\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 118, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2010/118\hfil Three positive periodic solutions]
{Existence of three positive periodic solutions for
 differential systems with feedback controls on time scales}

\author[Y. Li, T. Zhang, J. Suo\hfil EJDE-2010/118\hfilneg]
{Yongkun Li,  Tianwei Zhang, Jianfeng Suo}  % in alphabetical order

\address{Department of Mathematics,
Yunnan University\\
Kunming, Yunnan 650091, China} 
\email[Yongkun Li]{yklie@ynu.edu.cn}
\email[Tianwei Zhang]{1200801347@stu.ynu.edu.cn} 
\email[Jianfeng Suo]{jfsuo@ynu.edu.cn}

\thanks{Submitted March 25, 2010. Published August 20, 2010.}
\thanks{Supported by grant 10971183 from the National Natural Sciences
 Foundation of China}
\subjclass[2000]{34K13}
\keywords{Positive periodic solutions;
 feedback controls; time scales; \hfill\break\indent
 Leggett-Williams multiple fixed point
 theorem}

\begin{abstract}
 Using the Leggett-Williams multiple fixed point theorem,
 we establish criteria for the existence of three positive
 periodic solutions of a class of differential systems with
 feedback controls on time scales.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Recently, by using the Krasnosel'skii's fixed point theorem for
cones, Li and Zhu \cite{l3} studied the existence of positive periodic
solutions of the following functional differential systems with
feedback controls:
\begin{equation} \label{e1.1}
 \begin{gathered}
\dot{x}(t)=-A(t)x(t)+f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t))), \\
\dot{u}(t)=-B(t)u(t)+C(t)x(h(t,x(t))).
\end{gathered}
\end{equation}
Zeng and Zhou \cite{z1} considered a class of more general
functional differential systems with feedback controls of the form
\begin{equation} \label{e1.2}
\begin{gathered}
\dot{x}(t)=-A(t,x(t))x(t)+f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t))), \\
\dot{u}(t)=-B(t,x(t))u(t)+C(t,x(t))x(h(t,x(t))).
\end{gathered}
\end{equation}
By means of the
Krasnosel'skii's fixed point theorem, they obtained some criteria
for the existence of two positive periodic solutions of \eqref{e1.2}.

Also, by applying the continuation theorem of coincidence degree
theory, Li and Zhu \cite{l4} studied the existence of positive periodic
solutions to the difference equations with feedback control of the
form
\begin{equation}
\begin{gathered}
N(n+1)=N(n)\exp\Big[r(n)\Big(1-\frac{N(n-m)}{k(n)}-c(n)\mu(n)\Big)
\Big], \\
\Delta\mu(n)=-a(n)\mu(n)+b(n)N(n-m),
\end{gathered}
\end{equation}
where $a:\mathbb{Z}\to(0,1), c, k, r, b:\mathbb{Z}\to\mathbb{R^{+}}$
are all $\omega$-periodic
 functions and $m$ is a positive integer.

In the previous ten years, many authors \cite{c2,l2,l5,l10}
have argued that the discrete time model governed by
difference equations are more
appropriate than the continuous ones when the populations have
non-overlapping generations. Discrete time models can also provide
efficient computational models of continuous models for numerical
simulations. Consequently, the studies of dynamic systems governed
by difference equations have received great attention from more
scholars.

In fact, continuous and discrete systems are very important in
implementing and applications. It is well known that the theory of
time scales \cite{b1,b2} has received a lot of attention which was
introduced by
Stefan Hilger \cite{h1} in order to unify continuous and discrete analysis.
Therefore, it is meaningful to study dynamic systems on time scales
which can unify differential and difference systems. For the work
concerning with the existence of periodic solutions for dynamic
systems on time scales, we refer the reader to
\cite{a2,a3,c1,l6,l8,l9, l11,l12,z2}.

Motivated by above statement, in this paper, we will study the
following differential systems with
feedback controls on time scales:
\begin{equation} \label{e1.4}
\begin{gathered}
x^{\Delta}(t)=-A(t,x(t))x(\sigma(t))+\lambda
f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t,x(t)))), \\
u^{\Delta}(t)=-B(t,x(t))u(\sigma(t))+g(t,x_{t},x(h(t,x(t)))),\quad
t\in\mathbb{T},
\end{gathered}
\end{equation}
in which $\mathbb{T}$ is a periodic time scales which has the
subspace topology inherited from the standard topology on
$\mathbb{R}$, $\lambda>0$ is parameter,
\begin{gather*}
A(t,x(t))=\operatorname{diag}[a_1(t,x(t)),a_2(t,x(t)),\dots
,a_{n}(t,x(t))],\\
B(t,x(t)) =\operatorname{diag}[b_1(t,x(t)),b_2(t,x(t)),\dots,b_{n}
(t,x(t))],
\end{gather*}
$a_{i}(t,y), b_{i}(t,y), \in C(\mathbb{T}\times
\mathbb{R}^{n},\mathbb{R})$ satisfy
$a_{i}(t+\omega,y)=a_{i}(t,y)$,
$b_{i}(t+\omega,y)=b_{i}(t,y)$ for
 all $t\in \mathbb{T}, y\in \mathbb{R}^{n}$,
$i=1,2,\dots,n$, $t-\tau(t,y)$, $t-\alpha(t,y)$, $h(t,y) \in
C(\mathbb{T}\times \mathbb{R}^{n},\mathbb{T})$
satisfy
$\tau(t+\omega,y)=\tau(t,y)$, $
\alpha(t+\omega,y)=\alpha(t,y),h(t+\omega,y)=h(t,y)$ for
 all $t\in \mathbb{T}$, $y\in \mathbb{R}^{n}$, $\omega>0$ is a
constant, $f$ is a function defined on $\mathbb{T}\times BC\times
\mathbb{R}^{n}\times \mathbb{R}^{n}$,
 satisfying $f(t+\omega,x_{t+\omega},y,z)=f(t,x_{t},y,z)$ for all
$t\in \mathbb{T}$, $x\in BC,y,z\in \mathbb{R}^{n}$, where $BC$ denotes
the Banach space of all bounded
 continuous functions $\eta: \mathbb{T}\to \mathbb{R}^{n}$
with the norm
$\|\eta\|=\sum_{i=1}^{n}\max_{\theta\in \mathbb{T}}|\eta_{i}(\theta)|$,
where $\eta=(\eta_1,\eta_2,\dots,\eta_{n})^{T}$, and
$g$ is a function defined on $\mathbb{T}\times BC\times
 \mathbb{R}^{n}$, satisfying $g(t+\omega,x_{t+\omega},y)=g(t,x_{t},y)$
for all $t\in \mathbb{T}, x\in BC,y\in \mathbb{R}^{n}$.
If $x\in BC$, then
$x_{t}\in BC$ for any $t\in \mathbb{T}$ is defined by
$x_{t}(\theta)=x_{t}(t+\theta)$ for $\theta\in \mathbb{T}$. In the
sequel, we denote $f=(f_1,f_2,\dots,f_{n})^{T},
g=(g_1,g_2,\dots,g_{n})^{T}$. Let
$\mathbb{R}=(-\infty,+\infty), \mathbb{R}_{+}=[0,+\infty),
\mathbb{R}_{+}^{n}=\{(x_1,x_2,\dots,x_{n})^{T}:x_{i}\geq
0,1\leq i\leq n\}$,
 respectively. For each
$x=(x_1,x_2,\dots,x_{n})^{T}\in \mathbb{R}^{n}$, the norm of $x$
is defined as $|x|_{0}=\sum_{i=1}^{n}|x_{i}|$.

The main purpose of this paper is to study the existence of at
least three nonnegative periodic solutions of \eqref{e1.4} by
using the Leggett-Williams multiple fixed point theorem.


The organization of this paper is as follows. In Section $2$, we
make some preparations. In Section $3$, by using the Leggett-Williams
multiple fixed point theorem, we obtain the existence of at least
three nonnegative periodic solutions of \eqref{e1.4}. In Section $4$,
an example is also provided to illustrate the effectiveness of the
main results obtained in Section $3$.

