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Positive periodic solutions for nonlinear firstorder delayed differential equations at resonance
Boundary Value Problems volume 2018, Article number: 187 (2018)
Abstract
This paper studies the existence of positive periodic solutions of the following delayed differential equation:
where \(a, \tau \in C(\mathbb{R},\mathbb{R})\) are ωperiodic functions with \(\int_{0}^{\omega }a(t)\,dt=0\), \(f:\mathbb{R}\times [0, \infty)\to \mathbb{R}\) is continuous and ωperiodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems are established. Our main results enrich and complement those available in the literature.
Introduction
In the past few years, there has been considerable interest in the existence of positive periodic solutions for the firstorder equation
where \(a, b\in C(\mathbb{R},[0,\infty))\) are ωperiodic functions with
and τ is a continuous ωperiodic function. Note that when \(\lambda =0\), equation (1.1) reduces to \(u'=a(t)u\), which is well known in Malthusian population models. In real world applications, (1.1) has also been viewed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, as well as cardiac arrhythmias. See, for instance, [1,2,3,4,5,6,7,8,9,10] for some research works on this topic. Meanwhile, many researchers have also paid their attention to the differential systems corresponds to (1.1), namely,
where \(a_{i}, b_{i}\in C(\mathbb{R},[0,\infty))\) are ωperiodic functions satisfying
Here we refer the readers to [11,12,13] and the references listed therein.
Obviously, the basic assumption \(\int_{0}^{\omega }a(t)\,dt>0\) or \(\int_{0}^{\omega }a_{i}(t)\,dt>0\) (\(i=1,2,\dots,n\)), usually employed to ensure the linear boundary value problem
is nonresonant, has played a key role in the arguments of the above mentioned papers. In fact, this assumption ensures that a number of tools, such as fixed point theory, bifurcation theory and so on, could be employed to study the corresponding problems and establish the desired existence results. Here the linear problem (1.2) is called nonresonant if the unique solution of it is the trivial one. It is well known if (1.2) is nonresonant then, provided h is an \(L^{1}\)function, the Fredholm’s alternative theorem implies that the nonhomogeneous problem
always admits a unique solution which, moreover, can be written as
where \(G(t,s)\) is the Green’s function associated to (1.2), see [7,8,9,10,11,12,13] for more details.
Compared with the nonresonant problems, the research of resonant problems proceeds very slowly and the related results are few. And, of course, a natural and interesting question is whether or not the corresponding nonlinear equation possesses a positive periodic solution, provided that
which means a may change its sign on \(\mathbb{R}\) and the studied problem is resonant. In the present paper, we shall give a positive answer to the above question. More concretely, several new existence and multiplicity results will be established for the resonant equation
To the best of our knowledge, the above problem has not been studied so far, and our results shall fill this gap.
For simplicity, we say a function \(q\gg 0\) provided that \(q: \mathbb{R}\to (0,\infty)\) is ωperiodic and continuous. If \(q:\mathbb{R}\to [0,\infty)\) is ωperiodic and continuous with \(\int_{0}^{\omega }q(t)\,dt>0\), then it’s denoted as \(q\succ 0\). Thus, if we choose a function \(\chi \gg 0\) such that \(p:=a+\chi \succ 0\), then the linear differential operator \(Lu:=u'+p(t)u\) is invertible since
Moreover, it is not difficult to see that u is a positive periodic solution of (1.3) if and only if it is a positive periodic solution of
Therefore, we shall focus on (1.4).
Throughout the paper, we make the following assumptions:

(H1)
\(a, \tau \in C(\mathbb{R},\mathbb{R})\) are ωperiodic functions with \(\int_{0}^{\omega }a(t)\,dt=0\);

(H2)
There exists \(\chi \gg 0\) such that \(p:=a+\chi \succ 0\);

