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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011(2011), No. 21, pp. 1--7.\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-2011/21\hfil A priori estimates for solutions]
{A priori estimates for solutions to a four point boundary
value problem for singularly perturbed semilinear differential
equations}

\author[R. Vrabel\hfil EJDE-2011/21\hfilneg]
{Robert Vrabel}  % in alphabetical order

\address{Robert Vrabel \newline
Institute of Applied Informatics, Automation and Mathematics\\
Faculty of Materials Science and Technology,
Hajdoczyho 1, 917 01 Trnava, Slovakia}
\email{robert.vrabel@stuba.sk, epsilon.phi1@gmail.com}

\thanks{Submitted January 5, 2010. Published February 7, 2011.}
\subjclass[2000]{34K10, 34K26}
\keywords{Singular perturbation; four point boundary value
problem; \hfill\break\indent lower and upper solutions}

\begin{abstract}
 This article concerns the existence and asymptotic behavior
 of solutions to a singularly perturbed second-order
 four-point boundary-value problem for nonlinear differential
 equations. Our analysis relies on the method of lower and
 upper solutions. We give accurate  approximations of the
 solutions up to order $O(\epsilon)$.
\end{abstract}

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

\section{Preliminaries}

We consider the four (or three) point boundary value problem
\begin{gather}\label{def_DE}
\epsilon y''+ky=f(t,y),\quad t\in [a,b], \; k<0,\; 0<\epsilon\ll 1,\\
\label{def_BC}
y(c)-y(a)=0,\quad y(b)-y(d)=0,\quad  a<c\leq d<b.
\end{gather}
We focus our attention on the existence and asymptotic behavior
of the solutions  $y_\epsilon(t)$ for \eqref{def_DE}, \eqref{def_BC}
and on an estimate of the difference between  $y_\epsilon(t)$
and a solution $u(t)$ of the equation $ku=f(t,u)$.

The situation in the present case is complicated by the fact that
there are  inner points in the boundary conditions, in contrast to
``standard"  boundary conditions such as the Dirichlet, Neumann,
Robin, or periodic problem \cite{ChHo,CoHa,Vra1,Vra2}, for
example. We note that the equation $\epsilon
v_{\epsilon}''-mv_{\epsilon}=0$, $m>0$, $0<\epsilon$ such that
$\tilde v_\epsilon(c)-\tilde v_\epsilon(a)=u(c)-u(a)>0$ and
$\tilde v_\epsilon(t)\to0^+$ for $t\in (a,b]$ and
$\epsilon\to 0^+$, which could be used to solve this
problem by the method of lower and upper solutions. Instead we
compose barrier functions ($\alpha$, $\beta$) for two-endpoint
boundary conditions to construct barrier functions for
\eqref{def_DE}, \eqref{def_BC}, see e.g. \cite{ChHo}.

In recent years  multi-point boundary value problems have received
a great deal of attention (see e.g. \cite{GG}, \cite{Kha} and the
references therein). The reader is referred to \cite{Kha} where a
four-point  boundary value problem with boundary conditions
$y(c)-\nu_1 y(a)=0, y(b)-\nu_2 y(d)=0$ where the constants
$\nu_1$, $\nu_2$ are not simultaneously equal to $1$ and
$\epsilon=1$ is  studied.


We apply the method of lower and upper solutions to prove the
existence of a solution for problem \eqref{def_DE}, \eqref{def_BC}
and by taking $\epsilon\to 0^+$, the corresponding
solutions converge uniformly on compact subsets of $(a,b)$ to $u$,
the solution of  the reduced problem. Moreover, we prove that
these solutions converge to $u$ up to order $O(\epsilon)$.

