1. Introduction
Functional differential equations are not only an extension of ordinary
delay differential equations but also provide good models in many fields
including Biology, Mechanics, Economics and bio-mathematics. For example, in population dynamics [1], since a growing population
consumes more (or less) food than a matured one, depending on individual
species, this leads to delay functional equations. Positive periodic
solutions of differential equations have been studied extensively in recent
times. We refer to the references [1] in this article and
references therein for a wealth of information on this subject.
In this paper, we study the existence of positive periodic solutions of a system of
neutral differential equations. The study on the functional differential
equations is more intricate than ordinary delay differential equations. That
is why comparing plenty of results on the existence of positive periodic
solutions for various types of first-order or second-order ordinary delay
differential equations or studies on positive periodic solutions for delay
differential equations are relatively less, and most of them are confined to
first-order delay differential equations, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2] which are studied by
using some techniques of the Mawhin coincidence degree theory.
In this paper, we consider the following class of nonlinear neutral
differential system with several delays
\begin{align}
\dfrac{dx\left( t\right) }{dt}& =\beta x^{\prime }\left( t-\tau \left(
t\right) \right) +f\left( x\left( t-\tau \left( t\right) \right) \right)
+g\left( u\left( t-\tau \left( t\right) \right) \right) +p\left( t\right) ,
\label{0}
\end{align}
(1)
\begin{align}
\dfrac{du\left( t\right) }{dt}& =-a\left( t\right) u\left( t\right) +\frac{d
}{dt}F\left( t,u\left( t-\sigma \left( t\right) \right) \right) +c\left(
t\right) G\left( t,x\left( t-\tau \left( t\right) \right) ,u\left( t-\sigma
\left( t\right) \right) \right) , \label{1}
\end{align}
(2)
where \(\beta >0\) is a parameter, \(G\in C\left(\mathbb{R}^{3},\mathbb{R}\right) \), \(F\in C\left(\mathbb{R}^{2},\mathbb{R}\right) \), \(f\in C\left(\mathbb{R},\mathbb{R}\right) \) and \(a,c,p\in C\left(\mathbb{R},\mathbb{R}^{+}\right) \). All of the above functions are continuous, \(T\)--periodic with
\(T>0\) is a constant. Here, we obtain various sufficient conditions for the
existence of positive periodic solutions for the problem (1)--(2) by
employing two available operators and by applying the coincidence degree
theorem and the fixed point theorem. Special cases of (1)-(2)
have been considered and investigated by many other authors. For example,
very recently, in [
2], Huo and Li discussed the existence of a positive
periodic solutions of the delay differential system
\begin{equation*}
\left\{
\begin{array}{l}
\dfrac{dx\left( t\right) }{dt}=\pm x\left( t\right) G\left( t,x\left( t-\tau
_{1}\left( t\right) \right) ,...,x\left( t-\tau _{n}\left( t\right) \right)
,u\left( t-\delta \left( t\right) \right) \right) , \\
\\
\dfrac{du\left( t\right) }{dt}=-a\left( t\right) u\left( t\right) +b\left(
t\right) x\left( t-\sigma \left( t\right) \right) ,
\end{array}
\right.
\end{equation*}
where all of the above functions are \(T\)--periodic functions with \(T>0\) is a
constant and \(G\) satisfies some specific conditions. The main tool employed
in their study is based on some techniques of the Mawhin coincidence degree.
For details on the Mawhin technique, we refer the reader to Gaines and
Mawhin [
16].
2. Periodic solutions
Let us give some known notions and notations used in the theory of
coincidence degree theorem which are taken from [16, 17, 18] and
which we will apply in the present part. We seek conditions under which
there exists a \(T\)--periodic function \(x\) which can be solution of (1)
for all function \(u\in X\). Otherwise speaking, our result here of existence
of \(T\)--periodic solutions of equation (1) doesn't depend on the
choice of \(u\in X\). For that end some preparations and notations are needed.
For that purpose, let \(T>0\) and let \(X\) be the set of all continuously
differentiable scalar functions \(x\), periodic in \(t\) of period \(T\). Take
\begin{eqnarray*}
Z &:&=\left\{ x\in C\left(\mathbb{R},\mathbb{R}\right) :x\left( t+T\right) =x\left( t\right) \right\} . \\
X &:&=\left\{ x\in C^{1}\left(\mathbb{R},\mathbb{R}\right) :x\left( t+T\right) =x\left( t\right) \right\} ,
\end{eqnarray*}
and denote
\begin{eqnarray*}
\left\Vert x\right\Vert &=&\sup_{t\in \mathbb{R}}\left\vert x\left( t\right) \right\vert . \\
\left\vert x\right\vert _{0} &=&\max \left\{ \left\Vert x\right\Vert
,\left\Vert \dot{x}\right\Vert \right\} .
\end{eqnarray*}
Then, \(Z\) and \(X\) are Banach spaces when they are endowed with the norms \(
\left\Vert \cdot \right\Vert \), \(\left\vert \cdot \right\vert _{0}\)
respectively.
The method we use, for proving existence, in this paper involves the
applications of the continuous theorem of coincidence degree (see Gaines and
Mawhin [16]). This theorem needs some introduction. So, let \(X\) and \(Z\)
be two Banach spaces. Consider the operator equation
\begin{equation*}
Lx=\lambda N\left( x,\lambda \right) ,\ \lambda \in \left( 0,1\right) ,
\end{equation*}
where \(L:X\cap DomL\rightarrow Z\) is a linear operator and \(\lambda \) is a
parameter. Let \(P\) and \(Q\) denote two projectors such that
\begin{equation*}
P:X\cap DomL\rightarrow \ker L\text{ and }Q:Z\rightarrow Z/ImL.
\end{equation*}
Recall that a linear mapping \(L:X\cap DomL\rightarrow Z\) with \(\ker
L=L^{-1}\left( 0\right) \) and \(ImL=L\left( DomL\right) \), will be called a
Fredholm mapping if the following two conditions hold;
(i) \(\ker L\) has a finite dimension;
(ii) \(ImL\) is closed and has a finite codimension.
Recall also that the codimension of \(ImL\) is the dimension of \(Z/ImL\), i.e.,
the dimension of the cokernel \(co\ker L\) of \(L\). When \(L\) is a Fredholm
mapping, its index is the integer \(Ind\left( L\right) =\dim \ker L-co\dim
ImL \). We shall say that a mapping \(N\) is \(L\)--compact on \(\Omega \) if the
mapping \(QN:\bar{\Omega}\rightarrow Z\) is continuous, \(QN\left( \bar{\Omega}
\right) \) is bounded, and \(K_{P}\left( I-Q\right) N:\bar{\Omega}\rightarrow
X\) is compact, i.e., it is \(K_{P}\) is continuous and \(K_{P}\left( I-Q\right)
N\left( \bar{\Omega}\right) \) is relatively compact, where \(
K_{P}:ImL\rightarrow DomL\cap \ker P\) is the inverse of the restriction \(
L_{P}\) of \(L\) to \(DomL\cap \ker P\), so that \(LK_{P}=I\) and \(K_{P}L=I-P\).
Now, we state the continuous theorem of coincidence degree (Gaines and
Mawhin [
16]) which enables us to prove the existence of periodic
solutions to (1). For its proof we refer the reader to [
16].
Lemma 1.
