In this paper, we study a class of first-order neutral differential equations with time-varying delays and coefficients. Employing the fixed point method and differential inequality techniques, easily verifiable delay-independent criteria are established to ensure the existence and global exponential stability of pseudo-almost periodic solutions for the addressed equations. These theoretical results are also supported with numerical simulations.
Keywords
Pseudo-almost periodic solution First-order neutral differential equation Existence Global exponential stability
to describe the distributed networks containing lossless transmission lines. Here f is a continuous function on \(\mathbb{R}^{2}\), Q is a continuous function on \(\mathbb{R}\), \(x (t)\) represents the state variable, \(r_{1} \) and \(r_{2}\) designate the transmission delays. The detailed biological explanations of the parameters of (1.1) can be found in [1, 2]. It is well known that the variation of the environment plays an important role in many biological and ecological dynamical systems. In particular, periodically varying environment and almost periodically varying environment are foundations for the theory of nature selection. Furthermore, (1.1) has been naturally generalized as the following first-order neutral differential equations with time-varying delays and coefficients:
where \(Q,P\in C(\mathbb{R}, (0, +\infty)), \tau_{1}, \tau_{2}\in C(\mathbb{R}, [0, +\infty))\) are bounded, and \(f\in C (\mathbb{R}\times \mathbb{R}, \mathbb{R})\).
Recently, the existence and stability of periodic solutions or pseudo-almost periodic solutions of (1.2) and its generalized equations have been extensively studied. For example, criteria ensuring the existence of periodic solutions are established in [3, 4, 5, 6, 7, 8, 9] and some sufficient conditions for the existence of pseudo-almost periodic (mild) solutions are obtained in [10, 11]. On the other hand, the global exponential stability of pseudo-almost periodic solutions plays a key role in characterizing the dynamical behavior of biological and ecological dynamical systems since the exponential convergence rate can be unveiled [12, 13, 14, 15, 16, 17, 18, 19, 20]. However, to the best of our knowledge, no such work has been performed on the dynamic analysis of pseudo-almost periodic solution of first-order neutral differential equations with time-varying delays and coefficients. With this motivation, our goal is to study the existence, uniqueness and global exponential stability of pseudo-almost periodic solutions of (1.2). Here, a new approach will be developed to obtain the global exponential convergence for the pseudo-almost periodic solutions.
In this section, a few lemmas, notations and assumptions are cited which will be used in Sect. 3.
Assume that \(\mathbb{B}(\mathbb{R},\mathbb{R} )\) represents the set of all bounded and continuous functions from \(\mathbb{R}\) to \(\mathbb{R} \). Then \((\mathbb{B}(\mathbb{R},\mathbb{R} ), \Vert \cdot \Vert )\) is a Banach space, where \(\Vert \cdot \Vert \) denotes the supremum norm \(\Vert w \Vert := \sup_{ t\in \mathbb{R}} \vert w (t) \vert \).
\(u(t)\in \mathbb{B}(\mathbb{R},\mathbb{R} )\) is said to be almost periodic on \(\mathbb{R}\) if, for any \(\varepsilon>0\), there exists a real number \(l=l(\varepsilon)>0 \) with the property that, for any interval with length \(l(\varepsilon)\), it is possible to find a number \(\delta=\delta(\varepsilon)\) in this interval such that \(\vert u(t+\delta)-u(t) \vert <\varepsilon \) for all \(t\in \mathbb{R}\).
Let \(AP(\mathbb{R},\mathbb{R} )\) be the set of the almost periodic functions from \(\mathbb{R}\) to \(\mathbb{R} \), and
A function \(\varphi\in{\mathbb{B}(\mathbb{R},\mathbb{R} )}\) is called pseudo-almost periodic if it can be expressed as \(\varphi=h+g\), where \(h\in{AP(\mathbb{R},\mathbb{R} )}\) and \(g\in{\mathit{PAP}_{0}(\mathbb{R},\mathbb{R} )}\). Furthermore, \((\mathit{PAP}(\mathbb{R},\mathbb{R} ), \Vert \cdot \Vert )\) is a Banach space and \(AP(\mathbb{R},\mathbb{R} )\) is a proper subspace of \(\mathit{PAP}(\mathbb{R},\mathbb{R} )\) [22].
