A transfer theorem and stability of Levin-Nohel integro-differential equations
- Nguyen Tien Dung^{1}Email author
https://doi.org/10.1186/s13662-017-1122-5
© The Author(s) 2017
Received: 30 October 2016
Accepted: 23 February 2017
Published: 1 March 2017
Abstract
In this paper we show a connection between Levin-Nohel integro-differential equations and ordinary functional differential equations. Based on this connection, we obtain several new conditions for the stability of the solution, including the famous 3/2 stability criterion.
Keywords
MSC
1 Introduction
It is well known that the Levin-Nohel integro-differential equations have many applications in various fields of science and engineering. This class of equations was first studied by Volterra [1] in connection with a biological application. As is well known, the explicit solutions of the Levin-Nohel equations can rarely be obtained. Therefore, it is necessary to investigate stability properties of the solution.
For the Levin-Nohel equations of convolution type with a constant delay, the stability problem was solved by Levin and Nohel [2, 3] in the 1960s when they successfully constructed a Liapunov functional. About 15 years ago, a new technique of fixed points developed for studying stability of functional differential equations. Based on this technique, many very nice results have been obtained for asymptotic stability of the non-convolution Levin-Nohel equations with a variable delay (see, for instance, [4–7]).
The rest of this paper is organized as follows. In Section 2, we formulate and prove a transfer theorem for general Levin-Nohel equations. Section 3 is devoted to a study of the stability of linear Levin-Nohel equations with several delays. The stability of quasilinear Levin-Nohel equations is discussed in Section 4. Conclusion is given in Section 5.
2 Preliminaries and the transfer theorem
Definition 1
- (i)
stable if for each \(\varepsilon> 0\) and any \(t_{0}\geq0\) there exists a \(\delta> 0\) such that \(\Vert \varphi \Vert _{t_{0}}<\delta\) and \(t\geq t_{0}\) imply \(|x(t,t_{0},\varphi)|<\varepsilon\).
- (ii)
uniformly stable if it is stable and the above δ is independent of \(t_{0}\).
- (iii)
asymptotically stable if it is stable and \(\lim_{t\to \infty}|x(t,t_{0},\varphi)|=0\).
- (iv)
uniformly asymptotically stable if it is uniformly stable and \(\lim_{t\to\infty}|x(t,t_{0},\varphi)|=0\).
Definition 2
Proposition 1
Mean value theorem for integrals
The main results of this section are stated in the following theorem.
Theorem 1
Transfer theorem
- 1.There exist functions \(h_{k}(t),k\in I\) satisfying \(0\leq h_{k}(t)\leq r_{k}(t)\) for all \(t\geq0\) and \(x(t)\) is also the solution to the following equation:$$\begin{aligned} &\dot{x}(t)+\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)f_{k}\bigl(x\bigl(t-h_{k}(t)\bigr)\bigr)+\sum _{k \notin I} \int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds=0, \\ &\quad t\geq0. \end{aligned}$$(7)
- 2.In addition, we assume that, for each \(k\in I\), \(f_{k}(x)=x\ \forall x\in\mathbf{R}\) andThen there exists a function \(h(t)\) such that \(0\leq h(t)\leq\max_{k\in I}r_{k}(t)\) for all \(t\geq0\) and \(x(t)\) is also the solution to the following equation:$$ \int_{t-r_{k}(t)}^{t}a_{k}(t,s)\,ds\geq0 \quad \forall t\geq0. $$(8)$$\begin{aligned} &\dot{x}(t)+ \biggl[\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr) \biggr] x\bigl(t-h(t)\bigr)+\sum_{k \notin I} \int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds=0, \\ &\quad t\geq0. \end{aligned}$$(9)
Proof
- 1.We first note that equation (3) can be rewritten as follows:Since \(f_{k},k\in I\) are continuous functions and the kernel \(a_{k}(t,s)\) has the property (6), we can apply the mean value theorem for integrals to infer that, for each \(k\in I\), there exists \(c_{k}(t)\in (t-r_{k}(t),t)\) such that$$\begin{aligned} &\dot{x}(t)+\sum_{k\in I} \int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds+\sum_{k\notin I} \int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds=0, \\ &\quad t\geq0. \end{aligned}$$(10)Set \(h_{k}(t)=t-c_{k}(t)\), then \(0\leq h_{k}(t)\leq r_{k}(t)\) and the above relation becomes$$\int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds= \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)f_{k}\bigl(x\bigl(c_{k}(t)\bigr)\bigr). $$Inserting equalities (11) into (10) leads us to equation (7) and so the proof is complete.$$ \int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds= \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)f_{k}\bigl(x\bigl(t-h_{k}(t)\bigr)\bigr),\quad k\in I. $$(11)
