Positive periodic solutions for multiparameter nonlinear differential systems with delays

Abstract

We establish several criteria for the existence of positive periodic solutions of the multi-parameter differential systems

$$\left \{ \textstyle\begin{array}{l} u'(t)+a_{1}(t)g_{1}(u(t))u(t)=\lambda b_{1}(t)f(u(t-\tau_{1}(t)),v(t-\zeta_{1}(t))), \\ v'(t)+a_{2}(t)g_{2}(v(t))v(t)=\mu b_{2}(t)g(u(t-\tau_{2}(t)),v(t-\zeta_{2}(t))), \end{array}\displaystyle \right . $$

where the functions $g_{1}, g_{2}:[0,\infty)\to[0,\infty)$ are assumed to be unbounded. The analysis in the paper relies on the classical fixed point index theory. Our main findings improve and complement some existing results in the literature.

1 Introduction

Let $\omega>0$ be a constant. In this article we shall seek some criterion to guarantee that the multiparameter system

$$ \left \{ \textstyle\begin{array}{l} u'(t)=a_{1}(t)g_{1}(u(t))u(t)-\lambda b_{1}(t)f(u(t-\tau_{1}(t)),v(t-\zeta_{1}(t))), \\ v'(t)=a_{2}(t)g_{2}(v(t))v(t)-\mu b_{2}(t)g(u(t-\tau_{2}(t)),v(t-\zeta_{2}(t))) \end{array}\displaystyle \right . $$

(1.1)

admits a positive ω-periodic solution, where the functions $a_{i}, b_{i}, \tau_{i}, \zeta_{i}\in C(\mathbb{R},\mathbb{R})$ are ω-periodic, and $g_{i}\in C([0,\infty),[0,\infty))$ are unbounded, $i=1,2$. In addition, we assume that the nonlinear terms $f, g\in C([0,\infty)\times[0,\infty),[0,\infty))$ and λ, μ are positive parameters.

Here a positive periodic solution of (1.1) means a solution $(u,v)\in E:=X^{2}$ of (1.1) satisfying $u>0$, $v>0$ on $[0,\omega]$, where

$$X=\bigl\{ x\in C(\mathbb{R},\mathbb{R}): x(t+\omega)=x(t)\bigr\} $$

is a Banach space, and the norm of $x\in X$ is

$$\Vert x \Vert =\max_{t\in[0,\omega]} \bigl\vert x(t) \bigr\vert . $$

Moreover, for $(x,y)\in E$, we denote $\|(x,y)\|=\|x\|+\|y\|$, and write $(x,y)\geq(0,0)$ if $(x,y)\in E$ fulfills $x(t)\geq0$, $y(t)\geq0$, $t\in[0,\omega]$.

Obviously, the first equation of (1.1) reduces in some special circumstances to

$$ u'(t)=a(t)g\bigl(u(t)\bigr)u(t)-\lambda b(t)f\bigl(u\bigl(t-\tau(t) \bigr)\bigr), $$

(1.2)

and when $\lambda=0$, $g(u)\equiv1$, Eq. (1.2) becomes $u'(t)=a(t)u(t)$, which is famous in Malthusian population dynamics. In recent decades, (1.2) has also been extensively applied to describe various physiological processes emerging in practical applications, for instance, the production of blood cells, respiration, cardiac arrhythmias, etc. One may refer to [1–6] and references therein. Nevertheless, the research work in the above mentioned papers is mainly dependent on the condition that $g(u)$ is positive and bounded, that is, there are constants $L>l>0$ such that $0< l\leq g(u)\leq L$, $u\in[0,\infty)$. Jin and Wang [7] have recently studied the spectral problem

$$u'(t)=a(t)e^{u(t)}u(t)-\lambda b(t)f\bigl(u\bigl(t-\tau(t) \bigr)\bigr), $$

and they obtained some existence results on positive periodic solutions by means of the fixed point theory. It is worth noting the function $e^{u}$ is unbounded on $[0,\infty)$. Since then, Eq. (1.2) has been extensively investigated under the more general case that $g(u)$ is unbounded on $[0,\infty)$, by applying the lower and upper solutions method, fixed point theory, and so on. See, for example, [7–10].

