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New results on positive almost periodic solutions for first-order neutral differential equations

Advances in Difference Equations20182018:192

https://doi.org/10.1186/s13662-018-1648-1

  • Received: 1 January 2018
  • Accepted: 7 May 2018
  • Published:

Abstract

In this paper, a class of first-order neutral differential equations with time-varying delays and coefficients is considered. Some results on the existence of positive almost periodic solutions for the equations are obtained by using the contracting mapping principle and the differential inequality technique. In addition, an example is given to illustrate our results.

Keywords

  • Positive almost periodic solution
  • First order
  • Neutral differential equation

MSC

  • 34C25
  • 34K13

1 Introduction

In recent years, the following first-order neutral differential equations
$$ \bigl[x (t)-P(t)x(t-r)\bigr]' = -Q(t) x(t)+f\bigl(t,x(t-r)\bigr) $$
(1.1)
and
$$ \bigl[x (t)-cx\bigl(t-\tau(t)\bigr)\bigr]' = -Q(t) x(t)+f\bigl(t,x \bigl(t-\tau(t)\bigr)\bigr) $$
(1.2)
have been extensively used to describe the dynamic behaviors for the blood cell production models, population models, and control models (see, for example, [15] and the references therein). Here, \(Q, \tau\in C(\mathbb{R}, (0, +\infty))\), \(P\in C^{1}(\mathbb{R}, \mathbb{R})\), \(f\in C (\mathbb{R}\times\mathbb{R}, \mathbb{R})\), \(r>0\), and \(|c|<1\) are constants. In particular, assuming that P, Q are ω-periodic functions, f is ω-periodic with respect to the first variable, and
$$ \inf_{t\in\mathbb{R}}Q(t)>0, $$
(1.3)
some criteria ensuring the existence on the positive periodic solutions of (1.1) and (1.2) have been established in [6] and [7], respectively. It is well known that almost periodically variable coefficients and delays in differential equations of population and ecology problems are much more realistic in the real world. Therefore, in recent years, there has been increasing interest in the existence and stability of almost periodic type solutions for first-order functional differential equations in population models [812]. However, to the best of our knowledge, there are few papers published on positive almost periodic solutions of (1.1) and (1.2). Motivated by the above discussions, in this paper we aim to establish some sufficient conditions on the existence of positive almost periodic solutions of the following first-order neutral differential equations with time-varying delays and coefficients:
$$ \bigl[x (t)-P(t)x\bigl(t-\tau_{1}(t)\bigr)\bigr]' = -Q(t) x(t)+f\bigl(t,x\bigl(t-\tau_{2} (t)\bigr)\bigr), $$
(1.4)
where \(Q,P\in C(\mathbb{R}, (0, +\infty))\), \(\tau_{1}, \tau_{2}\in C(\mathbb{R}, [0, +\infty))\) are almost periodic functions, \(f\in C (\mathbb{R}\times\mathbb{R}, \mathbb{R})\) is an almost periodic function for t uniformly on \(\mathbb{R}\), and
$$M[Q]=\lim_{T\rightarrow+\infty}\frac{1}{T} \int_{t}^{t+T}Q(s)\,ds>0 , $$
where the limit above is independent of t. The contributions of this paper can be summarized as follows: (1) In this manuscript, all delays and coefficients of (1.4) are time-varying, and (1.1) and (1.2) are special cases of (1.4); (2) The sufficient conditions for the existence of positive almost periodic solution are derived in terms of its coefficients without (1.3), which has not been investigated till now.

Throughout this paper, we denote the set of almost periodic functions from \(\mathbb{R}\) to \(\mathbb{R}\) by \(\mathit{AP}(\mathbb{R},\mathbb{R})\). Then, \(( \mathit{AP}(\mathbb{R},\mathbb{R}), \|\cdot\|_{\infty})\) is a Banach space, where \(\|\cdot\|_{\infty}\) denotes the supremum \(\|u\|_{\infty} := \sup_{ t\in\mathbb{R}} |u (t)| \). For more details, we refer the reader to [13, 14].