As usual, we say that $\alpha_\epsilon \in C^2([a,b])$ is a lower
solution for problem \eqref{def_DE}, \eqref{def_BC} if for every
$t\in(a,b)$ we have $\epsilon
\alpha''_\epsilon(t)+k\alpha_\epsilon (t) \geq f(t,\alpha_\epsilon
(t) )$, and $\alpha_\epsilon (c)-\alpha_\epsilon (a)= 0$,
$\alpha_\epsilon (b)-\alpha_\epsilon (d)\leq 0$. An upper solution
$\beta_\epsilon \in C^2([ a,b])$ satisfies  $\epsilon
\beta''_\epsilon(t)+k\beta_\epsilon (t) \leq f(t,\beta_\epsilon
(t) )$ and $\beta_\epsilon (c)-\beta_\epsilon (a)= 0$,
$\beta_\epsilon (b)-\beta_\epsilon (d)\geq 0$ for every
$t\in(a,b)$.

\begin{lemma}[\cite{Ma}]\label{mainlemma}
If $\alpha_\epsilon ,\beta_\epsilon $ are respectively lower
and upper solutions for \eqref{def_DE}, \eqref{def_BC} such
that $\alpha_\epsilon \leq \beta_\epsilon, $ then there exists
solution $y_\epsilon$ of \eqref{def_DE}, \eqref{def_BC}
with $\alpha_\epsilon\leq y_\epsilon\leq\beta_\epsilon$.
\end{lemma}

 Consider the set
 $\mathcal{H}(u)=\{ (t,y): a\leq t\leq b,  \vert y-u(t)\vert <d(t)\}$,
where $d(t)$  is the positive continuous function on $[ a,b]$
defined by
 \[
d(t) =  \begin{cases}
\vert u(c)-u(a)\vert+\delta & \text{for }a\leq t\leq a+\frac\delta2\\
\delta  & \text{for }a+\delta\leq t\leq b-\delta \\
\vert u(b)-u(d)\vert+\delta & \text{for }b-\frac\delta2\leq t\leq b
\end{cases}
\]
where $\delta$ is a small positive constant and $u\in C^2$ is
a solution of the reduced problem $ky=f(t,y)$ defined on $[ a,b]$.
We write $s(\epsilon)={O}(r(\epsilon))$ when
$0<\lim_{\epsilon\to0^+ }
\vert\frac{s(\epsilon)}{ r(\epsilon)}\vert<\infty$.

\section{Main result}

\begin{theorem}\label{maintheorem}
Let $f\in C^1(\mathcal{H}(u))$ satisfy the condition
\[ %\label{podmienka}
\big|\frac{\partial f(t,y)}{\partial y}\big|
\leq w<-k \quad\text{for  every } (t,y)\in \mathcal{H}(u).
\]
Then there exists $\epsilon_0 $ such that for every
$\epsilon \in (0,\epsilon_0]$,
problem \eqref{def_DE}, \eqref{def_BC} has a unique solution
$y_\epsilon(t)$ satisfying the inequality
\[
-v^{\rm (corr)}_{\epsilon}(t)-\hat v_\epsilon(t)-C\epsilon\leq y_\epsilon(t)
-(u(t)+v_\epsilon(t))\leq\hat v_\epsilon(t)
+ C\epsilon\quad \text{if }  u(c)-u(a)\geq 0
\]
and
\[
-\hat v_\epsilon(t)-C\epsilon\leq y_\epsilon(t)-(u(t)
+v_\epsilon(t))\leq v^{\rm (corr)}_{\epsilon}(t)+\hat v_\epsilon(t)+ C\epsilon\quad \text{if }
  u(c)-u(a)\leq 0
\]
on $[ a,b]$ where
\begin{gather*}
v_\epsilon(t)  =  \frac{u(c)-u(a)}{D}\cdot
\left(e^{\sqrt{\frac m{\epsilon}}(b-t)}
 -e^{\sqrt{\frac m{\epsilon}}(t-b)}
 +e^{\sqrt{\frac m{\epsilon}}(t-d)}
 -e^{\sqrt{\frac m{\epsilon}}(d-t)}\right),
\\
\hat v_\epsilon(t)  = \frac{\vert u(b)-u(d)\vert}{D}\cdot
\left(e^{\sqrt{\frac m{\epsilon}}(t-a)}
-e^{\sqrt{\frac m{\epsilon}}(a-t)}
+e^{\sqrt{\frac m{\epsilon}}(c-t)}
-e^{\sqrt{\frac m{\epsilon}}(t-c)}\right),
\\
\begin{aligned}
D&= \left(e^{\sqrt{\frac m{\epsilon}}(b-a)}
  +e^{\sqrt{\frac m{\epsilon}}(d-c)}
  +e^{\sqrt{\frac m{\epsilon}}(c-b)}
  +e^{\sqrt{\frac m{\epsilon}}(a-d)}\right)\\
&\quad -\left(e^{\sqrt{\frac m{\epsilon}}(a-b)}
  +e^{\sqrt{\frac m{\epsilon}}(c-d)}
  +e^{\sqrt{\frac m{\epsilon}}(b-c)}
  +e^{\sqrt{\frac m{\epsilon}}(d-a)}\right),
\end{aligned}
\end{gather*}
$m=-k-w$, $C=\frac1m\max_{t\in[ a,b]}| u''(t)|$
and the positive function
\begin{align*}
v^{\rm (corr)}_{\epsilon}(t)
&=\frac{w\vert u(c)-u(a)\vert}{\sqrt{m\epsilon}}\cdot
\Big[-{ O}(1)\frac{v_\epsilon(t)}{(u(c)-u(a))}\\
&\quad + O\big(e^{\sqrt{\frac{m}{\epsilon}}(a-d)}\big)
\frac{\hat v_\epsilon(t)}{\vert u(b)-u(d)\vert}
+t{ O}\big(e^{\sqrt{\frac{m}{\epsilon}}\chi (t)}\big)\Big],
\end{align*}
$\chi (t)<0$ for $t\in(a,b]$ and $v^{\rm (corr)}_{\epsilon}(a)=v^{\rm (corr)}_{\epsilon}(c)$.
\end{theorem}