Besides, researchers have focused on the differential systems associated to (1.2), namely,

$$ u'_{i}(t)=a_{i}(t)g_{i} \bigl(u_{i}(t)\bigr)u_{i}(t)-\lambda b_{i}(t)f_{i} \bigl(u_{1}(t),u_{2}(t),\dots ,u_{n}(t)\bigr),\quad i=1,2,\dots,n. $$

(1.3)

One can see [11–14] for some related results. However, in [11–13], the authors have only dealt with the special case $g_{i}(u_{i})\equiv1$, $i=1,2,\dots,n$. Indeed in that case, the Green’s function corresponding to $u'_{i}(t)=a_{i}(t)u_{i}(t)$ is simple, and some suitable cones could be easily constructed. Furthermore, system (1.3) investigated in above papers includes only one positive parameter λ. Hence, it will be interesting to study the multiparameter systems (1.1) with $g_{i}$ ($i=1,2$) being unbounded. On the other hand, what is worth mentioning is that Zhang et al. [14] considered system (1.1) for the special case $g_{i}\equiv1$, $i=1,2$, where nonlinearities $f(u,v)$ and $g(u,v)$ were assumed to be nondecreasing, and only the case $f(0,0)>0$, $g(0,0)>0$ was treated. Therefore, we want to know whether or not (1.1) has a positive periodic solution under more relaxed assumption $f(0,0)=0$, $g(0,0)=0$. In view of above reasons, we shall concentrate on the existence of positive periodic solutions for system (1.1) in the current paper, to further improve and generalize tho results in the literature. For this purpose, we assume

• (C1) $a_{i}, b_{i}, \tau_{i}, \zeta_{i}\in C(\mathbb{R},[0,\infty))$ are ω-periodic with $\int_{0}^{\omega}a_{i}(t)\,dt>0$, $\int_{0}^{\omega}b_{i}(t)\,dt>0$, $i=1,2$.

• (C2) There is $l_{i}>0$ such that $0< l_{i}\leq g_{i}(s)<\infty$, $s\in [0,\infty)$.

• (C3) $f, g\in C([0,\infty)\times[0,\infty),[0,\infty))$ with $f(u,v)>0, g(u,v)>0$ for $(u,v)\neq(0,0)$.

Remark 1.1

For other research work on periodic solutions of functional differential equations and systems, we refer the readers to [15–17] and references therein.

The remainder of the paper is arranged as follows. In Sect. 2, we introduce some preliminaries needed in our proof. Section 3 is devoted to stating and proving our main findings. Meanwhile, some related results and remarks will be given.

2 Preliminaries

Recall that $E=X^{2}$ is the Banach space defined as in Sect. 1. We first give the following lemma.

Lemma 2.1

Assume (C1)–(C3). If $(u,v)\in E$is a solution of (1.1), then

$$\begin{gathered} u(t)=\lambda \int_{t}^{t+\omega}G_{1}(t,s)b_{1}(s)f \bigl(u\bigl(s-\tau_{1}(s)\bigr),v\bigl(s-\zeta _{1}(s) \bigr)\bigr)\,ds, \\ v(t)=\mu \int_{t}^{t+\omega}G_{2}(t,s)b_{2}(s)g \bigl(u\bigl(s-\tau_{2}(s)\bigr),v\bigl(s-\zeta _{2}(s)\bigr) \bigr)\,ds,\end{gathered} $$

where

$$G_{1}(t,s)=\frac{e^{-\int_{t}^{s} a_{1}(\theta)g_{1}(u(\theta))\,d\theta }}{1-e^{-\int_{0}^{\omega}a_{1}(\theta)g_{1}(u(\theta))\,d\theta}},\qquad G_{2}(t,s)=\frac{e^{-\int_{t}^{s} a_{2}(\theta)g_{2}(v(\theta))\,d\theta }}{1-e^{-\int_{0}^{\omega}a_{2}(\theta)g_{2}(v(\theta))\,d\theta}},\quad t\leq s\leq t+\omega. $$

Proof

Multiplying the both sides of the first equation of (1.1) with $e^{-\int_{0}^{t} a_{1}(s)g_{1}(u(s))\,ds}$, we can obtain

$$\bigl(u(t)e^{-\int_{0}^{t} a_{1}(s)g_{1}(u(s))\,ds} \bigr)'=-\lambda b_{1}(t)f \bigl(u\bigl(t-\tau_{1}(t)\bigr),v\bigl(t-\zeta_{1}(t)\bigr) \bigr)\cdot e^{-\int_{0}^{t} a_{1}(s)g_{1}(u(s))\,ds}. $$

