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New existence results for nonlinear delayed differential systems at resonance
Journal of Inequalities and Applications volume 2018, Article number: 312 (2018)
Abstract
This paper deals with the firstorder delayed differential systems
where a, b, τ, h are continuous ωperiodic functions with \(\int_{0}^{\omega }a(t)\,dt=0\) and \(\int_{0}^{\omega }b(t)\,dt>0\); \(f\in C(\mathbb{R}\times [0,\infty ),\mathbb{R})\) and \(g\in C( \mathbb{R}\times [0,\infty ),[0,\infty ))\) are ωperiodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established. Our main results enrich and complement those available in the literature.
1 Introduction
In the past few decades, there has been considerable interest in the existence of positive periodic solutions of the firstorder delayed equation
where \(a, b\in C(\mathbb{R},[0,\infty ))\) are ωperiodic 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 and cardiac arrhythmias. We refer the reader to [1,2,3,4,5,6,7,8,9,10] for some research work on this topic. Meanwhile, many authors have paid attention to the corresponding differential systems of (1.1), namely,
where \(a_{i}, b_{i}\in C(\mathbb{R},[0,\infty ))\) are ωperiodic functions, and also supposed to satisfy
See, for instance, Wang [11, 12], Chen et al. [13] and the references therein.
Obviously, the basic assumption \(\int_{0}^{\omega }a(t)\,dt>0\) or \(\int_{0}^{\omega }a_{i}(t)\,dt>0\) (\(i=1,2,\ldots,n\)), employed usually to guarantee the linear boundary value problem
is nonresonant, has played a key role in the arguments of the abovementioned papers. Indeed, it ensures a number of tools, such as fixed point theory, bifurcation theory and so on, could be applied to study the corresponding problems and establish desired existence results. Here (1.2) is called nonresonant if the unique solution of it is the trivial one. Moreover, if (1.2) is nonresonant then, provided that h is a \(L^{1}\)function, the Fredholm alternative theorem implies the nonhomogeneous problem
always admits a unique solution, which 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 the resonant problems proceeds very slowly and the related results are relatively little. Therefore inspired by the existing literature, we study the existence of positive periodic solutions of the following firstorder delayed differential system:
A natural and interesting question is whether or not (1.3) possesses a positive periodic solution provided that
which means a may change its sign on \(\mathbb{R}\) and the linear differential operator \(L_{a}u:=u'+a(t)u\) is resonant, \(L_{b}u:=u'+b(t)u\) is nonresonant.
Recently, Domoshnitsky et al. [14] studied the following system of periodic functional differential equations:
where \(x=\operatorname{col}(x_{1},\ldots,x_{n})\), \(B_{ij}: C[0,\omega ] \to L[0,\omega ]\) are linear bounded operator for \(i,j=1,\ldots,n\), \(f_{i}\in L[0,\omega ]\) and \(c_{i}\in \mathbb{R}\) for \(i=1,\ldots,n\). Note that if the corresponding homogeneous problem
has only trivial solution, then, for any \(f=\operatorname{col}(f_{1},\ldots,f _{n})\) and \(c=\operatorname{col}(c_{1},\ldots,c_{n})\), system (1.5), (1.6) admits a unique solution x defined as [15]
Here \(G(t,s)\) is a \(n\times n\) matrix and is called Green’s matrix of (1.5), the \(n\times n\) matrix \(X(t)\) is the fundamental matrix of (1.7) satisfying \(X(0)X(\omega )=E\), where E denotes the unit \(n\times n\) matrix. By an inspection of (1.9), It is not difficult to see all properties of solutions are determined by \(G(t,s)\) and \(X(t)\). Hence a series of excellent results on positivity and negativity of \(G(t,s)\) have been established in [14]. For instance, there are results on positivity (negativity) of \(G(t,s)\) when all nondiagonal operators \(B_{ij}\) (\(i\neq j\), \(i,j=1,\ldots,n\)) are negative (positive), and results about positivity or negativity of the elements in the nth row of \(G(t,s)\). To prove these results, the main idea and approach adopted by Domoshnitsky is to construct a corresponding firstorder scalar functional equation
for nth component of a solution vector, here \(B: C[0,\omega ]\to L[0, \omega ]\) is a linear continuous operator and \(f^{\ast }\in L[0, \omega ]\). This idea has also been applied by Domoshnitsky to a study of the Cauchy problem [16] and some twopoint boundary value problems [17].
