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Existence of solutions converging to zero for nonlinear delayed differential systems
Advances in Difference Equations volume 2015, Article number: 349 (2015)
Abstract
We present a result about an interesting asymptotic property of real twodimensional delayed differential systems satisfying certain sufficient conditions. We employ two previous results, which were obtained using a Razumikhintype modification of the Ważewski topological method for retarded differential equations and the method of a LyapunovKrasovskii functional. The result is illustrated by a nontrivial explanatory example.
1 Introduction
Various properties of solutions of differential equations with delay were extensively studied recently. Among others we mention [1–9] and the references therein. The results contained in this paper are a generalization of previous research published in [10–12] and [13].
Our aim here is to study the asymptotic behavior of solutions of the following system of differential equations:
where \(t\theta_{k}(t) \ge0\) are bounded nonconstant delays satisfying \(\lim_{t \to \infty} \theta_{k}(t) = \infty\), \(\theta_{k} (t)\) are real functions,
is a real vector function, where \(x=(x_{1},x_{2})\), \(y_{k}=(y_{1k},y_{2k})\), and
are real square matrices.
In this paper, we introduce an interesting result, which is a combination of two theorems presented in [14], one regarding the instability of solutions, the other one dealing with the existence of bounded solutions.
It is supposed that the function h satisfies the Carathéodory conditions on \([t_{0},\infty)\times\mathbb{R}^{2(m+1)}\), the functions \(b_{ijk}\) are locally Lebesgue integrable on \([t_{0},\infty)\), and the functions \(\theta_{k}\), \(a_{ij}\) are locally absolutely continuous on \([t_{0},\infty )\).
Since we study twodimensional systems, we use a transformation into complex variables to simplify the system (1) into one equation with complex coefficients.
The complex variables are defined as \(z=x_{1} + i x_{2}\), \(w_{1}=y_{11} +i y_{12}, \ldots, w_{m}=y_{m1} +i y_{m2}\). Using this transformation we get
where we assume (\(J=[t_{0}, \infty)\)):

\(A_{k}, B_{k} \in L_{\mathrm{loc}} (J, \mathbb{C})\) (i.e. locally Lebesgue integrable complexvalued functions on J) for \(k=1, \ldots, m\),

\(\theta_{k} \in AC_{\mathrm{loc}} (J, \mathbb{R})\) (i.e. locally absolutely continuous realvalued functions on J) for \(k=1, \ldots, m\),

\(a,b \in AC_{\mathrm{loc}} (J, \mathbb{C})\) (i.e. locally absolutely continuous complexvalued functions on J),

\(g \in K (J \times{\mathbb{C}}^{m+1}, \mathbb{C})\) (i.e. a complexvalued function which satisfies the Carathéodory conditions on \(J \times{\mathbb{C}}^{m+1}\)).
Obviously, the function g is in general dependent on z̄ as well as on every \(\bar{z}(\theta_{k})\). However, the fact that the function g satisfies the Carathéodory conditions enables us to significantly simplify the notation by using only z, since the validity of the Carathéodory conditions is not violated by composing with continuous functions z̄, \(\bar{w}_{k}\), and \(\theta_{k}\).
The relations between the functions are the following:
Conversely, putting
equation (2) can be written in the real form (1) as well.
The equivalence of the dynamical invariants and asymptotic properties of the solutions of the real system (1) and the complex equation (2) is shown in [15] for the simple case covering ordinary differential equations.
In this paper we consider (2) in the case when
and study the behavior of the solutions of (2) under this assumption, which generally means that \(\det\mathsf{A}(t) > 0\) for t sufficiently large. This situation corresponds to the case when the equilibrium point 0 of the autonomous homogeneous system
where A is supposed to be a regular constant matrix, is a center, a focus or a node. Such a situation has some geometrical aspects, which are used in an analysis of the transformed equation (2). See [15] for more details.
Further, we suppose that (2) satisfies the uniqueness property of solutions.
2 Preliminaries
Throughout this paper we will assume that
where \(r>0\) is a constant, which means that the delays \(\theta_{k}\) are bounded. This is the same case as considered in [14]. Similar results for a case different from (5) were obtained in [7].
Then there are numbers \(T \ge t_{0}+r\) and \(\mu>0\) such that
Denote
and
Moreover, we assume the following conditions to be valid:
(i) The numbers \(T \ge t_{0}+r\) and \(\mu>0\) satisfy condition (6).
(ii) There are functions \(\varrho, \varkappa, \kappa_{k} : [T,\infty)\to\mathbb{R}\), where ϱ is continuous on \([T,\infty)\), such that
for \(t\ge T\), \(z, w_{k} \in\mathbb{C}\) (\(k = 1, \ldots, m\)).