Let \(X\) and \(Z\) be two Banach spaces and \(L\) a Fredholm mapping of
index zero. Assume that \(N:\bar{\Omega}\times \left[ 0,1\right] \rightarrow
Z \) is \(L\)--compact on \(\bar{\Omega}\times \left[ 0,1\right] \) with \(\Omega \)
open bounded in \(X\). Furthermore, we assume that
- for each \(\lambda \in \left( 0,1\right) \), \(x\in \partial \Omega \cap
DomL\),
\begin{equation*}
Lx\neq \lambda N\left( x,\lambda \right) ,
\end{equation*}
- for each \(x\in \partial \Omega \cap \ker L\),
\begin{equation*}
QNx\neq 0,
\end{equation*}
and
\begin{equation*}
\deg \left\{ QNx,\Omega \cap \ker L,0\right\} \neq 0.
\end{equation*}
Then the equation \(Lx=N\left( x,1\right) \) has at least one solution in \(
\bar{\Omega}\).
As a first case, we consider the following nonlinear neutral equation with
delay
\begin{equation}
\frac{d}{dt}x\left( t\right) =\beta x^{\prime }\left( t-\tau \left( t\right)
\right) +f\left( x\left( t-\tau \left( t\right) \right) \right) +g\left(
u\left( t-\sigma \left( t\right) \right) \right) +p\left( t\right) ,\text{ }
x\in X,\text{ }t\in\mathbb{R}, \label{2}
\end{equation}
(3)
where
\begin{equation}
p\left( t+T\right) =p\left( t\right) , \label{3}
\end{equation}
(4)
and
\begin{equation}
\tau \left( t+T\right) =\tau \left( t\right) ,\sigma \left( t+T\right)
=\sigma \left( t\right) . \label{4}
\end{equation}
(5)
All of the above functions are continuous, \(T\)--periodic functions and \(T>0\)
is a constant. Here \(\beta >0\) is a parameter. Before we state the main
results we make the following basic assumptions on the delay function \(\tau
\left( t\right) \) of (3).
- (H0) The inverse of \(t-\tau \left( t\right) \) exists and we denote it by \(
r\left( t\right) \) such that
\begin{equation*}
\tau ^{\prime }\left( t\right) \neq 1\text{ for }t\in \left[ 0,T\right] .
\end{equation*}
Lemma 2.
Assume the condition (H0) holds. Suppose that in (1) the
following conditions hold
- (H1) there exists constants \(K\) such that \(\left\vert g\left( x\right)
\right\vert \leq K\) for all \(x\in\mathbb{R}\)
- (H2) \(xf\left( x\right) >0\) and there exists a constant \(M>0\) such that when
\(\left\Vert x\right\Vert \geq M\), then we have
\begin{equation*}
\left\vert f\left( x\right) \right\vert >K+\left\Vert p\right\Vert +R\beta
\left\Vert x\right\Vert ,
\end{equation*}
with,
\begin{equation*}
R=\underset{t\in \left[ 0,T\right] }{\sup }\left\vert \dfrac{1}{1-\tau
^{\prime }\left( t\right) }\right\vert \text{ and }R_{0}=\underset{t\in
\left[ -\tau \left( 0\right) ,T-\tau \left( T\right) \right] }{\sup }\frac{1
}{\left[ 1-\tau ^{\prime }\left( r\left( t\right) \right) \right] ^{2}}.
\end{equation*}
- (H3) There exists a positive constant \(H>0\) such that
\begin{equation*}
\underset{x\rightarrow -\infty }{\lim }\frac{f\left( x\right) }{x}\leq H,
\end{equation*}
Then, the equation (1) has at least one \(T\)--periodic solution if \(
1>2\beta R_{0}+2H\left( T+1\right) \).
-
Proof.
In order to apply Lemma 1. Set
\begin{equation*}
Lx\left( t\right) =\dfrac{dx\left( t\right) }{dt}=\dot{x}\left( t\right) ,\
x\in X,\ t\in\mathbb{R},
\end{equation*}
\begin{gather*}
N\left( x,\lambda \right) =\lambda \beta x^{\prime }\left( t-\tau \left(
t\right) \right) +f\left( x\left( t-\tau \left( t\right) \right) \right)
+\lambda g\left( u\left( t-\sigma \left( t\right) \right) \right) +\lambda
p\left( t\right) \text{,} \\
\text{for all }x\in X\text{ and }t\in\mathbb{R},
\end{gather*}
and
\begin{equation*}
Px=\frac{1}{T}\int_{0}^{T}x\left( t\right) dt,\text{\ }x\in X\text{ and }Qz=
\frac{1}{T}\int_{0}^{T}z\left( t\right) dt,\text{\ }z\in Z.
\end{equation*}
Obviously, \(\ker L=\left\{ x\mid x\in X,\ x=\xi ,\ \xi \in\mathbb{R}\right\} \), \(ImL=\left\{ y\mid y\in Z,\ \int_{0}^{T}y\left( t\right)
dt=0\right\} \) are closed in \(X\) and \(\dim \ker L=co\dim ImL\). Hence, \(L\) is
a Fredholm mapping of index zero. Furthermore, the generalized inverse (to \(
L \)) \(K_{P}:ImL\rightarrow \ker P\cap DomL\) has the form
\begin{equation*}
K_{P}\left( x\right) =\int_{0}^{t}x\left( s\right) ds-\frac{1}{T}
\int_{0}^{T}\int_{0}^{t}x\left( s\right) dsdt.
\end{equation*}
One has
\begin{equation*}
\left( QN\right) \left( x,\lambda \right) =-\frac{1}{T}\int_{0}^{T}\left[ \lambda \beta x^{\prime }\left( t-\tau
\left( t\right) \right) +f\left( x\left( t-\tau \left( t\right) \right)
\right) +\lambda g\left( u\left( t-\sigma \left( t\right) \right) \right)
+\lambda p\left( t\right) \right] dt,
\end{equation*}
and
\begin{align*}
K_{P}\left( I-Q\right) N\left( x,\lambda \right) & =-\frac{1}{T}\int_{0}^{T}\left[ \lambda \beta x^{\prime }\left( t-\tau
\left( t\right) \right) +f\left( x\left( t-\tau \left( t\right) \right)
\right) +\lambda g\left( u\left( t-\sigma \left( t\right) \right) \right)
+\lambda p\left( t\right) \right] dt. \\
& \,\,\,\,\,+\frac{1}{T}\int_{0}^{T}\int_{0}^{t}\left[ \lambda \beta x^{\prime }\left(
s-\tau \left( s\right) \right) +f\left( x\left( s-\tau \left( s\right)
\right) \right) +\lambda g\left( u\left( s-\sigma \left( s\right) \right)
\right) +\lambda p\left( s\right) \right] dsdt \\
& \,\,\,\,\,\,+\left( \frac{t}{T}-\frac{1}{2}\right) \int_{0}^{T}\left[ \lambda \beta
x^{\prime }\left( t-\tau \left( t\right) \right) +f\left( x\left( t-\tau
\left( t\right) \right) \right) +\lambda g\left( u\left( t-\sigma \left(
t\right) \right) \right) +\lambda p\left( t\right) \right] dt.