In Sects. 2 and 3 of the present paper, the key idea of [14] to use nonresonant second scalar equation to obtain the assertion of \(v(t)\), and then to set its representation in the first equation, will be employed to establish new existence theorems for system (1.3). To the best of our knowledge, the abovementioned problem has not been studied so far and our results shall fill this gap. For the simplicity of statement, let E be the Banach space composed of continuous ωperiodic functions with the norm \(\Vert u\Vert =\max_{t\in [0,\omega ]}\vert u(t)\vert \). For \(q\in E\), we say \(q\gg 0\) if it is strictly positive on \([0,\omega ]\), and \(q\succ 0\) if q is nonnegative and \(\int_{0}^{\omega }g(t)\,dt>0\). We denote by q̄ and \(\underline{q}\) the maximum and minimum of \(q\gg 0\) on \([0,\omega ]\). By a positive ωperiodic solution of (1.3), we mean a vector function \((u,v)\), such that \(u, v\gg 0\) are continuously differentiable everywhere and satisfy (1.3). Hence for the above question, if we choose \(\chi \gg 0\) such that \(p:=a+\chi \succ 0\), then \(\int_{0}^{\omega }p(t)\,dt>0\), so the linear differential operator \(Lu:=u'+p(t)u\) is invertible. In the following, we always assume \(\tau \in C(\mathbb{R},\mathbb{R})\) is ωperiodic, and:

(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 [18,19,20,21] 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
Let \(\tilde{G}(t,s)\) be the Green’s function of the linear boundary value problem
Then simple calculation gives the following.
Lemma 2.1
Let (H1) hold and \(\tilde{\delta }=e^{\int _{0}^{\omega }b(t)\,dt}\). Then
and
By Lemma 2.1 and (H4), it is not difficult to verify
is equivalent to the equation
and \(A:E\to E\) is completely continuous. Then system (1.3) can be equivalently written as the following integraldifferential equation:
Moreover, by (H2) we get
Clearly, if u is a positive ωperiodic solution of (2.2), then the original system (1.3) admits a positive ωperiodic solution \((u, v)\). In the following, we shall focus on studying (2.2).
Recall that (H2) ensures the linear differential operator \(Lu:=u'+p(t)u\) is invertible. Therefore, by an argument similar to obtain Lemma 2.1, the Green’s function of
can be expressed as
where \(\delta =e^{\int_{0}^{\omega }p(t)\,dt}\), and accordingly,
Consequently, (2.2) can be written as the equivalent operator equation
Setting \(\sigma :=\frac{m}{M}\) and define
Then \(\sigma <1\) and \(\mathcal{P}\) is a positive cone in E.
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])
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)
\(\Vert Tu\Vert \leq \Vert u\Vert \), \(u\in \mathcal{P}\cap \partial \varOmega _{1}\), 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 \(\Vert Tu\Vert \leq \Vert u\Vert \), \(u\in \mathcal{P}\cap \partial \varOmega _{2}\).
Then T has a fixed point in \(\mathcal{P}\cap (\bar{\varOmega } _{2}\setminus \varOmega_{1})\).
3 Main results
In this section, we shall state and prove our main findings. First we give the following notations:
Theorem 3.1
Let (H1)–(H4) hold. If \(g_{0}=0\), \(f_{\infty }=\infty \) and
then (1.3) admits at least one positive ωperiodic solution.
Proof
For \(0< r< R<\infty \), setting
then we have \(0\in \varOmega_{1}\), \(\bar{\varOmega }_{1}\subseteq \varOmega_{2}\).
It follows from (3.1) that there exists \(r_{1}>0\), such that, for any \(0< u\leq r_{1}\),
where c is a positive constant small such that \(cM\omega \leq \frac{1}{2}\). Therefore, for \(u\in \mathcal{P}\) with \(\Vert u\Vert \leq r_{1}\), we can obtain
Moreover, since \(g_{0}=0\), there exists \(r_{2}>0\) so that \(g(t,u) \leq du\) for any \(0< u\leq r_{2}\). Thus, for \(u\in \mathcal{P}\) with \(\Vert u\Vert \leq r_{2}\), simple estimation gives
where d is a positive constant small enough such that \(\omega \,dM \tilde{M}h_{0}\leq \frac{1}{2}\) and \(h_{0}=\int_{0}^{\omega }h(t)\,dt\). Let \(r=\min \{r_{1}, r_{2}\}\). Then, for \(u\in \mathcal{P}\) with \(\Vert u\Vert =r\), we get
which implies \(\Vert Tu\Vert \leq \Vert u\Vert \), \(\forall u\in \mathcal{P}\cap \partial \varOmega_{1}\).
On the other hand, \(f_{\infty }=\infty \) shows there exists \(\tilde{R}>0\) such that, for any \(u\geq \tilde{R}\),
where \(\eta >0\) is a constant large enough such that \(\sigma m\omega (\eta +\underline{\chi })\geq 1\) and \(\underline{\chi }=\min_{t\in [0,\omega ]}\chi (t)\). Fixing \(R>\max \{r, \frac{ \tilde{R}}{\sigma }\}\) and let \(u\in \mathcal{P}\) with \(\Vert u\Vert =R\), then
and
Consequently, for \(u\in \mathcal{P}\) with \(\Vert u\Vert =R\), by (H4) we can obtain
Hence \(\Vert Tu\Vert \geq \Vert u\Vert \), \(\forall u\in \mathcal{P}\cap \partial \varOmega _{2}\).
