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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 259, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/259\hfil Existence and uniqueness of solutions]
{Existence and uniqueness of solutions for first-order nonlinear
 differential equations with two-point and integral boundary conditions}

\author[M. J. Mardanov, Y. A. Sharifov, H. H. Molaei \hfil EJDE-2014/259\hfilneg]
{Misir J. Mardanov, Yagub A. Sharifov, Habib H. Molaei}  % in alphabetical order

\address{Misir J. Mardanov \newline
Institute of Mathematics and
Mechanics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan}
\email{misirmardanov@yahoo.com}

\address{Yagub A. Sharifov \newline
Baku State University, Institute of Control Systems,
National Academy of Sciences of Azerbaijan, Baku, Azerbaijan}
\email{sharifov22@rambler.ru}

\address{Habib H. Molaei \newline
Department of Mathematics, Technical and Vocational University,
Urmia, Iran}
\email{habibmolaei@gmail.com}

\thanks{Submitted November 1, 2014. Published December 11, 2014.}
\subjclass[2000]{34A37, 34G60, 34G20}
\keywords{Nonlocal boundary conditions; contraction principle;
\hfil\break\indent existence and uniqueness; fixed point theorem}

\begin{abstract}
 In this article, we study the existence of solutions to
 boundary-value problems for ordinary differential equations with
 two-point and integral boundary conditions. Existence and uniqueness
 results are obtained by using well known fixed point theorems.
 Some illustrative examples are also discussed.
\end{abstract}

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

\section{Introduction}

Many of the physical systems can be described by the
differential equations with integral boundary conditions.
Integral boundary conditions are encountered
in various applications, such as population dynamics, blood flow models,
chemical engineering and cellular systems. Moreover,
 boundary-value problems with integral conditions constitute
an interesting and important class of problems. They include two, three,
multi and nonlocal boundary-value problems as special cases.
For boundary-value problems with nonlocal boundary conditions and comments on
their importance, we refer the reader to \cite{a1,b1,b2,b3,b4,b5}
and the references therein.

 In this article, we study existence and uniqueness of the solutions of
 nonlinear differential equations of the type
\begin{equation} \label{e11}
 \dot{x}(t)=f(t,x(t)), \quad\text{for } t\in [0,T],
\end{equation}
 with two-point and integral boundary conditions
\begin{equation} \label{e12}
Ax(0)+\int _{0}^{T}m(s)x(s)ds+ Bx(T)=\int _{0}^{T}g(s,x(s))ds ,
\end{equation}
where $A,B\in R^{n\times n} $ are given matrices, $\det \big(A+\int
_{0}^{T}m(s)ds+ B\big)\ne 0$;
$f,g:[0,T]\times R^{n} \to R^{n} $, are given functions.
By $C([0,T]\mathbb{R}^n )$ we denote the Banach
space of all continuous functions from $[0,T]$ into
$R^{n} $ with the norm
\[
\| x\| =\max \{|x(t)|:t\in [0,T]\},
\]
where $|\cdot |$ is the norm in space $R^{n} $.

We prove some new existence and uniqueness results by using a
variety of fixed point theorems. In Theorem \ref{thm3.1} we prove an
existence and uniqueness result by using Banach's contraction
principle. In Theorem \ref{thm3.2} we prove the existence of a solution by
using Schaefer's fixed point theorem, while in Theorem \ref{thm3.3} we prove
the existence of a solution via Leray-Schauder nonlinear
alternative.

 It is worth mention that the methods used in this paper are standard.
Our impact is implementation of these methods to the solution of the
problem  \eqref{e11}, \eqref{e12}.