Integrating above equation from t to $t+\omega$ and by elementary calculation, we can easily get

$$u(t)=\lambda \int_{t}^{t+\omega}G_{1}(t,s)b_{1}(s)f \bigl(u\bigl(s-\tau_{1}(s)\bigr),v\bigl(s-\zeta _{1}(s) \bigr)\bigr)\,ds. $$

Similar evaluation shows

$$v(t)=\mu \int_{t}^{t+\omega}G_{2}(t,s)b_{2}(s)g \bigl(u\bigl(s-\tau_{2}(s)\bigr),v\bigl(s-\zeta _{2}(s)\bigr) \bigr)\,ds. $$

 □

Let $q>0$ be a fixed constant. Then we can establish a series of lemmas required in the subsequent discussion.

Lemma 2.2

Assume (C1)–(C3). Let $\sigma_{i}=e^{-\int_{0}^{\omega}a_{i}(\theta)\,d\theta}$, $i=1,2$. Then for any $(u,v)\in E$satisfying $(u,v)\geq(0,0)$and $\|(u,v)\|\leq q$,

$$ 0< \frac{\sigma_{i}^{g_{i}^{\ast}(q)}}{1-\sigma_{i}^{g_{i}^{\ast}(q)}}\leq G_{i}(t,s)\leq\frac{1}{1-\sigma_{i}^{g_{i\ast}(q)}},\quad i=1,2, $$

(2.1)

where

$$g_{i}^{\ast}(q)=\max_{0\leq s\leq q}g_{i}(s),\qquad g_{i\ast}(q)=\min_{0\leq s\leq q}g_{i}(s),\quad i=1,2. $$

Proof

Clearly, for $(u,v)\in E$ with $(u,v)\geq(0,0)$ and $\| (u,v)\|\leq q$, we have $0\leq u\leq\|u\|\leq q$. Thus,

$$g_{1\ast}(q)\leq g_{1}(u)\leq g_{1}^{\ast}(q), $$

and then simple estimation shows (2.1) holds for $i=1$. The case $i=2$ is similar. □

Defining for $i=1,2$,

$$m_{i}(q)=\frac{\sigma_{i}^{g_{i}^{\ast}(q)}}{1-\sigma_{i}^{g_{i}^{\ast}(q)}},\quad\quad M_{i}(q)=\frac{1}{1-\sigma_{i}^{g_{i\ast}(q)}},\qquad \eta_{i}(q)=\frac{m_{i}(q)}{M_{i}(q)}. $$

Then it is not hard to verify $\eta_{i}(q)\in(0,1)$, and accordingly,

$$\eta(q):=\min\bigl\{ \eta_{1}(q), \eta_{2}(q)\bigr\} \in(0,1). $$

Set

$$\begin{gathered} P= \bigl\{ (u,v)\in E: u(t)\geq0, v(t)\geq0, t\in[0,\omega] \bigr\} , \\ K_{q}= \bigl\{ (u,v)\in P: u(t)+v(t)\geq\eta(q) \bigl\Vert (u,v) \bigr\Vert , t\in[0,\omega] \bigr\} ,\end{gathered} $$

and for $r>0$,

$$\varOmega_{r}=\bigl\{ (u,v)\in K_{q}: \bigl\Vert (u,v) \bigr\Vert < r\bigr\} , \partial\varOmega_{r}=\bigl\{ (u,v)\in K_{q}: \bigl\Vert (u,v) \bigr\Vert =r\bigr\} . $$

Then P and $K_{q}$ are cones in E.

Lemma 2.3

Assume (C1)–(C3). Let $0< r\leq q$. Then for any $(u,v)\in\bar{\varOmega}_{r}$,

$$ \frac{\sigma_{i}^{g_{i}^{\ast}(q)}}{1-\sigma_{i}^{g_{i}^{\ast}(q)}}\leq\frac{\sigma _{i}^{g_{i}^{\ast}(r)}}{1-\sigma_{i}^{g_{i}^{\ast}(r)}}\leq G_{i}(t,s)\leq \frac {1}{1-\sigma_{i}^{g_{i\ast}(r)}}\leq\frac{1}{1-\sigma_{i}^{g_{i\ast}(q)}},\quad i=1,2. $$

(2.2)