\begin{remark} \label{rmk2.2}\rm
The function $v_{\epsilon}(t)$ satisfies
\begin{enumerate}
\item $\epsilon v''_{\epsilon}-mv_{\epsilon}=0$,
\item $v_{\epsilon}(c)-v_{\epsilon}(a)=-(u(c)-u(a))$,
$v_{\epsilon}(b)-v_{\epsilon}(d)=0$,
\item If $u(c)-u(a)\geq 0$ $(\leq 0)$ then $v_{\epsilon}(t)\geq 0$
$(\leq 0)$ and it is decreasing (increasing) for $a\leq t\leq\frac{b+d}{2}$ and increasing (decreasing) for $\frac{b+d}{2}\leq t\leq b$,
\item If $\epsilon\to 0^+$ then $v_{\epsilon}(t)$
converges uniformly to $0$ on every compact subset of $(a, b ]$,
\item $v_{\epsilon}(t)=(u(c)-u(a))
{O}\big(e^{\sqrt{\frac{m}{\epsilon}}\chi(t)}\big)$, where $\chi (t)=a-t$ for $a\leq t\leq\frac{b+d}{2}$ and  $\chi (t)=t-b+a-d$ for $\frac{b+d}{2}<t\leq b$.
\end{enumerate}
The function $\hat v_{\epsilon}(t)$ satisfies
\begin{enumerate}
\item $\epsilon\hat v_\epsilon''-m\hat v_\epsilon=0$,
\item $\hat v_{\epsilon}(c)-\hat v_{\epsilon}(a)=0$,
$\hat v_{\epsilon}(b)-\hat v_{\epsilon}(d)=\vert u(b)-u(d)\vert$,
\item $\hat v_{\epsilon}(t)\geq 0$ and it is decreasing for $a\leq t\leq\frac{a+c}{2}$ and increasing for $\frac{a+c}{2}\leq t\leq b$,
\item If $\epsilon\to 0^+$ then $\hat v_{\epsilon}(t)$
converges uniformly to $0$  on every compact subset of $[ a, b)$,
\item $\hat v_{\epsilon}(t)=\vert u(b)-u(d)\vert
{O}\big(e^{\sqrt{\frac{m}{\epsilon}}\hat\chi (t)}\big)$
where $\hat\chi (t)=c-b+a-t$ for $a\leq t<\frac{a+c}{2}$
and $\hat\chi (t)=t-b$ for $\frac{a+c}{2}\leq t\leq b$.
\end{enumerate}
The correction function $v^{\rm (corr)}_{\epsilon}(t)$ is  determined precisely in the
next section.
\end{remark}

