 Research
 Open Access
Positive periodic solutions for nonlinear firstorder delayed differential equations at resonance
 Ruipeng Chen^{1}Email author and
 Xiaoya Li^{1}
 Received: 12 July 2018
 Accepted: 30 November 2018
 Published: 11 December 2018
Abstract
Keywords
 Positive periodic solutions
 Existence
 Fixed point
 Resonance
MSC
 34B15
1 Introduction
 (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 [14–17] 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.
2 Preliminaries
Lemma 2.1
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 \([0,\omega ]\). □
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
The following lemma is crucial to prove our main results.
Lemma 2.4
(Guo–Krasnoselskii’s fixed point theorem [18])
 (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\\), \(u\in \mathcal{P}\cap \partial \varOmega _{2}\).
3 Main results
In this section, we state and prove our main findings.
Theorem 3.1
Proof
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
Proof
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). □
 (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)
Theorem 3.3
Assume that (H1)–(H4) hold. Then (1.3) admits at least two positive ωperiodic solutions.
Proof
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. □
 (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.
4 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.
Declarations
Acknowledgements
The authors are very grateful to the referees for their valuable suggestions. The authors thank for the help from the editor.
Availability of data and materials
Not applicable.
Funding
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).
Authors’ contributions
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.
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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