\section{Preliminaries}

In this section, we shall first recall some basic definitions,
lemmas which are used in what follows.

\begin{definition}[\cite{b1}] \label{def2.1} \rm
A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset
of the real set $\mathbb{R}$
with the topology and ordering inherited from $\mathbb{R}$.
The forward and backward jump operators
$\sigma$, $\rho:\mathbb{T}\to\mathbb{T}$ and the graininess
$\mu:\mathbb{T}\to\mathbb{R}^{+}$ are defined, respectively, by
\[
\sigma(t):=\inf\{s\in\mathbb{T}:s>t\},
\quad\rho(t):=\sup\{s\in\mathbb{T}:s<t\},
\quad\mu(t):=\sigma(t)-t.
\]
The point $t\in\mathbb{T}$ is called left-dense, left-scattered,
right-dense or right-scattered if
$\rho(t)=t$, $\rho(t)<t$, $\sigma(t)=t$ or $\sigma(t)>t$,
respectively.
Points that are right-dense and left-dense
at the same time are called dense.
If $\mathbb{T}$ has a left-scattered maximum $m$,
defined $\mathbb{T}^{k}=\mathbb{T}-\{m\}$;
otherwise, set $\mathbb{T}^{k}=\mathbb{T}$.
\end{definition}

\begin{definition}[\cite{k1}] \label{def2.2} \rm
 We say that a time scale $\mathbb{T}$ is periodic if there exists $p>0$
such that if $t\in \mathbb{T}$, then $t\pm p\in \mathbb{T}$.
For $\mathbb{T}\neq\mathbb{R}$, the smallest positive $p$ is called the period of the time scale.
Let $\mathbb{T}\neq\mathbb{R}$ be a periodic time scale with period p.
We say that the function $f:\mathbb{T}\to \mathbb{R}$ is periodic with period T if there
exists a natural number $n$ such that $T=np,f(t+T)=f(t)$ for all
$t\in \mathbb{T}$ and T is the smallest number such that $f(t+T)=f(t)$. If $\mathbb{T}=\mathbb{R}$,
we say that $f$ is periodic with period $T>0$ if $T$ is the smallest
positive number such that $f(t+T)=f(t)$ for all $t\in \mathbb{T}$.
\end{definition}

\begin{definition}[\cite{b1}] \label{def.2.3} \rm
 For $f:\mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}^{k}$, the delta
derivative of $f$ at $t$, denoted by $f^{\Delta}(t)$, is the number
(provided it exists) with the property that given any $\epsilon>0$,
there is a neighborhood $U\subset\mathbb{T}$ of $t$ such that
$$
|f(\sigma(t))-f(s)-f^{\Delta}(t)[\sigma(t)-t]|
\leq\epsilon|\sigma(t)-s|,\quad \forall s\in U .
$$
\end{definition}

\begin{definition}[\cite{b1}] \label{def2.4} \rm
 A function $f:\mathbb{T}\to\mathbb{R}$ is called
regulated provided its right-sided limits exist $($finite) at all right-dense points in $\mathbb{T}$ and
its left-sided limits exist $($finite) at all left-dense points in $\mathbb{T}$.
\end{definition}

\begin{definition}[\cite{b1}] \label{def2.5} \rm
 A continuous function $F:\mathbb{T}\to\mathbb{R}$ is called
pre-differentiable with (region of differentiation) $D$,
provided $D\subset\mathbb{T}^{k}$,
$\mathbb{T}^{k}\backslash D$ is countable and contains no
right-scattered elements of $\mathbb{T}$,
and $F$ is differentiable at each $t\in D$.
\end{definition}

\begin{definition}[\cite{b1}] \label{def2.6} \rm
Assume that $f:\mathbb{T}\to\mathbb{R}$ is a regulated function.
Suppose further that there exists a function $F$ which is
pre-differentiable with region of differentiation $D$ such that
$$
F^{\Delta}(t)=f(t)\quad\text{holds for all $t\in D$}.
$$
We define the Cauchy integral by
$$
\int_{a}^{b}f(s)\Delta s=F(b)-F(a)
\quad\text{for all $a, b\in \mathbb{T}$}.
$$
\end{definition}


\begin{definition}[\cite{b1}] \label{def2.7} \rm
A function $p:\mathbb{T}\to \mathbb{R}$ is said to be regressive
provided
$1+\mu(t)p(t)\neq0$ for all $t\in \mathbb{T}^{k}$, where
$\mu(t)=\sigma(t)-t$ is the graininess function. The set of all
regressive rd-continuous functions
$f:\mathbb{T}\to \mathbb{R}$ is denoted by $\mathcal{R}$
while the set $\mathcal{R}^{+}$ is given by $\{f\in \mathcal{R}:1+\mu(t)f(t)>0\}$ for
 all $t\in \mathbb{T}$. Let $p\in\mathcal{R}$. The exponential
 function is defined by
$$
e_{p}(t,s)=\exp\Big(\int_{s}^{t} \xi_{\mu(\tau)}(p(\tau))
\Delta\tau\Big),
$$
where $\xi_{h(z)}$ is the so-called cylinder transformation.
\end{definition}

\begin{lemma}[\cite{b1}] \label{lem2.1}
Let $p,q\in\mathcal{R}$. Then
\begin{itemize}
\item [(i)]
$e_{0}(t,s)\equiv 1$ and $e_{p}(t,t)\equiv 1$;
 \item [(ii)]
$e_{p}(\sigma(t),s)=(1+\mu(t)p(t))e_{p}(t,s)$;
 \item [(iii)]
$\frac{1}{e_{p}(t,s)}=e_{\ominus p}(t,s)$, where $\ominus p(t)=-\frac{p(t)}{1+\mu(t)p(t)}$;
 \item [(iv)]
$e_{p}(t,s)=\frac{1}{e_{p}(s,t)}=e_{\ominus p}(s,t)$;
 \item [(v)]
$e_{p}(t,s)e_{p}(s,r)=e_{p}(t,r)$;
 \item [(vi)]
$\frac{e_{p}(t,s)}{e_{q}(t,s)}=e_{p\ominus q}(t,s)$, where $p\ominus q=p\oplus(\ominus q)$;
 \item [(vii)]
$e_{p}^{\Delta}(\cdot,s)=pe_{p}(\cdot,s)$.
\end{itemize}
\end{lemma}

 For convenience, we introduce the following notation:
\begin{gather*}
r_1^{i}=\sup_{t\in [0,\omega]_{\mathbb{T}}}
 \frac{1}{|1-e_{\ominus a_{i}}(t,t-\omega)|}, \quad
r_2^{i}=\inf_{t\in [0,\omega]_{\mathbb{T}}}
\frac{1}{|1-e_{\ominus a_{i}}(t,t-\omega)|}, \\
\eta_1^{i}=\sup_{u\in [t-\omega,t]_{\mathbb{T}}}
 e_{\ominus a_{i}}(t,u),\quad
\eta_2^{i}=\inf_{u\in [t-\omega,t]_{\mathbb{T}}}
 e_{\ominus a_{i}}(t,u), \\
r^{M}=\max_{1\leq i\leq n}\{r_1^{i}\},\quad
r^{l} =\min_{1\leq i\leq n}\{r_2^{i}\},\\
\eta^{M}=\max_{1\leq i\leq n}\{\eta_1^{i}\},\quad
\eta^{l}=\min_{1\leq i\leq n} \{\eta_2^{i}\}, \\
\gamma^{i}=\sup_{t\in [0,\omega]_{\mathbb{T}}}
|\ominus a_{i}|,\quad
\kappa^{i}=\sup_{t\in [0,\omega]_{\mathbb{T}}}e_{\ominus
a_{i}}(\sigma(t),t),\\
\gamma= \max_{1\leq i\leq n}\{\gamma^{i}\}, \quad
\kappa=\max_{1\leq i\leq n}\{\kappa^{i}\}.
\end{gather*}

