\in \mathbb{R}^{n}\), \(V(\beta (t),y)<\psi (V(t,x))\)implies thatwhere\(\psi (s)>Ms\)for\(s>0\)with\(M=\prod^{\infty }_{k=1}(1+b_{k})\).$$ D^{+}V(t,x,y)\leq W\bigl(\vert x\vert \bigr), $$
Proof
Clearly, the conditions of Theorem 5 imply the uniform stability by Theorem 2.
In what follows, we show that, for arbitrary ε, \(0<\varepsilon <\rho \), there exists \(T=T(\varepsilon)>0\) such that \(x(t)\leq \varepsilon \) for \(t\geq t_{0}+T\) if \(x(t_{0})<\delta \).
Now, let \(t_{0}\neq \theta_{i}\) for any \(i\in \mathbb{Z}_{+}\). Similar to the arguments in Theorem 4, taking \(\delta_{1}=\frac{\delta }{K( \rho)}\), we obtain that \(x(t_{0})<\delta_{1}\) implies \(x(t)< \varepsilon\), \(t\geq t_{0}+T\). The proof of Theorem 5 is complete. □
Theorem 6
 (i)
\(u(x)\leq V(t,x)\leq v(x)\), \((t,x)\in [t_{0},\infty)\times \mathbb{R}^{n}\);
 (ii)for all\(t\in [\theta_{k},\theta_{k+1})\), \(k\in \mathbb{Z}_{+}\)and\(x, y\in \mathbb{R}^{n}\), \(e^{\eta \beta (t)}V(\beta (t),y)< \mu e^{ \eta t}V(t,x)\)implies thatwhere\(g:[t_{0},\infty)\rightarrow \mathbb{R}_{+}\)is locally integrable;$$ D^{+}V(t,x,y)\leq g(t)W\bigl(V(t,x)\bigr), $$
 (iii)
for all\(k\in \mathbb{Z}_{+}\{0\}\)and\(x\in \mathbb{R}^{n}\), \(V( \theta_{k},x+I_{k}(\theta_{k},x))\leq \mu V(\theta^{}_{k},x)\);
 (iv)
\(\inf_{t\in \mathbb{R}_{+}}g(t)\inf_{t\in \mathbb{R}_{+}} \frac{W(t)}{t}>\frac{1}{\tau }\ln \mu\), \(\tau =\inf_{k\in \mathbb{Z} _{+}}\{\theta_{k+1}\theta_{k}\}\).
Proof
4 Examples
In this section, we give some examples to illustrate the theoretical results obtained in the previous section.
Example 1
Example 2
Example 3
Remark 4
On one hand, the stability of the systems in the examples may not be obtained by the results in existing references due to the existence of both impulse and piecewise constant argument. So the results in this paper are more general than those in the references. On the other hand, it should be noted that in the three systems, the sequence \(\{\theta_{k}\}\) needs to satisfy conditions (A3), (A4), and (A5). I particular, in Examples 1 and 3, considering the presence of persisting impulses, we give the lower bound of θ (e.g., \(\theta_{k+1}\theta_{k}>\frac{\ln p}{abp}\) in Example 1), and, in fact, the range of the parameter θ is constructed for the stability of zero solution in Examples 1 and 3.
5 Conclusion
In this paper, we have derived several stability theorems for nonlinear systems with impulses and piecewise constant argument by employing the LyapunovRazumikhin method and impulsive control theory. Examples are also given to show the effectiveness and novelty of the results. The theoretical results obtained can be applied to study the stability problem of many nonlinear impulsive models with piecewise constant argument such as neural networks, population models, and other biological models. However, our results are based on the fact that the piecewise constant argument in systems is of retarded type, so it is interesting to develop the Lyapunov–Razumikhin technique to impulsive systems with piecewise constant argument of advanced type or hybrid type, which requires further research in the future.
Notes
Acknowledgements
The author wishes to express his sincere gratitude to the editors and reviewers for their useful suggestions, which helped to improve the paper. This work is supported by National Natural Science Foundation of China (11601269, 61503214), Natural Science Foundation of Shandong Province (ZR2017MA048), and A Project of Shandong Province Higher Educational Science and Technology Program (J15LI02). The paper has not been presented at any conference.
Availability of data and materials
Not applicable.
Authors’ information
Qiang Xi received the B.S. and M.S. degrees in applied mathematics from Liaocheng University, Liaocheng, China, in 2003 and Shandong Normal University, Ji’nan, China, in 2006, respectively, and the Ph.D. degree in fundamental mathematics from Shandong University, Ji’nan, China, in 2014. He is currently an associate professor with School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics. His current research interests include stability theory, delay systems, impulsive control theory, artificial neural networks, and applied mathematics
Authors’ contributions
The main idea of this paper was proposed by QX. QX prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Funding
National Natural Science Foundation of China (11601269); National Natural Science Foundation of China (61503214); Natural Science Foundation of Shandong Province (ZR2017MA048); the Project of Shandong Province Higher Educational Science and Technology Program (J15LI02).
Ethics approval and consent to participate
Not applicable.
Competing interests
The author declares that there is no competing interest regarding the publication of this paper.
Consent for publication
Not applicable.
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