\end{align*}
Clearly, \(QN\) and \(K_{P}\left( I-Q\right) N\) are continuous and, moreover, \(
QN\left( \bar{\Omega}\times \left[ 0,1\right] \right) \),
\(K_{P}\left( I-Q\right) N\left( \bar{\Omega}\times \left[ 0,1\right] \right)
\) are relatively compact for any open bounded set \(\Omega \subset X\). Hence,
\(N\) is \(L\)--compact on \(\bar{\Omega}\). Here \(\Omega \) is any open bounded
set in \(X\). Now we reach the position to search for an appropriate open
bounded subset \(X\) for the application of Lemma 1. The corresponding
differential equation for the operator \(Lx=\lambda N\left( x\left( t\right)
,\lambda \right) \), \(\lambda \in \left( 0,1\right) \), takes the form
\begin{equation}
\dot{x}\left( t\right) =\lambda ^{2}\beta x^{\prime }\left( t-\tau \left(
t\right) \right) +\lambda f\left( x\left( t-\tau \left( t\right) \right)
\right) +\lambda ^{2}g\left( u\left( t-\sigma \left( t\right) \right)
\right) +\lambda ^{2}p\left( t\right) \label{6}
\end{equation}
(6)
Let \(x\in X\) be a solution of (6) for a certain \(\lambda \in \left(
0,1\right) \). By integrating (6) over the interval \(\left[ 0,T\right] \)
, we obtain
\begin{equation}
\int_{0}^{T}\left[ \lambda \beta x^{\prime }\left( t-\tau \left( t\right)
\right) +f\left( x\left( t-\tau \left( t\right) \right) \right) +\lambda
g\left( u\left( t-\sigma \left( t\right) \right) \right) +\lambda p\left(
t\right) \right] dt=0, \label{7}
\end{equation}
(7)
Thus, there is a point \(\xi \in \left[ 0,T\right] \), such that
\begin{equation}
\lambda \beta x^{\prime }\left( \xi -\tau \left( \xi \right) \right)
+\lambda f\left( x\left( \xi -\tau \left( \xi \right) \right) \right)
+\lambda ^{2}g\left( u\left( \xi -\sigma \left( \xi \right) \right) \right)
+\lambda p\left( \xi \right) =0. \label{8}
\end{equation}
(8)
Moreover, in view of (H1) and (8)
\begin{align}
\left\vert f\left( x\left( \xi -\tau \left( \xi \right) \right) \right)
\right\vert \notag
& \leq \beta \left\vert x^{\prime }\left( \xi -\tau \left( \xi \right)
\right) \right\vert +\lambda ^{2}\left\vert g\left( u\left( \xi -\sigma
\left( \xi \right) \right) \right) \right\vert +\lambda \left\vert p\left(
\xi \right) \right\vert \notag \\
& \leq \beta \left[ \underset{\xi \in \left[ 0,T\right] }{\sup }\frac{1}{
\left\vert 1-\tau ^{\prime }\left( \xi \right) \right\vert }\right]
\left\vert (1-\tau ^{\prime }\left( \xi \right) )x^{\prime }\left( \xi -\tau
\left( \xi \right) \right) \right\vert \notag \\
& \,\,\,\,\,\,+\lambda ^{2}\left\vert g\left( u\left( \xi -\sigma \left( \xi \right)
\right) \right) \right\vert +\lambda \left\vert p\left( \xi \right)
\right\vert \notag \\
& \leq \beta R\left\vert \dot{x}\left( \xi -\tau \left( \xi \right) \right)
\right\vert +\left\vert g\left( u\left( \xi -\sigma \left( \xi \right)
\right) \right) \right\vert +\left\vert p\left( \xi \right) \right\vert
\notag \\
& \leq \beta R\left\Vert \dot{x}\right\Vert +K+\left\Vert p\right\Vert ,
\label{9}
\end{align}
(9)
We shall prove that there is a point \(t^{\ast }\in \left[ 0,T\right] \) such
that
\begin{equation*}
\left\vert x\left( t^{\ast }\right) \right\vert \leq \left\Vert \dot{x}
\right\Vert .
\end{equation*}
Otherwise, if \(\left\vert x\left( \xi -\tau \left( \xi \right) \right)
\right\vert \geq M\) and any \(u\in X\). Conditions (H1), (H2) and (9)
ensure that
\begin{equation*}
\beta R\left\vert x\left( \xi -\tau \left( \xi \right) \right) \right\vert
+K+\left\Vert p\right\Vert < \left\vert f\left( x\left( \xi -\tau \left( \xi
\right) \right) \right) \right\vert \leq \beta R\left\Vert \dot{x}
\right\Vert +K+\left\Vert p\right\Vert .
\end{equation*}
So that,
\begin{equation*}
\left\vert x\left( \xi -\tau \left( \xi \right) \right) \right\vert \leq
\left\Vert \dot{x}\right\Vert .
\end{equation*}
Denote \(\xi -\tau \left( \xi \right) =t^{\ast }+kT\), \(t^{\ast }\in \left[ 0,T
\right] \) with \(k\) being an integer. Then,
\begin{equation}
\left\vert x\left( \xi -\tau \left( \xi \right) \right) \right\vert
=\left\vert x\left( t^{\ast }+kT\right) \right\vert =\left\vert x\left(
t^{\ast }\right) \right\vert \leq \left\Vert \dot{x}\right\Vert , \label{10}
\end{equation}
(10)
so, from (10) we have
\begin{equation}
\left\Vert x\right\Vert \leq \left\vert x\left( t^{\ast }\right) \right\vert
+\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert dt< \left\Vert
\dot{x}\right\Vert +\int_{0}^{T}\left\vert \dot{x}\left( t\right)
\right\vert dt< \left\Vert \dot{x}\right\Vert \left( T+1\right) . \label{11}
\end{equation}
(11)
For such a small \(\varepsilon >0\), in view of assumption (H3), we find that
there must be a constant \(D>M\), which is independent of \(x\), \(u\) and \(
\lambda \), such that
\begin{equation*}
\frac{f\left( x\right) }{x}\leq \left( H+\varepsilon \right) ,\text{ for all
}x\in X\text{ and }u\in X.
\end{equation*}
Now let
\begin{eqnarray*}
E_{1} &=&\left\{ t:t\in \left[ 0,T\right] ,\ x\left( t-\tau \left( t\right)
\right) >D\right\} , \\
E_{2} &=&\left\{ t:t\in \left[ 0,T\right] ,\ x\left( t-\tau \left( t\right)
\right) < -D\right\} , \\
E_{3} &=&\left\{ t:t\in \left[ 0,T\right] ,\ \left\vert x\left( t-\tau
\left( t\right) \right) \right\vert \leq D\right\} ,
\end{eqnarray*}
and
\begin{equation*}
f_{D}=\sup \left\{ f\left( x\right) :\left\Vert x\right\Vert \leq D\right\} .