\section{The correction function}

 Consider the linear problem
 \begin{equation}\label{def_LDE}
\epsilon y''-my=-2w| v_{\epsilon}(t)|,\quad t\in[ a,b ], \; \epsilon>0
\end{equation}
with the boundary conditions \eqref{def_BC}.
We apply the method of lower and upper solutions in order to
obtain a solution. We define
$$
\alpha_\epsilon(t)=0
$$
and
$$
\beta_\epsilon(t)=\frac{2w}{m}\max\{| v_\epsilon(t)|, t\in[ a,b]\}
=\frac{2w}{m}| v_\epsilon(a)|.
$$
 Obviously, $| v_\epsilon(a)|=\vert u(c)-u(a)\vert
\big(1+{O}\big(e^{\sqrt{\frac{m}{\epsilon}}(a-c)}\big)\big)$
and the constant functions $\alpha, \beta$ are, respectively,
a lower and an upper solution for  \eqref{def_LDE}, \eqref{def_BC}.
Thus, in view of Lemma \ref{mainlemma}, for every $\epsilon>0$
there exists unique solution $v^{\rm (corr)}_{\epsilon}(t)$ of linear
problem \eqref{def_LDE}, \eqref{def_BC} such that
 $$
0\leq v^{\rm (corr)}_{\epsilon}(t)\leq \frac{2w}{m}| u(c)-u(a)|
\Big(1+{O}\big(e^{\sqrt{\frac{m}{\epsilon}}(a-c)}\big)\Big)
$$
 on $[ a,b]$. We compute $v^{\rm (corr)}_{\epsilon}(t)$ exactly:
 $$
v^{\rm (corr)}_{\epsilon}(t)=-\frac{\left(\psi_\epsilon (a)
-\psi_\epsilon(c)\right)}{(u(c)-u(a))}v_\epsilon(t)
+\frac{\left(\psi_\epsilon (d)-\psi_\epsilon(b)\right)}{\vert u(b)
-u(d)\vert}\hat v_\epsilon(t)+\psi_\epsilon(t),
$$
 where
$$
\psi_\epsilon(t)=\frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}t
\left(e^{\sqrt{\frac m{\epsilon}}(b-t)}+e^{\sqrt{\frac m{\epsilon}}
(t-b)}-e^{\sqrt{\frac m{\epsilon}}(d-t)}-e^{\sqrt{\frac m{\epsilon}}
(t-d)}\right).
$$
 Hence
\begin{align*}
&\psi_\epsilon(a)-\psi_\epsilon(c)\\
&= \frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}a
 \left(e^{\sqrt{\frac m{\epsilon}}(b-a)}
 +e^{\sqrt{\frac m{\epsilon}}(a-b)}
 -e^{\sqrt{\frac m{\epsilon}}(d-a)}
 -e^{\sqrt{\frac m{\epsilon}}(a-d)}\right)\\
&\quad - \frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}c
 \left(e^{\sqrt{\frac m{\epsilon}}(b-c)}
 +e^{\sqrt{\frac m{\epsilon}}(c-b)}
 -e^{\sqrt{\frac m{\epsilon}}(d-c)}
 -e^{\sqrt{\frac m{\epsilon}}(c-d)}\right)\\
&= \frac{w\vert u(c)-u(a)\vert }{\sqrt{m\epsilon}}{O}(1),
\end{align*}
\begin{align*}
\psi_\epsilon(d)-\psi_\epsilon(b)
&=\frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}d
 \left(e^{\sqrt{\frac m{\epsilon}}(b-d)}
 +e^{\sqrt{\frac m{\epsilon}}(d-b)}-2\right)\\
&\quad - \frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}b
 \left(2-e^{\sqrt{\frac m{\epsilon}}(d-b)}
 -e^{\sqrt{\frac m{\epsilon}}(b-d)}\right)\\
&= \frac{w\vert u(c)-u(a)\vert }{\sqrt{m\epsilon}}{O}
 \left(e^{\sqrt{\frac{m}{\epsilon}}(a-d)}\right),
\end{align*}
\[
\psi_\epsilon(t)=\frac{w\vert u(c)-u(a)\vert }{\sqrt{m\epsilon}}
{O}\left(e^{\sqrt{\frac{m}{\epsilon}}\chi (t)}\right).
\]
Thus, we obtain
\begin{equation} \label{corrfun}
\begin{aligned}
v^{\rm (corr)}_{\epsilon}(t)
&= \frac{w\vert u(c)-u(a)\vert}{\sqrt{m\epsilon}}\cdot
\Big[-{ O}(1)\frac{v_\epsilon(t)}{(u(c)-u(a))}\\
&\quad + O\left(e^{\sqrt{\frac{m}{\epsilon}}(a-d)}\right)
\frac{\hat v_\epsilon(t)}{\vert u(b)-u(d)\vert}+t{ O}
\left(e^{\sqrt{\frac{m}{\epsilon}}\chi (t)}\right)\Big].
\end{aligned}
\end{equation}
Hence, taking into consideration \eqref{corrfun} and the fact that
$v^{\rm (corr)}_{\epsilon}(a)=v^{\rm (corr)}_{\epsilon}(c)$, the correction function $v^{\rm (corr)}_{\epsilon}$
converges uniformly to $0$ on $[ a,b]$ as $\epsilon\to 0^+$.