2 Main results

Theorem 2.1

Let \(\tau_{1}(t)\not\equiv\tau_{2}(t)\) for all \(t\in\mathbb{R}\), and assume that the following conditions hold:
\((A_{1})\)
There exist positive constants \(F^{S}\), \(F^{i}\) and a bounded and continuous function \(Q^{*} :\mathbb{R}\rightarrow(0, +\infty)\) such that
$$ F^{i} e ^{ -\int_{s}^{t}Q^{*}(u)\,du}\leq e ^{ -\int_{s}^{t}Q(u)\,du}\leq F^{S} e ^{ -\int_{s}^{t}Q^{*}(u)\,du} \quad\textit{for all } t,s\in\mathbb{R} \textit{ and }t-s\geq0, $$
(2.1)
where \(Q^{*}\) has the lower bound different from zero.
\((A_{2})\)
There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such that
$$ \left \{ \textstyle\begin{array}{l} 0\leq p_{0}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{1},\\ \sup_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{S}[-Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\leq(1-p_{1})M,\\ \inf_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{i}[-Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\geq (1-p_{0})m. \end{array}\displaystyle \right . $$
(2.2)
\((A_{3})\)
There exist positive constants \(L^{f} \) and L such that \(L+p_{1}<1\),
$$\begin{gathered} \sup_{t\in \mathbb{R}}F^{S}\frac{ \vert Q(t)P(t) \vert +L^{f}}{Q^{*}(t)}\leq L\quad \textit{and}\\ \bigl\vert f(t,x_{1})-f(t,x_{2}) \bigr\vert \leq L^{f} \vert x_{1} - x_{2} \vert \quad\textit {for all } t,x_{1}, x_{2}\in\mathbb{R} .\end{gathered} $$
(2.3)
Then equation (1.4) has at least one positive almost periodic solution \(x^{*}\) such that \(x^{*}(t)\in[m, M]\) for all \(t \in\mathbb{R}\).

Proof

Set
$$B =\bigl\{ \varphi|\varphi\in \mathit{AP}(\mathbb{R},\mathbb{R}) , m\leq \varphi(t) \leq M \mbox{ for all } t\in\mathbb{R} \bigr\} . $$
Clearly, B is a closed subset of \(\mathit{AP}(\mathbb{R},\mathbb{R})\). For any \(\varphi\in B\), we consider an auxiliary equation
$$ x '(t)=-Q(t)x(t)-Q(t)P(t)\varphi\bigl(t-\tau_{1}(t) \bigr)+f\bigl(t, \varphi\bigl(t-\tau _{2}(t)\bigr)\bigr) . $$
(2.4)
In view of the fact that \(M[Q]>0\), it follows from Theorem 7.7 of [13] that system (2.4) has exactly one almost periodic solution
$$ x^{\varphi}(t) = \int_{-\infty}^{t}e^{-\int_{s}^{t}Q(u)\,du}\bigl[-Q(s)P(s)\varphi \bigl(s-\tau _{1}(s)\bigr)+f\bigl(s, \varphi\bigl(s- \tau_{2}(s)\bigr)\bigr)\bigr]\,ds, \quad\forall\varphi\in B, $$
(2.5)
where
$$\begin{aligned}[b] \bigl[ x^{\varphi}(t) \bigr]' &= \biggl\{ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q(u)\,du}\bigl[-Q(s)P(s)\varphi \bigl(s-\tau _{1}(s)\bigr)+f\bigl(s, \varphi\bigl(s- \tau_{2}(s)\bigr)\bigr)\bigr]\,ds \biggr\} ' \\ &= -Q(t)\biggl\{ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q(u)\, du}\bigl[-Q(s)P(s)\varphi \bigl(s-\tau_{1}(s)\bigr)+f\bigl(s, \varphi\bigl(s-\tau_{2}(s) \bigr)\bigr)\bigr]\, ds\biggr\} \\ &\quad{}-Q(t)P(t)\varphi\bigl(t-\tau_{1}(t)\bigr) +f\bigl(t, \varphi\bigl(t- \tau_{2}(t)\bigr)\bigr) \\ &= -Q(t)x^{\varphi}(t)-Q(t)P(t)\varphi\bigl(t-\tau_{1}(t) \bigr)+f\bigl(t, \varphi\bigl(t-\tau_{2}(t)\bigr)\bigr).\end{aligned} $$
(2.6)
In view of \(P,\tau_{1}\in \mathit{AP}(\mathbb{R},\mathbb{R})\) and Lemma 2.4 in [15], we obtain
$$P(t)\varphi\bigl(t-\tau_{1}(t)\bigr)\in \mathit{AP}(\mathbb{R},\mathbb{R}),\quad \forall\varphi\in B. $$
Now, we define a mapping \(T:B\rightarrow \mathit{AP}(\mathbb{R},\mathbb{R})\) as follows:
$$(T \varphi) (t) =P(t)\varphi\bigl(t-\tau_{1}(t)\bigr)+x^{\varphi}(t),\quad \forall\varphi\in B. $$

Next, we will prove that the mapping T is a contraction mapping on B.