\end{equation*}
From (7) and using condition (H1), we have
\begin{align*}
\int_{0}^{T}f\left( x\left( t-\tau \left( t\right) \right) \right) dt
&=
\left( \int_{E_{1}}+\int_{E_{2}}+\int_{E_{3}}\right) f\left( x\left( t-\tau
\left( t\right) \right) \right) dt \\
& \leq \beta \int_{0}^{T}\left\vert x^{\prime }\left( t-\tau \left( t\right)
\right) \right\vert dt+\int_{0}^{T}\left\vert g\left( u\left( t-\sigma
\left( t\right) \right) \right) \right\vert dt+\int_{0}^{T}\left\vert
p\left( t\right) \right\vert dt \\
& \leq \beta \int_{0}^{T}\left\vert \frac{1}{1-\tau ^{\prime }\left(
t\right) }\right\vert \left\vert \dot{x}\left( t-\tau \left( t\right)
\right) \right\vert dt+KT+T\left\Vert p\right\Vert \\
& \leq \beta \int_{-\tau \left( 0\right) }^{T-\tau \left( T\right)
}\left\vert \frac{1}{1-\tau ^{\prime }\left( r\left( s\right) \right) }
\right\vert \left\vert \dot{x}\left( s\right) \right\vert ds+KT+T\left\Vert
p\right\Vert \\
& \leq \beta \left( \underset{t\in \left[ -\tau \left( 0\right) ,T-\tau
\left( T\right) \right] }{\sup }\frac{1}{\left[ 1-\tau ^{\prime }\left(
r\left( t\right) \right) \right] ^{2}}\right) \int_{0}^{T}\left\vert \dot{x}
\left( t\right) \right\vert dt+KT+T\left\Vert p\right\Vert \\
& \leq \beta R_{0}\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert
dt+T\left( K+\left\Vert p\right\Vert \right) ,
\end{align*}
which implies that
\begin{equation}
\int_{0}^{T}f\left( x\left( t-\tau \left( t\right) \right) \right) dt\leq
\beta R_{0}\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert
dt+T\left( K+\left\Vert p\right\Vert \right) . \label{12}
\end{equation}
(12)
One can deduce from (12)
\begin{align}
\int_{E_{1}}\left\vert f\left( x\left( t-\tau \left( t\right) \right)
\right) \right\vert dt \notag
& \leq \left( \int_{E_{2}}+\int_{E_{3}}\right) f\left( x\left( t-\tau \left(
t\right) \right) \right) dt+\beta R_{0}\int_{0}^{T}\left\vert \dot{x}\left(
t\right) \right\vert dt+T\left( K+\left\Vert p\right\Vert \right) \notag \\
& \leq T\left( H+\varepsilon \right) \left\Vert x\right\Vert +Tf_{D}+\beta
R_{0}\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert dt+T\left(
K+\left\Vert p\right\Vert \right) \label{13}
\end{align}
(13)
Thus from (6), (11), (12) and (13), we have
\begin{align*}
& \int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert dt
\begin{array}{c}
\leq
\end{array}
\beta \int_{0}^{T}\left\vert x^{\prime }\left( t-\tau \left( t\right)
\right) \right\vert dt+\int_{0}^{T}\left\vert f\left( x\left( t-\tau \left(
t\right) \right) \right) \right\vert dt +\int_{0}^{T}\left\vert g\left( u\left( t-\tau \left( t\right) \right)
\right) \right\vert dt+\int_{0}^{T}\left\vert p\left( t\right) \right\vert dt
\\
& \leq \beta R_{0}\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert
dt+\int_{E_{1}}\left\vert f\left( x\left( t-\tau \left( t\right) \right)
\right) \right\vert dt+\left( \int_{E_{2}}+\int_{E_{3}}\right) \left\vert
f\left( x\left( t-\tau \left( t\right) \right) \right) \right\vert dt +T\left( K+\left\Vert p\right\Vert \right) \\
& \leq \beta R_{0}\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert
dt+\int_{E_{1}}\left\vert f\left( x\left( t-\tau \left( t\right) \right)
\right) \right\vert dt+T\left( H+\varepsilon \right) \left\Vert x\right\Vert
+Tf_{D}+T\left( K+\left\Vert p\right\Vert \right) \\
& \leq 2\beta R_{0}\int_{0}^{T}\left\vert \dot{x}\left( t\right) \right\vert
dt+2\left[ T\left( H+\varepsilon \right) \left\Vert x\right\Vert
+Tf_{D}+T\left( K+\left\Vert p\right\Vert \right) \right] ,
\end{align*}
we deduce that
\begin{equation*}
\left\Vert \dot{x}\right\Vert T\left( 1-2\beta R_{0}-2\left( H+\varepsilon
\right) \left( T+1\right) \right) \leq T\left[ 2f_{D}+K+\left\Vert
p\right\Vert \right]
\end{equation*}
that is
\begin{equation}
\left\Vert \dot{x}\right\Vert \leq \frac{\left[ 2f_{D}+K+\left\Vert
p\right\Vert \right] }{\left( 1-2\left[ \beta R_{0}+\left( H+\varepsilon
\right) \left( T+1\right) \right] \right) }:=J_{1}. \label{14}
\end{equation}
(14)
Substituting (14) in (11), we obtain
\begin{equation}
\left\Vert x\right\Vert \leq \frac{\left( T+1\right) \left[
2f_{D}+K+\left\Vert p\right\Vert \right] }{\left( 1-2\left[ \beta
R_{0}+\left( H+\varepsilon \right) \left( T+1\right) \right] \right) }
:=J_{2}. \label{15}
\end{equation}
(15)
Take
\begin{eqnarray*}
J &=&\max \left\{ J_{1},J_{2}\right\} \\
\Omega _{1} &=&\left\{ x\in X|\ \left\vert x\right\vert _{0}\leq J\right\} .
\end{eqnarray*}
Notice first that \(\Omega _{1}\) is a closed convex bounded subset of a
Banach space. So \(\Omega _{1}\) satisfies the condition (a) of the Lemma 1. When \(x\in \partial \Omega _{1}\cap \ker L=\partial \Omega _{1}\cap\mathbb{R}\), \(x\) is a constant in \(\mathbb{R}\) with \(\left\vert x\right\vert _{0}=J\). Then,
\begin{align*}
\left( QN\right) \left( x,0\right) & =-\frac{1}{T}\int_{0}^{T}f\left(
x\left( t-\tau \left( t\right) \right) \right) dt \\
& =-\frac{1}{T}\int_{0}^{T}f\left( \pm J\right) dt\neq 0.
\end{align*}
Finally, consider the mapping
\begin{equation*}
\Psi \left( x,\lambda \right) =\theta x+\frac{1-\theta }{T}
\int_{0}^{T}f\left( x\right) dt,\text{ }\theta \in \left[ 0,T\right] .
\end{equation*}
Since for every \(\theta \in \left[ 0,1\right] \) and \(x\) in the intersection
of \(\ker L\) and \(\partial \Omega _{1}\), we have
\begin{equation*}
x\Psi \left( x,\theta \right) =\theta x^{2}+\frac{\left( 1-\theta \right) }{T
}\int_{0}^{T}xf\left( x\right) dt>0.
\end{equation*}
It follows from the property of invariance under a homotopy that
\begin{eqnarray*}
\deg \left\{ QN\left( x,0\right) ,\Omega _{1}\cap \ker L,0\right\} &=&\deg
\left\{ -f\left( x\right) ,\Omega _{1}\cap \ker L,0\right\} \\
&=&\deg \left\{ -x,\Omega _{1}\cap \ker L,0\right\} \\
&=&\deg \left\{ -x,\Omega _{1}\cap\mathbb{R},0\right\} \neq 0.
\end{eqnarray*}
We know that \(\Omega _{1}\) verifies all the requirements of Lemma 1.
Then (1) has at least one \(T\)--periodic solution \(x\in \Omega _{1}\)
. The proof is complete.