Suppose that \(\Omega \subset \mathbb{R}\) and K is any compact subset of Ω. Let \(\mathit{PAP}_{0}( \mathbb{R}\times \Omega ,\mathbb{R})\) consist of all bounded and continuous functions ϕ such that
uniformly with respect to \(z\in K\). Let \(\mathit{PAP} ( \mathbb{R}\times \Omega ,\mathbb{R} )\) designate all functions η of the form
$$\eta(t,x)=\phi_{1}(t,x)+\phi_{2}(t,x), $$
where \(\phi_{2}\in \mathit{PAP}_{0}( \mathbb{R}\times \Omega ,\mathbb{R} )\), and \(\phi_{1}\) is an almost periodic function for t uniformly on Ω.
In order to study the pseudo-almost periodic solutions for (1.2), we shall assume that \(Q\in AP(\mathbb{R},\mathbb{R} )\), \(P , \tau_{1}, \tau_{2} \in \mathit{PAP}(\mathbb{R},\mathbb{R} )\), \(f\in \mathit{PAP} (\mathbb{R}\times \mathbb{R}, \mathbb{R})\), \(\tau_{1}\) and P are uniformly continuous on \(\mathbb{R}\), and
$$M[Q]=\lim_{T\rightarrow +\infty}\frac{1}{T} \int_{t}^{t+T}Q(s)\,ds>0 \quad \mbox{for all } t\in \mathbb{R}. $$
Hence, \(x (t )\) exists and is unique on \([0, r]\). Consequently, the lemma follows from the induction. □
3 Main results
In this section, we establish the existence and global exponential stability of pseudo-almost periodic solutions of (1.2) by using the fixed point theorem and Lyapunov functional method.
Theorem 3.1
Assume that the following conditions hold.
(\(A_{1}\))
There exist a positive constant\(K^{*}\)and a bounded and continuous function\(\tilde{Q} :\mathbb{R}\rightarrow (0, +\infty)\)such that
$$e ^{-\int_{s}^{t}Q(u)\,du}\leq K^{*} e ^{ -\int_{s}^{t}\tilde{Q} (u)\,du}\quad \textit{for all } t,s \in \mathbb{R} \textit{ and }t-s\geq 0. $$
(\(A_{2}\))
There exist positive constants\(L^{f} \)andLsuch that
Then Eq. (1.2) has a unique pseudo-almost periodic solution, and the solution\(x(t )\)of (1.2) with initial condition (1.4) converges exponentially to the pseudo-almost periodic solution as\(t\rightarrow+\infty\).
Proof
Set \(\varphi \in B \) and \(F(t,z)= \varphi(t-z) \). According to Theorem 5.3 in [22, p. 58] and Definition 2.2, the uniform continuity of φ entails that \(F \in \mathit{PAP}(\mathbb{R}\times \Omega, \mathbb{R})\) and F is continuous in \(z\in L\) and uniformly in \(t\in \mathbb{R}\) for all compact subset L of \(\Omega\subset \mathbb{R}\). This, together with \(\tau_{i} \in \mathit{PAP}(\mathbb{R},\mathbb{R})\) and Theorem 5.11 in [22, p. 60], involves
Now, we define a mapping \(T:B\longrightarrow AP(\mathbb{R},\mathbb{R})\) as follows:
$$(T \varphi) (t) =P(t)\varphi\bigl(t-\tau_{1}(t)\bigr)+x^{\varphi}(t), \quad \forall \varphi\in B. $$
Next, we prove that the mapping T is a contraction mapping on B.
For all \(t\in \mathbb{R}\), (3.6) entails that \([x^{\varphi}(t)]' \) is bounded on \(\mathbb{R}\), and \(x^{\varphi}(t) \) is uniformly continuous on \(\mathbb{R}\). This, together with the uniform continuities of \(\tau_{1}\) and P, implies that \(P(t)\varphi(t-\tau_{1}(t))\in B\), and the mapping T is a self-mapping from B to B.