For all \(t\in\mathbb{R}\), according to \((A_{1})\) and \((A_{2})\), we have
$$\begin{gathered} P(t)\varphi\bigl(t-\tau_{1}(t)\bigr)+ x^{\varphi}(t) \\ \quad\leq p_{1}M+ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du} F^{S} \bigl[-Q(s)P(s)\varphi\bigl(s-\tau_{1}(s)\bigr) \\ \qquad{}+f\bigl(s, \varphi\bigl(s-\tau_{2}(s)\bigr)\bigr) \bigr]\,ds \\ \quad\leq p_{1}M+ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du}(1- p_{1}) MQ^{*}(s) \,ds \leq M\end{gathered} $$
and
$$\begin{gathered} P(t)\varphi\bigl(t-\tau_{1}(t)\bigr)+ x^{\varphi}(t) \\ \quad\geq p_{0}m+ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du} F^{i} \bigl[-Q(s)P(s)\varphi\bigl(s-\tau_{1}(s)\bigr) \\ \qquad{}+f\bigl(s, \varphi\bigl(s-\tau_{2}(s)\bigr)\bigr) \bigr]\,ds \\ \quad\geq p_{0}m+ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du}(1- p_{0}) m Q^{*}(s) \,ds \geq m,\end{gathered} $$
which imply that the mapping T is a self-mapping from B to B.
Furthermore, for all \(\varphi, \psi\in B \), (2.5), \((A_{1})\) and \((A_{3})\) yield
$$\begin{gathered} \Vert T \varphi-T \psi \Vert _{\infty} \\ \quad \leq\sup_{t \in \mathbb{R}} \biggl\{ \bigl\vert P(t)\bigl[\varphi \bigl(t-\tau_{1}(t)\bigr)-\psi\bigl(t-\tau _{1}(t)\bigr)\bigr] \bigr\vert \\ \qquad{}+ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du} F^{S} \bigl\vert -Q(s)P(s) \bigl(\varphi\bigl(s-\tau_{1}(s)\bigr)-\psi\bigl(s- \tau_{1}(s)\bigr)\bigr) \\ \qquad{}+\bigl(f\bigl(s, \varphi\bigl(s-\tau_{2}(s)\bigr)\bigr)-f\bigl(s, \psi\bigl(s-\tau_{2}(s)\bigr)\bigr)\bigr) \bigr\vert \,ds \biggr\} \\ \quad\leq \Vert \varphi-\psi \Vert _{\infty} \biggl\{ p_{1}+\sup _{t \in \mathbb{R}} \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du} F^{S}\bigl[ \bigl\vert Q(s)P(s) \bigr\vert +L^{f}\bigr] \,ds \biggr\} \\ \quad\leq \Vert \varphi-\psi \Vert _{\infty}\biggl[p_{1}+\sup _{t \in \mathbb{R}} \int_{-\infty}^{t}e^{-\int_{s}^{t}Q^{*}(u)\,du}Q^{*}(s)L \,ds \biggr] \\ \quad\leq (p_{1}+L ) \Vert \varphi-\psi \Vert _{\infty} .\end{gathered} $$
Thus, the mapping T is a contraction on B. Using the classical contraction mapping principle of Banach–Caccioppoli, we obtain that the mapping T possesses a unique fixed point \(x^{*}\in B \), \(Tx^{*}=x^{*}\), i.e.,
$$\begin{aligned} x^{*}(t) &=P(t)x^{*}\bigl(t-\tau_{1}(t)\bigr)+x ^{x^{*}}(t) \\ &=P(t)x^{*}\bigl(t-\tau_{1}(t)\bigr)\\ &\quad{}+ \int_{-\infty}^{t}e^{-\int_{s}^{t}Q(u)\,du}\bigl[-Q(s)P(s)x^{*} \bigl(s-\tau _{1}(s)\bigr)+f\bigl(s, x^{*}\bigl(s- \tau_{2}(s)\bigr)\bigr)\bigr]\,ds,\end{aligned} $$
which together with (2.6) leads to
$$\bigl[x ^{*} (t)-P(t)x^{*}\bigl(t-\tau_{1}(t) \bigr)\bigr]' = -Q(t) x^{*}(t)+f\bigl(t,x^{*} \bigl(t-\tau_{2}(t)\bigr)\bigr). $$
This completes the proof. □

Remark 2.1

When \(\tau_{1}(t) \equiv\tau_{2}(t)\) for all \(t\in\mathbb{R}\), the statement of Theorem 2.1 remains valid if we replace \((A_{2})\) by the following condition:
\((A_{2}^{*})\)
There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such that
$$\left \{ \textstyle\begin{array}{rcl} &&0\leq p_{0}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{1},\\ &&\sup_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{S}[-Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\leq(1-p_{1})M,\\ &&\inf_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{i}[-Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\geq (1-p_{0})m. \end{array}\displaystyle \right . $$

Theorem 2.2

Suppose \((A_{1})\) and \((A_{3})\) hold. If \(\tau_{1}(t)\not\equiv\tau _{2}(t)\) for all \(t\in\mathbb{R}\), and the following condition holds:
\((\bar{A}_{2})\)
There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such that
$$\left \{ \textstyle\begin{array}{rcl} &&- p_{1}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)=- p_{0}\leq0,\\ &&\sup_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{S}[-Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\leq M+ p_{0}m,\\ &&\inf_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{i}[-Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\geq m+p_{1}M. \end{array}\displaystyle \right . $$
Then equation (1.4) has at least one positive almost periodic solution \(x^{*}\) such that \(x^{*}(t)\in[m, M]\) for all \(t \in\mathbb{R}\).