Proof

Similar to the proof of Lemma 2.2, we obtain for $t\leq s\leq t+\omega$,

$$\frac{\sigma_{i}^{g_{i}^{\ast}(r)}}{1-\sigma_{i}^{g_{i}^{\ast}(r)}}\leq G_{i}(t,s)\leq \frac{1}{1-\sigma_{i}^{g_{i\ast}(r)}},\quad i=1,2. $$

Moreover, since $\varphi(t):=\frac{\sigma_{i}^{t}}{1-\sigma_{i}^{t}}$ and $\psi (t):=\frac{1}{1-\sigma_{i}^{t}}$ are strictly decreasing on $[0,\infty)$, one can easily see that (2.2) holds true. □

Define, for given $(u,v)\in E$,

$$T_{\lambda,\mu}(u,v) (t)= \bigl(A_{\lambda}(u,v) (t), B_{\mu}(u,v) (t) \bigr), $$

where

$$A_{\lambda}(u,v) (t)=\lambda \int_{t}^{t+\omega}G_{1}(t,s)b_{1}(s)f \bigl(u\bigl(s-\tau _{1}(s)\bigr),v\bigl(s-\zeta_{1}(s) \bigr)\bigr)\,ds $$

and

$$B_{\mu}(u,v) (t)=\mu \int_{t}^{t+\omega}G_{2}(t,s)b_{2}(s)g \bigl(u\bigl(s-\tau _{2}(s)\bigr),v\bigl(s-\zeta_{2}(s)\bigr) \bigr)\,ds. $$

Then we have

Lemma 2.4

Assume (C1)–(C3) and $0< r\leq q$. Then $T_{\lambda ,\mu}(\bar{\varOmega}_{r})\subseteq K_{q}$and $T_{\lambda,\mu}:\bar{\varOmega }_{r}\to K_{q}$is completely continuous.

Proof

For $(u,v)\in\bar{\varOmega}_{r}$, we can deduce from Lemma 2.3 that

$$\begin{aligned} A_{\lambda}(u,v) (t)& =\lambda \int_{t}^{t+\omega}G_{1}(t,s)b_{1}(s)f \bigl(u\bigl(s-\tau_{1}(s)\bigr),v\bigl(s-\zeta _{1}(s) \bigr)\bigr)\,ds \\ &\leq\lambda\frac{1}{1-\sigma_{1}^{g_{1\ast}(r)}} \int_{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta_{1}(s)\bigr)\bigr)\,ds, \end{aligned} $$

which yields

$$\bigl\Vert A_{\lambda}(u,v) \bigr\Vert \leq\lambda\frac{1}{1-\sigma_{1}^{g_{1\ast}(r)}} \int _{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta_{1}(s)\bigr)\bigr)\,ds. $$

Meanwhile, (2.2) implies

$$ \begin{aligned}[b] A_{\lambda}(u,v) (t) &\geq\lambda \frac{\sigma_{1}^{g_{1}^{\ast}(r)}}{1-\sigma_{1}^{g_{1}^{\ast}(r)}} \int _{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta_{1}(s)\bigr)\bigr)\,ds \\ &=\lambda\frac{\sigma_{1}^{g_{1}^{\ast}(r)}(1-\sigma_{1}^{g_{1\ast }(r)})}{1-\sigma_{1}^{g_{1}^{\ast}(r)}}\cdot\frac{1}{1-\sigma_{1}^{g_{1\ast }(r)}} \int_{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta_{1}(s)\bigr)\bigr)\,ds\hspace{-12pt} \\ &\geq\frac{\sigma_{1}^{g_{1}^{\ast}(r)}(1-\sigma_{1}^{g_{1\ast}(r)})}{1-\sigma _{1}^{g_{1}^{\ast}(r)}} \bigl\Vert A(u,v) \bigr\Vert \\ &\geq\eta_{1}(q) \bigl\Vert A_{\lambda}(u,v) \bigr\Vert \\ &\geq\eta(q) \bigl\Vert A_{\lambda}(u,v) \bigr\Vert . \end{aligned} $$

(2.3)

In an analogous manner, we get

$$B_{\mu}(u,v) (t)\geq\eta(q) \bigl\Vert B_{\mu}(u,v) \bigr\Vert ,\quad (u,v)\in\bar{\varOmega}_{r}. $$

Hence $T_{\lambda,\mu}(\bar{\varOmega}_{r})\subseteq K_{q}$. The completely continuity of $T_{\lambda,\mu}$ is obvious. □

It is obvious that if $(u,v)$ is a fixed point of the completely continuous operator $T_{\lambda,\mu}$ in $K_{q}$, then $(u,v)$ is a positive periodic solution of (1.1). We conclude this section by giving the main tool employed in proving our main results.