Lemma 3.
Suppose the conditions (H1) (H2) of Lemma 2 and (H0) hold.
Suppose further that
- (H4) there exists a positive constant \(Q>0\) such that
\begin{equation*}
\underset{x\rightarrow +\infty }{\lim }\frac{f\left( x\right) }{x}\leq Q.
\end{equation*}
Then, the equation (1) has at least one \(T\)--periodic solution if \(
1>2\beta R_{0}+2H\left( T+1\right) \).
Proof.
By straightforward modification of the proof of Lemma 2 we may apply
the argument in the proof of Lemma 2 to obtain Lemma 3.
It is obvious that the existence problem of \(T\)--periodic solution of (1)-(2) is equivalent to that of \(T\)--periodic solutions of the
equation (2).
In fact equation (1) has a \(T\)--periodic solution for all \(T\)-periodic
function \(u\in X.\) So in this connection we offer existence criteria for the
periodic solutions of the (2).
Next recall that the problem of existence \(T\)--periodic solution with
feedback control system (1)--(2), then from the results of the
previous sections we derive what follows
\begin{equation*}
\frac{d}{dt}u\left( t\right) =-a\left( t\right) u\left( t\right) +\frac{d}{
dt}F\left( t,u\left( t-\sigma \left( t\right) \right) \right) +c\left(
t\right) G\left( t,x\left( t-\tau \left( t\right) \right) ,u\left( t-\sigma
\left( t\right) \right) \right)
\end{equation*}
In this part we use a different method which relies on the hybrid theorem of
Krasnoselskii to establish the existence of periodic solutions of (2). To get
round this, we will introduce some notations to simplify notations. We ask
that \(a\left( t\right) \) satisfies the average condition
\begin{equation}
\int_{0}^{T}a\left( v\right) dv>0, \label{16-2}
\end{equation}
(16)
Define the function \(\Phi \) by
\begin{equation}
\Phi \left( t,s\right) :=\dfrac{e^{-\int_{s}^{t}a\left( v\right) dv}}{
1-e^{-\int_{0}^{T}a\left( v\right) dv}},\ t\in \left[ 0,T\right] ,s\in \left[
0,t\right] . \label{16-1}
\end{equation}
(17)
In addition to (16)-(17), suppose
\begin{equation}
m:=\dfrac{\mu }{1-\mu }\leq \Phi \left( t,s\right) \leq \dfrac{1}{1-\mu }:=M,
\label{16-3}
\end{equation}
(18)
where
\begin{equation*}
\mu =\exp \left( -\int_{0}^{T}a\left( v\right) dv\right) .
\end{equation*}
Assume also that the functions \(a\left( t\right) ,c\left( t\right) ,\sigma
\left( t\right) ,F\left( t,u\right) \) and \(G\left( t,x,u\right) \) are
continuous and periodic in \(t\) with period \(T\), that is,
\begin{equation}
a\left( t+T\right) =a\left( t\right) \;,\; c\left( t+T\right) =c\left(
t\right) \;\;,\;\sigma \left( t+T\right) =\sigma \left( t\right) ,
\label{16}
\end{equation}
(19)
and
\begin{equation}
F\left( t,u\right) =F\left( t+T,u\right) \;\;,G\left( t,x,u\right)
=G\left( t+T,x,u\right) . \label{17}
\end{equation}
(20)
Recall that the Equation (2) can be rewritten as
\begin{align}
\dfrac{d}{dt}\left[ u\left( t\right) -F\left( t,u\left( t-\sigma \left(
t\right) \right) \right) \right]
&=-a\left( t\right) \left[ u\left( t\right) -F\left( t,u\left( t-\sigma
\left( t\right) \right) \right) \right] -a\left( t\right) F\left( t,u\left( t-\sigma \left( t\right) \right)
\right)\nonumber\\&\,\,\,\,\,\,\,\,\,\,\,\, +c\left( t\right) G\left( t,x\left( t-\tau \left( t\right) \right)
,u\left( t-\sigma \left( t\right) \right) \right) . \label{20}
\end{align}
(21)
Let \(u\in X\) be a solution of (2), multiply both sides of (21)
with \(e^{\int_{0}^{t}a\left( s\right) ds}\) and then integrate from \(t-T\) to \(
t\) to obtain
\begin{align*}
\int_{t-T}^{t}\left\{ \left[ u\left( s\right) -F\left( s,u\left( s-\sigma
\left( s\right) \right) \right) \right] e^{\int_{0}^{s}a\left( v\right)
dv}\right\} ^{\prime }ds
& =-\int_{t-T}^{t}e^{\int_{0}^{s}a\left( v\right) dv}\left[ a\left( s\right)
F\left( s,u\left( s-\sigma \left( s\right) \right) \right) \right] ds \\
& \,\,\,\,\,\,\,\,\,\,+\int_{t-T}^{t}e^{\int_{0}^{s}a\left( v\right) dv}c\left( s\right) G\left(
s,x\left( s-\tau \left( s\right) \right) ,u\left( s-\sigma \left( s\right)
\right) \right) ds.
\end{align*}
Which implies that
\begin{align*}
& \left[ u\left( t\right) -F\left( t,u\left( t-\sigma \left( t\right)
\right) \right) \right] \left[ e^{\int_{0}^{t}a\left( s\right)
ds}-e^{\int_{0}^{t-T}a\left( s\right) ds}\right] \\
& =\int_{t-T}^{t}e^{\int_{0}^{s}a\left( v\right) dv}\left[ c\left( s\right)
G\left( s,x\left( s-\tau \left( s\right) \right) ,u\left( s-\sigma \left(
s\right) \right) \right) -a\left( s\right) F\left( s,u\left( s-\sigma \left(
s\right) \right) \right) \right] ds.
\end{align*}
Dividing both sides of the above equation by \(e^{\int_{0}^{t}a\left(
s\right) ds}\) and due to the fact that \(u\left( t\right) =u\left( t-T\right)
\) and by conditions (19) and (20) we conclude that that the
solution of (2) is given by
\begin{align}
u\left( t\right)&=
F\left( t,u\left( t-\sigma \left( t\right) \right) \right)+\int_{t-T}^{t}\Phi \left( t,s\right) \left[ c\left( s\right) G\left(
s,x\left( s-\tau \left( s\right) \right) ,u\left( s-\sigma \left( s\right)
\right) \right) -a\left( s\right) F\left( s,u\left( s-\sigma \left( s\right)
\right) \right) \right] ds. \label{20-1}
\end{align}
(22)
Now, define a mapping \(A\) by
\begin{align}
\left( A\varphi \right) \left( t\right)=
F\left( t,\varphi \left( t-\sigma \left( t\right) \right) \right) +\int_{t-T}^{t}\Phi \left( t,s\right) \left[ c\left( s\right) G\left(
s,x\left( s-\tau \left( s\right) \right) ,\varphi \left( s-\sigma \left(
s\right) \right) \right) -a\left( s\right) F\left( s,\varphi \left( s-\sigma
\left( s\right) \right) \right) \right] ds. \label{21}
\end{align}
(23)
Due to the periodicity conditions (19) and (20), one can easily
check that \(\left( A\varphi \right) \left( t\right) \) is periodic of \(T\).
Lastly in this section, we state the Krasnoselskii Fixed point theorem which
enables us to prove the existence of positive periodic solutions to (2). For the proof of the Krasnoselskii fixed point theorem we refer the reader
to [
22].