\section{Proof of main theorem}

First we analyze the case $ {u(c)-u(a)\geq 0}$.
 Consider the lower solutions
$$
\alpha_\epsilon (t)=u(t)+v_\epsilon(t)-v^{\rm (corr)}_{\epsilon}(t)
-\hat v_\epsilon(t)-\Gamma_\epsilon
$$
and the upper solutions
$$
\beta_\epsilon (t)=u(t)+v_\epsilon(t)
 +\hat v_\epsilon(t)+\Gamma_\epsilon.
$$
Here $\Gamma_\epsilon =\epsilon\Delta /m$, where $\Delta$
is a constant to be defined below,
$\alpha_\epsilon\leq\beta_\epsilon$ on $[ a,b]$ and they satisfy
the correspondent prescribed boundary conditions.

Now we show that $\epsilon \alpha''_\epsilon(t)+k\alpha_\epsilon
(t) \geq f(t,\alpha_\epsilon (t) )$ and
$\epsilon \beta''_\epsilon(t)+k\beta_\epsilon (t) \leq f(t,\beta_\epsilon
(t) )$.
Denoting $h(t,y)=f(t,y)-ky$, by the Taylor we have
\begin{align*}
h(t,\alpha_\epsilon (t))
&= h(t,\alpha_\epsilon (t))-h(t,u(t))\\
&= \frac{\partial h(t,\theta_\epsilon (t))}{\partial y}
(v_\epsilon(t)-v^{\rm (corr)}_{\epsilon}(t) -\hat v_\epsilon(t)-\Gamma_\epsilon),
\end{align*}
where $\alpha_\epsilon (t)<\theta_\epsilon (t)<\beta_\epsilon (t)$
and $(t,\theta_\epsilon (t))\in \mathcal{H}(u)$ for $\epsilon$
sufficiently small.
Hence, from the inequalities
$m\leq\frac{\partial h(t,\theta_\epsilon (t))}{\partial y}\leq m+2w$
in $\mathcal{H}(u)$ we have
\begin{align*}
&\epsilon \alpha''_\epsilon(t)-h(t,\alpha_\epsilon(t))\\
&\geq \epsilon u''(t)+\epsilon v''_\epsilon(t)
  -\epsilon v^{(corr)''}_{\epsilon}(t)
  -\epsilon \hat v''_\epsilon(t)\\
&\quad -(m+2w)v_\epsilon(t)+mv^{\rm (corr)}_{\epsilon}(t)+m\hat v_\epsilon(t)
 +m\Gamma_\epsilon.
\end{align*}
Since $v_\epsilon(t)=| v_\epsilon(t)|$, we have
$-\epsilon v^{(corr)''}_{\epsilon}(t)-2wv_\epsilon(t)+mv^{\rm (corr)}_{\epsilon}(t)=0$ and
using (\eqref {def_LDE}, we obtain
$$
\epsilon \alpha''_\epsilon(t)-h(t,\alpha_\epsilon(t))
\geq \epsilon u''(t)+m\Gamma_\epsilon
\geq-\epsilon\vert u''(t)\vert+\epsilon\Delta.
$$
 For $\beta_\epsilon (t))$ we have the inequality
\begin{align*}
&h(t,\beta_\epsilon(t))-\epsilon \beta''_\epsilon(t)\\
&=\frac{\partial h(t,\tilde\theta_\epsilon (t))}{\partial y}
 (v_\epsilon(t)+\hat v_\epsilon(t)+\Gamma_\epsilon)
 -\epsilon \beta''_\epsilon(t)\\
&=m(v_\epsilon(t)+\hat v_\epsilon(t)+\Gamma_\epsilon)
 -\epsilon(u''(t)+ v''_\epsilon(t)+\hat v''_\epsilon(t))\\
&\geq \epsilon\Delta-\epsilon\vert u''(t)\vert,
\end{align*}
where $\alpha_\epsilon (t)<\tilde\theta_\epsilon (t)
<\beta_\epsilon (t)$ and
$(t,\tilde\theta_\epsilon (t))\in \mathcal{H}(u)$ for
$\epsilon$ sufficiently small.