Lemma 2.5

([18, 19])

AssumeEis a Banach space and $K\subseteq E$is a cone. For $r>0$, let $K_{r}=\{u\in K: \|u\|< r\}$and $\partial K_{r}=\{u\in K: \|u\|=r\}$. Suppose $T: \bar{K}_{r}\to K$is a completely continuous operator satisfying $Tu\neq u$, $u\in\partial K_{r}$. Then

• (i) If $\|Tu\|<\|u\|$, $u\in\partial K_{r}$, then $i(T, \bar{K}_{r}, K)=1$;

• (ii) If $\|Tu\|>\|u\|$, $u\in\partial K_{r}$, then $i(T, \bar{K}_{r}, K)=0$.

3 Main results

Let

$$f_{0}=\lim_{(u,v)\to0}\frac{f(u,v)}{u+v},\qquad g_{0}=\lim_{(u,v)\to0}\frac {g(u,v)}{u+v}. $$

Theorem 3.1

Assume (C1)–(C3) hold and $f_{0}=0=g_{0}$. Then for every $q>0$, there is a constant $\gamma_{q}>0$such that for all $\lambda , \mu>\gamma_{q}$, system (1.1) admits a positive periodic solution $(u,v)$satisfying $\|(u,v)\|\leq q$.

Proof

Choose $r_{1}=q$ and define

$$\begin{gathered} \psi_{f}(q)=\min \bigl\{ f(u,v): \eta(q)q\leq u+v\leq q \bigr\} , \\ \psi_{g}(q)=\min \bigl\{ g(u,v): \eta(q)q\leq u+v\leq q \bigr\} .\end{gathered} $$

Take

$$\gamma_{q}=q\cdot\max \biggl\{ \frac{1}{2\psi_{f}(q)m_{1}(q)\int_{0}^{\omega}b_{1}(s)\, ds}, \frac{1}{2\psi_{g}(q)m_{2}(q)\int_{0}^{\omega}b_{2}(s)\,ds} \biggr\} . $$

By Lemma 2.4, we know $T_{\lambda,\mu}(\bar{\varOmega}_{q})\subseteq K_{q}$ and $T_{\lambda,\mu}:\bar{\varOmega}_{q}\to K_{q}$ is completely continuous. Fix $\lambda, \mu>\gamma_{q}$. Then for $(u,v)\in\partial\varOmega_{q}$, we have $\eta(q)q\leq u+v\leq q$, and so

$$\begin{aligned} A_{\lambda}(u,v) (t) &\geq\lambda m_{1}(q) \int_{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta _{1}(s)\bigr)\bigr)\,ds \\ &=\lambda m_{1}(q)\psi_{f}(q)\cdot \int_{0}^{\omega}b_{1}(s)\,ds \\ &>\frac{q}{2}=\frac{ \Vert (u,v) \Vert }{2}, \end{aligned} $$

which implies

$$\bigl\Vert A_{\lambda}(u,v) \bigr\Vert >\frac{ \Vert (u,v) \Vert }{2},\quad (u,v)\in \partial\varOmega_{q}. $$

Similarly,

$$\bigl\Vert B_{\mu}(u,v) \bigr\Vert >\frac{ \Vert (u,v) \Vert }{2},\quad (u,v)\in \partial\varOmega_{q}. $$

Hence $\|T_{\lambda,\mu}(u,v)\|>\|(u,v)\|$ on $\partial\varOmega_{q}$, and then Lemma 2.5 gives $i(T_{\lambda,\mu}, \bar{\varOmega}_{q}, K_{q})=0$.