\begin{lemma} \label{lem2.2}
$(x(t),u(t))^{T}$ is an $\omega$-periodic solution of \eqref{e1.4}
if and only if it is an $\omega$-periodic solution of the
system
\begin{equation} \label{e2.1}
\begin{gathered}
x^{\Delta}(t)=-A(t,x(t))x(\sigma(t))+$$\lambda$$f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t,x(t)))), \\
u(t)=\int_{t-\omega}^{t}\overline{G}(t,s)g(s,x_{s},x(h(s,x(s))))\Delta
s :=(\Phi x)(t),
\end{gathered}
\end{equation}
 where
$$
\overline{G}(t,s)=\operatorname{diag}[\overline{G}_1(t,s),
\overline{G}_2(t,s),\dots,\overline{G}_{n}(t,s)]
$$
and
$$
\overline{G}_{i}(t,s)=\frac{e_{\ominus
b_{i}}(t,s)}{1-e_{\ominus b_{i}}(t,t-\omega)}, \quad
s\in[t-\omega,t]_{\mathbb{T}},\; i=1,2,\dots,n.
$$
\end{lemma}

\begin{proof}
First, assume that $(x(t),u(t))^{T}$ is an $\omega$-periodic
solution of \eqref{e1.4}.  From the second equation of \eqref{e1.4},
it follows that
\begin{equation} \label{e2.2}
u_{i}^{\Delta}(t)+b_{i}(t,x(t))u_{i}(\sigma(t))
=g_{i}(t,x_{t},x(h(t,x(t)))),\quad i=1,2,\dots,n.
\end{equation}
Multiply both sides of this equation by $e_{b_{i}}(t,0)$
and then integrate them from $t-\omega$ to $t$
to obtain
$$
\int_{t-\omega}^{t}[e_{b_{i}}(s,0)u_{i}(s)]^{\Delta}\Delta s
= \int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i}
(s,x_{s},x(h(s,x(s))))\Delta s,
$$
for $i=1,2,\dots,n$,
and then
$$
e_{b_{i}}(t,0)u_{i}(t)-e_{b_{i}}(t-\omega,0)u_{i}(t-\omega)
=\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i}
(s,x_{s},x(h(s,x(s))))\Delta s,
$$
for $i=1,2,\dots,n$.
Dividing both sides of the above equation by $e_{b_{i}}(t,0)$, we have
$$
u_{i}(t)=\int_{t-\omega}^{t}\frac{e_{\ominus b_{i}}(t,s)}
{1-e_{\ominus b_{i}}(t,t-\omega)}g_{i}(s,x_{s},
x(h(s,x(s))))\Delta s ,\quad i=1,2,\dots,n.
$$
So $(x(t),u(t))^{T}$ is an $\omega$-periodic solution of \eqref{e2.1}.

Conversely, assume that $(x(t),u(t))^{T}$ is an $\omega$-periodic
solution of \eqref{e2.1}. Then we have
$$
e_{b_{i}}(t,0)u_{i}(t)-e_{b_{i}}(t-\omega,0)u_{i}(t-\omega)
=\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i}
(s,x_{s},x(h(s,x(s))))\Delta s,
$$
for $i=1,2,\dots,n$; that is,
$$
\big(1+e_{b_{i}}(\omega,0)\big)e_{b_{i}}(t,0)u_{i}(t)
=\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i}
(s,x_{s},x(h(s,x(s))))\Delta s,
$$
for $i=1,2,\dots,n$.
Then
\begin{align*}
&\big[\big(1+e_{b_{i}}(\omega,0)\big)e_{b_{i}}(t,0)u_{i}(t)
\big]^{\Delta}\\
&= e_{b_{i}}(t,0)\big(1+e_{b_{i}}(\omega,0)\big)\big[u_{i}^{\Delta}(t)+b_{i}u_{i}(\sigma(t))\big]\\
&= \Big(\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i}
(s,x_{s},x(h(s,x(s))))\Delta s\Big)^{\Delta}\\
&= e_{b_{i}}(t,0)g_{i}(t,x_{t},x(h(t,x(t))))
 -e_{b_{i}}(t-\omega,0)g_{i}
(t-\omega,x_{t-\omega},x(h(t-\omega,x(t-\omega))))\\
&= e_{b_{i}}(t,0)\big(1+e_{b_{i}}(\omega,0)\big)
g_{i}(t,x_{t},x(h(t,x(t)))),
\end{align*}
which implies
\begin{align*}
u_{i}^{\Delta}(t)+b_{i}(t,x(t))u_{i}(\sigma(t))
=g_{i}(t,x_{t},x(h(t,x(t)))),\quad i=1,2,\dots,n.
\end{align*}
So $(x(t),u(t))^{T}$ is an $\omega$-periodic solution of \eqref{e1.4}.
The proof of the lemma is complete.
\end{proof}

At the same time, from the definition of $e_{p}(t,s)$ and the
periodicity of $b_{i}$, we have
$e_{\ominus b_{i}}(t+\omega,s+\omega)
=e_{\ominus b_{i}}(t,s),\,i=1,2,\dots,n$,
so it is clear that $\overline{G}(t,s)=\overline{G}(t+\omega,s+\omega)$
for all $(t,s)\in\mathbb{T}^{2}$
and $u(t+\omega)=u(t)$ when $x$ is $\omega$-periodic
solution.

Now, \eqref{e2.1} can be reformulated as
\begin{equation} \label{e2.3}
x^{\Delta}(t)=-A(t,x(t))x(\sigma(t))
+\lambda f(t,x_{t},x(t-\tau(t,x(t))),(\Phi x)(t-\alpha(t,x(t)))).
\end{equation}
We proceed from \eqref{e2.3} and obtain
\begin{equation}
x(t)=\lambda\int_{t-\omega}^{t}G(t,s)f(s,x_{s},x(s-\tau(s,x(s))),
(\Phi x)(s-\alpha(s,x(s))))\Delta s,
\end{equation}
where
$$
G(t,s)=\operatorname{diag}[G_1(t,s),G_2(t,s),\dots,{G}_{n}(t,s)]
$$
and
$$
G_{i}(t,s)=\frac{e_{\ominus a_{i}}(t,s)}{1-e_{\ominus
a_{i}}(t,t-\omega)}, \quad s\in[t-\omega,t]_{\mathbb{T}}, \quad
i=1,2,\dots,n.
$$
 By the periodicity of $a_{i}$, $i=1,2,\dots,n$, it is also obvious
that $G(t,s)=G(t+\omega,s+\omega)$ for all $(t,s)\in\mathbb{T}^{2}$.

To obtain our main results, we make the following assumptions
throughout this paper.
\begin{itemize}
\item [(H1)] $a_{i}(t,x(t))>0$ or $a_{i}(t,x(t))<0$ for all
$t\in \mathbb{T}$, $i=1,2,\dots,n$;

\item [(H2)] $f_{i}(t,\zeta,\xi,\Phi(\eta))a_{i}(t,x(t))\geq0$
 for all $(t,\zeta,\xi,\eta)\in \mathbb{T}\times
BC(\mathbb{T}\times\mathbb{R}_{+}^{n})\times\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n},\,i=1,2,\dots,n$;

\item [(H3)] $f(t,\phi_{t},\phi(t-\tau(t,\phi(t))),
(\Phi\phi)(t-\alpha(t,\phi(t))))$ is a continuous function
of $t$ for each $\phi\in BC(\mathbb{T}\times\mathbb{R}_{+}^{n})$;

\item [(H4)] for any $L>0$ and $\epsilon>0$, there exists a real
number $\delta>0$ such that $\phi,\psi\in BC,\|\phi\|\leq L$,
$\|\psi\|\leq L,\| \phi-\psi\|<\delta,0\leq s\leq \omega$
imply
\begin{align*}
&\|f(s,\phi_{s},\phi(s-\tau(s,\phi(s))),(\Phi\phi)
(s-\alpha(s,\phi(s))))\\
&-f(s,\psi_{s},\psi(s -\tau(s,\psi(s))),
 (\Phi\psi)(s-\alpha(s,\psi(s))))\|<\epsilon.
\end{align*}
\end{itemize}