Let us now analyse the case ${u(c)-u(a)\leq 0}$:
The lower solutions
$$
\alpha_\epsilon (t)=u(t)+v_\epsilon(t)
-\hat v_\epsilon(t)-\Gamma_\epsilon
$$
and the upper solutions
$$
\beta_\epsilon (t)=u(t)+v_\epsilon(t)+v^{\rm (corr)}_{\epsilon}(t)
+\hat v_\epsilon(t)+\Gamma_\epsilon
$$
satisfy
\begin{align*}
\epsilon \alpha''_\epsilon-h(t,\alpha_\epsilon)
&=\epsilon u''+\epsilon v''_\epsilon-\epsilon\hat v''_\epsilon
 -\frac{\partial h}{\partial y}(v_\epsilon-\hat v_\epsilon
 -\Gamma_\epsilon)\\
&=\epsilon u''+\epsilon v''_\epsilon-\epsilon\hat v''_\epsilon
 +\frac{\partial h}{\partial y}(-v_\epsilon
 +\hat v_\epsilon+\Gamma_\epsilon)\\
&\geq \epsilon u''+\epsilon v''_\epsilon-\epsilon\hat v''_\epsilon
 +m(-v_\epsilon+\hat v_\epsilon+\Gamma_\epsilon)\\
&=\epsilon u''+\epsilon\Delta\geq\epsilon\Delta-\epsilon\vert u''\vert
\end{align*}
and
\begin{align*}
h(t,\beta_\epsilon)-\epsilon \beta''_\epsilon
&=\frac{\partial h}{\partial y}
 \Big(v_\epsilon+v^{\rm (corr)}_{\epsilon}
 +\hat v_\epsilon+\Gamma_\epsilon\Big)-\epsilon u''
 -\epsilon v''_\epsilon-\epsilon v^{(corr)''}_{\epsilon}
 -\epsilon\hat v''_\epsilon\\
&\geq (m+2w)v_\epsilon+m\Big(v^{\rm (corr)}_{\epsilon}
 +\hat v_\epsilon+\Gamma_\epsilon\Big)-\epsilon u''
 -\epsilon v''_\epsilon-\epsilon v^{(corr)''}_{\epsilon}
 -\epsilon\hat v''_\epsilon\\
&=-2w| v_\epsilon|+mv^{\rm (corr)}_{\epsilon}
 -\epsilon v^{(corr)''}_{\epsilon}+\epsilon\Delta-\epsilon u''\\
&=\epsilon\Delta-\epsilon u''\geq\epsilon\Delta-\epsilon\vert u''\vert.
\end{align*}
Now, if we choose a constant $\Delta$ such that $\Delta\geq\vert
u''(t)\vert$, $t\in[ a,b]$ then $\epsilon \alpha''_\epsilon(t)
\geq h(t,\alpha_\epsilon (t) )$  and $\epsilon \beta''_\epsilon(t)
\leq h(t,\beta_\epsilon (t) )$ in $[ a,b]$.

