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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 104, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/104\hfil Existence of solutions]
{Existence of solutions of systems of Volterra integral equations
 via Brezis-Browder arguments}

\author[R.P. Agarwal, D. O'Regan,  P. J. Y. Wong \hfil EJDE-2011/104\hfilneg]
{Ravi P. Agarwal, Donal O'Regan,  Patricia J. Y. Wong}  % in alphabetical order

\address{Ravi P. Agarwal \newline
Department of Mathematics, Texas A\&M University - Kingsville,
Kingsville, Texas 78363-8202, USA 
\newline
KFUPM Chair Professor, Mathematics and Statistics Department, 
King Fahd University of Petroleum and Minerals, Dhahran 31261, 
Saudi Arabia}
\email{agarwal@tamuk.edu}

\address{Donal O'Regan \newline
Department of Mathematics, National University of Ireland,
Galway, Ireland}
\email{donal.oregan@nuigalway.ie}

\address{Patricia J. Y. Wong \newline
School of Electrical and Electronic Engineering,
Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798,
Singapore}
\email{ejywong@ntu.edu.sg}

\thanks{Submitted April 26, 2011. Published August 16, 2011.}
\subjclass[2000]{45B05, 45G15, 45M20}
\keywords{System of Volterra integral equations;
 Brezis-Browder argument}

\begin{abstract}
 We consider two systems of Volterra integral equations
 $$
 u_i(t)=h_i(t) + \int_{0}^{t}g_i(t,s)f_i(s,u_1(s),u_2(s),\dots,
 u_n(s))ds, \quad 1\leq i\leq n
 $$
 where $t$ is in the closed interval $[0,T]$, or in
 the half-open interval $[0,T)$. By an argument originated from
 Brezis and Browder \cite{BB}, criteria are offered for the existence
 of solutions of the systems of Volterra integral equations. We
 further establish the existence of \emph{constant-sign} solutions,
 which include \emph{positive} solutions (the usual consideration) as
 a special case.  Some examples are also presented to illustrate the
 results obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article, we shall consider the system of Volterra integral
equations
\begin{equation}
 u_i(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u_1(s),u_2(s),\dots,
 u_n(s))ds,  \label{e1.1}
\end{equation}
for $t\in [0,T]$, $1\leq i\leq n$, where $0<T<\infty$;
and  the following system on a half-open interval
\begin{equation} \label{e1.2}
u_i(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u_1(s),u_2(s),\dots, u_n(s))ds,
\end{equation}
for $t\in [0,T)$, $1\leq i\leq n$,
where $0<T\leq \infty$. Throughout, let $u=(u_1,u_2,\dots,u_n)$.
We are interested in establishing the existence of solutions
$u$ of the systems \eqref{e1.1} and
\eqref{e1.2}, in $(C[0,T])^n=C[0,T]\times C[0,T]\times \dots \times
C[0,T]$ ($n$ times), and $(C[0,T))^n$, respectively. In addition, we
shall tackle the existence of \emph{constant-sign} solutions of \eqref{e1.1}
and \eqref{e1.2}. A solution $u$ of \eqref{e1.1} (or \eqref{e1.2}) is said to be of \emph{
constant sign} if for each $1\leq i\leq n$, we have
$\theta_iu_i(t)\geq 0$ for all $t\in [0,T]$ (or $t\in [0,T)$), where
$\theta_i\in\{-1,1\}$ is fixed. Note that when $\theta_i=1$ for all
$1\leq i\leq n$, a constant-sign solution reduces to a
\emph{positive} solution, which is the usual consideration in the
literature.

System \eqref{e1.1} when $h_i=0,\;1\leq i\leq n$ reduces to
\begin{equation}
u_i(t)=\int_{0}^{t}g_i(t,s)f_i(s,u_1(s),u_2(s),\dots,
u_n(s))ds,\quad t\in [0,T],\; 1\leq i\leq n. \label{e1.3}
\end{equation}
This equation when $n=1$ has received a lot of attention in the
literature \cite{B1, B2, B3, D1, G1, G2, paper15}, since it arises in
real-world problems. For instance, astrophysical problems (e.g., the
study of the density of stars) give rise to the Emden differential
equation
\begin{equation}
\begin{gathered}
y''-t^ry^q=0, \quad t\in [0,T] \\
y(0)=y'(0)=0,\quad  r\geq 0,\; 0<q<1
\end{gathered} \label{e1.4}
\end{equation}
which reduces to \eqref{e1.3} with $n=1$ when $g_1(t,s)=(t-s)s^r$ and
$f_1(t,y)=y^q$. Other examples occur in nonlinear diffusion and
percolation problems (see \cite{B2, B3} and the references cited
therein), and here we obtain \eqref{e1.3} where $g_i$ is a convolution
kernel; i.e.,
\begin{equation}
u_i(t)=\int_{0}^{t}g_i(t-s)f_i(s,u_1(s),u_2(s),\dots,
u_n(s))ds,\quad t\in [0,T],\;1\leq i\leq n.
 \label{e1.5}
\end{equation}
In particular, Bushell and Okrasi\'nski \cite{B2} investigated a
special case of the above system given by
\begin{equation}
y(t)=\int_{0}^{t}(t-s)^{\gamma-1}f(y(s))ds,\quad t\in [0,T]
 \label{e1.6}
\end{equation}
where $\gamma>1$.

Using an argument originated from Brezis and Browder \cite{BB}, we
shall establish the existence of solutions as well as constant-sign
solutions of the systems \eqref{e1.1} and \eqref{e1.2}. Our results extend,
improve and complement the existing theory in the literature
\cite{AOW, C2, 6, 7, d90, D}. We have generalized the problems to
(i) \emph{systems}, (ii) more \emph{general} form of nonlinearities
$f_i,\;1\leq i\leq n$, and (iii) existence of \emph{constant-sign}
solutions. Other related work on systems of integral equations can
be found in \cite{sys14, sys16, sys18, sys20, sys27, sys30}. Note
that the technique employed in Volterra integral equations
\cite{sys20, sys27, sys30} is entirely different from the present
work. The paper is outlined as follows. In Section 2, we present an
existence result for a system of Fredholm integral equations which
will be used in Section 3 to develop existence criteria for \eqref{e1.1}
and \eqref{e1.2}. The existence of constant-sign solutions is tackled in
Section 4. Finally, some examples are included in Section 5 to
illustrate the results obtained.

\section{Preliminary result}

We shall obtain an existence result for the following system of
Fredholm integral equations which will be used later in Section 3:
\begin{equation}
 u_i(t)=h_i(t)+\int_{0}^{T}g_i(t,s)f_i(s,u_1(s),u_2(s),\dots,
u_n(s))ds,  \label{e2.1}
\end{equation}
for $t\in [0,T]$, $1\leq i\leq n$.
Let the Banach space $B=(C[0,T])^{n} $ be equipped with the norm
$$
\| u\|=\max_{1\leq i\leq n}~\sup_{t\in [0,T]}|u_i(t)|
=\max_{1\leq i\leq n}|u_i|_0
$$
where we let $|u_i|_0:=\sup_{t\in [0,T]}|u_i(t)|$, $1\leq i\leq n$.

\begin{theorem} \label{thm2.1}
For each $1\leq i\leq n$, let $1 \leq p_i \leq \infty$ be an
integer and $q_i$ be such that
$\frac{1}{p_i}+\frac{1}{q_i}=1$. Assume the following conditions
hold for each $1\leq i\leq n$:
\begin{equation}
h_i\in C[0,T]; \label{e2.2}
\end{equation}
\begin{equation}
f_i:[0,T] \times \mathbb{R}^n \to \mathbb{R} \quad\text{is an
$L^{q_i}$-Carath\'eodory function;} \label{e2.3}
\end{equation}
 i.e.,
\begin{itemize}
\item[(i)]  the map $u \mapsto f_i(t,u)$ is continuous for almost
all $t\in [0,T]$,
\item[(ii)] the map $t \mapsto f_i(t,u)$ is measurable for all
$u\in\mathbb{R}^n$,
\item[(iii)] for any $r>0$,  there exists
$\mu_{r,i} \in L^{q_i}[0,T]$ such that
$|u| \leq r$  implies $|f_i(t,u)| \leq \mu_{r,i}(t)$  for
almost all $t\in [0,T]$;
\end{itemize}

\begin{equation}
g^t_i(s):=g_i(t,s) \in L^{p_i}[0,T]\quad \text{for each }t\in [0,T]
\label{e2.4}
\end{equation}
and
\begin{equation}
\text{the map }t \mapsto g^t_i\text{ is continuous from } [0,T]
\text{ to }L^{p_i}[0,T]. \label{e2.5}
\end{equation}
In addition, suppose there is a constant $M>0$, independent of
$\lambda$, with $\|u\|\neq M$ for any solution $u\in (C[0,T])^n$ to
\begin{equation}
u_i(t)=\lambda\Big(h_i(t)+ \int^T_0 g_i(t,s)f_i(s,u(s))ds\Big),\quad t\in
[0,T],\;1\leq i\leq n \label{e2.6lambda}
\end{equation}
for each $\lambda \in (0,1)$. Then \eqref{e2.1} has at least one
solution in $(C[0,T])^n$.
\end{theorem}

\begin{proof}
 Let the operator $S$ be defined by
\begin{equation}
Su(t)=\big(S_1u(t),S_2u(t),\dots,S_nu(t)\big),\quad
t\in [0,T] \label{e2.7}
\end{equation}
where
\begin{equation}
S_iu(t)=h_i(t)+\int^T_0 g_i(t,s)f_i(s,u(s))ds,\quad t\in [0,T],\;1\leq
i\leq n. \label{e2.8}
\end{equation}
Clearly, system \eqref{e2.1} is equivalent to
$u=Su$, and \eqref{e2.6lambda} is the same as $u=\lambda Su$.