On the other hand, since $f_{0}=g_{0}=0$, there exists a constant $r_{2}$ with $0< r_{2}< q$, such that for $(u,v)$ satisfying $0< u+v\leq r_{2}$,

$$f(u,v)\leq\varepsilon(u+v),\qquad g(u,v)\leq\varepsilon(u+v), $$

where $\varepsilon>0$ is a constant satisfying

$$ \frac{2\lambda\varepsilon\int_{0}^{\omega}b_{1}(s)\,ds}{1-\sigma_{1}^{g_{1}^{\ast}(q)}}< 1,\qquad \frac{2\mu\varepsilon\int_{0}^{\omega}b_{2}(s)\,ds}{1-\sigma _{2}^{g_{2}^{\ast}(q)}}< 1. $$

(3.1)

For $(u,v)\in\partial\varOmega_{r_{2}}$, we can deduce by (2.2) and (3.1) that

$$\begin{aligned} A_{\lambda}(u,v) (t)& \leq\lambda \frac{1}{1-\sigma_{1}^{g_{1\ast}(r_{2})}} \int_{t}^{t+\omega }b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta_{1}(s)\bigr)\bigr)\,ds \\ &\leq\frac{\lambda\varepsilon}{1-\sigma_{1}^{g_{1\ast}(q)}}\cdot \int _{0}^{\omega}b_{1}(s)\,ds\cdot \bigl\Vert (u,v) \bigr\Vert \\ &< \frac{ \Vert (u,v) \Vert }{2}, \end{aligned} $$

and hence

$$\bigl\Vert A_{\lambda}(u,v) \bigr\Vert < \frac{ \Vert (u,v) \Vert }{2}, \quad(u,v)\in \partial\varOmega_{r_{2}}. $$

In an analogous way, we get

$$\bigl\Vert B_{\mu}(u,v) \bigr\Vert < \frac{ \Vert (u,v) \Vert }{2}, \quad(u,v)\in \partial\varOmega_{r_{2}}. $$

Thus $\|T_{\lambda,\mu}(u,v)\|<\|(u,v)\|$ on $\partial\varOmega_{r_{2}}$. Lemma 2.5 ensures $i(T_{\lambda,\mu}, \bar{\varOmega}_{r_{2}}, K_{q})=1$.

Consequently, $i(T_{\lambda,\mu}, \bar{\varOmega}_{q}\setminus\varOmega _{r_{2}}, K_{q})=-1$. Therefore, $T_{\lambda,\mu}$ possesses a fixed point $(u,v)$ in $\bar{\varOmega}_{q}\setminus\varOmega_{r_{2}}$, and system (1.1) has a positive periodic solution $(u,v)$ with $\| (u,v)\|\leq q$. □

Theorem 3.2

Assume (C1)–(C3) hold and $f_{0}=\infty$. Then for every $q>0$, there is a constant $\gamma_{q}>0$such that for all $\lambda , \mu<\gamma_{q}$, system (1.1) admits a positive periodic solution $(u,v)$satisfying $\|(u,v)\|\leq q$.

Proof

Fix $r_{1}=q$ and set

$$\begin{gathered} \varPsi_{f}(q)=\max \bigl\{ f(u,v): \eta(q)q\leq u+v\leq q \bigr\} , \\ \varPsi_{g}(q)=\max \bigl\{ g(u,v): \eta(q)q\leq u+v\leq q \bigr\} . \end{gathered}$$

Define

$$\gamma_{q}=q\cdot\min \biggl\{ \frac{1}{2\varPsi_{f}(q)M_{1}(q)\int_{0}^{\omega}b_{1}(s)\, ds}, \frac{1}{2\varPsi_{g}(q)M_{2}(q)\int_{0}^{\omega}b_{2}(s)\,ds} \biggr\} . $$

By Lemma 2.4, $T_{\lambda,\mu}(\bar{\varOmega}_{q})\subseteq K_{q}$ and $T_{\lambda,\mu}:\bar{\varOmega}_{q}\to K_{q}$ is completely continuous. Thus, for fixed $\lambda, \mu<\gamma_{q}$ and $(u,v)\in\partial\varOmega_{q}$,

$$\begin{aligned} A_{\lambda}(u,v) (t) &\leq\lambda M_{1}(q) \int_{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta _{1}(s)\bigr)\bigr)\,ds \\ &=\lambda M_{1}(q)\psi_{f}(q)\cdot \int_{0}^{\omega}b_{1}(s)\,ds \\ &< \frac{q}{2}=\frac{ \Vert (u,v) \Vert }{2}, \end{aligned} $$

and then

$$\bigl\Vert A_{\lambda}(u,v) \bigr\Vert < \frac{ \Vert (u,v) \Vert }{2},\quad (u,v)\in \partial\varOmega_{q}. $$

By a similar argument, we can also obtain

$$\bigl\Vert B_{\mu}(u,v) \bigr\Vert < \frac{ \Vert (u,v) \Vert }{2},\quad (u,v)\in \partial\varOmega_{q}. $$

Therefore, $\|T_{\lambda,\mu}(u,v)\|<\|(u,v)\|$ for $(u,v)\in\partial \varOmega_{q}$. Using Lemma 2.5 again, we can easily get $i(T_{\lambda,\mu}, \bar{\varOmega}_{q}, K_{q})=1$.