\section{Preliminaries}


We define a solution of the problem
\eqref{e11}-\eqref{e12} as follows:

\noindent\textbf{Definition.}
A function $x\in C([0,T]\mathbb{R}^n )$ is said to be a solution of
problem \eqref{e11}-\eqref{e12} if
$\dot{x}(t)=f(t,x(t))$, for each
$t\in [0,T]$,  and the boundary conditions
\eqref{e12} are satisfied.

\begin{lemma} \label{lem2.1}
Let $y,g\in C([0,T]\mathbb{R}^n )$. Then the unique solution of
the boundary-value problem for the differential equation
\begin{equation} \label{e21}
\dot{x}(t)=y(t), t\in [0,T]
\end{equation}
with boundary condition
\begin{equation} \label{e22}
Ax(0)+\int _{0}^{T}m(s)x(s)ds+ Bx(T)=\int _{0}^{T}g(s)ds
\end{equation}
is given by
\begin{equation} \label{e23}
x(t)=C+\int _{0}^{T}K(t,\tau )y(\tau )d\tau  ,
\end{equation}
where
\begin{gather*}
K(t,\tau )=\begin{cases}
\Gamma ^{-1} \big(A+\int _{0}^{t}m(\tau )d\tau \big) & 0\le \tau \le t \\
-\Gamma ^{-1} (\int _{t}^{T}m(\tau )d\tau +B ), & t\le \tau \le T
 \end{cases}, \\
 C=\Gamma ^{-1} \int _{0}^{T}g(s)ds ,\\
\Gamma =(A+\int _{0}^{T}m(t)dt +B).
\end{gather*}
\end{lemma}

\begin{proof}
 If $x=x(\cdot )$ is a solution
of the differential equation \eqref{e21}, then for $t\in (0,T)$,
\begin{equation} \label{e24}
x(t)=x(0)+\int _{0}^{t}y(\tau )d\tau   ,
\end{equation}
where $x(0)$ is an arbitrary constant vector. In order to
determine $x(0)$ we require that the function in equality
\eqref{e21} should satisfy condition
\eqref{e22}, i.e.,
\[
\Gamma x(0)=\int _{0}^{T}g(t)dt-\int _{0}^{T}m(t)\int _{0}^{t}y(\tau )d\tau dt
-B\int _{0}^{T}y(t)\,dt .
\]
Since $\det \Gamma \ne 0$, we have
\begin{equation} \label{e25}
x(0)=C+\Gamma ^{-1} \int _{0}^{T}\int _{t}^{T}m  (\tau )d\tau y(t)dt
-\Gamma ^{-1} B\int _{0}^{T}y(t)dt .
\end{equation}
Now in \eqref{e24} we take into account the value
$x(0)$ determined from the equality \eqref{e25}
and obtain
\[
x(t)=C+\int _{0}^{T}K(t,\tau )y(\tau )d\tau  .
\]

Thus we have proved that one can write the boundary-value problem
\eqref{e21}, \eqref{e22} as the integral
equation \eqref{e23}. One can immediately verify that a
solution to the integral equation \eqref{e23} also
satisfies the boundary-value problem \eqref{e21},
\eqref{e22}.
\end{proof}


\begin{lemma} \label{lem2.2}
Assume that $f,g\in C([0,T]\times R^{n} \mathbb{R}^n )$. Then the
function $x(t)$ is a solution of the boundary-value problem
\eqref{e11}-\eqref{e12} if and only if
$x(t)$ is a solution of the integral equation
\begin{equation} \label{e27}
x(t)=\int _{0}^{T}K(t,s)f(s,x(s))ds+\Gamma ^{-1} \int _{0}^{T}g(s,x(s))ds  .
\end{equation}
\end{lemma}

\begin{proof}
 Let $x(t)$ be a solution of the
boundary-value problem \eqref{e11}, \eqref{e12}.
Then in the same way as in Lemma \ref{lem2.1}, we can prove that it is also a
solution of the integral equation \eqref{e27}. By direct
verification we can show that the solution of the integral equation
\eqref{e27} also satisfies equation \eqref{e11}
and nonlocal boundary condition \eqref{e12}. Lemma \ref{lem2.2} is
proved.
\end{proof}

\section{Main results}


 Define the operator $P:C([0,T]\mathbb{R}^n )\to P([0,T]\mathbb{R}^n )$ as
\begin{equation} \label{e31}
Px(t)=\Gamma ^{-1} \int _{0}^{T}g(t,x(t))dt
+\int _{0}^{T}K(t,\tau )f(\tau ,x(\tau ))d\tau .
\end{equation}
Obviously, the problem \eqref{e11}, \eqref{e12}
is equivalent to the fixed point problem $x=Px$. In consequence,
problem \eqref{e11}, \eqref{e12} has a solution
if and only if the operator $P$ has a fixed point.