Moreover, for the sake of simplicity, let
$$
f(t,\phi ,\Phi)=f(t,\phi_{t},\phi(t-\tau(t,\phi(t))),
(\Phi\phi)(t-\alpha(t,\phi(t))))
$$
and
$$
f(t,x ,\Phi)=f(t,x_{t},x(t-\tau(t,x(t))),
(\Phi x)(t-\alpha(t,x(t)))).
$$

Then by (H2),
\begin{equation} \label{e2.5}
G_{i}(t,s)f_{i}(t,x ,\Phi)\geq 0 \quad\text{for }
(t,s)\in\mathbb{T}^{2},\; i=1,2,\dots,n.
\end{equation}

 Let $X$ be a Banach space and $K$ be
a cone in $X$. A mapping $\psi$ is said to be a concave nonnegative
continuous functional on $K$
if $\psi:K\to\mathbb{R}_{+}$ is continuous and
$$
\psi(\mu x+(1-\mu)y)\geq \mu\psi(x)+(1-\mu)\psi(y),\quad
 x,y\in K, \;\mu\in[0,1].
$$
Let $a,b,c>0$ be constants with $K$ and $X$ as defined above. Define
$$
K_{a}=\{x\in K:\|x\|<a\},\quad K(\psi,b,c)
=\{x\in K:\psi(y)\geq b,\|x\|\leq c\}.
$$

\begin{theorem}[Leggett-Williams multiple fixed point
theorem \cite{l1}] \label{thm2.1}
 Let $X=(X,\|\cdot\|)$ be a Banach space and $K\subset X$ a cone,
and $c_{4}>0$ a constant. Suppose there exists a concave nonnegative
continuous function $\psi$ on $K$ with $\psi(u)\leq u$ for
$u\in\overline{K}_{c_{4}}$ and let
$T:\overline{K}_{c_{4}}\to\overline{K}_{c_{4}}$ be a
continuous compact map. Assume that there are numbers $c_1,c_2$
and $c_{3}$ with $0<c_1<c_2<c_{3}\leq c_{4}$ such that
\begin{itemize}

\item [(i)]
$\{u\in K(\psi,c_2,c_{3}):\psi(u)>c_2\}\neq\emptyset$ and
$\psi(Tu)>c_2$ for all $u\in K(\psi,c_2,c_{3})$;

\item [(ii)]
 $\|Tu\|<c_1$ for all $u\in\overline{K}_{c_1}$;

\item [(iii)]
$\psi(Tu)>c_2$ for all $u\in K(\psi,c_2,c_{4})$ with $\|Tu\|>c_{3}$.

\end{itemize}
Then $T$ has at least three fixed points $u_1,u_2$ and
$u_{3}$ in $\overline{K}_{c_{4}}$. Furthermore,
 e have $u_1\in\overline{K}_{c_1}$,
$u_2\in\{u\in K(\psi,c_2,c_{4}):\psi(u)>c_2\}$,
$u_{3}\in\overline{K}_{c_{4}}\backslash\{K(\psi,c_2$,
$c_{4})\cup\overline{K}_{c_1}\}$.
\end{theorem}

Let $X=\{x(t)=(x_1,x_2,\dots,x_{n})^{T}\in
C(\mathbb{T},\mathbb{R}^{n}):x(t)=x(t+\omega)\}$ with the norm
$\|x\|=\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}} |x_{i}(t)|$,
then $X$ is a Banach space with the norm $\|\cdot\|$. Define a cone
$K$ in $X$ by
$$
K=\{x(t)=(x_1,x_2,\dots,x_{n})^{T}\in X:x_{i}(t)\geq 0,\forall
t\in [0,\omega]_{\mathbb{T}},\,i=1,2,\dots,n\}
$$
and an operator $T_{\lambda}$ on $X$ by
$$
(T_{\lambda}x)(t)=\lambda\int_{t-\omega}^{t}G(t,s)f(s,x,\Phi)\Delta s .
$$
And let
\[
T_{\lambda}x=(T_{\lambda}^{1}x,T_{\lambda}^{2}x,
\dots,T_{\lambda}^{n}x)^{T}.
\]

\begin{lemma} \label{lem2.3}
 $T_{\lambda}(K)\subset K$ and $T_{\lambda}:K\to K$ is well-defined.
\end{lemma}

\begin{proof}
For each $x\in K $, by $(H_3)$, we have
$T_{\lambda}x\in C(\mathbb{T},\mathbb{R}^{n})$,
with the periodicity of $G(t,s)$ and $(\Phi x)(t)$, then
\begin{align*}
 (T_{\lambda}x)(t+\omega)
&= \lambda\int_{t}^{t+\omega}G(t+\omega,s)f(s,x,\Phi)\Delta s\\
&= \lambda\int_{t-\omega}^{t}G(t+\omega,s+\omega)
 f(s+\omega,x,\Phi)\Delta s\\ &= \lambda\int_{t-\omega}^{t}G(t,s)f(s,x,\Phi)\Delta s\\
&= (T_{\lambda}x)(t) ,
\end{align*}
and by \eqref{e2.5}, $T_{\lambda}x\in K$. The proof
 is complete.
\end{proof}


\begin{lemma} \label{lem2.4}
$T_{\lambda}:K\to K$ is completely continuous.
\end{lemma}

\begin{proof}
We first show that $T_{\lambda}$ is continuous. By (H4),
for any $L>0$ and $\epsilon>0$, there exists a $\delta>0$ such that
$\phi,\psi\in BC$, $\|\phi\|\leq L$,
$\|\psi\|\leq L$, $\| \phi-\psi\|<\delta$, $0\leq s\leq \omega$
imply
\begin{align*}
 &\|f(s,\phi_{s},\phi(s-\tau(s,\phi(s))),(\Phi\phi)(s-\alpha(s,\phi(s))))\\
 & -f(s,\psi_{s},\psi(s-\tau(s,\psi(s))),
(\Phi\psi)(s-\alpha(s,\psi(s))))\|<\frac{\epsilon}
{\lambda r^{M}\eta^{M}}.
\end{align*}
If $x,y\in K$ with $\|x\| \leq L$, $\|y\| \leq L$, and
$\| x-y\|<\delta$, then
\begin{align*}
&\|(T_{\lambda}x)(t)-(T_{\lambda}y)(t)\|\\
&\leq\lambda\int_{t-\omega}^{t}\max_{1\leq i\leq n}
|G_{i}(t,s)|\|f(s,x_{s},x(s-\tau(s,x(s))),
(\Phi x)(s-\alpha(s,x(s))))\\
&\quad -f(s,y_{s},y(s-\tau(s,y(s))),(\Phi y)(s-\alpha(s,y(s))))\|\Delta s\\
&\leq \lambda r^{M}\eta^{M}\int_{0}^{\omega}\|f(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,x(s))))\\
&\quad -f(s,y_{s},y(s-\tau(s,y(s))),(\Phi y)(s-\alpha(s,y(s))))\|\Delta s\\
&< \lambda r^{M}\eta^{M}\frac{\epsilon}{\lambda r^{M}\eta^{M}}
= \epsilon
\end{align*}
for all $t\in[0,\omega]_{\mathbb{T}}$.
This yields $\|T_{\lambda}x -T_{\lambda}y\|<\epsilon$.
Thus, $T_{\lambda}$ is continuous.

