Existence of solutions converging to zero for nonlinear delayed differential systems
 Josef Rebenda^{1} and
 Zdeněk Šmarda^{1, 2}Email author
https://doi.org/10.1186/s1366201506870
© Rebenda and Šmarda 2015
Received: 15 September 2015
Accepted: 5 November 2015
Published: 14 November 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.
Keywords
MSC
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].
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.

\(A_{k}, B_{k} \in L_{\mathrm{loc}} (J, \mathbb{C})\) (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 \(J \times{\mathbb{C}}^{m+1}\)).
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.
Further, we suppose that (2) satisfies the uniqueness property of solutions.
2 Preliminaries
Moreover, we assume the following conditions to be valid:
(i) The numbers \(T \ge t_{0}+r\) and \(\mu>0\) satisfy condition (6).
(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 \(\varTheta_{n} (t) = \alpha(t) \operatorname {Re}a(t) + \vartheta(t)  \varkappa_{n} (t) + m \beta_{n} (t)\).
3 Main results
First of all, we recall the two results from [14].
Lemma 1
Lemma 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
Proof
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
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 \(\gamma(t) = a(t)+ \sqrt {a(t)^{2}b(t)^{2}} \equiv5+2 \sqrt{6}\), \(c(t)=\bar{a}(t)b(t)/a(t) \equiv\frac{3+4i}{5}\).
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.
Declarations
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.
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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