Our first result is based on the Banach fixed point theorem.
It uses the assumptions:
\begin{itemize}
\item[(H1)]  There exists a continuous function $N(t)>0$ such that
\[
|f(t,x)-f(t,y)|\le N(t)|x-y|,
\]
for each $t\in [0,T]$ and all $x,y\in R^{n} $;

\item[(H2)] There exists a continuous function $M(t)>0$ such that
\[
|g(t,x)-g(t,y)|\le M(t)|x-y|,
\]
 for each $t\in [0,T]$ and all $x,y\in R^{n} $.
\end{itemize}

\begin{theorem} \label{thm3.1}
Assume {\rm (H1), (H2)} hold, and
\begin{equation} \label{e32}
L=T[SN+M]\| \Gamma ^{-1} \| <1  \,.
\end{equation}
Then the boundary-value problem
\eqref{e11}-\eqref{e12} has a unique solution on
$[0,T]$, where
\begin{gather*}
N=\max_{[0,T]}N(t), \quad
M=\max_{[0,T]} M(t),\\
S=\max \{\| (A+\int _{0}^{t}m(\tau ) d\tau )\| ,
 \| (\int _{t}^{T}m(\tau )d\tau +B )\| \}.
\end{gather*}
\end{theorem}

\begin{proof}
Setting  $\max_{[0,T]} |f(t,0)|=M_{f}$, $\max_{[0,T]} |g(t,0)|=M_{g} $ and
 choosing
  $$
r\ge [1-\| \Gamma ^{-1} \|T (SN+M)]^{-1} \| \Gamma ^{-1} \| (M_{f} +M_{g} ),
$$
   we show that $PB_{r} \subset B_{r} $, where
$$
B_{r} =\{x\in C([0,T]\mathbb{R}^n ):\| x\| \le    r\}.
$$
For $x\in B_{r} $, we have
\begin{align*}
 \| {( {Px} )( t )} \|
&\le \max_{[0,T]} \Big[ {\int_0^T {| {K( {t,s} )} || {f( {s,x( s )} )} |ds} } \Big]
+ \Big[ {\| {\Gamma ^{ - 1} } \|\int_0^T {| {g( {s,x( s )} )} |ds} } \Big]
\\
&\le   \max_{[0,T]} \Big[ {\int_0^T {| {K( {t,s} )} |} ( {| {f( {s,x( s )} )
- f( {s,0} )} | + | {f( {s,0} )} |} )ds} \Big] \\
&\quad +   \| {\Gamma ^{ - 1} } \|\int_0^T {( {| {g( {s,x( s )} ) - g( {s,0} )} |
+ | {g( {s,0} )} |} )ds} \\
& \leq   \| {\Gamma ^{ - 1} } \|S( {Nr + M_f } )T
 + \| {\Gamma ^{ - 1} } \|( {Mr + M_g } )T \le r.
 \end{align*}
 Now, for any $u,v\in B_{r} $ we have
\begin{align*}
&| {( {Pu} )( t ) - ( {Pv} )( t )} | \\
&\le \| {\Gamma ^{ - 1} } \|\int_0^T {| {g( {t,u( t )} )
 - g( {t,v( t )} )} |} dt
 +   \int_0^T {| {K( {t,\tau } )} |} | {f( {\tau ,u( \tau  )} )
 - f( {\tau ,v( \tau  )} )} |d\tau  \\
&\le    \| {\Gamma ^{ - 1} } \|\int_0^T {M( t )| {u( t ) - v( t )} |dt}
  + \| {\Gamma ^{ - 1} } \|S\int_0^T {N( t )| {u( t ) - v( t )dt} |dt} \\
&\le  \| {\Gamma ^{ - 1} } \|[ {M + NS} ]T\| {u - v} \|,
 \end{align*}
 or
\begin{equation} \label{e33}
\| Pu-Pv\| \le L\| u-v\| .
\end{equation}