(H3)
\(f\in C(\mathbb{R}\times [0,\infty),\mathbb{R})\) is ωperiodic with respect to t and \(f(t,u)\geq \chi (t)u\).
Remark 1.1
Obviously, since a and f are signchanging, equation (1.3) is more general than corresponding ones studied in the existing literature. For other existence results on nonlinear differential equations at resonance, we refer the readers to [14,15,16,17] and the references listed therein.
The rest of the paper is arranged as follows. In Sect. 2, we introduce some preliminaries. Finally, in Sect. 3, we shall state and prove our main results. In addition, several remarks will be given to demonstrate the feasibility of our main results.
Preliminaries
Let us consider the linear boundary value problem
where p is defined as in (H2). If we denote by \(K(t,s)\) the Green’s function of (2.1), then a simple calculation gives
Lemma 2.1
Suppose (H1) and (H2) hold. Let \(\delta =e^{ \int_{0}^{\omega }p(t)\,dt}\). Then
Moreover,
Let E be the Banach space of continuous ωperiodic functions equipped with the norm \(\u\=\max_{t\in [0,\omega ]}u(t)\). For \(h\in E\), define
Then we have
Lemma 2.2
Suppose (H1) and (H2) hold. Then \(A:E\to E\) is a completely continuous linear operator. Moreover, if \(h\succ 0\), then \((Ah)(t)>0\) on \([0,\omega ]\).
Proof
By a standard argument, we can easily show that A is a linear completely continuous operator. In addition, Lemma 2.1 yields \(K(t,s)>0\) for any \((t,s)\), which, together with \(h\succ 0\), implies \((Ah)(t)>0\) on \([0,\omega ]\). □
Let
and
Then \(\mathcal{P}\) is a positive cone in E. Moreover, it is not difficult to check that (1.4) can be rewritten as an equivalent operator equation
Lemma 2.3
Suppose (H1), (H2) and (H3) hold. Then \(T(\mathcal{P})\subseteq \mathcal{P}\) and \(T:\mathcal{P}\to \mathcal{P}\) is completely continuous.
Proof
Assume \(u\in \mathcal{P}\), then \(u(t)\geq \sigma \u\\). It follows from (H3) that \(\chi (s)u(s)+f(s,u(s\tau (s)))\) is nonnegative, and therefore
Applying (H3) again, we get
This, together with (2.3), yields \(T(\mathcal{P})\subseteq \mathcal{P}\). Finally, by Lemma 2.2 and an argument similar to that of [12, Lemmas 2.2, 2.3] with obvious changes, we can prove that T is a completely continuous operator. □
The following lemma is crucial to prove our main results.
Lemma 2.4
(Guo–Krasnoselskii’s fixed point theorem [18])
Let E be a Banach space, and let \(\mathcal{P}\subseteq E\) a cone. Assume \(\varOmega_{1}\), \(\varOmega_{2}\) are two open bounded subsets of E with \(0\in \varOmega_{1}, \bar{\varOmega }_{1}\subseteq \varOmega_{2}\), and let \(T:\mathcal{P}\cap (\bar{\varOmega }_{2}\setminus \varOmega_{1})\to \mathcal{P}\) be a completely continuous operator such that

(i)
\(\Tu\\leq \u\\), \(u\in \mathcal{P}\cap \partial \varOmega _{1}\), and \(\Tu\\geq \u\\), \(u\in \mathcal{P}\cap \partial \varOmega _{2}\); or