Theorem 4.
Let \(\Omega \) be a closed bounded convex nonempty subset of a
Banach space \(\left( X,\left\Vert \cdot \right\Vert \right) \). Suppose that \(
A_{1}\) and \(A_{2}\) map \(\Omega \) into itself satisfying
- (i) \(x,y\in \Omega \), implies \(A_{1}x+A_{2}y\in \Omega \)
- (ii) \(A_{1}\) is completely continuous,
- (iii) \(A_{2}\) is a contraction mapping.
Then there exists \(z\in \Omega \) with \(z=A_{1}z+A_{2}z\).
To apply Theorem 4, we need to define a Banach space \(X\), a closed
convex subset \(\Omega _{2}\) of \(X\) and construct two mappings, one is a
contraction and the other is compact. So, we let \(\left( X,\left\Vert \cdot
\right\Vert \right) \) and \(\Omega _{2}=\left\{ \varphi \in X:l\leq \varphi
\leq L\right\} \), where \(l\) is non-negative constant and \(L\) is positive
constant. We express equation (23) as
\begin{equation}
\left( A\varphi \right) \left( t\right) :=\left( A_{1}\varphi \right) \left(
t\right) +\left( A_{2}\varphi \right) \left( t\right) \label{21-0}
\end{equation}
(24)
where \(A_{1},A_{2}:\Omega _{2}\rightarrow X\) are defined as follows
\begin{align}
\left( A_{1}\varphi \right) \left( t\right)
:= \int_{t-T}^{t}\Phi \left( t,s\right) \left[ c\left( s\right) G\left(
s,x\left( s-\tau \left( s\right) \right) ,\varphi \left( s-\sigma \left(
s\right) \right) \right) -a\left( s\right) F\left( s,\varphi \left( s-\sigma
\left( s\right) \right) \right) \right] ds, \label{21-1}
\end{align}
(25)
and
\begin{equation}
\left( A_{2}\varphi \right) \left( t\right) :=F\left( t,\varphi \left(
t-\sigma \left( t\right) \right) \right) . \label{21-2}
\end{equation}
(26)
Comparing (24) to (22), it is easy to see that the existence
of periodic solutions for (2) is equivalent to the existence of
solutions \(u\in \Omega _{2}\) for the operator equation
\begin{equation*}
u=A_{1}u+A_{2}u.
\end{equation*}
In this section we obtain the existence of a periodic solution of (2)
by considering the two cases; \(F\left( t,u\right) \geq 0\) and \(F\left(
t,u\right) \leq 0\) for all \(t\in
\mathbb{R}
\), \(u\in \Omega _{2}\). We assume that function \(F\left( t,u\right) \) is
locally Lipschitz continuous in \(u\). That is, there exists a positive
constant \(\alpha \) such that
\begin{equation}
\left\vert F\left( t,u\right) -F\left( t,y\right) \right\vert \leq \alpha
\left\Vert u-y\right\Vert \text{ for all }t\in \left[ 0,T\right] \text{, }
u,y\in \Omega _{2}. \label{22}
\end{equation}
(27)
In the case \(F\left( t,u\right) \geq 0\), we assume that there exist a
non-negative constant \(k_{1}\) and positive constant \(k_{2}\) such that
\begin{equation}
k_{1}u\leq F\left( t,u\right) \leq k_{2}u\text{ for all }t\in \left[ 0,T
\right] \text{, }u\in \Omega _{2}, \label{23}
\end{equation}
(28)
\begin{equation}
k_{2}< 1, \label{24}
\end{equation}
(29)
and for all \(u\in \Omega _{2}\), \(\left\Vert x\right\Vert \leq J\)
\begin{equation}
\frac{l\left( 1-k_{1}\right) }{Tm}\leq c\left( t\right) G\left( t,x,u\right)
-a\left( t\right) F\left( t,u\right) \leq \frac{L\left( 1-k_{2}\right) }{MT}
\text{ for all }t\in \left[ 0,T\right] . \label{25}
\end{equation}
(30)
Lemma 5.
Suppose that the conditions (19), (20) and (28)-(30) hold. Then \(A_{1}:\Omega _{2}\rightarrow X\) is completely
continuous.
Proof.
Let \(\varphi \in \Omega _{2}\). Obviously, \(A_{1}\varphi \) is continuous and
it is easy to show that \(\left( A_{1}\varphi \right) \left( t+T\right)
=\left( A_{1}\varphi \right) \left( t\right) \). For \(t\in \left[ 0,T\right] \)
and for \(\varphi \in \Omega _{2}\), we have
\begin{align*}
\left\vert \left( A_{1}\varphi \right) \left( t\right) \right\vert
&= \left\vert \int_{t-T}^{t}\Phi \left( t,s\right) \left[ c\left( s\right)
G\left( s,x\left( s-\tau \left( s\right) \right) ,\varphi \left( s-\sigma
\left( s\right) \right) \right) -a\left( s\right) F\left( s,\varphi \left(
s-\sigma \left( s\right) \right) \right) \right] ds\right\vert \\
& \leq TM\frac{L\left( 1-k_{2}\right) }{TM}=L\left( 1-k_{2}\right) .
\end{align*}
Thus from the estimation of \(\left\vert \left( A_{1}\varphi \right) \left(
t\right) \right\vert \) we have
\begin{equation*}
\left\vert \left( A_{1}\varphi \right) \left( t\right) \right\vert \leq
L\left( 1-k_{2}\right) .
\end{equation*}
This shows that \(A_{1}\left( \Omega _{2}\right) \) is uniformly bounded. To
show that \(A_{1}\left( \Omega _{2}\right) \) is equicontinuous, let \(\left\{
\varphi _{n}\right\} \in \Omega _{2}\) where \(n\) is a positive integer. Next
we calculate \(\dfrac{d}{dt}\left( A_{1}\varphi _{n}\right) \) and show that
it is uniformly bounded. By making use of (19) and (20) we
obtain by taking the derivative in (25) that
\begin{align*}
\dfrac{d}{dt}\left( A_{1}\varphi _{n}\right) \left( t\right)&
=-a\left( t\right) \left( A_{1}\varphi _{n}\right) \left( t\right) \\
&\,\,\,\,\,\,\, +\left[ \Phi \left( t,t\right) -\Phi \left( t,t-T\right) \right] \left[
c\left( t\right) G\left( t,x\left( t-\tau \left( t\right) \right) ,\varphi
_{n}\left( t-\sigma \left( t\right) \right) \right) -a\left( t\right)
F\left( t,\varphi _{n}\left( t-\sigma \left( t\right) \right) \right) \right]
.
\end{align*}
Consequently, by invoking (30), we obtain
\begin{eqnarray*}
\left\vert \dfrac{d}{dt}\left( A_{1}\varphi _{n}\right) \left( t\right)
\right\vert &\leq &\left\Vert a\right\Vert L\left( 1-k_{2}\right)
+\left\vert \dfrac{1}{1-\mu }-\dfrac{\mu }{1-\mu }\right\vert \frac{L\left(
1-k_{2}\right) }{MT} \\
&\leq &\left\Vert a\right\Vert L\left( 1-k_{2}\right) +\frac{L\left(
1-k_{2}\right) }{MT}\leq \xi ,
\end{eqnarray*}
for some positive constant \(\xi \). Hence the sequence \(\left( A_{1}\varphi
_{n}\right) \) is equicontinuous. The Ascoli-Arzela theorem implies that a
subsequence \(\left( A_{1}\varphi _{n_{k}}\right) \) of \(\left( A_{1}\varphi
_{n}\right) \) converges uniformly to a continuous \(T\)--periodic function.