Note that $S$ maps $(C[0,T])^n$ into $(C[0,T])^n$; i.e.,
$S_i:(C[0,T])^n\to C[0,T]$, $1\leq i\leq n$. To see this, note that
for any $u\in (C[0,T])^n$, there exits $r>0$ such that $\|u\|<r$.
Since $f_i$ is a $L^{q_i}$-Carath\'eodory function, there exists
$\mu_{r,i} \in L^{q_i}[0,T]$ such that
$|f_i(s,u)| \leq \mu_{r,i}(s)$ for almost all $s\in [0,T]$.
Hence, for any $t_1,t_2\in [0,T]$, we find for $1\leq i\leq n$,
\begin{equation}
\begin{split}
|S_iu(t_1)-S_iu(t_2)|
&\leq |h_i(t_1)-h_i(t_2)|\\
&\quad + \Big[\int_0^T |g_i^{t_1}(s)-g_i^{t_2}(s)|^{p_i}\,ds\Big]^{1/p_i}
\|\mu_{r,i}\|_{q_i}
\to 0
\end{split}\label{e2.9}
\end{equation}
as $t_1\to t_2$, where we have used \eqref{e2.2} and
\eqref{e2.4}. This shows that $S:(C[0,T])^n\to (C[0,T])^n$.

Next, we shall prove that $S:(C[0,T])^n\to (C[0,T])^n$ is
continuous. Let $u^m=(u^m_1,u^m_2,\dots$, $u^m_n)\to u$ in
$(C[0,T])^n$; i.e., $u^m_i\to u_i$ in $C[0,T]$, $1\leq i\leq n$. We
need to show that $Su^m\to Su$ in $(C[0,T])^n$, or equivalently
$S_iu^m\to S_iu$ in $C[0,T]$, $1\leq i\leq n$. There exists  $r>0$
such that $\|u^m\|,\|u\|<r$. Since $f_i$ is a
$L^{q_i}$-Carath\'eodory function, there exists
$\mu_{r,i} \in L^{q_i}[0,T]$ such that
$|f_i(s,u^m)|, |f_i(s,u)| \leq \mu_{r,i}(s)$ for almost all
$s\in [0,T]$. Using a similar
argument as in \eqref{e2.9}, we obtain for any $t_1,t_2\in [0,T]$ and
$1\leq i\leq n$,
\begin{equation}
|S_iu^m(t_1)-S_iu^m(t_2)|
\to 0 \quad\text{and}\quad |S_iu(t_1)-S_iu(t_2)| \to 0\label{e2.10}
\end{equation}
as $t_1\to t_2$. Furthermore, $S_iu^m(t)\to S_iu(t)$
pointwise on $[0,T]$, since, by the Lebesgue dominated convergence
theorem,
\begin{equation}
\begin{split}
|S_iu^m(t)-S_iu(t)|&\leq \sup_{t\in
[0,T]}\|g_i^t\|_{p_i}\Big[\int^T_0|f_i(s,u^m(s))-f_i(s,u(s))
|^{q_i}ds\Big]^{1/q_i}\\
&\to 0
\end{split}\label{e2.11}
\end{equation}
as $m\to \infty$. Combining \eqref{e2.10} and \eqref{e2.11} and
using the fact that $[0,T]$ is compact, gives for all $t\in [0,T]$,
\begin{equation}
\begin{split}
|S_iu^m(t)-S_iu(t)|
&\leq |S_iu^m(t)-S_iu^m(t_1)|+|S_iu^m(t_1)-S_iu(t_1)|\\
&\quad +|S_iu(t_1)-S_iu(t)| \to 0
\end{split} \label{e2.12}
\end{equation}
 as $m\to \infty$. Hence, we have proved that
$S:(C[0,T])^n\to (C[0,T])^n$ is continuous.

Finally, we shall show that $S:(C[0,T])^n\to (C[0,T])^n$ is
completely continuous. Let $\Omega$ be a bounded set in
$(C[0,T])^n$ with $\|u\|\leq r $ for all $u\in \Omega$. We need to
show that $S_i\Omega$ is relatively compact for $1\leq i\leq n$.
Clearly, $S_i\Omega$ is uniformly bounded, since there exists
$\mu_{r,i} \in L^{q_i}[0,T]$ such that $|f_i(s,u)| \leq
\mu_{r,i}(s)$ for all $u\in \Omega$ and $a.e.~s\in [0,T]$, and
hence
\[
|S_iu|_0\leq |h_i|_0 + \sup_{t\in [0,T]}\|g_i^t\|_{p_i}
\cdot\|\mu_{r,i}\|_{q_i} \equiv K_i,\quad u\in \Omega.
\]
Further, using a similar argument as in \eqref{e2.9}, we see that
$S_i\Omega$ is equicontinuous. It follows from
the Arz\'ela-Ascoli theorem \cite[Theorem 1.2.4]{D}
 that $S_i\Omega$ is relatively compact.

We now apply the Nonlinear Alternative \cite[Theorem 1.2.1]{D}
with $\tilde N=S$, $U=\{u\in (C[0,T])^n:\|u\|<M\}$,
$C=E=(C[0,T])^n$ and $p^*=0$ to obtain the conclusion of the
theorem.
\end{proof}

\section{Existence of solutions}

In this section, we shall establish the existence of solutions of
the systems \eqref{e1.1} and \eqref{e1.2}, in $(C[0,T])^n$ and
$(C[0,T))^n$ respectively. We shall first apply
Theorem \ref{thm2.1} to obtain an
existence result for \eqref{e1.1}.



\begin{theorem} \label{thm3.1}
For each $1\leq i\leq n$, let $1 \leq p_i
\leq \infty$ be an integer and $q_i$ be such that
$\frac{1}{p_i}+\frac{1}{q_i}=1$. Assume the following conditions
hold for each $1\leq i\leq n$:
\begin{gather}
h_i\in C[0,T]; \label{e3.1}
\\
f_i:[0,T] \times \mathbb{R}^n \to \mathbb{R} \text{ is an }
L^{q_i}\text{-Carath\'eodory function;} \label{e3.2}
\end{gather}
\begin{equation}
\begin{gathered}
g^t_i(s):=g_i(t,s) \in L^{p_i}[0,t]\text{ for each }t\in [0,T],
\\
 \sup_{t\in [0,T]} \int_0^t|g_i^t(s)|^{p_i}\,ds<\infty,
  \quad 1\leq p_i<\infty, \\
\sup_{t\in [0,T]}\operatorname{ess\,sup}_{s\in [0,t]} |g_i^t(s)|<\infty,
\quad p_i=\infty
\end{gathered} \label{e3.3}
\end{equation}
and for any $t,t'\in [0,T]$  with $t^*=\min\{t,t'\}$, we have
\begin{equation}
\begin{gathered}
\int_0^{t^*}|g^t_i(s)-g^{t'}_i(s)|^{p_i}\,ds \to 0 \quad
\text{as }t\to t',\; 1\leq p_i<\infty\\
 \operatorname{ess\,sup}_{s\in[0,t^*]}|g^t_i(s)-g^{t'}_i(s)| \to 0
\quad \text{as }t\to t',\;  p_i=\infty.
\end{gathered} \label{e3.4}
\end{equation}
In addition, suppose there is a constant $M>0$, independent of
$\lambda$, with $\|u\|\neq M$ for any solution $u\in (C[0,T])^n$ to
\begin{equation}
u_i(t)=\lambda\Big(h_i(t)+ \int^t_0 g_i(t,s)f_i(s,u(s))ds\Big),\quad t\in
[0,T],\;1\leq i\leq n \label{e3.5lambda}
\end{equation}
for each $\lambda \in (0,1)$. Then \eqref{e1.1} has at least one
solution in $(C[0,T])^n$.
\end{theorem}

\begin{proof}
 For each $1\leq i\leq n$, define
$$
g_i^*(t,s)=\begin{cases}
g_i(t,s), & 0\leq s\leq t\leq T\\
0, & 0\leq t\leq s\leq T.
\end{cases}\
$$
Then \eqref{e1.1} is equivalent to
\begin{equation}
u_i(t)=h_i(t)+\int^T_0 g_i^*(t,s)f_i(s,u(s))ds,\quad
t\in [0,T],\;1\leq i\leq n. \label{e3.6}
\end{equation}
In view of \eqref{e3.3} and \eqref{e3.4}, $g_i^*$ satisfies
\eqref{e2.4} and \eqref{e2.5}.
Hence, by Theorem \ref{thm2.1} the system \eqref{e3.6}
(or equivalently \eqref{e1.1}) has
at least one solution in $(C[0,T])^n$.
\end{proof}

\begin{remark} \label{rmk3.1}\rm
If \eqref{e3.4} is changed to:
for any $t,t'\in[0,T]$ with $t^*=\min\{t,t'\}$
and $t^{**}=\max\{t,t'\}$, we have
\begin{equation} \begin{gathered}
\int_0^{t^*}|g_i(t,s)-g_i(t',s)|^{p_i}\,ds
 +\int_{t^*}^{t^{**}}|g_i(t^{**},s)|^{p_i}\,ds
\to 0 \quad \text{as } t\to t',\quad 1\leq p_i<\infty, \\
\operatorname{ess\,sup}_{s\in[0,t^*]}|g_i(t,s)-g_i(t',s)|
+\operatorname{ess\,sup}_{s\in[t^*,t^{**}]}|g_i(t^{**},s)|
\to 0 \end{gathered}\label{e3.4p}
\end{equation}
as $t\to t'$, $p_i=\infty$;
then automatically we have the inequalities in \eqref{e3.3}.
\end{remark}