Next, we show that $T_{\lambda}$ maps any bounded sets in $K$
into relatively compact sets. Now, we first prove
that $f$ maps bounded sets into bounded sets. Indeed,
let $\epsilon=1$. By (H4), for any $\mu>0$, there exists
$\delta>0$ such that $x, y\in BC$, $\|x\|\leq \mu$,
$\|y\|\leq \mu$, $\|x-y\|<\delta$, $0\leq s\leq \omega$ imply
\begin{align*}
&\|f(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,x(s))))\\
& -f(s,y_{s},y(s-\tau(s,y(s))),(\Phi y)(s-\alpha(s,y(s))))\|<1.
\end{align*}
Choose a positive integer $N$ such that $\frac{\mu}{N} < \delta$.
 Let $x\in BC$ and define $x^{k}(t)=\frac{k}{N}x(t)$
for $k=0,1,2,\dots,N$. If $\|x\|\leq\mu$, then
\begin{align*}
 \|x^{k}-x^{k-1}\|=\|\frac{k}{N}x(t)-\frac{k-1}{N}x(t)\|\leq\frac{1}{N} \|x\|\leq\frac{\mu}{N}
 <\delta.
\end{align*}
Thus,
\begin{align*}
&\|f(s,x_{s}^{k},x^{k}(s-\tau(s,x^{k}(s))),(\Phi x^{k})
 (s-\alpha(s,x^{k}(s))))\\
& -f(s,x_{s}^{k-1},x^{k-1}(s-\tau(s,x^{k-1}(s))),
 (\Phi x^{k-1})(s-\alpha(s,x^{k-1}(s))))\|<1
\end{align*}
for all $s\in [0,\omega]_{\mathbb{T}}$. This yields
\begin{align*}
&\|f(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,x(s))))\|\\
&=\|f(s,x_{s}^{N},x^{N}(s-\tau(s,x^{N}(s))),(\Phi x^{N})
 (s-\alpha(s,x^{N}(s))))\|\\
&\leq \sum_{k=1}^{N}\|f(s,x_{s}^{k},x^{k}(s-\tau(s,x^{k}(s))),
 (\Phi x^{k})(s-\alpha(s,x^{k}(s))))\\
&\quad -f(s,x_{s}^{k-1},x^{k-1}(s-\tau(s,x^{k-1}(s))),
 (\Phi x^{k-1})(s-\alpha(s,x^{k-1}(s))))\|\\
&\quad +\|f(s,0,0,0\|\\
&< N+\|f(s,0,0,0\|=:Q.
\end{align*}
For $t\in [0,\omega]_{\mathbb{T}}$, we have
\begin{align*}
\|T_{\lambda}x\|
&= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|(T_{\lambda}^{i}x)(t)|\\
&\leq \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\int_{0}^{\omega}|f_{i}(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,
x(s))))|\Delta s\\
&\leq \lambda r^{M}\eta^{M}\omega Q.
\end{align*}
Finally, for $t\in \mathbb{T}$,
\begin{align*}
(T_{\lambda}^{i}x)^{\Delta}(t)
&= [\lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s
 ]^{\Delta}\\
&= [\lambda\int_{t-\omega}^{t}\frac{e_{\ominus a_{i}}(t,s)}
 {1-e_{\ominus a_{i}}(t, t-\omega)}f_{i}(s,x,\Phi)\Delta s]^{\Delta}\\
&= \lambda\frac{e_{\ominus a_{i}}(\sigma(t),t)-e_{\ominus a_{i}}(\sigma(t),t-\omega)}
 {1-e_{\ominus a_{i}}(t,t-\omega)}f_{i}(t,x,\Phi)\\
&\quad +\lambda\frac{1}{1-e_{\ominus a_{i}}(t,t-\omega)}\ominus a_{i}
\int_{\tilde{a}}^{t} e_{\ominus a_{i}}(t,s)f_{i}(s,x,\Phi)\Delta s\\
&\quad -\lambda\frac{1}{1-e_{\ominus a_{i}}(t,t-\omega)}\ominus a_{i}
 \int_{\tilde{a}}^{t-\omega}e_{\ominus a_{i}}(t,s)f_{i}(s,x,\Phi)
 \Delta s\\
&= \ominus a_{i}(T_{\lambda}^{i}x)(t)
 +\lambda e_{\ominus a_{i}}(\sigma(t),t) f_{i}(t,x,\Phi),
\end{align*}
where $\tilde{a}\in [0,\omega]_{\mathbb{T}}$ is an arbitrary constant,
$i=1,2,\dots,n$.
So we obtain
\[
|(T_{\lambda}^{i}x)^{\Delta}(t)|\leq\gamma^{i}|T_{\lambda}^{i}x|
+\lambda\kappa^{i}|f_{i}(t,x,\Phi)|,\quad i=1,2,\dots,n.
\]
Then
\begin{align*}
\|(T_{\lambda}x)^{\Delta}(t)\|
&= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|
(T_{\lambda}^{i}x)^{\Delta}(t)|\\
&\leq \gamma\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|T_{\lambda}^{i}x|
+\lambda\kappa\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|f_{i}(t,x,\Phi)|\\
&\leq \gamma\|T_{\lambda}x\|+\lambda\kappa\|f(t,x,\Phi)\|\\
&\leq \lambda\gamma r^{M}\eta^{M}\omega Q+\lambda\kappa Q.
\end{align*}
Hence $\{T_{\lambda}x: x\in K,\|x\|\leq\mu\}$ is a family of
uniformly bounded and equicontinuous functions on
$[0,\omega]_{\mathbb{T}}$. Applying Arzela-Ascoli theorem on
time scales \cite{a1}, the function $T_{\lambda}$ is completely
continuous. The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.5}
Existence of nonnegative periodic solutions of \eqref{e1.4}
is equivalent to the existence of fixed point problem of
$ T_{\lambda}$ in $K$.
\end{lemma}

The proof of the above lemma is straight forward and we will omit it.


\section{Main results}

Let
$$
f_{i}^{h}=\limsup_{\phi_{i}\to h}
\max_{t\in[0,\omega]_{\mathbb{T}}}
\frac{|f_{i}(t,\phi,\Phi)|}{|a_{i}(t,\phi)|\phi_{i}},\quad i=1,2,\dots,n,
$$
where $\phi(t)=(\phi_1(t),\phi_2(t),\dots,\phi_{n}(t))^{T}\in
C(\mathbb{T},\mathbb{R}^{n})$,
$|\phi|_1=\sum_{i=1}^{n}\min_{t\in
[0,\omega]_{\mathbb{T}}}|\phi_{i}(t)|$ and
$\delta=\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}$.

 From the definitions of $\eta^{M}$ and $\eta^{l}$, it is obvious
that $\delta>1$.

\begin{theorem} \label{thm3.1}
Assume that {\rm (H1)-(H4)} hold, there are constants
$0<c_1<c_2$ such that the following conditions hold:
\begin{itemize}
 \item [(H5)] $r^{M}\eta^{M}>1$;

 \item [(H6)] $f_{i}^{\infty}<\omega$, $i=1,2,\dots,n $;

 \item [(H7)] $\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s
 \geq\delta c_2\omega$ for $c_2\leq|x|_1\leq\|x\|\leq\delta c_2$;