 From condition \eqref{e32} it follows that $\| Pu-Pv\| <\| u-v\| $.
Therefore, $P$ is a contraction in $B_{r} $. Therefore,
  in view of the contraction principle the operator $P$ defined by
 \eqref{e31} has a unique fixed point in $C([0,T]\mathbb{R}^n )$.
Consequently, the integral equation \eqref{e27} (or the boundary-value
problem \eqref{e11}, \eqref{e12}) has a unique solution.
\end{proof}

 The second result is based on Schaefer's fixed point theorem.
It uses the assumptions:
\begin{itemize}

\item[(H3)]  The function $f:[0,T]\times R^{n} \to R^{n} $ is continuous;

\item[(H4)]  There exists a constant $N_1 >0$ such that
$|f(t,x)|\le N_1 $ for each $t\in [0,T]$ and all $x\in R^{n} $;

\item[(H5)]  The function $g:[0,T]\times R^{n} \to R^{n} $ is continuous;

\item[(H6)]  There exists a constant $N_2 >0$ such that
 $|g(t,x)|\le N_2 $ for each $t\in [0,T]$ and all $x\in R^{n} $.
\end{itemize}

 \begin{theorem} \label{thm3.2}  Assume {\rm (H3)--(H6)}
Then the boundary-value problem
\eqref{e11}-\eqref{e12} has at least one
solution on $[0,T]$.
\end{theorem}

\begin{proof}
 We  divide the proof into several main steps in which we  show that under
the assumptions of the theorem, the operator $P$ has a fixed point.
\smallskip

\noindent \textbf{Step 1.}
The operator $P$ under the assumptions of
the  theorem is continuous. Let $\{x_n\}$ be a
sequence such that $x_n \to x$ in $C([0,T]\mathbb{R}^n)$. Then for any $t\in (0,T)$,
\begin{align*}
 &| {P( {x_n } )( t ) - P( x )( t )} |\\
& \le \| {\Gamma ^{ - 1} } \|\int_0^T {| {g( {t,x_n ( t )} ) - g( {t,x( t )} )} |dt}
 +  \int_0^T {| {K( {t,\tau } )} || {f( {\tau ,x_n ( \tau  )} )
 - f( {\tau ,x( \tau  )} )} |d\tau }  \\
&\le TM\| {\Gamma ^{ - 1} } \|\max_{[0,T]} | {g( {t,x_n ( t )} )
 - g( {t,x( t )} )} |\\
&\quad +   TNS\| {\Gamma ^{ - 1} } \|\max_{[0,T]} | {f( {t,x_n ( t )} )
 - f( {t,x( t )} )} |.
 \end{align*}
Since $f$ and $g$ are continuous functions, we have
\[
\| P(x_n )(t)-P(x)(t)\| \to 0, \quad\text{as }n\to \infty .
\]

\noindent\textbf{Step 2.} The operator $P$ maps bounded sets into
$C([0,T]\mathbb{R}^n )$. Indeed, it is sufficient to
show that for any $\eta >0$, there exists a positive constant $l$
such that for each
$x\in B_{\eta } =\{x\in C([0,T]\mathbb{R}^n ):\| x\| \le \eta\}$,
we have $\| P(x)\| \le l$. By (H4)
and (H6) we have for each $t\in [0,T]$,
\[
|P(x)(t)|\le \int _{0}^{T}|K(t,s)||f(s,x(s))|ds
+\| \Gamma ^{-1} \| \int _{0}^{T}|g(s,x(s))|ds  .
\]
 Hence,
\[
|P(x)(t)|\le \| \Gamma ^{-1} \| STN_1 +\| \Gamma ^{-1} \| TN_2 .
\]
Thus,
\[
\| P(x)(t)\| \le \| \Gamma ^{-1} \| STN_1 +\| \Gamma ^{-1} \| TN_2=l.
\]