Our subsequent results use an argument originated from Brezis and
Browder \cite{BB}.



\begin{theorem} \label{thm3.2}
Let the following conditions be satisfied:
for each $1\leq i\leq n$, \eqref{e3.1}, \eqref{e3.2}--\eqref{e3.4}
with $p_i=\infty$ and $q_i=1$,
there exist $B_i> 0$ such that for any
$u\in (C[0,T])^n$,
\begin{equation}
\int_0^T[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds]dt\leq B_i;
\label{e3.7}
\end{equation}
and there exist $r>0$ and $\alpha_i>0$  with
$r\alpha_i>H_i\equiv \sup_{t\in [0,T]}|h_i(t)|$
such that for any $u\in (C[0,T])^n$,
\begin{equation}
u_i(t)f_i(t,u(t))\geq r\alpha_i |f_i(t,u(t))|\quad
\text{for a. e. $t\in[0,T]$ such that $\|u(t)\|>r$,} \label{e3.8}
\end{equation}
where we denote $\|u(t)\|:=\max_{1\leq i\leq n}|u_i(t)|$.
Then \eqref{e1.1} has at least one solution in $(C[0,T])^n$.
\end{theorem}

\begin{proof}
 We shall employ Theorem \ref{thm3.1}, so let
$u=(u_1,u_2,\dots,u_n)\in (C[0,T])^n$ be any solution of
\eqref{e3.5lambda} where $\lambda\in (0,1)$.
For each $z\in [0,T]$, define
\begin{equation}
I_z=\{t\in[0,z]:\|u(t)\|\leq r\}, \quad
J_{z}=\{t\in[0,z]:\|u(t)\|> r\}. \label{e3.9}
\end{equation}
Clearly, $[0,z]=I_z\cup J_z$ and hence
\begin{equation}
\int_0^z = \int_{I_z}+\int_{J_z} .\label{e3.10}
\end{equation}

Let $1\leq i\leq n$. For a.e. $t\in I_z$, by \eqref{e3.2}
there exists $\mu_{r,i}\in L^1[0,T]$ such that
$|f_i(t,u(t))|\leq \mu_{r,i}(t)$.
Thus, we obtain
\begin{equation}
\int_{I_z}|f_i(t,u(t))|dt\leq \int_{I_z}\mu_{r,i}(t)dt\leq
\int_0^T\mu_{r,i}(t)dt= \|\mu_{r,i}\|_1. \label{e3.11}
\end{equation}
On the other hand, if $t\in J_{z}$, then it is clear from
\eqref{e3.8} that
$u_i(t)f_i(t,u(t))\geq 0$ for a.e. $t\in [0,T]$. It follows that
\begin{equation}
\int_{J_z}u_i(t)f_i(t,u(t))dt=\int_{J_z}|u_i(t)f_i(t,u(t))|dt \geq
r\alpha_i\int_{J_{z}}|f_i(t,u(t))|dt. \label{e3.12}
\end{equation}

Let $z\in [0,T]$. We now multiply \eqref{e3.5lambda} by $f_i(t,u(t))$,
then integrate from 0 to $z$, and use \eqref{e3.7} to obtain
\begin{equation}
\begin{aligned}
& \int_0^z u_i(t)f_i(t,u(t))dt\\
&= \lambda\int_0^z h_i(t)f_i(t,u(t))dt +
\lambda\int_0^z\Big[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds\Big]dt \\
&\leq   H_i\int_0^z|f_i(t,u(t))|dt+B_i.
\end{aligned}\label{e3.13}
\end{equation}
Splitting the integrals in \eqref{e3.13} using \eqref{e3.10}, and
applying \eqref{e3.12}, we obtain
\begin{align*}
&\int_{I_z}u_i(t)f_i(t,u(t))dt +
r\alpha_i\int_{J_{z}}|f_i(t,u(t))|dt\\
&\leq H_i\int_{I_z}|f_i(t,u(t))|dt
 + H_i\int_{J_z}|f_i(t,u(t))|dt +B_i
\end{align*}
or
\begin{align*}
(r\alpha_i-H_i)\int_{J_z} |f_i(t,u(t))|dt
&\leq   H_i\int_{I_z}|f_i(t,u(t))|dt
+\int_{I_z}|u_i(t)f_i(t,u(t))|dt + B_i \\
&\leq   (H_i+r)\|\mu_{r,i}\|_1 + B_i
\end{align*}
where we have used \eqref{e3.11} in the last inequality.
It follows that
\begin{equation}
\int_{J_z}|f_i(t,u(t))|dt\leq
\frac{(H_i+r)\|\mu_{r,i}\|_1+B_i}{r\alpha_i-H_i}\equiv c_i.
\label{e3.14}
\end{equation}
Now, it is clear from \eqref{e3.5lambda} that for $t\in [0,T]$ and
$1\leq i\leq n$,
\begin{align*}
|u_i(t)|
&\leq  H_i+\int_0^t|g_i(t,s)f_i(s,u(s))|ds\\
&=  H_i+\Big(\int_{I_t}+\int_{J_{t}}\Big)|g_i(t,s)f_i(s,u(s))|ds\\
&\leq   H_i+\big(\sup_{t\in [0,T]}\operatorname{ess\,sup}_{s\in
[0,t]}|g_i(t,s)|\big)(\|\mu_{r,i}\|_1+c_i) \equiv\,d_i
\end{align*}
where we have applied \eqref{e3.11} and \eqref{e3.14} in the last
inequality.
Thus, $|u_i|_0\leq d_i$ for $1\leq i\leq n$ and $\|u\|\leq
\max_{1\leq i\leq n}d_i\equiv D$. It follows from
Theorem \ref{thm3.1} (with $M=D+1$) that \eqref{e1.1} has
a solution $u^*\in(C[0,T])^n$.
 \end{proof}

Our next result replaces condition \eqref{e3.7} with condition
\eqref{e3.15} which involves the integral of $f_i$ in the right side.

\begin{theorem} \label{thm3.3}
Let the following conditions be satisfied
for each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.2}--\eqref{e3.4}
with $p_i=\infty$ and $q_i=1$,
there exist constants $a_i\geq 0$  and $b_i$ such
that for any $z\in [0,T]$,
\begin{equation}
\int_0^z\Big[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds\Big]dt\leq
a_i\int_0^z|f_i(t,u(t))|dt+b_i; \label{e3.15}
\end{equation}
and
there exist $r>0$ and $\alpha_i>0$  with
$r\alpha_i>H_i+a_i$ such that
for any $u\in (C[0,T])^n$,
\begin{equation}
u_i(t)f_i(t,u(t))\geq r\alpha_i |f_i(t,u(t))|\quad
\text{for a.e. $t\in[0,T]$  such that }\|u(t)\|>r.
\label{e3.16}
\end{equation}
Then \eqref{e1.1} has at least one solution
in $(C[0,T])^n$.
\end{theorem}


 \begin{proof}
 The proof is the same as that of Theorem \ref{thm3.2} until
\eqref{e3.12}. Let $z\in [0,T]$ and $1\leq i\leq n$. Multiplying
\eqref{e3.5lambda} by $f_i(t,u(t))$ and then integrating from 0 to $z$,
we use \eqref{e3.15} to get
\begin{equation}
\begin{split}
&\int_0^z u_i(t)f_i(t,u(t))dt \\
&\leq  \int_0^z |h_i(t)f_i(t,u(t))|dt +
\lambda\int_0^z\Big[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds\Big]dt \\
&\leq   (H_i+a_i)\int_0^z|f_i(t,u(t))|dt+|b_i|.
\end{split}\label{e3.17}
\end{equation}
Splitting the integrals in \eqref{e3.17} and applying \eqref{e3.12},
we obtain
\begin{align*}
&(r\alpha_i-H_i-a_i)\int_{J_z} |f_i(t,u(t))|dt \\
&\leq   (H_i+a_i)\int_{I_z}|f_i(t,u(t))|dt
 +\int_{I_z}|u_i(t)f_i(t,u(t))|dt + |b_i| \\
&\leq   (H_i+a_i+r)\|\mu_{r,i}\|_1 + |b_i|
\end{align*}
where we have used \eqref{e3.11} in the last inequality.
It follows that
\begin{equation}
\int_{J_z}|f_i(t,u(t))|dt\leq
\frac{(H_i+a_i+r)\|\mu_{r,i}\|_1+ |b_i|}{r\alpha_i-H_i-a_i}\equiv
c^*_i. \label{e3.18}
\end{equation}
The rest of the proof proceeds as in the proof
of Theorem \ref{thm3.2}.
 \end{proof}


The next result is for general $p_i$, $q_i$ (i.e., $1\leq p_i\leq
\infty$ and $\frac{1}{p_i}+\frac{1}{q_i}=1$), it also replaces
condition \eqref{e3.8} or \eqref{e3.16} with conditions \eqref{e3.19}
and \eqref{e3.20}. Note
that in Theorems \ref{thm3.2} and  \ref{thm3.3} the conditions
\eqref{e3.2}--\eqref{e3.4} hold for
$p_i=\infty$, whereas in Theorem \ref{thm3.4} the conditions
\eqref{e3.2}--\eqref{e3.4}
hold for $1\leq p_i\leq \infty$.