 \item [(H8)] $\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s
\leq\frac{c_1\omega}{r^{M}\eta^{M}}$
 for $0\leq|x|_1\leq \|x\|\leq c_1$
\end{itemize}
 Then \eqref{e1.4} has at least three nonnegative $\omega$-periodic
solutions for
$$
\frac{1}{r^{M}\eta^{M}\omega}<\lambda<\frac{1}{\omega}.
$$
\end{theorem}

\begin{proof}
Since $f_{i}^{\infty}<\omega$ holds for $1\leq i\leq n$,
there exist $\varepsilon\in (0,\omega)$ and $\theta>0$
 such that
$|f_{i}(t,x,\Phi)|\leq \varepsilon|a_{i}(t,x)|x_{i}$ for
$x_{i}\geq \theta$, $t\in [0,\omega]_{\mathbb{T}}$,
 $i=1,2,\dots,n$.
Let
$$
\xi_{i}=\max_{0\leq x_{i}\leq \theta,0\leq t\leq
\omega}|f_{i}(t,x,\Phi)|, \quad i=1,2,\dots,n,
 \quad \xi=\sum_{i=1}^{n}\xi_{i}.
$$
Then $|f_{i}(t,x,\Phi)|\leq \varepsilon|a_{i}(t,x)|x_{i}+\xi_{i}$
for $x_{i}\geq 0$, $t\in [0,\omega]_{\mathbb{T}}$,
$i=1,2,\dots,n$. Choose
$$
c_{4}>\max\Big\{\frac{r^{M}\eta^{M}\xi\omega}{\omega-\varepsilon},
\delta c_2\Big\}.
$$
Then for $x\in \overline{K}_{c_{4}}$, we have
\begin{align*}
 \|T_{\lambda}x\|
&= \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}|T_{\lambda}^{i}x|\\
&= \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}
 \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\
&= \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}\lambda
 \int_{t-\omega}^{t}|G_{i}(t,s)||f_{i}(s,x,\Phi)|\Delta s\\
&\leq \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}
 \lambda\int_{t-\omega}^{t}|G_{i}(t,s)|(\varepsilon|a_{i}(s,x)|x_{i}+\xi_{i})\Delta s\\
&\leq \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}
 \lambda\int_{t-\omega}^{t}|G_{i}(t,s)|(\varepsilon|a_{i}(s,x)||x_{i}|+\xi_{i})\Delta s\\
&\leq \lambda[\varepsilon \max_{0\leq i\leq n}
 \{\int_{t-\omega}^{t}G_{i}(t,s)a_{i}(s,x)\Delta s\}\sum_{i=1}^{n}
 \max_{s\in[0,\omega]_{\mathbb{T}}}|x_{i}(s)|\\
&\quad +\sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}\int_{t-\omega}^{t}
 |G_{i}(t,s)|\xi_{i}\Delta s]\\
&\leq \lambda[\varepsilon\|x\|+\sum_{i=1}^{n}r_1^{i}\eta_1^{i}\xi_{i}\omega]\\
&\leq \lambda[\varepsilon c_{4}+r^{M}\eta^{M}\xi\omega ]\\
&< \frac{1}{\omega}[\varepsilon c_{4}+ r^{M}\eta^{M}\xi\omega]
< c_{4}.
\end{align*}
Hence $T_{\lambda}:\overline{K}_{c_{4}}\to\overline{K}_{c_{4}}$.

Next, we define a concave nonnegative continuous function $\psi$
on $K$ by
$\psi(x)=\sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}}\\
|x_{i}(t)|$,
then $\psi(x)\leq \|x\|$. Let $c_{3}=\delta c_2
=\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}c_2$
and $\phi_{0}(t)=\{\phi_{0},0,\dots,0\}^{T}, \phi_{0}$ is any
given number satisfying $c_2<\phi_{0}<c_{3}$.
Then $\phi_{0}(t)\in \{x\in K(\psi,c_2,c_{3}):
\psi(x)>c_2\}\neq\emptyset$. Further,
for $x\in K(\psi,c_2,c_{3})$, by (H7)
\begin{equation}
\begin{aligned}
\psi(T_{\lambda}x)
&= \sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}}
 |(T_{\lambda}^{i}x)(t)| \\
 &= \sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}}
 \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\
&\geq \lambda r^{l}\eta^{l}\sum_{i=1}^{n}\int_{0}^{\omega}
 |f_{i}(s,x,\Phi)|\Delta s \\
 &= \lambda r^{l}\eta^{l}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s \\
 &\geq \lambda r^{l}\eta^{l}\delta c_2\omega \\
 &= \lambda r^{l}\eta^{l}\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}
 c_2\omega \\
 &\geq\lambda r^{M}\eta^{M} c_2\omega
 > c_2,
\end{aligned} \label{e3.6}
\end{equation}
so condition (i) of Theorem \ref{thm2.1} holds.

Now, let $x\in\overline{K}_{c_1}$, by (H8)
\begin{align*}
\|T_{\lambda}x\|
&= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}
\lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s\\
 &\leq \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\int_{0}^{\omega}|f_{i}(s,x,\Phi)|\Delta s \\
 &\leq \lambda r^{M}\eta^{M}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s\\
 &< \frac{1}{\omega} r^{M}\eta^{M}\frac{c_1\omega}{r^{M}\eta^{M}}\
 = c_1,
\end{align*}
then $T_{\lambda}x\in\overline{K}_{c_1}$.

Finally, for $x\in K(\psi,c_2,c_{4})$ and
$\|T_{\lambda}x\|>c_{3}$, so
\begin{align*}
c_{3}<\|T_{\lambda}x\|
&\leq  \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\max_{t\in
[0,\omega]_{\mathbb{T}}} \int_{t-\omega}^{t}|f_{i}(s,x,\Phi)|\Delta s\\
 &= \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\int_{0}^{\omega}
 |f_{i}(s,x,\Phi)|\Delta s \\
 &= \lambda r^{M}\eta^{M}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s,
\end{align*}
which implies
\begin{equation}
\begin{aligned}
\psi(T_{\lambda}x)
&= \sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}}
 \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\
&\geq \lambda r^{l}\eta^{l}\sum_{i=1}^{n}\int_{0}^{\omega}|f_{i}(s,x,\Phi)|\Delta s \\
&= \lambda r^{l}\eta^{l}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s \\
&> \lambda r^{l}\eta^{l}\frac{c_{3}}{\lambda r^{M}\eta^{M}} \\
&= \frac{c_{3}}{\delta}
 = c_2.
\end{aligned} \label{e3.7}
\end{equation}
So all the conditions of Theorem \ref{thm2.1} are satisfied. Consequently,
 \eqref{e1.4}
has at least three nonnegative $\omega$-periodic solutions.
This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.2}
Let $f_{i}^{0}<\omega,\,i=1,2,\dots,n$. Assume that there
exists a constant $c_2>0$ such that {\rm (H1)-(H7)} holds,
then \eqref{e1.4} has at least three nonnegative $\omega$-periodic
solutions for
$$
\frac{1}{r^{M}\eta^{M}\omega}<\lambda<\frac{1}{\omega}.
$$
\end{theorem}

\begin{proof}
Since $f_{i}^{0}<\omega$ holds for $1\leq i\leq n$, there exist
$\rho$, $\zeta$, $0<\rho<\omega$ and $0<\zeta<c_2$ such that
$$
|f_{i}(t,x,\Phi)|\leq \rho|a_{i}(t,x)|x_{i} ,\quad
0\leq x_{i}\leq \frac{\zeta}{n},
\quad t\in [0,\omega]_{\mathbb{T}}, \; i=1,2,\dots,n.
$$
 Set $c_1=\zeta$. For $x\in \overline{K}_{c_1}$,
\begin{align*}
\|T_{\lambda}x\|
&= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}
\lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\
&\leq \lambda \rho\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}
 \int_{t-\omega}^{t}G_{i}(t,s)a_{i}(s,x)x_{i}\Delta s \\
&\leq \lambda \rho \max_{0\leq i\leq n}
 \{\int_{t-\omega}^{t}G_{i}(t,s)a_{i}(s,x)\Delta s\}\sum_{i=1}^{n}
 \max_{s\in[0,\omega]_{\mathbb{T}}}|x_{i}(s)|\\
&\leq \lambda \rho \|x(t)\| \\
&< \frac{1}{\omega}\rho c_1
 < c_1 .
\end{align*}
Then condition (ii) of Theorem \ref{thm2.1} is satisfied.
In view of conditions (H6)-(H7),
using a similar proof to Theorem \ref{thm3.1},
it can be shown that \eqref{e3.6}
and \eqref{e3.7} hold. That is,
conditions (i) and (iii) of Theorem \ref{thm2.1}
are satisfied. By Theorem \ref{thm2.1}, there exist at least three
nonnegative $\omega$-periodic solutions of \eqref{e1.4}.
Thus the theorem is proved.
\end{proof}


\begin{theorem} \label{thm3.3}
Assume that there are constants $0<c_1<c_2$ such that
{\rm (H1)-(H6)} and the following two conditions hold:
\begin{itemize}
 \item [(H9)] $\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta
 s\geq 2\delta c_2\omega$
 for $c_2\leq|x|_1\leq\|x\|\leq\delta c_2$;