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
Let (H1)–(H4) hold. If \(f_{0}=\infty \), \(g_{\infty }=0\) 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 for \(r>0\) sufficiently small,
It follows from \(f_{0}=\infty \) 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 \(\sigma m\omega ( \beta +\underline{\chi })\geq 1\). Therefore, for \(0< r\leq \tilde{r}\), if \(u\in \mathcal{P}\) and \(\Vert u\Vert =r\), then
and
Thus, (3.3) is true.
Secondly, we show for \(R>0\) sufficiently large,
It follows from (3.2) that there exists \(\tilde{R}>0\) so that
for any \(u\geq \tilde{R}\), where \(\mu >0\) satisfies \(\mu M\omega \leq \frac{1}{2}\). Let \(R_{1}>\max \{\tilde{r}, \frac{\tilde{R}}{ \sigma }\}\), then if \(u\in \mathcal{P}\) and \(\Vert u\Vert \geq R_{1}\), we get
and then
Moreover, \(g_{\infty }=0\) implies there exists \(R_{2}>0\) so that \(g(t,u)\leq \gamma u\) for any \(u\geq R_{2}\) and \(t\in [0,\omega ]\). Therefore, for \(u\in \mathcal{P}\) with \(\Vert u\Vert \geq R_{2}\), we have
where γ is a positive constant small enough such that \(\omega \gamma M\tilde{M}h_{0}\leq \frac{1}{2}\). Let \(R=\max \{R_{1}, R_{2}\}\). Then, for \(u\in \mathcal{P}\) with \(\Vert u\Vert =R\), we can obtain
which implies (3.4) is true.
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. □
In the following, we investigate the multiplicity of positive ωperiodic solutions of system (1.3). To the end, we suppose:

(H5)
\(g_{0}=0=g_{\infty }\), and (3.1) and (3.2) hold. In addition, there exists \(\alpha >0\), such that
$$\min \bigl\{ f(t,u): \sigma \alpha \leq u\leq \alpha , t\in [0,\omega ]\bigr\} > \bigl( \mu \sigma \chi (t)\bigr)\alpha , $$where \(\mu >0\) satisfies \(m\omega \mu \geq 1\).
Theorem 3.3
Let (H1)–(H5) hold. Then (1.3) admits at least two positive ωperiodic solutions.
Proof
Define
Let \(\varOmega_{1}\) and \(\varOmega_{2}\) be the same as in the proofs of Theorems 3.1 and 3.2. Then, for \(0< r<\alpha <R\), we have \(\bar{\varOmega }_{1}\subseteq \varOmega_{3}\), \(\bar{\varOmega }_{3}\subseteq \varOmega_{2}\).
Applying \(g_{0}=0\) and (3.1), and by an argument similar to the first part of the proof of Theorem 3.1, we can obtain
Similarly, combining \(g_{\infty }=0\) and (3.2), we conclude
Obviously, the proof is finished if we prove
Suppose \(u\in \mathcal{P}\) and \(\Vert u\Vert =\alpha \), then \(\sigma \alpha \leq \sigma \Vert u\Vert \leq u(t)\leq \Vert u\Vert =\alpha \), which together with (H5) yields
and then
Thus, similar to the first part of the proof of Theorem 3.2, we get
and (3.5) is true. By Lemma 2.3, 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. Hence, (1.3) admits at least two positive ωperiodic solutions. □
Conversely, if (H5) is replaced with:
 \((\mathrm{H}5)'\) :

\(f_{0}=\infty =f_{\infty }\), and there exists \(\alpha >0\) such that
$$\begin{aligned}& \max \bigl\{ f(t,u): \sigma \alpha \leq u\leq \alpha , t\in [0,\omega ]\bigr\} < \bigl( \epsilon \chi (t)\bigr)\alpha , \\& \max \bigl\{ g(t,u): \sigma \alpha \leq u\leq \alpha , t\in [0,\omega ]\bigr\} < \varepsilon \alpha , \end{aligned}$$where \(\epsilon >0\), \(\varepsilon >0\) 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.
Theorem 3.4
Let (H1)–(H4) and \((\mathrm{H}5)'\) hold. Then (1.3) admits at least two positive ωperiodic solutions.
Remark 3.1
We would like to point out that the results of Theorems 3.1–3.4 remain true for the special case \(a(\cdot )\equiv 0\), i.e., for system
Remark 3.2
It is worth remarking Theorems 3.1–3.4 apply to some equations which cannot be treated by the results of [7,8,9,10], and thus our main results are novel.
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.
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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).
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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.
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Chen, R., Li, X. New existence results for nonlinear delayed differential systems at resonance. J Inequal Appl 2018, 312 (2018). https://doi.org/10.1186/s1366001819127
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DOI: https://doi.org/10.1186/s1366001819127