The existence of a solution for \eqref{def_DE}, \eqref{def_BC}
satisfying the above inequality follows from Lemma \ref{mainlemma}.
The uniqueness  follows from the fact that the function $h(t,y)$
is increasing in the variable $y$ on the set $\mathcal{H}$.


\begin{remark} \label{rmk4.1} \rm
 Theorem \ref{maintheorem} implies
$y_\epsilon(t)=u(t)+O(\epsilon)$ on every compact subset
of $(a,b)$ and  $\lim_{\epsilon\to0^+ }y_\epsilon(a)=u(c)$,
$\lim_{\epsilon\to 0^+ }y_\epsilon(b)=u(d)$.
The boundary layer effect occurs at the point $a$ ($b$)
whenever $u(a)\neq u(c)$ ($u(b)\neq u(d)$).
\end{remark}

\section{Approximation of the solutions for \eqref{def_DE},
\eqref{def_BC}}

In this section we consider only the case $u(c)-u(a)\leq 0$ as
the other case could be treated analogously. We define
the approximate solution $\tilde y_\epsilon(t)$
of \eqref{def_DE}, \eqref{def_BC} by
\begin{equation}\label{approx}
\tilde y_\epsilon(t)=\frac{1}{2}\left(\alpha_\epsilon (t)
+\beta_\epsilon (t)\right)=u(t)+v_\epsilon(t)+\frac{v^{\rm (corr)}_{\epsilon}(t)}{2}.
\end{equation}
Taking into consideration the conclusions of
Theorem \ref{maintheorem}, in both cases we obtain the following
estimate for the solution $y_\epsilon$ of problem
\eqref{def_DE}, \eqref{def_BC}
\[
| y_\epsilon(t)-\tilde y_\epsilon(t)|
\leq\hat v_\epsilon(t)+\frac{v^{\rm (corr)}_{\epsilon}(t)}{2}+\frac{\epsilon}{m}
\max\{| u''(t)|,t\in[ a,b]\}.
\]

\begin{example} \label{exa5.1} \rm
Consider the nonlinear differential equation
\begin{equation}\label{example}
\epsilon y''+ky=y^2+g(t),\quad k<0,\; g\in C([ a,b])
\end{equation}
subject to the boundary conditions \eqref{def_BC}.
The assumptions of Theorem \ref{maintheorem} are satisfied
if and only if there exists $w>0$ such that
\begin{gather}
\frac14\big(k^2-(w-k)^2\big)
< g(t) <\frac14\big(k^2-(w+k)^2\big)\quad\text{on }[ a,b],\label{c1}\\
| g(c)-g(a)|<\frac18\big(w-k-\zeta(a)\big)
 \big(\zeta(a)+\zeta(c)\big),\label{c2}\\
| g(b)-g(d)|<\frac18\big(w-k-\zeta(b)\big)
 \big(\zeta(b)+\zeta(d)\big),\label{c3}\\
| g(c)-g(a)|<\frac18\big(w+k+\zeta(a)\big)
 \big(\zeta(a)+\zeta(c)\big),\label{c4}\\
| g(b)-g(d)|<\frac18\big(w+k+\zeta(b)\big)
 \big(\zeta(b)+\zeta(d)\big),\label{c5}
\end{gather}
where $\zeta(t)=\sqrt{k^2-4g(t)}$.

As an illustrative example we consider the problem
\eqref{example}, \eqref{def_BC} with $k=-2$, $g(t)=t$, $a=0$,
$b=1/2$, $c=d=1/4$. It is not difficult to verify that the
solution $u(t)=-1+\sqrt{1-t}$ of the reduced problem satisfies
conditions \eqref{c1}--\eqref{c5}  for every
$w\in\big(\frac2{\sqrt2+\sqrt3}+2-\sqrt2,2\big)$. Thus, on the
basis of Theorem \ref{maintheorem}, there exists
$\epsilon_0=\epsilon_0(w)$ such that for every  $\epsilon \in
(0,\epsilon_0]$ the problem $\epsilon y''-2y=y^2+t$,
\eqref{def_BC} has a unique solution which is $O(\epsilon)$ close
to approximate solution \eqref{approx}; i.e.,
\[
\tilde y_\epsilon(t)=-1+\sqrt{1-t}+v_\epsilon(t)+\frac{v^{\rm (corr)}_{\epsilon}(t)}{2}.
\]
\end{example}


\subsection*{Acknowledgments}
This research was supported by grant  1/0068/08 from the
Slovak Grant Agency, Ministry of Education of Slovak Republic.


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