 \item [(H10)] $\|f(t,x,\Phi)\|\leq \|x\|$ for
$0\leq|x|_1\leq \|x\|\leq c_1$
 \end{itemize}
 Then \eqref{e1.4} has at least three nonnegative $\omega$-periodic
solutions for
$$
\frac{1}{2r^{M}\eta^{M}\omega}<\lambda<\frac{1}{r^{M}\eta^{M}\omega}.
$$
\end{theorem}

\begin{proof}
 From (H10), for $x\in\overline{K}_{c_1}$, we have
\begin{align*}
\|T_{\lambda}x\|
&= \|\lambda\int_{t-\omega}^{t}G(t,s)f(s,x,\Phi)\Delta s \|\\
 &\leq \lambda r^{M}\eta^{M}\int_{0}^{\omega}\|f(s,x,\Phi)\|\Delta s \\
 &\leq \lambda r^{M}\eta^{M}c_1\omega\\
 &< \frac{1}{r^{M}\eta^{M}\omega}r^{M}\eta^{M}c_1\omega
 = c_1.
\end{align*}
Then condition (ii) of Theorem \ref{thm2.1} is satisfied. With (H9)
and Theorem \ref{thm3.1}, the proof of conditions (i) and (iii)
of Theorem \ref{thm2.1}
is easy and hence we will omit it.
This completes the proof.
\end{proof}

\section{Examples}

\begin{example} \label{exa4.1} \rm
When $\mathbb{T}=\mathbb{R}$, the following system
has at least three nonnegative
$2\pi$-periodic solutions:
\begin{equation} \label{e4.1}
\begin{gathered}
\dot{x}_1(t)=-\frac{1}{10\pi}(2+\sin
t)x_1(t)+\frac{e^{8}}{24\pi}[x_1(t+1)+x_2(t)]^{2}
e^{-x_1(t-1)x_2(t)}|2+\sin u_1(t)|, \\
\dot{u}_1(t)
=-(0.85-0.05\sin t)u_1(t)+0.001x_1(t),\\
\dot{x}_2(t)=-\frac{1}{20\pi}(2-\cos
t)x_2(t)+\frac{e^{8}}{36\pi}[x_1(t)+x_2(t+1)]^{2}
e^{-x_1(t)x_2(t-1)}|2+\cos u_2(t)|,\\
\dot{u}_2(t)
=-(0.85-0.05\sin t)u_2(t)+0.001x_2(t)
\end{gathered}
\end{equation}
\end{example}

\begin{proof}
Corresponding to system \eqref{e1.4}, we have
$a_1(t)=\frac{1}{10\pi}(2+\sin t)$,
$a_2(t)=\frac{1}{20\pi}(2-\cos t)$,
$ b_1(t)=b_2(t)=0.85-0.05\sin t$,
$g_{i}(t)=0.001x_{i}(t)$, $i=1,2$,
$f_1(t,x,\Phi)=\frac{e^{8}}{2\pi}[x_1(t+\theta)+x_2(t)]^{2}
e^{-x_1(t-\theta)x_2(t)}|2+\sin u_1(t)|$,
$f_2(t,x,\Phi)=\frac{e^{8}}{3\pi}[x_1(t)+x_2(t+\theta)]^{2}
e^{-x_1(t)x_2(t-\theta)}|2+\cos u_2(t)|$,
$\lambda=\frac{1}{12}$
and $\omega=2\pi$. So we obtain
\begin{gather*}
e_{\ominus a_1}(t,t-\omega)
=\exp\Big\{-\int_{t-2\pi}^{t}\frac{1}{10\pi}(2+\sin s)\,
\mathrm{d}s\Big\}=e^{-0.4},\\
e_{\ominus a_2}(t,t-\omega)
 =\exp\Big\{-\int_{t-2\pi}^{t}\frac{1}{20\pi}(2-\cos t)\,
 \mathrm{d}s\Big\}=e^{-0.2},\\
r_1^{1}=\sup_{t\in
[0,\omega]_{\mathbb{T}}}\frac{1}{|1-e_{\ominus a_1}(t,t-\omega)|}
=\frac{e^{-0.4}} {e^{-0.4}-1}=3.033244,\\
r_1^{2}=\sup_{t\in [0,\omega]_{\mathbb{T}}}\frac{1}{|1-e_{\ominus
a_2}(t,t-\omega)|}=\frac{e^{-0.2}}
{e^{-0.2}-1}=5.516650,\\
r^{M}=\max\{3.033244,5.516650\}=5.516650,\\
r^{l}=\min\{3.033244,50516650\}=3.033244,\\
\eta_1^{1}=\sup_{u\in
[t-\omega,t]_{\mathbb{T}}}e_{\ominus a_1}(t,u)=\sup_{u\in
[t-\omega,t]_{\mathbb{T}}}\exp\Big\{-\int_{u}^{t}\frac{1}{10\pi}(2+\sin s)\,\mathrm{d}s\Big\}=1,\\
\\
\eta_1^{2}=\sup_{u\in
[t-\omega,t]_{\mathbb{T}}}e_{\ominus a_2}(t,u)= \sup_{u\in
[t-\omega,t]_{\mathbb{T}}}
\exp\Big\{-\int_{u}^{t}\frac{1}{20\pi}(2-\cos s)\,\mathrm{d}s\Big\}=1,\\
\eta^{M}=\max\{1,1\}=1,\\
\begin{aligned}
\eta_2^{1}&=\inf_{u\in
[t-\omega,t]_{\mathbb{T}}}e_{\ominus a_1}(t,u)=\inf_{u\in
[t-\omega,t]_{\mathbb{T}}}
\exp\Big\{-\int_{u}^{t}\frac{1}{10\pi}(2+\sin s)\,\mathrm{d}s\Big\}\\
&=e^{-0.4}=0.670320,
\end{aligned}\\
\begin{aligned}
\eta_2^{2}&=\inf_{u\in
[t-\omega,t]_{\mathbb{T}}}e_{\ominus a_2}(t,u)=\inf_{u\in
[t-\omega,t]_{\mathbb{T}}}
\exp\Big\{-\int_{u}^{t}\frac{1}{20\pi}(2-\cos s)\,\mathrm{d}s\Big\}\\
&=e^{-0.2}=0.818731,
\end{aligned}\\
\eta^{l}=\min\{0.670320,0.818731\}=0.670320.
\end{gather*}
Then
$r^{M}\eta^{M}=5.516650$, $\delta=\frac{5.516650\times
1}{3.033244\times 0.670320}=2.713226$,
 it is easy to verify that
$\frac{1}{r^{M}\eta^{M}\omega}<\lambda<\frac{1}{\omega}$; that is,
 $\frac{1}{6.132488\pi}<\frac{1}{12}<\frac{1}{2\pi}$.
Furthermore, $f_{i}^{\infty}<2\pi$ holds for $i=1,2$, so conditions
(H5) and (H6) of Theorem \ref{thm3.1} is
 satisfied. Choose $c_1=\frac{1}{100000}, c_2=\frac{1}{2}$, then
$c_{3}=\delta c_2=1.356613$.

For $c_2\leq|x|_1\leq\|x\|\leq\delta c_2$, we obtain
\begin{align*}
\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\mathrm{d}s
&= \int_{0}^{\omega}\frac{e^{8}}{2\pi}[x_1(t
+\theta)+x_2(t)]^{2}e^{-x_1(t-\theta)x_2(t)}
 |2+\sin u_1(t)|\mathrm{d} s\\
&\quad +\int_{0}^{\omega}\frac{e^{8}}{3\pi}[x_1(t)+x_2(t
 +\theta)]^{2}e^{-x_1(t)x_2(t-\theta)}|2+\cos u_2(t)|
 \mathrm{d} s\\
&\geq \frac{2\omega}{3\pi}e^{8-\delta^{2}c_2^{2}}c_2^{2}\\
&>24c_2\omega >\delta c_2\omega;
\end{align*}
that is, (H7) holds.