\begin{theorem} \label{thm3.4}
 Let the following conditions be satisfied:
for each $1\leq i\leq n$: \eqref{e3.1}--\eqref{e3.4},  \eqref{e3.7},
 there exist $r>0$  and $\beta_i>0$  such that
for any $u\in (C[0,T])^n$,
\begin{equation}
\begin{gathered}
u_i(t)f_i(t,u(t))\geq \beta_i|u_i|_0 \cdot |f_i(t,u(t))|\\
\text{for a.e. $t\in[0,T]$  such that }\|u(t)\|>r,
\end{gathered} \label{e3.19}
\end{equation}
where we denote $|u_i|_0:=\max_{t\in [0,T]}|u_i(t)|$;
and there exist $\eta_i>0$, $\gamma_i\geq q_i-1>0$  and
$\phi_i\in L^{p_i}([0,T],\mathbb{R})$ such that for any
$u\in (C[0,T])^n$,
\begin{equation}
|u_i|_0\geq \eta_i|f_i(t,u(t)|^{\gamma_i}+\phi_i(t)\quad
\text{for a.e. $t\in[0,T]$ such that }\|u(t)\|>r. \label{e3.20}
\end{equation}
Then \eqref{e1.1} has at least one solution in $(C[0,T])^n$.
\end{theorem}

 \begin{proof} As in the proof of Theorem \ref{thm3.2}, we consider the
sets $I_z$ and $J_z$ where $z\in [0,T]$ (see \eqref{e3.9}).
Let $1\leq i\leq n$. If $t\in I_z$, then by \eqref{e3.2} there exists
$\mu_{r,i}\in L^{q_i}[0,T]$ such that $|f_i(t,u(t))|\leq
\mu_{r,i}(t)$. Consequently, we have
\begin{equation}
\int_{I_z}|f_i(t,u(t))|dt\leq \int_{I_z}\mu_{r,i}(t)dt\leq
\int_0^T\mu_{r,i}(t)dt\leq T^{1/p_i}\|\mu_{r,i}\|_{q_i}.
\label{e3.21}
\end{equation}
On the other hand, if $t\in J_{z}$, then noting
\eqref{e3.19} we have $u_i(t)f_i(t,u(t))\geq 0$ for a.e. $t\in [0,T]$,
and so
\begin{equation}
\begin{split}
\int_{J_z}u_i(t)f_i(t,u(t))dt
&= \int_{J_z}|u_i(t)f_i(t,u(t))|dt \\
&\geq  \beta_i\int_{J_{z}}|u_i|_0\cdot |f_i(t,u(t))|dt \\
&\geq   \beta_i\eta_i\int_{J_{z}} |f_i(t,u(t))|^{\gamma_i+1}dt +
\beta_i\int_{J_{z}} \phi_i(t)|f_i(t,u(t))|dt
\end{split}\label{e3.22}
\end{equation}
where we have used \eqref{e3.20} in the last inequality.

Let $z\in [0,T]$. Multiplying \eqref{e3.5lambda} by $f_i(t,u(t))$ and
then integrating from 0 to $z$, we use \eqref{e3.7} to get
 \eqref{e3.13}.
Splitting the integrals in \eqref{e3.13} and applying \eqref{e3.22},
we find
\begin{align*}
&\int_{I_z}u_i(t)f_i(t,u(t))dt+\beta_i\eta_i\int_{J_{z}}
|f_i(t,u(t))|^{\gamma_i+1}dt
+ \beta_i\int_{J_{z}} \phi_i(t)|f_i(t,u(t))|dt
\\
& \leq H_i\int_{I_z}|f_i(t,u(t))|dt +
H_i\int_{J_z}|f_i(t,u(t))|dt + B_i
 \end{align*}
or
\begin{equation}
\begin{aligned}
&\beta_i\eta_i\int_{J_{z}} |f_i(t,u(t))|^{\gamma_i+1}dt \\
&\leq \beta_i\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt +
H_i\int_{J_z}|f_i(t,u(t))|dt + B_i \\
&\quad +\int_{I_z}(|u_i(t)|+H_i)|f_i(t,u(t))|dt \\
&\leq  \beta_i\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt +
H_i\int_{J_z}|f_i(t,u(t))|dt + B_i \\
&\quad + (r+H_i)T^{1/p_i}\|\mu_{r,i}\|_{q_i} \\
&=  \beta_i\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt +
H_i\int_{J_z}|f_i(t,u(t))|dt + B'_i
\end{aligned}\label{e3.23}
\end{equation}
where \eqref{e3.21} has been used in the last inequality and
$B'_i\equiv B_i+(r+H_i)T^{1/p_i}\|\mu_{r,i}\|_{q_i}$.

Next, an application of H\"{o}lder's inequality gives
\begin{equation}
\begin{aligned}
&\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt\\
&\leq \Big[\int_0^T
|\phi_i(t)|^{(\gamma_i+1)/\gamma_i}dt\Big]
^{\gamma_i/(\gamma_i+1)}\cdot
\Big[\int_{J_z}|f_i(t,u(t))|^{\gamma_i+1}dt\Big]
^{1/\gamma_i+1}.
\end{aligned}
\label{e3.24}
\end{equation}
Another application of H\"{o}lder's inequality yields
$$
\int_0^T|\phi_i(t)|^{\frac{\gamma_i+1}{\gamma_i}}dt\leq
T^{\frac{\gamma_ip_i-\gamma_i-1}{p_i\gamma_i}}\Big[\int_0^T|
\phi_i(t)|^{p_i}dt\Big]^{\frac{\gamma_i+1}{\gamma_ip_i}},
$$
which upon substituting into \eqref{e3.24} leads to
\begin{equation}
\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt\leq
T^{\frac{\gamma_ip_i-\gamma_i-1}{p_i(\gamma_i+1)}}
\|\phi_i\|_{p_i}\Big[\int_{J_z}|f_i(t,u(t))|^{\gamma_i+1}dt\Big]^{1/(\gamma_i+1)}.
\label{e3.25}
\end{equation}
Similarly, we have
\begin{equation}
\int_{J_{z}}|f_i(t,u(t))|dt\leq
T^{\frac{\gamma_ip_i-\gamma_i-1}{p_i(\gamma_i+1)}
+\frac{1}{p_i}}\Big[\int_{J_z}|f_i(t,u(t))|^{\gamma_i+1}dt\Big]
^{1/(\gamma_i+1)}.
\label{e3.26}
\end{equation}

Substituting \eqref{e3.25} and \eqref{e3.26} into \eqref{e3.23},
we obtain
\begin{equation}
\beta_i\eta_i\int_{J_{z}}
|f_i(t,u(t))|^{\gamma_i+1}dt \leq
A_i\Big[\int_{J_z}|f_i(t,u(t))|^{\gamma_i+1}dt\Big]^{1/(\gamma_i+1)}+B'_i
\label{e3.27}
\end{equation}
where
$$
A_i=T^{\frac{\gamma_ip_i-\gamma_i-1}{p_i(\gamma_i+1)}}
\big(\beta_i\|\phi_i\|_{p_i}+H_iT^{1/p_i}\big).
$$
Since $\frac{1}{\gamma_i+1}<1$, from \eqref{e3.27} there exists a
constant $c^{**}_i$ such that
\begin{equation}
\int_{J_{z}}|f_i(t,u(t))|^{\gamma_i+1}dt \leq
c^{**}_i. \label{e3.28}
\end{equation}
Now, it is clear from \eqref{e3.5lambda} that for $t\in [0,T]$ and
$1\leq i\leq n$,
\begin{align*}
|u_i(t)|
&\leq  H_i+\int_0^t|g_i(t,s)f_i(s,u(s))|ds\\
&=  H_i+\int_{I_t}|g_i(t,s)f_i(s,u(s))|ds
 +\int_{J_{t}}|g_i(t,s)f_i(s,u(s))|ds\\
&\leq H_i+\big(\sup_{t\in[0,T]}\|g_i^t\|_{p_i}\big)\|\mu_{r,i}\|_{q_i}\\
&\quad +T^{\frac{\gamma_ip_i-\gamma_i-1}{p_i(\gamma_i+1)}}
\big(\sup_{t\in[0,T]}\|g_i^t\|_{p_i}\big)
\Big[\int_{J_t}|f_i(s,u(s))|^{\gamma_i+1}ds\Big]^{1/(\gamma_i+1)}\\
&\leq  d^*_i \quad \text{(a constant)},
\end{align*}
where in the second last inequality a similar argument as in
\eqref{e3.25} is used and in the last inequality we have
used \eqref{e3.28}. Thus, $|u_i|_0\leq d^*_i$ for $1\leq i\leq n$
and $\|u\|\leq \max_{1\leq i\leq n}d^*_i\equiv D^*$.
It follows from Theorem \ref{thm3.1} (with $M=D^*+1$)
that \eqref{e1.1} has a solution $u^*\in(C[0,T])^n$.
\end{proof}

The next result is also for general $p_i$, $q_i$, and here the
condition \eqref{e3.7} is replaced by \eqref{e3.15}.

\begin{theorem} \label{thm3.5}
Let the following conditions be satisfied
for each $1\leq i\leq n$: \eqref{e3.1}--\eqref{e3.4},
\eqref{e3.15}, \eqref{e3.19} and \eqref{e3.20}. Then \eqref{e1.1}
 has at least one solution in $(C[0,T])^n$.
\end{theorem}

\begin{proof}
 The proof is similar to that of Theorem \ref{thm3.4} until
\eqref{e3.22}. Let $z\in [0,T]$ and $1\leq i\leq n$. Multiplying
\eqref{e3.5lambda} by $f_i(t,u(t))$ and then integrating from 0
to $z$, we use \eqref{e3.15} to get \eqref{e3.17}.