(ii_{n}) There are numbers \(R_{n}\ge0\) and functions \(\varkappa_{n}, \kappa_{nk} : [T,\infty)\to\mathbb{R}\) satisfying the inequality
for \(t\ge\tau_{n}\ge T\), \(z+\sum_{k=1}^{m} w_{k}>R_{n}\).
(iii) The function \(\beta\in AC_{\mathrm{loc}} ([T,\infty),\mathbb{R}_{}^{0})\) is such that
where \(\lambda_{k}\) is given for \(t\ge T\) by
(iii_{n}) The function \(\beta_{n}\in AC_{\mathrm{loc}}( [T,\infty),\mathbb{R}_{}^{0})\) is such that
where \(\lambda_{nk}\) is given for \(t\ge T\) by
(iv_{n}) The function \(\varLambda_{n}\) is real locally Lebesgue integrable and the inequalities \(\beta'_{n}(t)\ge\varLambda_{n}(t) \beta_{n}(t)\), \(\varTheta_{n}(t)\ge\varLambda_{n}(t)\) are satisfied for almost all \(t\in [\tau_{n},\infty)\), where \(\varTheta_{n}\) is given by \(\varTheta_{n} (t) = \alpha(t) \operatorname {Re}a(t) + \vartheta(t)  \varkappa_{n} (t) + m \beta_{n} (t)\).
Furthermore, denote
3 Main results
First of all, we recall the two results from [14].
Lemma 1
Let the assumptions (i), (ii_{0}), (iii_{0}), (iv_{0}) be fulfilled for some \(\tau_{0}\ge T\). Suppose there exist \(t_{1}\ge\tau_{0}\) and \(\nu\in(\infty,\infty)\) such that
If \(z(t)\) is any solution of (2) satisfying
where
then
for all \(t\ge t_{1}\) for which \(z(t)\) is defined.
Lemma 2
Let the conditions (i), (ii), (iii) be fulfilled and Λ, \(\theta'_{k} \) (\(k=1,\ldots,m\)) be continuous functions such that inequality \(\varLambda(t)\le \varTheta(t)\) holds a.e. on \([T,\infty)\), where Θ is defined by (13). Suppose that \(\xi: [Tr,\infty)\to\mathbb{R}\) is a continuous function such that
for \(t\in[T,\infty]\) and some constant \(C>0\). Then there exists a \(t_{2}>T\) and a solution \(z_{0}(t)\) of (2) satisfying
for \(t\ge t_{2}\).
If we combine the previous two results, we are able to prove the following theorem, which is the fundamental result of this paper.
Theorem 1
Assume that the hypotheses (i), (ii), (\(\mathrm{ii}_{\mathrm{n}}\)), (iii), (\(\mathrm{iii}_{\mathrm{n}}\)), (\(\mathrm{iv}_{\mathrm{n}}\)) are valid for \(T \le\tau_{n}\), where \(0 < R_{n}\), \(n\in\mathbb{N}\), \(\inf_{n\in\mathbb{N}} R_{n}=0\). Suppose that \(\theta'_{k}\), Λ are continuous functions such that inequality \(\varTheta(t) \ge\varLambda(t)\) is satisfied almost everywhere on \([T,\infty)\), where \(\varTheta(t) = \alpha(t) \operatorname {Re}a(t) + \vartheta(t)  \varkappa(t)\). Let \(\xi: [Tr,\infty)\to\mathbb{R}\) be a continuous function satisfying the inequality
for some constant \(C>0\) and \(t\in[T,\infty)\). Assume
for \(n\in\mathbb{N}\), where \(\nu\in(\infty,\infty)\) and \(\theta(t)=\min_{k=1,\ldots,m} \theta_{k} (t)\). Then there is a solution \(z_{0}(t)\) of (2) with the property
Proof
Using Lemma 2 we obtain the existence of \(T \le t_{2}\) and a solution \(z_{0}(t)\) of (2) satisfying for \(t\ge t_{2}\) the inequality
From (20) we get
Lemma 1 yields
for \(\tau\le t\), where Ψ is given by
Assume that (23) does not hold. This implies the existence of \(\varepsilon_{0}>0\) satisfying \(\limsup_{t\to\infty}\min_{\theta(t)\le s\le t}z_{0}(s)>\varepsilon_{0}\). We take \(N\in\mathbb{N}\) such that \(\max \{R_{N},\frac{2}{\mu}R_{N} e^{\nu} \}<\varepsilon_{0}\). Then
holds for some \(\tau>\max\{T,\tau_{N},t_{2}\}\). Taking (22) into account we may assume that
Hence, with respect to (5), (7), (24), (26), (27), and the nonpositiveness of \(\beta_{N}\), we obtain
The inequalities (24) and (25) give the estimation
which means that
for \(\tau\le t\), which is in contradiction to (21). The proof is complete. □
Remark 1
Theorem 1 covers more general situations than Theorem 3 in [7], where the different fundamental assumption \(\liminf_{t \to\infty} ( \operatorname {Im}a(t)  b(t) ) > 0\) is supposed to hold. Indeed, if we take for example \(a(t) \equiv7+i\) and \(b(t) \equiv2i\), then condition (5) in this paper is satisfied but the condition \(\liminf_{t \to\infty } ( \operatorname {Im}a(t)  b(t) ) > 0\) is not valid.