(ii)
\(\Tu\\geq \u\\), \(u\in \mathcal{P}\cap \partial \varOmega _{1}\), and \(\Tu\\leq \u\\), \(u\in \mathcal{P}\cap \partial \varOmega _{2}\).
Then T has a fixed point in \(\mathcal{P}\cap (\bar{\varOmega } _{2}\setminus \varOmega_{1})\).
Main results
In this section, we state and prove our main findings.
Theorem 3.1
Let (H1)–(H3) hold. If
and
then (1.3) admits at least one positive ωperiodic solution.
Proof
For \(0< r< R<\infty \), setting
we have \(0\in \varOmega_{1}\), \(\bar{\varOmega }_{1}\subseteq \varOmega_{2}\).
It follows from (3.1) that there exists \(r>0\) so that for any \(0< u\leq r\),
where c is a positive constant satisfying \(cM\omega <1\). Therefore, for \(u\in \mathcal{P}\) with \(\u\=r\),
Moreover, \(0<\sigma \u\\leq u(t)\leq \u\=r\). Thus,
which implies \(\Tu\\leq \u\\), \(\forall u\in \mathcal{P}\cap \partial \varOmega_{1}\).
On the other hand, (3.2) yields the existence of \(\tilde{R}>0\) such that for any \(u\geq \tilde{R}\),
where \(\eta >0\) is a constant large enough such that \(\sigma m\omega (\eta +\underline{\chi })>1\) and \(\underline{\chi }=\min_{t\in [0,\omega ]}\chi (t)\). Fixing \(R>\max \{r, \frac{ \tilde{R}}{\sigma }\}\) and letting \(u\in \mathcal{P}\) with \(\u\=R\), we get \(u(t)\geq \sigma \u\=\sigma R>\tilde{R}\), and therefore
Consequently, for \(u\in \mathcal{P}\) with \(\u\=R\), we can conclude
Hence \(\Tu\\geq \u\\), \(\forall u\in \mathcal{P}\cap \partial \varOmega _{2}\).
Consequently, by Lemma 2.4(i), T has a fixed point in \(\mathcal{P} \cap (\bar{\varOmega }_{2}\setminus \varOmega_{1})\), which is just a positive ωperiodic solution of (1.3). □
Theorem 3.2
Let (H1)–(H3) hold. If
and
then (1.3) admits at least one positive ωperiodic solution.
Proof
We follow the same strategy and notations as in the proof of Theorem 3.1. Firstly, we show that for \(r>0\) sufficiently small,
From (3.3) it follows that there exists \(\tilde{r}>0\) such that \(f(t,u)\geq \beta u\) for any \(0< u\leq \tilde{r}\), where \(\beta >0\) is a constant large enough such that \(\sigma m\omega (\beta +\underline{ \chi })>1\). Therefore, for \(0< r\leq \tilde{r}\), if \(u\in \mathcal{P}\) and \(\u\=r\), then
Furthermore, we obtain
Thus, (3.5) is true.
Next we show for \(R>0\) sufficiently large,
It follows from (3.4) that there exists \(\tilde{R}>0\) so that for any \(u\geq \tilde{R}\),
where \(\mu >0\) satisfies \(\mu M\omega <1\). Let \(R>\max \{\tilde{r}, \frac{ \tilde{R}}{\sigma }\}\), then if \(u\in \mathcal{P}\) and \(\u\=R\), we can obtain
and therefore,
Thus for \(u\in \mathcal{P}\) with \(\u\=R\), we have
which means that (3.6) is also true.
Finally, it follows from Lemma 2.4(ii) that T has a fixed point in \(\mathcal{P}\cap (\bar{\varOmega }_{2}\setminus \varOmega_{1})\), which is just a positive ωperiodic solution of (1.3). □
In the rest of this section, we shall investigate the multiplicity of positive ωperiodic solutions of (1.3). To the end, we assume:

(H4)
\(\lim_{u\to 0+}\frac{f(t,u)}{u}=\infty \) and \(\lim_{u\to +\infty }\frac{f(t,u)}{u}=\infty \) uniformly for \(t\in [0,\omega ]\). In addition, there is a constant \(\alpha >0\) such that
$$ \max \bigl\{ f(t,u): \sigma \alpha \leq u\leq \alpha, t\in [0,\omega ]\bigr\} \leq \bigl(\epsilon \chi (t)\bigr)\alpha, $$(3.7)
where \(\epsilon >0\) satisfies \(\epsilon M\omega <1\).
Theorem 3.3
Assume that (H1)–(H4) hold. Then (1.3) admits at least two positive ωperiodic solutions.
Proof
Define
Let \(\varOmega_{1}\) and \(\varOmega_{2}\) be as in the proof of Theorems 3.1 and 3.2. Then for \(0< r<\alpha <R\), we have \(\bar{\varOmega }_{1}\subseteq \varOmega_{3}\), \(\bar{\varOmega }_{3}\subseteq \varOmega_{2}\).
Since \(\lim_{u\to 0+}\frac{f(t,u)}{u}=\infty \) uniformly for \(t\in [0,\omega ]\), by an argument similar to the proof of Theorem 3.2, we can obtain
Similarly, we can show by (H4) that
Clearly, the proof is completed if we prove
Suppose \(u\in \mathcal{P}\) and \(\u\=\alpha \), then \(\sigma \alpha \leq \sigma \u\\leq u(t)\leq \u\=\alpha\), and from (3.7) it follows
Thus, we get
and so (3.8) is satisfied.
Consequently, Lemma 2.4 implies that T has at least two fixed points \(u_{1}\) and \(u_{2}\), located in \(\mathcal{P}\cap (\bar{\varOmega }_{3} \setminus \varOmega_{1})\) and \(\mathcal{P}\cap (\bar{\varOmega }_{2}\setminus \varOmega_{3})\), respectively. And accordingly, (1.3) admits at least two positive ωperiodic solutions. □
If we replace (H4) with
 (H4)′:

\(\lim_{u\to 0+}\frac{f(t,u)}{u}=\chi (t)\), \(\lim_{u\to +\infty }\frac{f(t,u)}{u}=\chi (t)\), and there exists a constant \(\alpha >0\) such that
$$ \min \bigl\{ f(t,u): \sigma \alpha \leq u\leq \alpha, t\in [0,\omega ]\bigr\} \geq \bigl(\mu \sigma \chi (t)\bigr)\alpha, $$(3.9)
Then we can obtain the following:
Theorem 3.4
Let (H1)–(H3) and (H4)′ hold. Then (1.3) admits at least two positive ωperiodic solutions.
Proof
For \(0< r<\alpha <R\), let \(\varOmega_{i}\) (\(i=1,2,3\)) be as in the proof of Theorems 3.1 and 3.3. Then \(\bar{\varOmega }_{1}\subseteq \varOmega_{3}\), \(\bar{\varOmega }_{3}\subseteq \varOmega_{2}\). We shall follow the same strategy as in the proof of Theorem 3.3.
By (H4)′ and an argument similar to the proof of Theorems 3.1 and 3.2, we can conclude
Now, to apply Lemma 2.4, we only need to show
Let \(u\in \mathcal{P}\) with \(\u\=\alpha \), then \(\sigma \alpha \leq \sigma \u\\leq u(t)\leq \u\=\alpha\), by (3.9) we get
and then
and therefore (3.10) is true. Using Lemma 2.4 again, we know T has two fixed points \(u_{1}\) and \(u_{2}\), located in \(\mathcal{P}\cap (\bar{ \varOmega }_{3}\setminus \varOmega_{1})\) and \(\mathcal{P}\cap (\bar{\varOmega }_{2}\setminus \varOmega_{3})\), respectively. Consequently, (1.3) admits at least two positive ωperiodic solutions. □
Remark 3.1
We would like to point out the results of Theorems 3.1–3.4 remain true for the special resonant equation \(u'=f(t,u(t \tau (t)))\), where \(a(\cdot)\equiv 0\).
Remark 3.2
It is worth remarking that Theorems 3.1–3.4 apply to equations which could not be treated by the existing results of [7,8,9,10], and therefore our main results are novel.
Conclusion
By applying the fixed point theorem in cones, some new existence theorems are established for a class of firstorder delayed differential equations. Our main results enrich and complement those available in the literature.
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Acknowledgements
The authors are very grateful to the referees for their valuable suggestions. The authors thank for the help from the editor.
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The first author is supported by National Natural Science Foundation of China (No. 61761002; No. 11761004), the Scientific Research Funds of North Minzu University (No. 2018XYZSX03) and The key project of North Minzu University (No. ZDZX201804).
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RC carried out the analysis and proof the main results, and was a major contributor in writing the manuscript. XL participated in checking the processes of proofs, English grammar as well as typing errors in the text. All authors read and approved the final manuscript.
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Correspondence to Ruipeng Chen.
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DOI
MSC
 34B15
Keywords
 Positive periodic solutions
 Existence
 Fixed point
 Resonance