\noindent\textbf{Step 3.}
The operator $P$ maps bounded sets into equicontinuous sets of
$C([0,T]\mathbb{R}^n )$.
 Let $t_1 ,t_2 \in (0,T]$, $t_1 <t_2 $,
$B_{\eta } $ be a bounded set of $C([0,T]\mathbb{R}^n
)$  as in Step 2, and let $x\in B_{\eta } $. Then
\begin{align*}
&|P(x)(t_2 )-P(x)(t_1 )|\\
&=|\int _{0}^{T}[K(t_2 ,s)-K(t_1 ,s)]f(s,x(s))ds |\\
&\le   |\Gamma ^{-1} \int _{t_1 }^{t_2 }[A+\int _{0}^{t}m(\tau )d\tau  ]
f(t,x(t))dt+\Gamma ^{-1} \int _{t_1 }^{t_2 }
[B+\int _{t}^{T}m(\tau )d\tau  ]f(t,x(t))dt  |\\
&\le  2S\| \Gamma ^{-1} \| \int _{t_1 }^{t_2 }|f(t,x(t))|dt .
 \end{align*}
As $t_1 \to t_2 $, the right-hand side of the above inequality tends to zero.
As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem,
we can conclude that
 $P:C([0,T]\mathbb{R}^n )\to C([0,T]\mathbb{R}^n )$ is completely continuous.
\smallskip

\noindent\textbf{Step 4.}
A priori bounds. Now it remains to show that the set
\[
\Delta =\{x\in C([0,T]\mathbb{R}^n ) :x=\lambda P(x), \text{ for some }
 0<\lambda <1\}
\]
is bounded.

 Let  $x=\lambda (Px)$ for some $0<\lambda <1$. Thus, for each $t\in [0,T]$
we have
\[
x(t)=\lambda \Big[\int _{0}^{T}K(t,s)f(s,x(s))ds
+\Gamma ^{-1} \int _{0}^{T}g(s,x(s))ds  \Big].
\]
This implies by (H4) and (H6) (as in Step 2) that for each
$t\in [0,T]$,
\[
|P(x)(t)|\le \| \Gamma ^{-1} \| (SN_1 +N_2)T.
\]
Thus, for every $t\in [0,T]$ we have
\[
\| x\| \le \| \Gamma ^{-1} \| (SN_1 +N_2)T.
\]
This shows that the set $\Delta $ is bounded. As a consequence of
Schaefer's fixed point theorem, we deduce that $P$ has a fixed point
which is a solution of the problem
\eqref{e11}-\eqref{e12}.
\end{proof}

In the following theorem we give an existence result for the problem
\eqref{e11}-\eqref{e12} by means of an
application of  the Leray-Schauder type nonlinear alternative, where
the conditions (H4) and  (H6) are weakened.
\begin{itemize}

\item[(H7)]  There exist $\theta _{f} \in L_1 ([0,T],R^{+} )$ and continuous
and nondecreasing $\psi _{f} :[0,\infty )\to [0,\infty )$
such that
\[
|f(t,x)|\le \theta _{f} (t)\psi _{f} (|x|),
\]
 for each $t\in [0,T]$ and all $x\in R$;

\item[(H8)]  There exist $\theta _{g} \in L^{1} ([0,T],R^{+} )$ and continuous
and nondecreasing $\psi _{g} :[0,\infty )\to [0,\infty )$ such that
\[
|g(t,x)|\le \theta _{g} (t)\psi _{g} (|x|),
\]
 for each $t\in [0,T]$ and all $x\in R$.