Splitting the integrals in \eqref{e3.17} and applying \eqref{e3.22},
we find
\begin{equation}
\begin{split}
&\beta_i\eta_i\int_{J_{z}} |f_i(t,u(t))|^{\gamma_i+1}dt\\
&\leq \beta_i\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt +
(H_i+a_i)\int_{J_z}|f_i(t,u(t))|dt + |b_i| \\
&\quad +\int_{I_z}(|u_i(t)|+H_i+a_i)|f_i(t,u(t))|dt \\
&\leq  \beta_i\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt +
(H_i+a_i)\int_{J_z}|f_i(t,u(t))|dt + |b_i| \\
&\quad + (r+H_i+a_i)T^{1/p_i}\|\mu_{r,i}\|_{q_i} \\
&= \beta_i\int_{J_{z}}|\phi_i(t)|\cdot|f_i(t,u(t))|dt +
(H_i+a_i)\int_{J_z}|f_i(t,u(t))|dt + B''_i
\end{split}\label{e3.29}
\end{equation}
where $B''_i\equiv |b_i|+(r+H_i+a_i)T^{1/p_i}\|\mu_{r,i}
\|_{q_i}$.
Substituting \eqref{e3.25} and \eqref{e3.26} into \eqref{e3.29}
then leads to
\begin{equation}
\beta_i\eta_i\int_{J_{z}} |f_i(t,u(t))|^{\gamma_i+1}dt
 \leq A'_i\Big[\int_{J_z}|f_i(t,u(t))|^{\gamma_i+1}dt\Big]
^{1/(\gamma_i+1)}+B''_i
\label{e3.30}
\end{equation}
where
$$
A'_i=T^{\frac{\gamma_ip_i-\gamma_i-1}{p_i(\gamma_i+1)}}
\big[\beta_i\|\phi_i\|_{p_i}+(H_i+a_i)T^{1/p_i}\big].
$$
Since $\frac{1}{\gamma_i+1}<1$, from \eqref{e3.30} we  obtain
\begin{equation}
\int_{J_{z}}|f_i(t,u(t))|^{\gamma_i+1}dt \leq \bar c_i \label{e3.31}
\end{equation}
where $\bar c_i$ is a constant. The rest of the proof proceeds as
in that of Theorem \ref{thm3.4}.
\end{proof}

We shall now tackle the system \eqref{e1.2}. Our next theorem is a
variation of an existence principle of Lee and O'Regan \cite{lee}.

\begin{theorem} \label{thm3.6}
For each $1\leq i\leq n$, let $1 \leq p_i \leq
\infty$ be an integer and $q_i$ be such that
$\frac{1}{p_i}+\frac{1}{q_i}=1$. Assume the following conditions
hold for each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3},
\eqref{e3.4} and
\begin{equation}
f_i:[0,T) \times \mathbb{R}^n \to \mathbb{R} \text{ is a locally }
L^{q_i}\text{-Carath\'eodory function;} \label{e3.32}
\end{equation}
i.e., the conditions (i)--(iii) in \eqref{e2.3} hold when
$f_i$ is restricted to $I\times \mathbb{R}^n$,
where $I$ is any compact subinterval of $[0,T)$.
Also let $\{t_k\}$ be a positive and increasing sequence such that
$\lim_{k\to\infty}t_k=T$. For each $k=1,2,\dots$, suppose there
exists $u^k=(u_1^k,u_2^k,\dots,u_n^k)\in (C[0,t_k])^n$ that
satisfies
\begin{equation}
 u_i^k(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u_1^k(s),u_2^k(s),\dots,
u_n^k(s))ds, \label{e3.33}
\end{equation}
for $t\in [0,t_k]$, $1\leq i\leq n$.
Further, for $1\leq i\leq n$ and $\ell=1,2,\dots$, there are
bounded sets $B_\ell\subseteq \mathbb{R}$ such that $k\geq \ell$ implies
$u_i^k(t)\in B_\ell$ for each $t\in [0,t_\ell]$.
Then \eqref{e1.2} has a solution $u^*\in (C[0,T))^n$ such that
for $1\leq i\leq n$,
$u_i^*(t)\in \overline B_\ell$ for each $t\in [0,t_\ell]$.
\end{theorem}

\begin{proof} First we shall show that
for each $1\leq i\leq n$  and
$\ell=1,2,\dots$,
\begin{equation}
\text{the sequence $\{u^k_i\}_{k\geq \ell}$ is uniformly bounded
and equicontinuous on $[0,t_\ell]$}.
\label{e3.34}
\end{equation}
The uniform boundedness of $\{u^k_i\}_{k\geq \ell}$ follows
immediately from the hypotheses, therefore we only need to prove
that $\{u^k_i\}_{k\geq \ell}$ is equicontinuous. Let $1\leq i\leq
n$. Since for all $k\geq \ell$, $u_i^k(t)\in B_\ell$ for each
$t\in [0,t_\ell]$, there exists $\mu_{B_\ell}\in L^{q_i}[0,t_\ell]$
such that $|f_i(s,u^k(s))|\leq \mu_{B_\ell}(s)$ for almost
every $s\in [0,t_\ell]$. Fix $t,t'\in [0,t_\ell]$ with $t<t'$.
Then, noting \eqref{e3.4}, from \eqref{e3.33} we find
\begin{align*}
&|u^k_i(t)-u^k_i(t')|\\
&\leq  |h_i(t)-h_i(t')| + \int_0^t
|g_i^{t}(s)-g_i^{t'}(s)|\cdot|f_i(s,u^k(s))|ds \\
&\quad +\int_t^{t'}|g_i^{t'}(s)|\cdot|f_i(s,u^k(s))|ds
\\
&\leq   |h_i(t)-h_i(t')| +
\Big[\int_0^t|g_i^{t}(s)-g_i^{t'}(s)|^{p_i}ds\Big]^{1/p_i}
\Big[\int_0^t \big(\mu_{B_\ell}(s)\big)^{q_i}ds\Big]^{1/q_i}
\\
& \quad + \Big[\int_t^{t'}|g_i^{t'}(s)|^{p_i}ds\Big]^{1/p_i}
\Big[\int_t^{t'} \big(\mu_{B_\ell}(s)\big)^{q_i}ds\Big]^{1/q_i}
\to  0
\end{align*}
as $t\to t'$. Therefore, $\{u^k_i\}_{k\geq
\ell}$ is equicontinuous on $[0,t_\ell]$.

Let $1\leq i\leq n$. Now, \eqref{e3.34} and the Arz\'{e}la-Ascoli
Theorem yield a subsequence $N_1$ of $\mathbb{N}=\{1,2,\dots\}$ and a
 function
$z_i^1\in C[0,t_1]$ such that $u_i^k\to z_i^1$ uniformly on
$[0,t_1]$ as $k\to \infty $ in $N_1$. Let
$N_2^*=N_1\backslash\{1\}$. Then \eqref{e3.34} and the
Arz\'{e}la-Ascoli Theorem yield a subsequence $N_2$ of $N_2^*$
and a function $z_i^2\in C[0,t_2]$ such that $u_i^k\to z_i^2$
uniformly on $[0,t_2]$ as $k\to \infty $ in $N_2$.
Note that $z_i^2=z_i^1$ on $[0,t_1]$ since $N_2\subseteq N_1$.
Continuing this process, we obtain subsequences of integers
$N_1,N_2,\dots$ with
$$
N_1\supseteq N_2 \supseteq \dots \supseteq N_\ell \supseteq
\dots, \quad \text{where }N_\ell\subseteq \{\ell, \ell+1,\dots\},
$$
and functions $z_i^\ell\in C[0,t_\ell]$ such that
$u_i^k\to z_i^\ell$ uniformly on $[0,t_\ell]$  as
$k\to \infty$ in $N_\ell$, and
$z_i^{\ell+1}=z_i^\ell$  on $[0,t_\ell]$, $ell=1,2,\dots$.

Let $1\leq i\leq n$. Define a function $u_i^*:[0,T)\to \mathbb{R}$ by
\begin{equation}
u_i^*(t)=z_i^\ell(t), \quad t\in [0,t_\ell]. \label{e3.35}
\end{equation}
Clearly, $u_i^*\in C[0,T)$ and $u_i^*(t)\in \overline B_\ell$ for
each $t\in [0,t_\ell]$. It remains to prove that
$u^*=(u_1^*,u_2^*,\dots,u_n^*)$ solves \eqref{e1.2}. Fix $t\in [0,T)$.
Then choose and fix $\ell$ such that $t\in [0,t_\ell]$. Take $k\geq
\ell$. Now, from \eqref{e3.33} we have
\begin{equation}
 u_i^k(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u_1^k(s),u_2^k(s),\dots,
 u_n^k(s))ds, \quad
 t\in [0,t_\ell]. \label{e3.36}
\end{equation}
Since $f_i$ is a locally
$L^{q_i}$-Carath\'eodory function and $u_i^k(t)\in B_\ell$ for each
$t\in [0,t_\ell]$, there exists $\mu_{B_\ell}\in L^{q_i}[0,t_\ell]$
such that $|f_i(s,u^k(s))|\leq \mu_{B_\ell}(s)$ for almost every
$s\in [0,t_\ell]$. Hence, we have
$$
|g_i(t,s)f_i(s,u_1^k(s),u_2^k(s),\dots, u_n^k(s))|\leq
|g_i^t(s)|\mu_{B_\ell}(s), ~~a.e.~s\in [0,t]
$$
and
$|g_i^t|\mu_{B_\ell}\in L^1[0,t]$. Let $k\to \infty$ in \eqref{e3.36}.
Since $u_i^k\to z_i^\ell$ uniformly on $[0,t_\ell]$, an application
of Lebesgue Dominated Convergence Theorem gives
$$
z_i^\ell(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,z_1^\ell(s),z_2^\ell(s),
\dots, z_n^\ell(s))ds,
\quad  t\in [0,t_\ell]
$$
or equivalently (noting \eqref{e3.35})
\begin{equation}
u_i^*(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u_1^*(s),u_2^*(s),
\dots,u_n^*(s))ds,
\quad t\in [0,t_\ell]. \label{e3.37}
\end{equation}
Finally, letting $\ell\to \infty$ in \eqref{e3.37} yields
$$
u_i^*(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u_1^*(s),u_2^*(s),
\dots,u_n^*(s))ds, \quad t\in [0,T).
$$
Hence, $u^*=(u_1^*,u_2^*,\dots,u_n^*)$ is a
solution of  \eqref{e1.2}.
 \end{proof}