Furthermore, for all \(\varphi, \psi \in B \), (3.5), \((A_{1})\) and \((A_{2})\) yield
Thus, the mapping T is a contraction on B. Using Theorem 0.3.1 of [24], we see that the mapping T possesses a unique fixed point \(x^{*}\in B \), \(Tx^{*}=x^{*}\), i.e.,
and \(x^{*}(t)\) is a pseudo almost periodic solution of equation (1.2).
Finally, we prove that \(x^{*}(t)\) is globally exponentially stable.
Suppose that \(x(t) \) is an arbitrary solution of (1.2) associated with initial value \(\phi\in C([-\tau, 0], \mathbb{R}) \). We trivially extend \(x(t) \) to \(\mathbb{R}\) by letting \(x(t) = x(-\tau) \) for \(t\in (-\infty, -\tau]\). Let
which contradicts the fact that \(\vert Y (\theta) \vert = M( \Vert \phi \Vert _{\xi}+\varepsilon)e^{-\lambda \theta}\). Hence, (3.13) holds. Letting \(\varepsilon\longrightarrow 0^{+}\) entails that
$$\begin{aligned} \bigl\Vert Y(t) \bigr\Vert \leq M \Vert \phi \Vert _{\xi}e^{-\lambda t} \quad \mbox{for all } t > 0 . \end{aligned}$$
(3.18)
Then, arguing as in the proof of (3.15) and (3.16), in view of (3.18), we can show
$$e^{\lambda t} \bigl\vert y (t) \bigr\vert \leq \sup_{s\in(-\infty, t]} e^{\lambda s} \bigl\vert y (s) \bigr\vert \leq \frac{M \Vert \varphi \Vert _{\xi} }{1-P ^{+} e^{\lambda \tau_{1} ^{+} }}, $$
and
$$\bigl\vert y (t) \bigr\vert \leq \frac{M \Vert \phi \Vert _{\xi} }{1-P ^{+} e^{\lambda \tau_{1} ^{+} }}e^{-\lambda t} \quad \mbox{for all } t > 0 , $$
which completes the proof. □
Remark 3.1
The results on periodic solutions or almost periodic solutions of (1.2) in references [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] are established under the condition that the decay term coefficient function \(Q(t)\) is not oscillating. In this paper, the assumption \((A_{1})\) relaxes the above technical condition. In fact, one can see Example 4.1 and Remark 4.1 for details.
4 An example and its numerical simulations
Example 4.1
Consider the following first-order neutral differential equations with time-varying delays and coefficients:
which imply that (4.1) satisfies all conditions in Theorem 3.1. Hence, Eq. (4.1) has exactly one positive almost periodic solution \(x^{*}(t)\). The corresponding simulation results of the solutions are seen in Fig. 1.
Numerical solutions \(x(t) \) of system (4.1) for initial value \(\varphi(s)=15,5,-5 \), respectively, where \(s\in [-6, 0]\)
Remark 4.1
It should be mentioned that there is no research on the global exponential convergence of the pseudo-almost periodic solution for first-order neutral differential equations with time-varying delays and coefficients. Moreover, because \(Q(t)=1 +2 \sin 200 t\) is oscillating on \(\mathbb{R}\), one can see that all results in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] cannot be applied to illustrate that all solutions for (4.1) converge exponentially to \(x^{*}(t)\). We all know that the pseudo-almost periodic functions contain almost periodic functions, thus, the derived results are still novel if we reduce all time-varying delays and coefficients of (1.2) to periodic functions or almost periodic functions.
Notes
Acknowledgements
We would like to thank the anonymous referees for their valuable comments, which led to an improvement in the presentation. This work was supported by the Natural Scientific Research Fund of Hunan Provincial of China (Grant Nos. 2018JJ2087, 2018JJ2372), Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 17C1076), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010019) and Zhejiang Provincial Education Department Natural Science Foundation of China (Y201533862).
Authors’ contributions
YHY and SHG worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
References
1.
Komanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic Press, London (1986)
Google Scholar
2.
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamical System. Academic Press, New York (1993)
MATHGoogle Scholar
3.
Liu, B., Huang, L.: Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations. J. Math. Anal. Appl. 322(1), 121–132 (2006)
MathSciNetCrossRefMATHGoogle Scholar
4.