\item[(H9)] There exists a number $K>0$ such that
\[
\frac{K}{\| \Gamma ^{-1} \| S\psi _{f} (K)\|
 \theta _{f} \| _{L_1 } +\psi _{g} (K)
\| \Gamma ^{-1} \| \| \theta \| _{L_1 } } >1.
\]
\end{itemize}

\begin{theorem} \label{thm3.3}
Assume that {\rm (H3), (H5), (H7)--(H9)} hold.
Then the boundary-value problem \eqref{e11}-\eqref{e12} has at least one
solution on $[0,T]$.
\end{theorem}

\begin{proof}
Consider the operator $P$ defined above. It can be easily shown that $P$ is
continuous and completely continuous.
For $\lambda \in [0,1]$ let $x$ be such that for each
$t\in [0,T]$ we have $x(t) = \lambda (Px)(t)$. Then from
(H7) and (H8), for each $t\in [0,T]$ we have
\begin{align*}
|x(t)|
&\le \int _{0}^{t}|K(t,s)|\theta _{f} (s)\psi (|x(s)|)ds
+  \| \Gamma ^{-1} \| \int _{0}^{T}\theta _{g} (s)\psi _{g} (|x(s)|)ds\\
&\le \| \Gamma ^{-1} \| S\psi _{f} (\| x\| ) \int _{0}^{T}\theta _{f} (s) ds+
 \psi _{g} (\| x\| )\| \Gamma ^{-1} \| \int _{0}^{T}\theta _{g} (s)ds.
\end{align*}
Thus,
\[
\frac{\| x\| }{\| \Gamma ^{-1} \| S\psi _{f} (\| x\| )\| \theta _{f} \| _{L_1 }
+\psi _{g} (\| x\| )\| \Gamma ^{-1} \| \| \theta _{g} \| _{L_1 } } \le 1.
\]
Then, in view of (H9), there exists $K$ such that  $\|
x\| \ne K$. Let us set
\[
U=\{x\in C([0,T],R):\| x\| <K\}.
\]

Note that the operator $P:\overline{U}\to C([0,T],\mathbb{R})$
is continuous and completely continuous. From the choice of  $U$,
there is no $x\in \partial U$ such that $x=\lambda P(x)$ for some
$\lambda \in (0,1)$. Consequently, by the nonlinear alternative of
Leray-Schauder type \cite{g1}, we deduce that $P$ has a fixed point $x$ in
$\overline{U}$ which is a solution of the problem
\eqref{e11}-\eqref{e12}. This completes the
proof.
\end{proof}

\section{Examples}

In this section, we give examples to illustrate the usefulness of our main results.

\begin{example} \label{examp4.1}  \rm
Let us consider the following
nonlocal boundary-value problem for the system of differential
equations
\begin{equation} \label{e41}
\begin{gathered}
\dot{x}_1 (t)=0.1\sin x_2 ,\quad t\in [0,1], \\
\dot{x}_2 (t)=\frac{|x_1 |}{(9+e^{t} )(1+|x_1 |)} ,
\end{gathered}
\end{equation}
with
\begin{equation} \label{e42}
x_1 (0)=1, \quad
x_2 (1)=\int _{0}^{1}2tx_1 (t)dt .
\end{equation}
We can rewrite the boundary conditions \eqref{e42} in the
equivalent form:
\begin{align*}
&\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
\begin{pmatrix} x_1 (0) \\ x_2 (0) \end{pmatrix}
-\int _{0}^{1}\begin{pmatrix} 0 & 0 \\ 2t & 0 \end{pmatrix}
\begin{pmatrix} x_1 (t) \\ x_2 (t) \end{pmatrix} dt
+\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} x_1 (1) \\ x_2 (1) \end{pmatrix}\\
&=\begin{pmatrix} 1 \\ 0 \end{pmatrix}.
\end{align*}
Obviously,
\[
\Gamma =\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
-\int _{0}^{1}\begin{pmatrix} 0 & 0 \\ 2t & 0 \end{pmatrix}dt
+\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
=\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix},
\]
and the matrix $\Gamma $ is invertible.