- 2.Under the assumption \(f_{k}(x)=x\), \(k\in I\), equation (7) now reduces toIt follows from the facts \(0\leq h_{k}(t)\leq r_{k}(t),k\in I\), for all \(t\geq0\), that \(t-\max_{k\in I}r_{k}(t)\leq t-h_{k}(t)\leq t\) for all \(t\geq0\). As a consequence, we have$$\begin{aligned} &\dot{x}(t)+\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)x \bigl(t-h_{k}(t)\bigr)+\sum_{k \notin I} \int_{t-r_{k}(t)}^{t} a_{k}(t,s)f_{k} \bigl(x(s)\bigr)\,ds=0, \\ &\quad t\geq0. \end{aligned}$$(12)This, combined with the condition \(\int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds\geq 0,k\in I\), gives us$$\min_{t-\max_{k\in I}r_{k}(t)\leq u\leq t}x(u)\leq x\bigl(t-h_{k}(t)\bigr)\leq\max _{t-\max_{k\in I}r_{k}(t)\leq u\leq t}x(u),\quad t\geq0. $$and$$\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)x \bigl(t-h_{k}(t)\bigr)\geq \biggl[\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr) \biggr] \min_{t-\max_{k\in I}r_{k}(t)\leq u\leq t}x(u) $$We therefore can obtain the estimate$$\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)x \bigl(t-h_{k}(t)\bigr)\leq \biggl[\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr) \biggr] \max_{t-\max_{k\in I}r_{k}(t)\leq u\leq t}x(u). $$By the intermediate value theorem there exists \(c(t)\in(t-\max_{k\in I}r_{k}(t),t)\) such that$$\begin{aligned} \min_{t-\max_{k\in I}r_{k}(t)\leq u\leq t}x(u)&\leq\frac{\sum_{k\in I} (\int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds )x(t-h_{k}(t))}{\sum_{k\in I} (\int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds )}\\ &\leq\max _{t-\max_{k\in I}r_{k}(t)\leq u\leq t}x(u),\quad t\geq0. \end{aligned}$$Set \(h(t)=t-c(t)\), then \(0\leq h(t)\leq\max_{k\in I}r_{k}(t)\) and the above relation becomes$$x\bigl(c(t)\bigr)= \frac{\sum_{k\in I} (\int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds )x(t-h_{k}(t))}{\sum_{k\in I} (\int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds )},\quad t\geq0. $$So we can finish the proof by inserting (13) into (12).$$\begin{aligned} &\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr)x \bigl(t-h_{k}(t)\bigr) \\ &\quad = \biggl[\sum_{k\in I} \biggl( \int_{t-r_{k}(t)}^{t} a_{k}(t,s)\,ds \biggr) \biggr]x\bigl(t-h(t)\bigr),\quad t\geq0. \end{aligned}$$(13)
3 Linear Levin-Nohel equations
Proposition 2
Myshkis [14] and Yoneyama [15]
- 1.Ifthen the zero solution of (16) is uniformly stable.$$\sup_{t\geq0} \int_{t}^{t+\max_{t\geq0}h(t)}b(s)\,ds\leq\frac{3}{2} $$
- 2.Ifthen the zero solution of (16) is uniformly asymptotically stable.$$\liminf_{t\to0}b(t)>0,\qquad \sup_{t\geq0} \int_{t}^{t+\max_{t\geq 0}h(t)}b(s)\,ds< \frac{3}{2} $$
When the delay \(h(t)\) is infinite we have the following results.
Proposition 3
Graef et al. [13]
Let us now apply Propositions 2 and 3 to establish the stability conditions for the zero solution of (1).
Theorem 2
Proof
The proof of uniformly asymptotical stability can be done similarity and, hence, we omit it here. □
Remark 2
If the kernel \(a_{k}(t,s)\) satisfies the two conditions (6) and (8), then \(a_{k}(t,s)\geq0\) because of its continuity. That is why we imposed the condition \(a_{k}(t,s)\geq0\ \forall (t,s)\) in Theorem 2.
Theorem 3
Proof
By using the same arguments presented in the proof of Theorem 2, the desired result follows directly from Proposition 3. □
Remark 3
The results of Theorems 2 and 3 are well known; see e.g. [16]. We restate these results here to illustrate the usefulness of the transfer theorem. By using the transfer theorem, the proofs of Theorems 2 and 3 are very simple.
Proposition 4
Berezansky and Braverman [17]
As a product of Proposition 4, we can obtain new conditions for the exponential stability of equation (1) in the next theorem.
Theorem 4
Remark 4
Remark 5
The exponential stability of equation (1) has been recently discussed in [9]. To the best of our knowledge, that paper seems to be the first one studying the exponential stability of Levin-Nohel integro-differential equations. Theorem 4 thus is an interesting and important contribution because it provides us one more sufficient condition for the exponential stability.