Liu, B., Huang, L.: Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations with a deviating argument. Taiwan. J. Math. 11(2), 497–510 (2007)
MathSciNetCrossRefMATHGoogle Scholar
5.
Luo, Y., Wang, W., Shen, J.: Existence of positive periodic solutions for two kinds of neutral functional differential equations. Appl. Math. Lett. 21, 581–587 (2008)
MathSciNetCrossRefMATHGoogle Scholar
6.
Graef, J.R., Kong, L.: Periodic solutions of first order functional differential equations. Appl. Math. Lett. 24, 1981–1985 (2011)
MathSciNetCrossRefMATHGoogle Scholar
7.
Olach, R.: Positive periodic solutions of delay differential equations. Appl. Math. Lett. 26, 1141–1145 (2013)
MathSciNetCrossRefMATHGoogle Scholar
8.
Peng, L., Wang, L.: Periodic solutions for first order neutral functional differential equations with multiple deviating arguments. Ann. Pol. Math. 111(2), 197–213 (2014)
MathSciNetCrossRefMATHGoogle Scholar
9.
Candan, T.: Existence of positive periodic solutions of first order neutral differential equations with variable coefficients. Appl. Math. Lett. 52, 142–148 (2016)
MathSciNetCrossRefMATHGoogle Scholar
10.
Diagana, T., Hernández, E.M.: Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional differential equations and applications. J. Math. Anal. Appl. 327(2), 776–791 (2007)
MathSciNetCrossRefMATHGoogle Scholar
11.
Diagana, T., Hernández, E.M., José dos Santos, P.C.: Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Anal. 71(1–2), 248–257 (2009)
MathSciNetCrossRefMATHGoogle Scholar
12.
Liu, B.: Pseudo almost periodic solutions for neutral type CNNs with continuously distributed leakage delays. Neurocomputing 148, 445–454 (2015)
CrossRefGoogle Scholar
13.
Liu, B.: Pseudo almost periodic solutions for CNNs with continuously distributed leakage delays. Neural Process. Lett. 42, 233–256 (2015)
CrossRefGoogle Scholar
14.
Shao, J.: Pseudo almost periodic solutions for a Lasota–Wazewska model with an oscillating death rate. Appl. Math. Lett. 43, 90–95 (2015)
MathSciNetCrossRefMATHGoogle Scholar
15.
Yu, Y.: Exponential stability of pseudo almost periodic solutions for cellular neural networks with multi-proportional delays. Neural Process. Lett. 45, 141–151 (2017)
CrossRefGoogle Scholar
16.
Xiong, W.: New results on positive pseudo-almost periodic solutions for a delayed Nicholson’s blowflies model. Nonlinear Dyn. 85, 563–571 (2016)
MathSciNetCrossRefMATHGoogle Scholar
17.
Huang, Z.: Almost periodic solutions for fuzzy cellular neural networks with time-varying delays. Neural Comput. Appl. 28, 2313–2320 (2017)
CrossRefGoogle Scholar
18.
Huang, Z.: Almost periodic solutions for fuzzy cellular neural networks with multi-proportional delays. Int. J. Mach. Learn. Cybern. 8, 1323–1331 (2017)
CrossRefGoogle Scholar
19.
Xu, Y.: Exponential stability of pseudo almost periodic solutions for neutral type cellular neural networks with D operator. Neural Process. Lett. 46, 329–342 (2017)
CrossRefGoogle Scholar
20.
Zhang, H.: New results on the positive pseudo almost periodic solutions for a generalized model of hematopoiesis. Electron. J. Qual. Theory Differ. Equ. 2014(24), 1 (2014)
MathSciNetGoogle Scholar
21.
Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics, vol. 377. Springer, Berlin (1974)
MATHGoogle Scholar
22.
Zhang, C.: Almost Periodic Type Functions and Ergodicity. Kluwer Academic, Beijing (2003)
CrossRefMATHGoogle Scholar
23.
Zhang, C.: Pseudo almost periodic solutions of some differential equations II. J. Math. Anal. Appl. 192, 543–561 (1995)
MathSciNetCrossRefMATHGoogle Scholar
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