By the assumption $f_{0}=\infty$, there exists a constant $r_{2}\in(0,q)$, such that for $(u,v)$ satisfying $0< u+v\leq r_{2}$,

$$f(u,v)\geq\varUpsilon(u+v), $$

where $\varUpsilon>0$ satisfies

$$ \lambda\varUpsilon\eta(q)\frac{\sigma_{1}^{g_{1}^{\ast}(q)}}{1-\sigma_{1}^{g_{1}^{\ast}(q)}} \int_{0}^{\omega}b_{1}(s) \,ds>1. $$

(3.2)

Thus for $(u,v)\in\partial\varOmega_{r_{2}}$, we get by (2.2) and (3.2) that

$$\begin{aligned} A_{\lambda}(u,v) (t) &\geq\lambda \frac{\sigma_{1}^{g_{1}^{\ast}(r_{2})}}{1-\sigma_{1}^{g_{1}^{\ast}(r_{2})}} \int_{0}^{\omega}b_{1}(s)f\bigl(u\bigl(s- \tau_{1}(s)\bigr),v\bigl(s-\zeta_{1}(s)\bigr)\bigr)\,ds \\ &\geq\lambda\varUpsilon\eta(q)\frac{\sigma_{1}^{g_{1}^{\ast}(q)}}{1-\sigma _{1}^{g_{1}^{\ast}(q)}} \int_{0}^{\omega}b_{1}(s)\,ds\cdot \bigl\Vert (u,v) \bigr\Vert \\ &> \bigl\Vert (u,v) \bigr\Vert , \end{aligned} $$

which means $\|A_{\lambda}(u,v)\|>\|(u,v)\|$ on $\partial\varOmega_{r_{2}}$. Hence

$$\bigl\Vert T_{\lambda,\mu}(u,v) \bigr\Vert \geq \bigl\Vert A_{\lambda}(u,v) \bigr\Vert > \bigl\Vert (u,v) \bigr\Vert ,\quad (u,v)\in \partial\varOmega_{r_{2}}, $$

and Lemma 2.5 again implies $i(T_{\lambda,\mu}, \bar{\varOmega}_{r_{2}}, K_{q})=0$.

Consequently, $i(T_{\lambda,\mu}, \bar{\varOmega}_{q}\setminus\varOmega _{r_{2}}, K_{q})=1$. Thus, $T_{\lambda,\mu}$ has a fixed point $(u,v)$ in $\bar{\varOmega}_{q}\setminus\varOmega_{r_{2}}$, and (1.1) has a positive periodic solution $(u,v)$ with $\|(u,v)\|\leq q$. □

Similarly to Theorems 3.1 and 3.2, we can prove the following

Theorem 3.3

Assume (C1)–(C3) and $g_{0}=\infty$. Then for every $q>0$, there is a constant $\gamma_{q}>0$such that for all $\lambda , \mu<\gamma_{q}$, system (1.1) admits a positive periodic solution $(u,v)$satisfying $\|(u,v)\|\leq q$.

Remark 3.1

Clearly, the results of Theorems 3.1–3.3 generalize and complement the corresponding ones in [7, 9, 12–14].

To illustrate our main findings, we may choose $\omega=2\pi$ and $\tau _{i}\equiv0$, $\zeta_{i}\equiv0$ ($i=1,2$) in the subsequent discussion. Let

$$\begin{gathered} a_{1}(t)=\sin t+1,\qquad a_{2}(t)=\sin t+2,\quad t\in[0,2\pi], \\ b_{1}(t)=\cos t+2,\qquad b_{2}(t)=\cos t+1,\quad t\in[0,2\pi].\end{gathered} $$

Then it is not hard to check that (C1) is satisfied. Moreover, define

$$g_{1}(s)=e^{s},\qquad g_{2}(s)=2e^{s},\quad s \in[0,\infty), $$

then there are constants $l_{1}=1$ and $l_{2}=2$ such that

$$0< 1=l_{1}\leq g_{1}(s)< \infty,\qquad 0< 2=l_{2}\leq g_{2}(s)< \infty,\quad s\in[0,\infty). $$

Hence (C2) is also satisfied.