Evidently, $\Gamma ^{-1} =\begin{pmatrix} {1} & {0} \\ {1} & {1} \end{pmatrix}$,
and $\| \Gamma ^{-1} \| =2$.

 Hence, the conditions (H1)--(H2) hold with
$N=0.1$, $M=0$, $S=2$. We can easily see that the condition
\eqref{e32} is satisfied. Indeed,
\begin{equation} \label{e43}
L=\| \Gamma ^{-1} \| SNT=4\times 0.1=0.4<1.
\end{equation}
Then, by Theorem \ref{thm3.1} the boundary-value problem \eqref{e41}-\eqref{e42}
has a unique solution on $[0,1]$.
\end{example}

\begin{example}\label{examp4.2} \rm
 On $[0,1]$ we consider the  boundary-value problem
\begin{equation} \label{e44}
\begin{gathered}
\dot{x}_1 =\sin x_2 , \\
\dot{x}_2 =\cos x_1 , \\
\dot{x}_{3} =\frac{1}{3} \big(\frac{1}{1+x_1^{2} } +\frac{1}{1+x_2^{2} }
+\frac{1}{1+x_{3}^{2} } \big).
\end{gathered}
\end{equation}
with
\begin{equation} \label{e4_5_}
x_1 (0)=0,\quad x_2 (0)+\int _{0}^{1}tx_{3} (t)dt =2,x_{3} (1)=1.
\end{equation}
Obviously, the matrix
\[
\Gamma =\begin{pmatrix} {1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {0}
\end{pmatrix}+\int _{0}^{1}\begin{pmatrix} {0} & {0} & {0} \\ {0} & {0} & {t} \\
 {0} & {0} & {0} \end{pmatrix}dt
+\begin{pmatrix} {0} & {0} & {0} \\ {0} & {0} & {0} \\ {0} & {0} & {1}
\end{pmatrix}
=\begin{pmatrix} {1} & {0} & {0} \\ {0} & {1} & {0.5} \\ {0} & {0} & {1}
\end{pmatrix}
\]
is invertible, and the function
\[
\begin{pmatrix} {f_1 } \\ {f_2 } \\ {f_{3} } \end{pmatrix}
=\begin{pmatrix} {\sin x_1 } \\ {\cos x_2 } \\
{\frac{1}{3} (\frac{1}{1+x_1^{2} } +\frac{1}{1+x_2^{2} }
+\frac{1}{1+x_{3}^{2} } )} \end{pmatrix}
\]
is continuous and bounded.
 Hence, by Theorem \ref{thm3.2} the boundary-value problem \eqref{e44},
\eqref{e4_5_} has at least one solution on $[0,1]$.
\end{example}


\subsection*{Conclusion}

The boundary condition considered in this article is general enough
to cover a wide class of boundary-value problems.
To illustrate this point, we consider the following three cases:
\begin{itemize}
\item[(1)]  If $g\equiv 0$, $A=E$ ($E$ is a unit matrix), $m(t)\equiv \theta $  and
$B=\theta$, ($\theta$ is a zero matrix), then we
get the Cauchy problem.

\item[(2)]  If $g\equiv 0$, $m(t)\equiv \theta$, we get a
two-point boundary-value problem.

\item[(3)] If $g\equiv 0$, $A=\theta$,
$B=\theta$, we get a boundary problem with integral condition.

\end{itemize}
Also note that the methods of this article may be used for more general case
of \eqref{e12},
\[
\sum_{i = 0}^n A_i x(t_i ) + \int_0^T {m(t)x(t)dt}=C,
\]
where $ 0 = t_0  < t_1  < \dots < t_n  = T$,
$ \det \big({\sum_{i = 0}^n {A_i  + \int_0^T {m(t)dt} } } \big)\ne 0$.



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\end{document}