Remark 2.2

When \(\tau_{1}(t) \equiv\tau_{2}(t)\) for all \(t\in\mathbb{R}\), the statement of Theorem 2.2 holds if we substitute \((\bar{A}_{2})\) into the following condition:
\((\bar{A}^{*}_{2})\)
There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such that
$$\left \{ \textstyle\begin{array}{rcl} &&- p_{1}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)=- p_{0}\leq0,\\ &&\sup_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{S}[-Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\leq M+ p_{0}m,\\ &&\inf_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{i}[-Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\geq m+p_{1}M. \end{array}\displaystyle \right . $$

3 An example

Figure 1
Figure 1

Numerical solutions \(x(t) \) of systems (3.1) for initial values \(x(0)=11,23,32\), respectively

Example 3.1

Consider the following first-order neutral differential equations with time-varying delays and coefficients:
$$\begin{aligned}[b] &\biggl[ x (t)-\frac{ \sin^{2} t }{100}x\bigl(t-\bigl(1+\sin^{2}t\bigr) \bigr)\biggr]' \\ &\quad = -(1 +2 \sin400 t) x (t)+ 20 +e^{\cos\sqrt{2}t}+\frac{1}{100}\cos x \bigl(t-\bigl(1+\sin^{2}\sqrt{3}t\bigr)\bigr), \end{aligned} $$
(3.1)
where
$$ \left \{ \textstyle\begin{array}{l} P (t)= \frac{ \sin^{2} t}{100},\qquad p_{0}=0,\qquad p_{1}=\frac{1}{100},\\ \tau_{1}(t)=1+\sin^{2}t, \qquad\tau_{2}(t)=1+\sin^{2}\sqrt{3}t,\\ Q(t)=1 +2 \sin400 t,\qquad Q^{*}(t) = 1 ,\qquad M[Q] =1,\\ F^{S}= e^{ \frac{1}{100}},\qquad F^{i}= e^{-\frac{1}{100}}, \\ f(t, x)= 20 +e^{\cos \sqrt{2}t}+\frac{1}{100}\cos x ,\qquad L^{f}=\frac{1}{100},\qquad L= \frac{1}{25}. \end{array}\displaystyle \right . $$
(3.2)
Taking \(m=10\), \(M=40\), we can easily show that (3.2) implies that (3.1) satisfies \((A_{1})\), \((A_{2})\), and \((A_{3})\). Hence, equation (3.1) has exactly one positive almost periodic solution \(x^{*}(t)\).

Remark 3.1

In equation (3.1), \(\tau_{1}(t)=1+\sin^{2}t\) and \(\tau _{2}(t)=1+\sin ^{2}\sqrt{3}t\) are two different time-varying functions, and \(Q(t)=1 +2 \sin400 t\) fails to satisfy (1.3). One can see that all the results obtained in [112, 15] are invalid for (3.1). Note that the space of almost periodic functions contains the space of periodic functions. If we reduce all time-varying delays and coefficients of (1.4) to ω-periodic functions, the derived results are still novel.

4 Conclusion

It is well known that the existence of positive almost periodic solutions plays an important role in characterizing the behavior of nonlinear differential equations. Thus it has been extensively investigated by numerous scholars in recent years. In this article, we have investigated a class of first-order neutral differential equations with time-varying delays and coefficients. With the aid of the contraction mapping fixed point theorem and differential inequality theory, some sufficient conditions for the existence of positive almost periodic solutions of the system were established. In order to demonstrate the usefulness of the presented results, an illustrative example was given. The established results were compared with those of recent methods existing in the literature. We expect to extend this work to more types of neutral differential equations with almost periodic delays and coefficients.

Declarations

Acknowledgements

Our deepest gratitude goes to the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially.

Funding

This work was supported by the Natural Scientific Research Fund of Hunan Provincial of China (Grant Nos. 2018JJ2087, 2018JJ2372), the Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 17C1076), the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010019), and the 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.

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

(1)
College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, P.R. China
(2)
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, P.R. China

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