Our subsequent results make use of Theorem \ref{thm3.6} and an argument
originated from Brezis and Browder \cite{BB}.

\begin{theorem} \label{thm3.7}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4}
and \eqref{e3.32} with
$p_i=\infty$ and $q_i=1$. Moreover, suppose the following conditions
hold for each $1\leq i\leq n$ and any $w\in (0,T)$:
there exist $B_i> 0$ such that for any $u\in (C[0,w])^n$,
\begin{equation}
\int_0^w\Big[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds\Big]dt\leq B_i\,;
\label{e3.38}
\end{equation}
and there exist $r>0$ and $\alpha_i>0$ with
$r\alpha_i>H_i(w)\equiv \sup_{t\in [0,w]}|h_i(t)|$
such that for any $u\in (C[0,w])^n$,
\begin{equation}
u_i(t)f_i(t,u(t))\geq r\alpha_i |f_i(t,u(t))|\quad
\text{for a.e. $t\in[0,w]$  such that }\|u(t)\|>r, \label{e3.39}
\end{equation}
where we denote $\|u(t)\|:=\max_{1\leq i\leq n}|u_i(t)|$.
 Then \eqref{e1.2} has at least one solution
in $(C[0,T))^n$.
\end{theorem}

\begin{proof} We shall establish the existence of `local'
solutions before we can apply Theorem \ref{thm3.6}. Indeed, we shall show
that the system
\begin{equation}
u_i(t)=h_i(t)+\int_{0}^{t}g_i(t,s)f_i(s,u(s))ds, \quad t\in
[0,w],\;1\leq i\leq n \label{e3.40}
\end{equation}
has a solution for any $w\in (0,T)$. Let $w\in (0,T)$ be fixed.
From the hypotheses, we see that \eqref{e3.1}--\eqref{e3.4} are
satisfied with $T$ replaced by $w$. We shall employ
a similar technique as in the proof of Theorem \ref{thm3.2}, with $T$
replaced by $w$. Let $u=(u_1,u_2,\dots,u_n)\in (C[0,w])^n$ be any
solution of
\begin{equation}
u_i(t)=\lambda\Big(h_i(t)+ \int^t_0 g_i(t,s)f_i(s,u(s))ds\Big),\quad
t\in [0,w],\;1\leq i\leq n \label{e3.41lambda}
\end{equation}
where $\lambda\in (0,1)$. We define for each $z\in [0,w]$,
$$
I_z=\{t\in[0,z]:\|u(t)\|\leq r\}, \quad
J_{z}=\{t\in[0,z]:\|u(t)\|> r\}.
$$
 Following the proof of Theorem \ref{thm3.2}, we obtain, corresponding
to \eqref{e3.14},
\begin{equation}
\int_{J_z}|f_i(t,u(t))|dt\leq
\frac{[H_i(w)+r]\int_0^w \mu_{r,i}(s)ds
+B_i}{r\alpha_i-H_i(w)}\equiv c_i(w),\quad
1\leq i\leq n. \label{e3.42}
\end{equation}
Consequently,  from \eqref{e3.41lambda} it follows that for
$t\in [0,w]$ and $1\leq i\leq n$,
\begin{equation}
\begin{split}
|u_i(t)|&\leq H_i(w)+[\sup_{t\in [0,w]}\operatorname{ess\,sup}_{s\in
[0,t]}|g_i(t,s)|]\Big[\int_0^w\mu_{r,i}(s)ds+c_i(w)\Big]\\
& \equiv d_i(w).
\end{split} \label{e3.43}
\end{equation}
Thus, $|u_i|_0=\sup_{t\in [0,w]}|u_i(t)|\leq d_i(w)$ for
$1\leq i\leq n$ and $\|u\|=\max_{1\leq i\leq n}|u_i|_0
\leq \max_{1\leq i\leq n}d_i(w)$ $\equiv D(w)$. It follows
from Theorem \ref{thm3.1} (with $M=D(w)+1$) that \eqref{e3.40} has a solution
$u^*\in(C[0,w])^n$. Hence, we have shown that \eqref{e3.40}
has a solution for any $w\in (0,T)$.

Now, let $\{t_k\}$ be a positive and increasing sequence such that
$\lim_{k\to\infty}t_k=T$. For each $k=1,2,\dots$, let
$u^k=(u_1^k,u_2^k,\dots,u_n^k)\in (C[0,t_k])^n$ be a solution of
\eqref{e3.33}. If we restrict $z\in [0,t_\ell]$ and $k\geq \ell$, then
using the same arguments as before, we can obtain \eqref{e3.42}
and \eqref{e3.43} with $w=t_\ell$ and $u=u^k$. So for $k\geq \ell$
we have
$$
|u_i^k(t)|\leq d_i(t_\ell),\quad t\in [0,t_\ell],\;1\leq i\leq n.
$$
All the conditions of Theorem \ref{thm3.6} are satisfied and hence it follows
that \eqref{e1.2} has at least one solution in $(C[0,T))^n$.
 \end{proof}

Our next result replaces condition \eqref{e3.38} with condition
\eqref{e3.44}
which involves the integral of $f_i$ in the right side.



\begin{theorem} \label{thm3.8}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4}
and \eqref{e3.32} with
$p_i=\infty$ and $q_i=1$. Moreover, suppose the following conditions
hold for each $1\leq i\leq n$ and any $w\in (0,T)$:
there exist constants $a_i\geq 0$  and $b_i$ such
that for any $z\in [0,w]$,
\begin{equation}
\int_0^z\Big[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds\Big]dt\leq
a_i\int_0^z|f_i(t,u(t))|dt+b_i; \label{e3.44}
\end{equation}
and there exist $r>0$ and $\alpha_i>0$  with
$r\alpha_i>H_i(w)+a_i$ such that
for any $u\in (C[0,w])^n$,
\begin{equation}
u_i(t)f_i(t,u(t))\geq r\alpha_i |f_i(t,u(t))|\quad \text{for
a.e. $t\in[0,w]$  such that }\|u(t)\|>r.
 \label{e3.45}
\end{equation}
Then \eqref{e1.2} has at least one solution
in $(C[0,T))^n$.
\end{theorem}

\begin{proof} As in the proof of Theorem \ref{thm3.7},  we shall first
show that the system \eqref{e3.40} has a solution for any
$w\in (0,T)$. Let $w\in (0,T)$ be fixed and let
$u=(u_1,u_2,\dots,u_n)\in (C[0,w])^n$
be any solution of \eqref{e3.41lambda}. Using a similar
argument as in the proof of Theorem \ref{thm3.3}, with $T$ replaced by $w$,
we obtain, corresponding to \eqref{e3.18},
\begin{equation}
\int_{J_z}|f_i(t,u(t))|dt\leq
\frac{[H_i(w)+a_i+r]\int_0^w\mu_{r,i}(s)ds+
|b_i|}{r\alpha_i-H_i(w)-a_i}\equiv c^*_i(w),
\label{e3.46}
\end{equation}
for $1\leq i\leq n$, and subsequently $\|u\|\leq D^*(w)$ (a constant). Then, it
follows from Theorem \ref{thm3.1} that \eqref{e3.40} has a solution for
any $w\in (0,T)$. The rest of the proof proceeds as in the proof
of Theorem \ref{thm3.7}.
\end{proof}

The next result is for general $p_i$, $q_i$, it also replaces
condition \eqref{e3.39} or \eqref{e3.45} with conditions
\eqref{e3.47} and \eqref{e3.48}.

\begin{theorem} \label{thm3.9}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4}
 and \eqref{e3.32}. Moreover,
suppose the following conditions hold for each $1\leq i\leq n$ and
any $w\in (0,T)$: \eqref{e3.38},
there exist $r>0$ and $\beta_i>0$ such that
for any $u\in (C[0,w])^n$,
\begin{equation}
u_i(t)f_i(t,u(t))\geq \beta_i|u_i|_0 \cdot |f_i(t,u(t))|\quad
\text{for a.e. $t\in[0,w]$  such that }\|u(t)\|>r, \label{e3.47}
\end{equation}
where we denote $|u_i|_0:=\max_{t\in [0,w]}|u_i(t)$|;
and there exist $\eta_i>0$, $\gamma_i\geq q_i-1>0$  and
$\phi_i\in L^{p_i}([0,w],\mathbb{R})$ such that for any $u\in (C[0,w])^n$,
\begin{equation}
|u_i|_0\geq \eta_i|f_i(t,u(t)|^{\gamma_i}+\phi_i(t)\quad
\text{for a.e. $t\in[0,w]$  such that }\|u(t)\|>r.
 \label{e3.48}
\end{equation}
Then \eqref{e1.2} has at least one solution in $(C[0,T))^n$.
\end{theorem}

 \begin{proof}
 Once again  we shall employ Theorem \ref{thm3.1} to show the
existence of `local' solutions; i.e.,  the system \eqref{e3.40} has a
solution for any $w\in (0,T)$. For this, we use a similar argument
as in the proof of Theorem \ref{thm3.4}, with $T$ replaced by $w$, to get an
analog of \eqref{e3.28}, viz.,
\begin{equation}
\int_{J_{z}}|f_i(t,u(t))|^{\gamma_i+1}dt \leq c^{**}_i(w),\quad
1\leq i\leq n \label{e3.49}
\end{equation}
which leads to $\|u\|\leq D^*(w)$ (a constant). The rest of the
proof follows as in the proof of Theorem \ref{thm3.7}.
 \end{proof}

