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New results on positive almost periodic solutions for firstorder neutral differential equations
Advances in Difference Equations volumeÂ 2018, ArticleÂ number:Â 192 (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.
1 Introduction
In recent years, the following firstorder neutral differential equations
and
have been extensively used to describe the dynamic behaviors for the blood cell production models, population models, and control models (see, for example, [1â€“5] 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 Ï‰periodic functions, f is Ï‰periodic with respect to the first variable, and
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 firstorder functional differential equations in population models [8â€“12]. 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 firstorder neutral differential equations with timevarying delays and coefficients:
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
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 timevarying, 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
$$ F^{i} e ^{ \int_{s}^{t}Q^{*}(u)\,du}\leq e ^{ \int_{s}^{t}Q(u)\,du}\leq F^{S} e ^{ \int_{s}^{t}Q^{*}(u)\,du} \quad\textit{for all } t,s\in\mathbb{R} \textit{ and }ts\geq0, $$(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
$$ \left \{ \textstyle\begin{array}{l} 0\leq p_{0}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{1},\\ \sup_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{S}[Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\leq(1p_{1})M,\\ \inf_{t\in\mathbb{R}, x,y\in[m, M]}\frac{F^{i}[Q(t)P(t)x +f(t, y) ]}{Q^{*}(t)}\geq (1p_{0})m. \end{array}\displaystyle \right . $$(2.2)  \((A_{3})\) :

There exist positive constants \(L^{f} \) and L such that \(L+p_{1}<1\),
$$\begin{gathered} \sup_{t\in \mathbb{R}}F^{S}\frac{ \vert Q(t)P(t) \vert +L^{f}}{Q^{*}(t)}\leq L\quad \textit{and}\\ \bigl\vert f(t,x_{1})f(t,x_{2}) \bigr\vert \leq L^{f} \vert x_{1}  x_{2} \vert \quad\textit {for all } t,x_{1}, x_{2}\in\mathbb{R} .\end{gathered} $$(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
Clearly, B is a closed subset of \(\mathit{AP}(\mathbb{R},\mathbb{R})\). For any \(\varphi\in B\), we consider an auxiliary equation
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
where
In view of \(P,\tau_{1}\in \mathit{AP}(\mathbb{R},\mathbb{R})\) and Lemma 2.4 in [15], we obtain
Now, we define a mapping \(T:B\rightarrow \mathit{AP}(\mathbb{R},\mathbb{R})\) as follows:
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
and
which imply that the mapping T is a selfmapping from B to B.
Furthermore, for all \(\varphi, \psi\in B \), (2.5), \((A_{1})\) and \((A_{3})\) yield
Thus, the mapping T is a contraction on B. Using the classical contraction mapping principle of Banachâ€“Caccioppoli, we obtain that the mapping T possesses a unique fixed point \(x^{*}\in B \), \(Tx^{*}=x^{*}\), i.e.,
which together with (2.6) leads to
This completes the proof.â€ƒâ–¡
Remark 2.1
When \(\tau_{1}(t) \equiv\tau_{2}(t)\) for all \(t\in\mathbb{R}\), the statement of TheoremÂ 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
$$\left \{ \textstyle\begin{array}{rcl} &&0\leq p_{0}=\inf_{t\in\mathbb{R}}P(t)\leq \sup_{t\in\mathbb{R}}P(t)= p_{1},\\ &&\sup_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{S}[Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\leq(1p_{1})M,\\ &&\inf_{t\in\mathbb{R}, x \in[m, M]}\frac{F^{i}[Q(t)P(t)x +f(t, x) ]}{Q^{*}(t)}\geq (1p_{0})m. \end{array}\displaystyle \right . $$
Theorem 2.2
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 TheoremÂ 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
Example 3.1
Consider the following firstorder neutral differential equations with timevarying delays and coefficients:
where
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 timevarying functions, and \(Q(t)=1 +2 \sin400 t\) fails to satisfy (1.3). One can see that all the results obtained in [1â€“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 Ï‰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 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.
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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).
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YHY and SHG worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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Yu, Y., Gong, S. New results on positive almost periodic solutions for firstorder neutral differential equations. Adv Differ Equ 2018, 192 (2018). https://doi.org/10.1186/s1366201816481
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DOI: https://doi.org/10.1186/s1366201816481