For $0\leq|x|_1\leq \|x\|\leq c_1$,
\begin{align*}
\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\,\mathrm{d}s
&= \int_{0}^{\omega}\frac{e^{8}}{2\pi}[x_1(t+\theta)+x_2(t)]^{2}
e^{-x_1(t-\theta)x_2(t)}|2+\sin u_1(t)|\mathrm{d} s\\
&\quad +\int_{0}^{\omega}\frac{e^{8}}{3\pi}[x_1(t)+x_2(t+\theta)]^{2}
e^{-x_1(t)x_2(t-\theta)}|2+\cos u_2(t)|\mathrm{d} s\\
 &\leq  6e^{8}c_1^{2}
 < 0.0000006\\
 &< \frac{c_1\omega}{r^{M}\eta^{M}}
 = 0.000006066488\pi,
\end{align*}
hence (H8) holds,
it is obvious that (H1)-(H4) hold. By Theorem \ref{thm3.1}, \eqref{e4.1} has
at least three nonnegative
 $2\pi$-periodic solutions.
\end{proof}

\begin{example} \label{exa4.2} \rm
When $\mathbb{T}=\mathbb{Z}$, the following system
has at least three nonnegative
$2$-periodic solutions for $1/4<\lambda<1/2$:
\begin{equation} \label{e4.2}
\begin{gathered}
\Delta x(n)=(1-e^{\sin n\pi-\ln\frac{\sqrt{2}}{2}}) x(n+1)
+\lambda \frac{18x^{2}(n)(3+\sin(x(n-1))+\cos(u(n)))}{e^{x(n)}}, \\
\Delta u(n) =-(0.85-0.05\sin n\pi)u(n)+0.001x(n),\quad n\in\mathbb{Z}
\end{gathered}
\end{equation}
\end{example}

\begin{proof}
Corresponding to system \eqref{e1.4}, we have
$a(n)=e^{\sin n\pi-\log\frac{\sqrt{2}}{2}}-1$,
$b(n)=0.85-0.05\sin n\pi, g(n)=0.001x(n)$,
$f(n,x,\Phi)=\frac{18x^{2}(n)(3+\sin(x(n-1))+\cos(u(n)))}{e^{x(n)}}$,
$\omega=2$. So we obtain
\begin{align*}
e_{\ominus a}(n,n-\omega)
&= \exp\Big\{\int_{n-\omega}^{n}\text{Log}\Big(1-\frac{a(\tau)}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\
&= \exp\Big\{\int_{n-2}^{n}\log\Big(\frac{1}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\
&= \exp\Big\{-\int_{n-2}^{n}\Big(\sin \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\,\Delta\tau\Big\}\\
&= \exp\Big\{-\sum_{\tau=n-2}^{n-1}\Big(\sin \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\Big\}\\
&= \exp\big\{\log\frac{1}{2}\big\}
= \frac{1}{2}.
\end{align*}
Then $r^{M}=r^{l}=\frac{1}{1-\frac{1}{2}}=2$. In a similar argument as
the above process, it is not difficult to calculate that $\eta^{M}=1$,
$\eta^{l}=\frac{1}{2}$. Then $r^{M}\eta^{M}=2,
\delta=\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}=\frac{2}{2\times
\frac{1}{2}}=2$. Furthermore, $f^{\infty}=0<2$ and $f^{0}=0<2\pi$
hold, so conditions (H5) and (H6) are
 satisfied. Choose $c_2=1$, then
$c_{3}=\delta c_2=2$.

For $c_2\leq\|x\|\leq2 c_2$, we obtain
\begin{align*}
\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\,\Delta s
&= \int_{0}^{2}\Big|\frac{18x^{2}(s)(3+\sin(x(s-1))+\cos(u(s)))}{e^{x(s)}}\Big|\,\Delta s\\
&= \sum_{s=0}^{1}\Big|\frac{18x^{2}(s)(3+\sin(x(s-1))+\cos(u(s)))}{e^{x(s)}}\Big|\\
&\geq \sum_{s=0}^{1}\Big|\frac{18x^{2}(s)}{e^{x(s)}}\Big|\\
&\geq \sum_{s=0}^{1}\frac{18c_2^{2}}{e^{2c_2}}
= \frac{36}{e^{2}}\\
&\geq 4
= c_2\delta\omega;
\end{align*}
that is, (H7) holds.

In addition, it is obvious that (H1)-(H4) hold. By
Theorem \ref{thm3.2}, \eqref{e4.2} has at least three nonnegative
 $2$-periodic solutions.
\end{proof}

\begin{example} When $\mathbb{T}=\mathbb{Z}$, the following system
has at least three nonnegative
 $2$-periodic solutions for $\frac{1}{4}<\lambda<\frac{1}{2}$:
\begin{equation} \label{e4.3}
\begin{gathered}
\Delta x(n)=(1-e^{\cos n\pi-\ln\frac{\sqrt{2}}{2}}) x(n+1)
+\lambda \frac{18x^{2}(n)(3+\cos(x(n-1))+\sin^{2}(u(n)))}{1+x^{2}(n)}, \\
\Delta u(n) =-(0.85-0.05\cos n\pi)u(n)+0.001x(n),\quad n\in\mathbb{Z}\,.
\end{gathered}
\end{equation}
\end{example}

\begin{proof}
Corresponding to system \eqref{e1.4}, we have
$a(n)=e^{\cos n\pi-\log\frac{\sqrt{2}}{2}}-1, b(n)=0.85-0.05\cos
n\pi, g(n)=0.001x(n)$,
$f(n,x,\Phi)=\frac{18x^{2}(n)
(3+\cos(x(n-1))+\sin^{2}(u(n)))}{1+x^{2}(n)}$,
$\omega=2$.
So we obtain
\begin{align*}
e_{\ominus a}(n,n-\omega)
&= \exp\Big\{\int_{n-\omega}^{n}\text{Log}\Big(1-\frac{a(\tau)}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\
&= \exp\Big\{\int_{n-2}^{n}\log\Big(\frac{1}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\
&= \exp\Big\{-\int_{n-2}^{n}\Big(\cos \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\,\Delta\tau\Big\}\\
&= \exp\Big\{-\sum_{\tau=n-2}^{n-1}\Big(\cos \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\Big\}\\
&= \exp\big\{\log\frac{1}{2}\big\}
= \frac{1}{2}.
\end{align*}
In a similar argument as Example \ref{exa4.2}, it is not difficult
to get that $r^{M}\eta^{M}= \delta=2$. Furthermore, $f^{\infty}=0<2$
holds, so conditions (H5) and (H6) are
satisfied.

Choose $c_2=1$, then
$c_{3}=\delta c_2=2$.
For $c_2\leq\|x\|\leq2 c_2$, we obtain
\begin{align*}
\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\,\Delta s
&= \int_{0}^{2}\Big|\frac{18x^{2}(s)(3+\cos(x(s-1))
  +\sin^{2}(u(s)))}{1+x^{2}(s)}\Big|\,\Delta s\\
&= \sum_{s=0}^{1}\Big|\frac{18x^{2}(s)(3+\cos(x(s-1))
  +\sin^{2}(u(s)))}{1+x^{2}(s)}\Big|\\
&\geq\sum_{s=0}^{1}\Big|\frac{36}{5}\Big|\\
&\geq 8
= 2c_2\delta\omega;
\end{align*}
that is, (H9) holds.

Choose $c_1=0.01$.
For $0\leq\|x\|\leq c_1$, we have
$\|f(n,x,\Phi)\|\leq 90\|x\|^{2}\leq\|x\|$;
that is, (H10) holds.
In addition, it is obvious that (H1)-(H4) hold. By
Theorem \ref{thm3.3}, \eqref{e4.3} has at least three nonnegative
 $2$-periodic solutions.
\end{proof}

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\end{document}