 Research
 Open Access
New existence results for nonlinear delayed differential systems at resonance
 Ruipeng Chen^{1}Email authorView ORCID ID profile and
 Xiaoya Li^{1}
https://doi.org/10.1186/s1366001819127
© The Author(s) 2018
 Received: 12 July 2018
 Accepted: 13 November 2018
 Published: 16 November 2018
Abstract
Keywords
 Positive periodic solution
 Existence
 Delay
 Resonance
 Fixed point
MSC
 34B15
1 Introduction
 (H1)
\(a\in C(\mathbb{R},\mathbb{R})\) is ωperiodic with \(\int_{0}^{\omega }a(t)\,dt=0\) and \(b\succ 0\), \(h\succ 0\);
 (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 \(f(t,u)\geq \chi (t)u\);
 (H4)
\(g\in C(\mathbb{R}\times [0,\infty ),[0,\infty ))\) is ωperiodic with respect to t.
Remark 1.1
Note that a and f are assumed to be signchanging, and therefore system (1.3) is more general than corresponding ones studied in the existing literature. For other research work on nonlinear differential equations at resonance, we refer the reader to [18–21] and the references listed therein.
The rest of the paper is arranged as follows. In Sect. 2, we introduce some preliminaries. And finally in Sect. 3, we 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
Let (H1)–(H4) hold. Then \(T(\mathcal{P}) \subseteq \mathcal{P}\) and \(T:\mathcal{P}\to \mathcal{P}\) is compact and continuous.
Proof
Using (H3), and similarly to the proof of [12, Lemmas 2.2, 2.3] with obvious changes, we can easily get the conclusion. □
The following lemma is crucial to prove our main results.
Lemma 2.3
(Guo–Krasnoselskii’s fixed point theorem [22, 23])
 (i)
\(\Vert Tu\Vert \leq \Vert u\Vert \), \(u\in \mathcal{P}\cap \partial \varOmega _{1}\), and \(\Vert Tu\Vert \geq \Vert u\Vert \), \(u\in \mathcal{P}\cap \partial \varOmega _{2}\); or
 (ii)
\(\Vert Tu\Vert \geq \Vert u\Vert \), \(u\in \mathcal{P}\cap \partial \varOmega _{1}\), and \(\Vert Tu\Vert \leq \Vert u\Vert \), \(u\in \mathcal{P}\cap \partial \varOmega _{2}\).
3 Main results
Theorem 3.1
Proof
By Lemma 2.3(i), T has a fixed point \(u^{\ast }\in \mathcal{P} \cap (\bar{\varOmega }_{2}\setminus \varOmega_{1})\), which is just a positive ωperiodic solution of (2.2). Subsequently, (1.3) admits at least one positive ωperiodic solution. □
Theorem 3.2
Proof
Consequently, it follows from Lemma 2.3(ii) that T has a fixed point \(u^{\ast }\) in \(\mathcal{P}\cap (\bar{\varOmega }_{2}\setminus \varOmega _{1})\), which is just a positive ωperiodic solution of (2.2), Subsequently, (1.3) admits at least one positive ωperiodic solution. □
 (H5)
Theorem 3.3
Let (H1)–(H5) hold. Then (1.3) admits at least two positive ωperiodic solutions.
Proof
 \((\mathrm{H}5)'\) :

\(f_{0}=\infty =f_{\infty }\), and there exists \(\alpha >0\) such thatwhere \(\epsilon >0\), \(\varepsilon >0\) satisfies \(\epsilon M\omega \leq \frac{1}{2}\) and \(\omega \varepsilon \tilde{M}Mh_{0}\leq \frac{1}{2}\), respectively, then similar to the proof of Theorems 3.1–3.3, we can prove the following.$$\begin{aligned}& \max \bigl\{ f(t,u): \sigma \alpha \leq u\leq \alpha , t\in [0,\omega ]\bigr\} < \bigl( \epsilon \chi (t)\bigr)\alpha , \\& \max \bigl\{ g(t,u): \sigma \alpha \leq u\leq \alpha , t\in [0,\omega ]\bigr\} < \varepsilon \alpha , \end{aligned}$$
Theorem 3.4
Let (H1)–(H4) and \((\mathrm{H}5)'\) hold. Then (1.3) admits at least two positive ωperiodic solutions.
Remark 3.1
4 Conclusions
We establish several novel existence theorems on positive periodic solutions for delayed differential systems (1.3), via fixed point theorem in cones. Our main findings Theorems 3.1–3.4 not only enrich and complement those available in the literature, but they also apply to some systems (equations) which cannot be dealt with by the results appeared in the existing literature.
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Funding
The first author is supported by National Natural Science Foundation of China (No. 11761004; No. 61761002), the Scientific Research Funds of North Minzu University (No. 2018XYZSX03), and FirstClass Disciplines Foundation of Ningxia (Grant No. NXYLXK2017B09).
Authors’ contributions
RC analyzed and proved the main results, and was a major contributor in writing the manuscript. XL checked the English grammar and typing errors in the full 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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