The following nontrivial example was constructed to illustrate an application of the theoretical result presented in Theorem 1.
Example 1
Consider the twodimensional system of the nonlinear delayed differential equations
This system can be written in matrix form (1), where
Following our approach, we use a transformation into the complex plane and obtain the delayed differential equation (2) with complexvalued coefficients,
where \(a(t) \equiv4+3i\), \(b(t) \equiv i\), \(A_{k}(t) \equiv0\), \(B_{k}(t) \equiv0\), \(\theta_{k} (t) =t  \mathrm {e}^{kt}\) for \(k=1,\ldots,m\), \(g(t,z,w_{1},\ldots,w_{m})=\frac{1}{t^{2}} z + \sum_{k=1}^{m} \frac {1}{4m} \mathrm {e}^{2t} w_{k} + \mathrm {e}^{t}\).
Obviously \(t  1 \le\theta_{k} (t) \le t\) and \(\theta'_{k} (t) = 1 + k \mathrm {e}^{kt} \ge1>0\) for \(t \ge0\).
Suppose \(t_{0}=0\) and \(T \ge1\). Then \(\gamma(t) = a(t)+ \sqrt {a(t)^{2}b(t)^{2}} \equiv5+2 \sqrt{6}\), \(c(t)=\bar{a}(t)b(t)/a(t) \equiv\frac{3+4i}{5}\).
Further,
Thus conditions (i) and (ii) are fulfilled with \(\varkappa(t) \equiv \frac{\sqrt{3}}{t^{2} \sqrt{2}}\), \(\kappa_{k} (t) = \frac{\mathrm {e}^{2t} \sqrt {3}}{4m \sqrt{2}}\), and \(\varrho(t) = \mathrm {e}^{t}\). To meet condition (ii_{n}), we estimate for \(R_{n} = \frac{1}{n}\), \(\tau_{n} = n\), and \(z>R_{n}\)
where \(\varkappa_{n}(t) =\frac{\sqrt{3}}{\sqrt{2}} [ \frac{1}{t^{2}} + n \mathrm {e}^{t} ]\) and \(\kappa_{nk}(t) = \frac{\sqrt{3}}{\sqrt{2}} \frac {\mathrm {e}^{2t}}{4m}\).
Condition (iii) holds with
Condition (iii_{n}) is satisfied for
We get condition (iv_{n}) by setting
Further, we put \(\Lambda(t) = \varTheta(t) = \frac{16}{5}  \frac {\sqrt{3}}{t^{2} \sqrt{2}}\).
Then condition (19) holds for
and \(\xi(t) >0\) for \(t \ge T = 1\).
Now it is not difficult to verify conditions (21) and (20), since \(\Lambda_{n} (t)  \xi(t) >0\) and \(\Lambda_{n} (t) >0\) for \(n \in\mathbb{N}\). Investigating the factors of the product in parentheses in (22), we come to the conclusion that
and
Consequently, the product of these factors is asymptotically equal to \(O(\mathrm {e}^{\delta t})\), where \(\delta>0\) and thus condition (22) is satisfied. All assumptions of Theorem 1 are fulfilled and we can conclude that there exist \(t_{2}>1\) and a solution \(z_{0}(t)\) of (30) satisfying (23) for \(t \ge t_{2}\).
Remark 2
This result is slightly surprising. However, it is in good agreement with the wellknown fact that introducing delay into an unstable system without delay can cause a change of behavior of the system. Such situations are described and corresponding results are formulated e.g. in [16].
4 Conclusion
We proved an interesting result about the stability of twodimensional systems with bounded delays. Sufficient conditions for the stability of an originally unstable system were presented. The result is in perfect agreement with the results stated and proved in the established literature. An example showed how this result can be used in practice.
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Acknowledgements
The work of the authors was realized in CEITEC  Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund and the second author was supported by the project FEKTS112921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This support is gratefully acknowledged.
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Rebenda, J., Šmarda, Z. Existence of solutions converging to zero for nonlinear delayed differential systems. Adv Differ Equ 2015, 349 (2015). https://doi.org/10.1186/s1366201506870
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DOI: https://doi.org/10.1186/s1366201506870