The next result is also for general $p_i$, $q_i$, and here the
condition \eqref{e3.38} is replaced by \eqref{e3.44}.

\begin{theorem} \label{thm3.10}
Let the following conditions be satisfied
for each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4}
and \eqref{e3.32}. Moreover,
suppose the following conditions hold for each $1\leq i\leq n$ and
any $w\in (0,T)$: \eqref{e3.44},  \eqref{e3.47} and \eqref{e3.48}.
Then \eqref{e1.2} has at
least one solution in $(C[0,T))^n$.
\end{theorem}

 \begin{proof}
To prove that the system \eqref{e3.40} has a solution for
any $w\in (0,T)$, we use a similar argument as in the proof of
Theorem \ref{thm3.5}, with $T$ replaced by $w$, to get an analog of
\eqref{e3.31}, viz.,
\begin{equation}
\int_{J_{z}}|f_i(t,u(t))|^{\gamma_i+1}dt
\leq \bar c_i(w),\;1\leq i\leq n \label{e3.50}
\end{equation}
and subsequently $\|u\|\leq D^*(w)$ (a constant). The rest of the
proof proceeds as in the proof of Theorem \ref{thm3.7}.
\end{proof}


\section{Existence of constant-sign solutions}

In this section, we shall establish the existence of
\emph{constant-sign} solutions of the systems \eqref{e1.1}
 and \eqref{e1.2}, in $(C[0,T])^n$ and $(C[0,T))^n$ respectively.
Once again we shall employ an argument originated from Brezis
and Browder \cite{BB}.

Throughout, let $\theta_i\in\{-1,1\}, ~1\leq i\leq n$ be fixed. For
each $1\leq j\leq n$, we define
$$
[0,\infty)_j = \begin{cases}
[0,\infty), & \theta_j=1\\
(-\infty,0], & \theta_j=-1.
\end{cases}
$$
Our first result is for the system \eqref{e1.1} and is `parallel' to
Theorem \ref{thm3.2}.

\begin{theorem} \label{thm4.1}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.2}--\eqref{e3.4}
with $p_i=\infty$ and
$q_i=1$, \eqref{e3.7}, \eqref{e3.8},
\begin{gather}
\theta_ih_i(t)\geq 0,\quad t\in [0,T]; \label{e4.1}\\
g_i(t,s)\geq 0, \quad 0\leq s\leq t\leq T ;\label{e4.2}\\
\theta_if_i(t,u)\geq 0,\quad  (t,u)\in [0,T]\times
\prod_{j=1}^n[0,\infty)_j. \label{e4.3}
\end{gather}
Then \eqref{e1.1} has at least
one constant-sign solution in $(C[0,T])^n$.
\end{theorem}

\begin{proof}
First, we shall show that the system
\begin{equation}
u_i(t)=h_i(t)+\int^t_0 g_i(t,s)f_i^*(s,u(s))ds,\quad
t\in [0,T],\;1\leq i\leq n \label{e4.4}
\end{equation}
has a solution in $(C[0,T])^n$. Here,
\begin{equation}
f_i^*(t,u_1,\dots,u_n)=f_i(t,v_1,\dots,v_n),\quad
t\in [0,T],\;1\leq i\leq n \label{e4.5}
\end{equation}
where
$$
v_j=\begin{cases}
u_j, & \theta_ju_j\geq 0\\
0, & \theta_ju_j\leq 0 .
\end{cases}
$$
Clearly, $f_i^*(t,u):[0,T]\times \mathbb{R}^n  \to \mathbb{R}$ and $f_i^*$
satisfies \eqref{e3.2}.

We shall employ Theorem \ref{thm3.1}, so let $u=(u_1,u_2,\dots,u_n)\in
(C[0,T])^n$ be any solution of
\begin{equation}
u_i(t)=\lambda\Big(h_i(t)+\int^t_0 g_i(t,s)f_i^*(s,u(s))ds\Big),\quad
t\in [0,T],\;1\leq i\leq n \label{e4.6lambda}
\end{equation}
where $\lambda\in (0,1)$. Using \eqref{e4.1}--\eqref{e4.3},
we have for $t\in [0,T]$ and $1\leq i\leq n$,
$$
\theta_iu_i(t)=\lambda\Big(\theta_ih_i(t)+\int^t_0
g_i(t,s)\theta_if_i^*(s,u(s))ds\Big)\geq 0.
$$
Hence, $u$ is a \emph{constant-sign} solution of \eqref{e4.6lambda},
and it follows that
\begin{equation}
f_i^*(t,u(t))=f_i(t,u(t)),\quad t\in [0,T],\;1\leq i\leq
n. \label{e4.7}
\end{equation}
For each $z\in [0,T]$, define $I_z$ and $J_{z}$ as in \eqref{e3.9}.
Noting \eqref{e4.7}, we see that \eqref{e4.6lambda} is the same
as \eqref{e3.5lambda}. Therefore, using a similar technique
as in the proof of Theorem \ref{thm3.2}, we obtain
\eqref{e3.11}--\eqref{e3.14} and subsequently $|u_i|_0\leq d_i$ for
$1\leq i\leq n$. Thus, $\|u\|\leq \max_{1\leq i\leq n}d_i\equiv D$.
It now follows from Theorem \ref{thm3.1} (with $M=D+1$)
that \eqref{e4.4} has a
solution $u^*\in(C[0,T])^n$.

Noting \eqref{e4.1}--\eqref{e4.3}, we have for $t\in [0,T]$ and
$1\leq i\leq n$,
$$
\theta_iu_i^*(t)=\theta_ih_i(t)+\int^t_0
g_i(t,s)\theta_if_i^*(s,u^*(s))ds\geq 0.
$$
So $u^*$ is of \emph{constant sign}.
 From \eqref{e4.5}, it is then clear that
$$
f_i^*(t,u^*(t))=f_i(t,u^*(t)),\quad t\in [0,T],\;1\leq i\leq n.
$$
Hence, the system \eqref{e4.4} is actually \eqref{e1.1}.
This completes the proof.
\end{proof}

Based on the proof of Theorem \ref{thm4.1}, we can develop parallel results
to Theorems \ref{thm3.3}--\ref{thm3.5} as follows.

\begin{theorem} \label{thm4.2}
 Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.2}--\eqref{e3.4}
with $p_i=\infty$ and $q_i=1$, \eqref{e3.15}, \eqref{e3.16}
and \eqref{e4.1}--\eqref{e4.3}. Then \eqref{e1.1} has at least
one constant-sign solution in $(C[0,T])^n$.
\end{theorem}


\begin{theorem} \label{thm4.3}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}--\eqref{e3.4},  \eqref{e3.7},
 \eqref{e3.19}, \eqref{e3.20} and
\eqref{e4.1}--\eqref{e4.3}. Then \eqref{e1.1} has at
least one constant-sign solution in
$(C[0,T])^n$.
\end{theorem}

\begin{theorem} \label{thm4.4}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}--\eqref{e3.4},  \eqref{e3.15},
\eqref{e3.19}, \eqref{e3.20} and
\eqref{e4.1}--\eqref{e4.3}. Then \eqref{e1.1} has at least
one constant-sign solution in $(C[0,T])^n$.
\end{theorem}

We shall now establish the existence of constant-sign solutions of
the system \eqref{e1.2}. The next result is `parallel' to
Theorem \ref{thm3.7}.

\begin{theorem} \label{thm4.5}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4}
and \eqref{e3.32} with $p_i=\infty$ and $q_i=1$, and
\eqref{e4.1}--\eqref{e4.3}. Moreover, suppose the
following conditions hold for each $1\leq i\leq n$ and any
$w\in (0,T)$: \eqref{e3.38} and \eqref{e3.39}.
Then \eqref{e1.2} has at least one
constant-sign solution in $(C[0,T))^n$.
\end{theorem}


\begin{proof}
To apply Theorem \ref{thm3.6},  we should show the
existence of `local' solutions by considering the following analog
to \eqref{e3.40},
\begin{equation}
u_i(t)=h_i(t)+\int^t_0 g_i(t,s)f_i^*(s,u(s))ds,\quad
t\in [0,w],\;1\leq i\leq n \label{e4.8}
\end{equation}
where $w\in (0,T)$ and $f_i^*$ is given in \eqref{e4.5}.
The rest of the proof models that of Theorems \ref{thm4.1}
 and \ref{thm3.7}.
\end{proof}

Based on the proof of Theorem \ref{thm4.5}, parallel results to
Theorems \ref{thm3.8}--\ref{thm3.10} are established as follows.

