 Research
 Open Access
New results on positive almost periodic solutions for firstorder neutral differential equations
 Yuehua Yu^{1} and
 Shuhua Gong^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366201816481
© The Author(s) 2018
 Received: 1 January 2018
 Accepted: 7 May 2018
 Published: 21 May 2018
Abstract
In this paper, a class of firstorder neutral differential equations with timevarying 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
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
 \((A_{1})\) :

There exist positive constants \(F^{S}\), \(F^{i}\) and a bounded and continuous function \(Q^{*} :\mathbb{R}\rightarrow(0, +\infty)\) such thatwhere \(Q^{*}\) has the lower bound different from zero.$$ F^{i} e ^{ \int_{s}^{t}Q^{*}(u)\,du}\leq e ^{ \int_{s}^{t}Q(u)\,du}\leq F^{S} e ^{ \int_{s}^{t}Q^{*}(u)\,du} \quad\textit{for all } t,s\in\mathbb{R} \textit{ and }ts\geq0, $$(2.1)
 \((A_{2})\) :

There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such that$$ \left \{ \textstyle\begin{array}{l} 0\leq p_{0}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{1},\\ \sup_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{S}[Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\leq(1p_{1})M,\\ \inf_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{i}[Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\geq (1p_{0})m. \end{array}\displaystyle \right . $$(2.2)
 \((A_{3})\) :

There exist positive constants \(L^{f} \) and L such that \(L+p_{1}<1\),Then equation (1.4) has at least one positive almost periodic solution \(x^{*}\) such that \(x^{*}(t)\in[m, M]\) for all \(t \in\mathbb{R}\).$$\begin{gathered} \sup_{t\in \mathbb{R}}F^{S}\frac{ \vert Q(t)P(t) \vert +L^{f}}{Q^{*}(t)}\leq L\quad \textit{and}\\ \bigl\vert f(t,x_{1})f(t,x_{2}) \bigr\vert \leq L^{f} \vert x_{1}  x_{2} \vert \quad\textit {for all } t,x_{1}, x_{2}\in\mathbb{R} .\end{gathered} $$(2.3)
Proof
Next, we will prove that the mapping T is a contraction mapping on B.
Remark 2.1
 \((A_{2}^{*})\) :

There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such that$$\left \{ \textstyle\begin{array}{rcl} &&0\leq p_{0}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{1},\\ &&\sup_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{S}[Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\leq(1p_{1})M,\\ &&\inf_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{i}[Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\geq (1p_{0})m. \end{array}\displaystyle \right . $$
Theorem 2.2
 \((\bar{A}_{2})\) :

There exist positive constants \(p_{0}\), \(p_{1}\), m, and M such thatThen equation (1.4) has at least one positive almost periodic solution \(x^{*}\) such that \(x^{*}(t)\in[m, M]\) for all \(t \in\mathbb{R}\).$$\left \{ \textstyle\begin{array}{rcl} && p_{1}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{0}\leq0,\\ &&\sup_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{S}[Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\leq M+ p_{0}m,\\ &&\inf_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{i}[Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\geq m+p_{1}M. \end{array}\displaystyle \right . $$
Remark 2.2
 \((\bar{A}^{*}_{2})\) :

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
Example 3.1
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 timevarying functions, and \(Q(t)=1 +2 \sin400 t\) fails to satisfy (1.3). One can see that all the results obtained in [1–12, 15] are invalid for (3.1). Note that the space of almost periodic functions contains the space of periodic functions. If we reduce all timevarying delays and coefficients of (1.4) to ωperiodic 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 firstorder neutral differential equations with timevarying 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
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