Example 3.1

For $(u,v)\in[0,\infty)\times[0,\infty)$, let

$$f(u,v)=3(u+v)^{2}\bigl(u^{2}+v^{2}+1 \bigr)^{2},\qquad g(u,v)=2(u+v)^{4}\bigl(u^{2}+v^{2}+5 \bigr)^{2}. $$

Then $f, g\in C([0,\infty)\times[0,\infty),[0,\infty))$ with $f(u,v)>0$, $g(u,v)>0$ for $(u,v)\neq(0,0)$. Thus (C3) holds true. Furthermore, simple calculation gives $f_{0}=0=g_{0}$. Consequently, the results of Theorem 3.1 are valid.

Example 3.2

We shall follow the same notations and definitions as before. Let us redefine

$$f(u,v)=\sqrt{u+v}\cdot\bigl(u^{2}+v^{2}+1 \bigr)^{2},\quad (u,v)\in[0,\infty)\times[0,\infty). $$

Clearly, f verifies (C3). Moreover, it is not difficult to see $f_{0}=\infty$, and accordingly the results of Theorem 3.2 are also valid.

At the end of the section, we list some related results and remarks.

Let us consider the multiparameter differential systems

$$ \left \{ \textstyle\begin{array}{l} u'(t)=-a_{1}(t)g_{1}(u(t))u(t)+\lambda b_{1}(t)f(u(t-\tau_{1}(t)),v(t-\zeta_{1}(t))), \\v'(t)=-a_{2}(t)g_{2}(v(t))v(t)+\mu b_{2}(t)g(u(t-\tau_{2}(t)),v(t-\zeta_{2}(t))), \end{array}\displaystyle \right . $$

(3.3)

where $\lambda, \mu>0$ are parameters. Under the same assumptions as before, one can check that system (3.3) is equivalent to

$$\begin{gathered} u(t)=\lambda \int_{t}^{t+\omega}G_{1}(t,s)b_{1}(s)f \bigl(u\bigl(s-\tau_{1}(s)\bigr),v\bigl(s-\zeta _{1}(s) \bigr)\bigr)\,ds, \\ v(t)=\mu \int_{t}^{t+\omega}G_{2}(t,s)b_{2}(s)g \bigl(u\bigl(s-\tau_{2}(s)\bigr),v\bigl(s-\zeta _{2}(s)\bigr) \bigr)\,ds,\end{gathered} $$

where

$$G_{1}(t,s)=\frac{e^{\int_{t}^{s} a_{1}(\theta)g_{1}(u(\theta))\,d\theta}}{e^{\int _{0}^{\omega}a_{1}(\theta)g_{1}(u(\theta))\,d\theta}-1}, \qquad G_{2}(t,s)= \frac{e^{\int_{t}^{s} a_{2}(\theta)g_{2}(v(\theta))\,d\theta}}{e^{\int _{0}^{\omega}a_{2}(\theta)g_{2}(v(\theta))\,d\theta}-1}, \quad t\leq s\leq t+\omega. $$

Furthermore, by a similar argument as above, it is not difficult to see that the results of Theorems 3.1–3.3 remain true for system (3.3).

Remark 3.2

It is worth remarking that, under some reasonable assumptions, the results of the paper are still valid for the more general coupled systems

$$u'_{i}(t)+a_{i}(t)g_{i} \bigl(u_{i}(t)\bigr)u_{i}(t)=\lambda_{i}b_{i}(t)f_{i} \bigl(u_{1}\bigl(t-\tau _{i1}(t)\bigr),\dots,u_{n} \bigl(t-\tau_{in}(t)\bigr)\bigr),\quad i=1,2,\dots,n $$

and

$$u'_{i}(t)=a_{i}(t)g_{i} \bigl(u_{i}(t)\bigr)u_{i}(t)-\lambda_{i}b_{i}(t)f_{i} \bigl(u_{1}\bigl(t-\tau _{i1}(t)\bigr),\dots,u_{n} \bigl(t-\tau_{in}(t)\bigr)\bigr),\quad i=1,2,\dots,n. $$