Thus \(A_{1}\) is continuous and \(A_{1}\left( \Omega _{2}\right) \) is
contained in a compact subset of \(X\).
Lemma 6.
\label{l4}Suppose that (27) holds. If \(A_{2}\) is given by (26)
with
\begin{equation}
\alpha < 1. \label{26}
\end{equation}
(31)
Then \(A_{2}:\Omega _{2}\rightarrow X\) is a contraction.
Proof.
Let \(A_{2}\) be defined by (26). Obviously, \(A_{2}\varphi \) is
continuous and it is easy to show that \(\left( A_{2}\varphi \right) \left(
t+T\right) =\left( A_{2}\varphi \right) \left( t\right) \). So, for any \(\phi
,\varphi \in \Omega _{2}\) we have
\begin{eqnarray*}
\left\vert \left( A_{2}\varphi \right) \left( t\right) -\left( A_{2}\phi
\right) \left( t\right) \right\vert &\leq &\left\vert F\left( t,\phi \left(
t-\sigma \left( t\right) \right) \right) -F\left( t,\varphi \left( t-\sigma
\left( t\right) \right) \right) \right\vert \\
&\leq &\alpha \left\Vert \phi -\varphi \right\Vert .
\end{eqnarray*}
This yields \(\left\Vert A_{2}\varphi -A_{2}\phi \right\Vert \leq \alpha
\left\Vert \phi -\varphi \right\Vert \). Thus \(A_{2}:\Omega _{2}\rightarrow X\)
is a contraction by (31).
Lemma 7.
Under the hypotheses of Lemmas 5 and 6. Then,
equation (2) has at least one positive \(T\)--periodic solution \(u\) in
the subset \(\Omega _{2}\) of \(X\).
Proof.
By Lemma 5, the operator \(A_{1}:\Omega _{2}\rightarrow X\) is compact
and continuous. Also, from Lemma 6, the operator \(A_{2}:\Omega
_{2}\rightarrow X\) is a contraction. Moreover, let \(\phi ,\varphi \in \Omega
_{2}\) then
\begin{align*}
\left( A_{1}\varphi \right) \left( t\right) +\left( A_{2}\phi \right)
\left( t\right)
& =\int_{t-T}^{t}\Phi \left( t,s\right) \left[ c\left( s\right) G\left(
s,x\left( s-\tau \left( s\right) \right) ,\varphi \left( s-\sigma \left(
s\right) \right) \right) -a\left( s\right) F\left( s,\varphi \left( s-\sigma
\left( s\right) \right) \right) \right] ds \\
&\,\,\,\,\,\,\,\, +F\left( t,\phi \left( t-\sigma \left( t\right) \right) \right) \\
& \leq L\left( 1-k_{2}\right) +k_{2}\left\Vert \phi \right\Vert \leq L\left(
1-k_{2}\right) +k_{2}L=L
\end{align*}
On the other hand,
\begin{align*}
\left( A_{1}\varphi \right) \left( t\right) +\left( A_{2}\phi \right)
\left( t\right)
& =\int_{t-T}^{t}\Phi \left( t,s\right) \left[ c\left( s\right) G\left(
s,x\left( s-\tau \left( s\right) \right) ,\varphi \left( s-\sigma \left(
s\right) \right) \right) -a\left( s\right) F\left( s,\varphi \left( s-\sigma
\left( s\right) \right) \right) \right] ds \\
& \,\,\,\,\,\,\,+F\left( t,\phi \left( t-\sigma \left( t\right) \right) \right) \\
& \geq l\left( 1-k_{1}\right) +k_{1}l=l.
\end{align*}
Clearly, all the hypotheses of the Krasnoselskii theorem are satisfied. Thus
there exists a fixed point \(u\in \Omega _{2}\) such that \(u=A_{1}u+A_{2}u\).
By (21), (22) and (23) we claim that this fixed point is
a solution of (2) for all continuous \(T\)--periodic functions \(
\left\Vert x\right\Vert \leq J\).
In the case \(F\left( t,u\right) \leq 0\), we substitute conditions (28)-(30) with the following conditions respectively. We assume that
there exist a negative constant \(k_{3}\) and a non-positive constant \(k_{4}\)
such that
\begin{equation}
k_{3}u\leq F\left( t,u\right) \leq k_{4}u\text{ for all }t\in \left[ 0,T
\right] \text{, }u\in \Omega \label{27}
\end{equation}
(32)
\begin{equation}
-k_{3}< 1 \label{28}
\end{equation}
(33)
and for all \(u\in \Omega _{2}\) and \(\left\Vert x\right\Vert \leq J\)
\begin{equation}
\frac{l-k_{3}L}{Tm}\leq c\left( t\right) G\left( t,x,u\right) -a\left(
t\right) F\left( t,u\right) \leq \frac{L-k_{4}l}{MT}\text{ for all }t\in
\left[ 0,T\right] , \label{29}
\end{equation}
(34)
Note that the proof of Lemmas 7 and 8 differ only by
conditions (16)--(20) and (32)-(33). So the
treatment is the same as in the first case. So, we have the following lemma
which can be proved by a similar argumentation.
Lemma 8.
Assume that (16)-(20) and (32)--(33)
hold. Then equation (2) has a positive \(T\)--periodic solution \(u\in
\Omega _{2}\).
Theorem 9.
Assume that either all hypotheses of Lemmas 2 and 7
or, 2 and 8 or, 3 and 7 or, 3 and 8
hold true. Then, system (1)-(2) has at least one \(T\)--periodic
solution \(\left( x\left( t\right) ,u\left( t\right) \right) \in \Omega
_{1}\times \Omega _{2}\).
Example 1.