\begin{theorem} \label{thm4.6}
 Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4}
and \eqref{e3.32} with
$p_i=\infty$ and $q_i=1$, and \eqref{e4.1}--\eqref{e4.3}.
Moreover, suppose the
following conditions hold for each $1\leq i\leq n$ and any $w\in
(0,T)$: \eqref{e3.44} and \eqref{e3.45}. Then \eqref{e1.2}
has at least one constant-sign solution in $(C[0,T))^n$.
\end{theorem}


\begin{theorem} \label{thm4.7}
Let the following conditions be satisfied for
each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.4},
 \eqref{e3.32} and \eqref{e4.1}--\eqref{e4.3}.
Moreover, suppose the following conditions hold for each $1\leq
i\leq n$ and any $w\in (0,T)$: \eqref{e3.38}, \eqref{e3.47}
 and \eqref{e3.48}. Then
\eqref{e1.2} has at least one constant-sign solution in $(C[0,T))^n$.
\end{theorem}


\begin{theorem} \label{thm4.8.} Let the following conditions be
satisfied for each $1\leq i\leq n$: \eqref{e3.1}, \eqref{e3.3},
\eqref{e3.4}, \eqref{e3.32} and \eqref{e4.1}--\eqref{e4.3}.
Moreover, suppose the following conditions hold for each
$1\leq i\leq n$ and any $w\in (0,T)$: \eqref{e3.44},
\eqref{e3.47} and \eqref{e3.48}.
Then \eqref{e1.2} has at least one constant-sign solution
in $(C[0,T))^n$.
\end{theorem}


\section{Examples}


We shall now illustrate the results obtained through some
examples.



 \begin{example} \label{examp5.1}\rm
 Consider system \eqref{e1.1} where for $1\leq i\leq n$,
\begin{equation}
\begin{split}
&f_i(t,u_1(t),u_2(t),\dots,u_n(t))\\
&=\begin{cases}
 \rho_i(t,u_1(t),u_2(t),\dots,u_n(t)),
& u_1(t),u_2(t),\dots,u_n(t)>\delta \\
0, & \text{otherwise.}
\end{cases}
\end{split} \label{e5.1}
\end{equation}
 Here, $\delta>0$ is a given constant, and $\rho_i$ is such that
\begin{itemize}
\item[(a)] the map $u \mapsto f_i(t,u)$ is continuous for almost
 all $t\in [0,T]$;
\item[(b)] the map $t\mapsto f_i(t,u)$ is measurable for all
 $u\in \mathbb{R}^n$;
\item[(c)] $\rho_i(t,u(t))\in L^1[0,T]$ and
 $u_i(t)\rho_i(t,u(t))\geq 0$ for any $u\in K$ where
$$
K=\{u\in (C[0,T])^n: u_1(t), u_2(t),\dots,u_n(t)>\delta,
\; t\in[0,T]\}.
$$
\end{itemize}
Moreover, suppose $h_i\in C[0,T],\;1\leq i\leq n$ fulfills
\begin{equation}
H_i\equiv \sup_{t\in [0,T]}|h_i(t)|<\delta. \label{e5.2}
\end{equation}
Clearly, conditions \eqref{e3.1} and \eqref{e3.2} with $q_i=1$
are fulfilled. We shall check that condition \eqref{e3.8}
is satisfied. Pick $r>\delta$ and
$\alpha_i=\frac{\delta}{r},\;1\leq i\leq n$. Then,
from \eqref{e5.2},  we have $r\alpha_i=\delta>H_i$.

Let $u\in K$. Then, from \eqref{e5.1}, we have $f_i(t,u)=\rho_i(t,u)$.
Consider $\|u(t)\|>r$ where $t\in [0,T]$.
If $\|u(t)\|=|u_i(t)|$, then
\begin{equation}
\begin{split}
u_i(t)f_i(t,u(t))
=|u_i(t)|\cdot|f_i(t,u(t))|
&=\|u(t)\|\cdot|f_i(t,u(t))|\\
&> r|f_i(t,u(t))|\\
&> r\cdot\frac{\delta}{r}\cdot|f_i(t,u(t))|\\
&= r\alpha_i |f_i(t,u(t))|.
\end{split}\label{e5.3}
\end{equation}
If $\|u(t)\|=|u_k(t)|$ for some $k\neq i$, then
\begin{equation}
\begin{split}
u_i(t)f_i(t,u(t))=|u_i(t)|\cdot|f_i(t,u(t))|
&= r\cdot\frac{|u_i(t)|}{r}\cdot |f_i(t,u(t))|\\
&> r\cdot \frac{\delta}{r}\cdot|f_i(t,u(t))|\\
&= r\alpha_i |f_i(t,u(t))|.
\end{split}\label{e5.4}
\end{equation}
Therefore, from
\eqref{e5.3} and \eqref{e5.4} we see that condition \eqref{e3.8}
 holds for $u\in K$.

For $u\in (C[0,T])^n\backslash K$, we have $f_i(t,u)=0$
and \eqref{e3.8} is trivially true. Hence, we have shown that
condition \eqref{e3.8} is satisfied.
\end{example}

The next example considers an  $g_i(t,s)$ of which the particular
case when $n=1$ (see \eqref{e1.6}) has been investigated by
Bushell and Okrasi\'nski \cite{B2}.

\begin{example} \label{examp5.2}\rm
 Consider system \eqref{e1.1} with \eqref{e5.1}, \eqref{e5.2}, and
for $1\leq i\leq n$,
\begin{equation}
g_i(t,s)=(t-s)^{\gamma_i-1} \label{e5.5}
\end{equation}
where $\gamma_i>1$.

Clearly, $g_i$ satisfies \eqref{e3.3} and \eqref{e3.4}
with $p_i=\infty$. Next,
for $u\in K$ ($K$ is given in Example \ref{examp5.1})  we have
\begin{equation}
\begin{split}
&\int_0^T\Big[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds\Big]dt\\
&=\int_0^T\Big[\rho_i(t,u(t))\int_0^t(t-s)^{\gamma_i-1}
\rho_i(s,u(s))ds\Big]dt \\
&\leq T^{\gamma_i-1}\int_0^T\Big[\rho_i(t,u(t))
 \int_0^t\rho_i(s,u(s))ds\Big]dt
\leq  B_i
\end{split}\label{e5.6}
\end{equation}
since $\rho_i(t,u(t))\in L^1[0,T]$ for any $u\in K$.
This shows that condition \eqref{e3.7} holds
for  $u\in K$. For $u\in (C[0,T])^n\backslash K$, we have
$f_i(t,u)=0$ and \eqref{e3.7} is trivially true. Therefore,
condition \eqref{e3.7} is satisfied.
It now follows from Theorem \ref{thm3.2} that the system \eqref{e1.1}
with \eqref{e5.1}, \eqref{e5.2} and \eqref{e5.5} has at
least one solution in $(C[0,T])^n$.
\end{example}

The next example considers an  $g_i(t,s)$ of which the particular
case when $n=1$ comes from the Emden differential equation \eqref{e1.4}.

\begin{example} \label{examp5.3}\rm
 Consider system \eqref{e1.1} with \eqref{e5.1}, \eqref{e5.2}, and
for $1\leq i\leq n$,
\begin{equation}
g_i(t,s)=(t-s)s^{r_i} \label{e5.7}
\end{equation}
where $r_i\geq 0$.

Clearly, $g_i$ satisfies \eqref{e3.3} and \eqref{e3.4} with
$p_i=\infty$. Next, for $u\in K$ ($K$ is given in Example \ref{examp5.1}),
corresponding to \eqref{e5.6} we have
\begin{equation}
\begin{split}
&\int_0^[f_i(t,u(t))\int_0^tg_i(t,s)f_i(s,u(s))ds]dt\\
&= \int_0^T[\rho_i(t,u(t))\int_0^t(t-s)s^{r_i}\rho_i(s,u(s))ds]dt\\
&\leq T^{r_i+1}\int_0^T[\rho_i(t,u(t))\int_0^t\rho_i(s,u(s))ds]dt
\leq  B_i.
\end{split}\label{e5.8}
\end{equation}
Hence, by Theorem \ref{thm3.2} the system \eqref{e1.1} with \eqref{e5.1},
\eqref{e5.2} and \eqref{e5.7}
has at least one solution in $(C[0,T])^n$.
\end{example}

\begin{example} \label{examp5.4} \rm
 Let $\theta_i=1,\;1\leq i\leq n$. Consider
system \eqref{e1.1} with \eqref{e5.1}, \eqref{e5.2}, and
 for $1\leq i\leq n$,
\begin{equation}
h_i(t)\geq 0,\quad t\in [0,T]. \label{e5.9}
\end{equation}
Clearly, conditions \eqref{e4.1} and \eqref{e4.3} are fulfilled.
Moreover, both $g_i(t,s)$ in \eqref{e5.5} and \eqref{e5.7}
satisfy condition \eqref{e4.2}.
From Examples \ref{examp5.1}--\ref{examp5.3}, we see that all the conditions of
Theorem \ref{thm4.1} are met.
Hence, we conclude that  system \eqref{e1.1}
with \eqref{e5.1}, \eqref{e5.2},
\eqref{e5.5} and \eqref{e5.9} and
 system \eqref{e1.1} with \eqref{e5.1}, \eqref{e5.2},
\eqref{e5.7} and \eqref{e5.9} each has at least
one \emph{positive} solution in
$(C[0,T])^n$.
\end{example}

We remark that  Examples \ref{examp5.1}--\ref{examp5.4} can easily be extended to
the system \eqref{e1.2}.

\subsection*{Acknowledgements}
 The authors would like to thank the anonymous
referee for his/her comments which help us improve this article.


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\end{document}