Applying Theorem 4 for \(J=\{1\},\{2\}\) and \(\{1,2\}\) yields the following.
Corollary 1
- 1.\(\int_{0}^{r_{1}} a_{1}(s)\,ds>0\) and there exists \(r(t) \geq0 \), such that for sufficiently large t$$\begin{aligned} &\biggl( \int_{0}^{r_{1}} a_{1}(s)\,ds \biggr) r(t)\leq \frac{1}{e}, \\ &\limsup_{t\to\infty} \biggl( \biggl( \int_{0}^{r_{1}} a_{1}(s)\,ds+\biggl\vert \int _{0}^{r_{2}} a_{2}(s)\,ds\biggr\vert \biggr)\bigl\vert r(t)-r_{1}\bigr\vert +\frac{\vert \int_{0}^{r_{2}} a_{2}(s)\,ds\vert }{\int_{0}^{r_{1}} a_{1}(s)\,ds} \biggr)< 1. \end{aligned}$$
- 2.\(\int_{0}^{r_{2}} a_{2}(s)\,ds>0\) and there exists \(r(t) \geq0 \) such that for sufficiently large t$$\begin{aligned} &\biggl( \int_{0}^{r_{2}} a_{2}(s)\,ds \biggr) r(t)\leq \frac{1}{e}, \\ &\limsup_{t\to\infty} \biggl( \biggl(\biggl\vert \int_{0}^{r_{1}} a_{1}(s)\,ds\biggr\vert + \int_{0}^{r_{2}} a_{2}(s)\,ds \biggr)\bigl\vert r(t)-r_{2}\bigr\vert +\frac{\vert \int _{0}^{r_{1}} a_{1}(s)\,ds\vert }{\int_{0}^{r_{2}} a_{2}(s)\,ds} \biggr)< 1. \end{aligned}$$
- 3.\(\int_{0}^{r_{1}} a_{1}(s)\,ds+\int_{0}^{r_{2}} a_{2}(s)\,ds>0\) and there exists \(r(t) \geq0 \), such that for sufficiently large tThen equation (25) is exponentially stable.$$\begin{aligned} &\biggl( \int_{0}^{r_{1}} a_{1}(s)\,ds+ \int_{0}^{r_{2}} a_{2}(s)\,ds \biggr) r(t)\leq \frac{1}{e}, \\ &\limsup_{t\to\infty} \biggl(\biggl\vert \int_{0}^{r_{1}} a_{1}(s)\,ds\biggr\vert \bigl\vert r(t)-r_{1}\bigr\vert + \biggl\vert \int_{0}^{r_{2}} a_{2}(s)\,ds\biggr\vert \bigl\vert r(t)-r_{2}\bigr\vert \biggr)\\ &\quad < \frac {\int_{0}^{r_{1}} a_{1}(s)\,ds+\int_{0}^{r_{2}} a_{2}(s)\,ds}{ \vert \int_{0}^{r_{1}} a_{1}(s)\,ds\vert +\vert \int_{0}^{r_{2}} a_{2}(s)\,ds\vert }. \end{aligned}$$
4 Quasilinear Levin-Nohel equations
- (a)For each \(k\in\{N+1,\ldots,M\}\), \(g_{k}(0)=0\) and \(g_{k}(x)\) is a Lipschitz function, i.e.where \(K_{k}\) is a finite positive constant.$$ \bigl\vert g_{k}(x)-g_{k}(y)\bigr\vert \leq K_{k}\vert x-y\vert ,\quad\forall x,y\in \mathbf{R} , $$(26)
- (b)
The kernel \(a_{k}(t,s)\) is continuous and fulfils the condition (6) for any \(k\in I=\{1,2,\ldots,M\}\).
- (c)
The delays \(r_{k}(t)\), \(k\in I\) are continuous with \(t-r_{k}(t)\to\infty\) as \(t\to\infty\).
Proposition 5
Dung [18]
Theorem 5
Proof
Once again, the proof of this theorem is similar to that of Theorem 2. So we omit it here. □
We conclude this section with an example.
Example
5 Conclusion
In this paper, we proved a transfer theorem for the Levin-Nohel equations. Because of its simplicity, the transfer theorem provides us with an effective method to investigate the stability of the Levin-Nohel equations. Each of the known stability conditions for the corresponding functional differential equations may give us a stability condition for the original Levin-Nohel equations.
By using the transfer theorem, we obtain simple proofs for some well-known results (Theorems 2 and 3) and new stability conditions (Theorems 4 and 5) for the Levin-Nohel equations.
Declarations
Acknowledgements
The author would like to thank the anonymous referees for their valuable comments and for the reference to [14, 16].
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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