Suppose that \(\dfrac{1}{\left( 6+4\left( \pi +1\right) \right) }>\beta >0\)
and let \(K\) be a positive constant. Consider the following the neutral
differential system equation
\begin{align}
\dfrac{dx\left( t\right) }{dt}& =\beta x^{\prime }\left( t-\tau \left(
t\right) \right) +f\left( x\left( t-\tau \left( t\right) \right) \right)
+Ke^{-u^{2}\left( t-\sigma \left( t\right) \right) }+p\left( t\right)
\label{30}
\end{align}
(35)
\begin{align}
\dfrac{du\left( t\right) }{dt}& =-a\left( t\right) u\left( t\right) +\frac{d
}{dt}F\left( t,u\left( t-\sigma \left( t\right) \right) \right) +c\left(
t\right) G\left( t,x\left( t-\tau \left( t\right) \right) ,u\left( t-\sigma
\left( t\right) \right) \right) , \label{31}
\end{align}
(36)
where
\begin{equation*}
\tau \left( t\right) =\sigma \left( t\right) =0.4\cos ^{2}\left( t\right) ,
\end{equation*}
and the functions \(a\), \(p\), \(c\), \(f\), \(F\) and \(G\) are defined as follows \(
a\left( t\right) =\frac{1}{2}\sin ^{2}\left( t\right) \), \(p\left( t\right) =
\frac{1+\cos \left( 2t\right) }{4}\), \(c\left( t\right) =\frac{1}{2}\sin
^{2}\left( t\right) +\frac{1}{5}\), \(F\left( t,u\left( t-\sigma \left(
t\right) \right) \right) =0.4\sin \left( u\left( t-\sigma \left( t\right)
\right) \right) ,\)
\begin{equation*}
G\left( t,x\left( t-\tau \left( t\right) \right) ,u\left( t-\sigma \left(
t\right) \right) \right) =\frac{1}{5}\frac{\cos \left( u\left( t-\sigma
\left( t\right) \right) x\left( t-\tau \left( t\right) \right) \right) +2}{
u^{2}\left( t-\sigma \left( t\right) \right) +1}+\sin u\left( t-\sigma
\left( t\right) \right) ,
\end{equation*}
and
\begin{equation*}
f\left( x\right) =\left\{
\begin{array}{c}
\dfrac{1}{3+2\left( \pi +1\right) }x+K+\dfrac{1}{2}+\arctan x,\;\;x>1,
\\
\dfrac{1}{3+2\left( \pi +1\right) }+K+\dfrac{1}{2}+\dfrac{\pi }{4},\;\;\;\;\left\vert x\right\vert \leq 1, \\
\dfrac{1}{3+2\left( \pi +1\right) }x-K-\dfrac{1}{2}+\arctan x,\;\;x< -1.
\end{array}
\right.
\end{equation*}
Then system (35)-(36) has a \(\pi \)-periodic solution.
To show this, we first remark that \(H=\dfrac{1}{3+2\left( \pi +1\right) }\).
A simple calculation yields
\begin{eqnarray}
R_{0} &=&\underset{t\in \left[ -\tau \left( 0\right) ,T-\tau \left( T\right)
\right] }{\sup }\frac{1}{\left[ 1-\tau ^{\prime }\left( r\left( t\right)
\right) \right] ^{2}}\leq \frac{1}{\left[ 1-0.4\right] ^{2}}=2.7778< 2\sqrt{2}
\label{32}
\end{eqnarray}
(37)
\begin{eqnarray}
R &=&\underset{t\in \left[ 0,\pi \right] }{\sup }\left\vert \frac{1}{1-\tau
^{\prime }\left( t\right) }\right\vert \leq \frac{1}{1-0.4}=1.6667\leq 2.
\label{33}
\end{eqnarray}
(38)
In fact \(\beta < \dfrac{1}{6+4\left( \pi +1\right) }\) and from (37) we
obtain
\begin{equation*}
\beta 4\sqrt{2}+\dfrac{2\left( \pi +1\right) }{3+2\left( \pi +1\right) }<
\dfrac{4\sqrt{2}}{6+4\left( \pi +1\right) }+\dfrac{2\left( \pi +1\right) }{
3+2\left( \pi +1\right) }=\dfrac{2\sqrt{2}+2\left( \pi +1\right) }{3+2\left(
\pi +1\right) }< 1.
\end{equation*}
Thus
\begin{equation*}
1>2\beta R_{0}+2H\left( \pi +1\right) .
\end{equation*}
Moreover, for all \(\left\Vert x\right\Vert \geq 1\) we have
\begin{equation*}
\left\vert f\left( x\right) \right\vert \geq \dfrac{1}{3+2\left( \pi
+1\right) }x+K+\dfrac{1}{2}>K+\left\Vert p\right\Vert +2\beta \left\Vert
x\right\Vert \geq K+\left\Vert p\right\Vert +R\beta \left\Vert x\right\Vert .
\end{equation*}
Consequently, for a any positive number \(K\), we can choose \(J>0\) so that
\begin{equation*}
J:=\frac{\left( \pi +1\right) \left[ 3K+\dfrac{2}{3+2\left( \pi +1\right) }+
\dfrac{\pi +1}{2}+1\right] }{1-\left( \beta 4\sqrt{2}+\dfrac{2\left( \pi
+1\right) }{3+2\left( \pi +1\right) }\right) }.
\end{equation*}
On the other hand since the function \(\sin \left( u\right) \) is a strictly
increasing on \(\left[ 0,\pi /2\right] \supset \left[ 0.1,1.5\right] \) we have
\begin{equation*}
\left\vert F\left( t,u\right) -F\left( t,y\right) \right\vert \leq \alpha
\left\Vert u-y\right\Vert \text{ for all }t\in \left[ 0,\pi \right] \text{, }
u,y\in \Omega \ \text{with}\ \alpha =0.4
\end{equation*}
and \(0.1u\leq F\left( t,u\right) \leq 0.4u\) with \(0.4=k_{2}\) ,\(k_{1}=0.1\).
Also
\begin{equation*}
2.1\geq m=\frac{\mu }{1-\mu }\geq 2\text{ and }3.1\geq M=\frac{1}{1-\mu }
\geq 3
\end{equation*}
where \(\mu \) is given by \(\mu =\exp \left( -\int_{0}^{\pi }a\left( t\right)
dt\right) =\exp \left( -\int_{0}^{\pi }\frac{1}{2}\sin ^{2}\left( t\right)
dt\right) \).
Let \(\Omega _{2}=\left[ 0.1,1.5\right] \). We have
\begin{equation*}
k_{1}u\leq F\left( t,u\right) \leq k_{2}u\text{ for all }t\in \left[ 0,\pi
\right] ,\ u\in \Omega _{2},\ k_{2}< 1,
\end{equation*}
and for all \(u\in \Omega _{2}\), \(\left\Vert x\right\Vert \leq J\),
\begin{align*}
c\left( t\right) G\left( t,x,u\right) -a\left( t\right) F\left( t,u\right)
& =0.4\left( \frac{1}{2}\sin ^{2}\left( t\right) +\frac{1}{5}\right) \left(
\frac{1}{5}\frac{\cos \left( u\left( t-\sigma \left( t\right) \right)
x\left( t-\sigma \left( t\right) \right) \right) +2}{u^{2}\left( t-\sigma
\left( t\right) \right) +1}\right) \\
&\,\,\,\,\,\, +0.4\frac{1}{5}\sin u\left( t-\sigma \left( t\right) \right) .
\end{align*}
Thus, we deduce that
\begin{equation*}
0.039987\leq c\left( t\right) G\left( t,x,u\right) -a\left( t\right) F\left(
t,u\right) \leq 0.062.
\end{equation*}
Furthermore,
\begin{eqnarray*}
\frac{l\left( 1-k_{1}\right) }{m\pi } &\leq &\frac{0.1\left( 1-0.1\right) }{
2\pi }=0.01432\,4 \\
0.092413 &=&\frac{1.5\left( 1-0.4\right) }{3.1\pi }\leq \frac{L\left(
1-k_{2}\right) }{M\pi }.
\end{eqnarray*}
These calculations, prove that
\begin{equation*}
\frac{l\left( 1-k_{1}\right) }{m\pi }\leq c\left( t\right) G\left(
t,x,u\right) -a\left( t\right) F\left( t,u\right) \leq \frac{L\left(
1-k_{2}\right) }{M\pi }.
\end{equation*}
Thus, under these hypotheses on the system (35)-(36), all the
conditions of Theorem 9 are satisfied. Hence, the system (35)-(
36) has at least one positive \(\pi \)--periodic solution.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
