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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 99, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/99\hfil Solvability of a singular problem]
{Solvability of a second-order singular boundary-value problem}

\author[P. S. Kelevedjiev\hfil EJDE-2012/99\hfilneg]
{Petio S. Kelevedjiev}  % in alphabetical order

\dedicatory{Dedicated to Professor Stepan Tersian on his 60th birthday}

\address{Petio S. Kelevedjiev \newline
Technical University of Sliven, Sliven, Bulgaria}
\email{keleved@lycos.com}

\thanks{Submitted February 21, 2012. Published June 12, 2012.}
\subjclass[2000]{34B15, 34B16}
\keywords{Boundary value problem;
 second order differential equation; \hfill\break\indent 
singularity; sign conditions}

\begin{abstract}
 Using the barrier strips technique, we study the  existence of
 solutions to the boundary-value problem
 \begin{gather*}
 x''=f(t,x,x'),\quad t\in(0,1),\\
 x'(0)=A,\quad x(1)=Bx'(1)+C,
 \end{gather*}
 where the scalar function $f$ may be singular at $t=0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

 In this article we give sufficient conditions that  guarantee the
solvability of the boundary-value problem (BVP) 
\begin{equation}
\begin{gathered}
 x''=f(t,x,x'),\quad t\in(0,1),\\
 x'(0)=A,\quad x(1)=Bx'(1)+C,
\end{gathered}\label{e1.1}
\end{equation}
where $f(t,x,p)$ is a scalar function defined for
$(t,x,p)\in(0,1]\times D_x\times D_p$, with
$D_x,D_p\subseteq \mathbb{R}$, and $f$ may be unbounded at $t=0$.

Our work is motivated by Ferguson and Finlagson \cite{f1}, Klokov \cite{k2}
and Vasil'ev and Klokov \cite{v1}. 
The work \cite{f1} is devoted to the solvability of the BVP
\begin{equation}
\begin{gathered}
 x''=-\frac{k}{t}x'+g(t,x),\quad t\in(0,1),\;k=0,1,2,\\
 x'(0)=0,\quad x(1)=Bx'(1)+C,
\end{gathered}\label{e1.2}
\end{equation}
which arises as a model for  processes in chemical reactors.
 The more general problem
\begin{gather*}
 x''=-\frac{k}{t}x'+g(t,x,x'),\quad t\in(0,1),\\
 x'(0)=0,\quad x'(1)=C-Bx(1),
\end{gather*}
where $k\geq0$ and $g\in C([0,1]\times R^2)$ is considered in \cite{k2}.
In \cite{v1}, existence and uniqueness results are given  for the   problem
\begin{equation}
\begin{gathered}
x''=-a(t)x'+g(t,x,x'),\quad t\in(0,1),\\
 x'(0)=0,\quad x(1)=Bx'(1)+C,
\end{gathered}\label{e1.3}
\end{equation}
where  $g\in C([0,1]\times R^2)$, and $a\in C((0,1],[0,+\infty))$
is such that $\lim_{t\to 0^+} a(t)=+\infty$.
The  results obtained in \cite{k2} and \cite{v1} rely on the assumption that
the considered problem has lower and upper solutions.

The literature devoted to the solvability of BVPs for second order ordinary 
differential equations with various singularities is too vast. 
We quote here only Agarwal and O'Regan \cite{a1}, Cabada and Pouso \cite{c1}, 
De Coster and Habets \cite{d1}, Heikkil\"{a} and Lakshmikantham \cite{h1},
 Kiguradze \cite{k1}, O'Regan \cite{o1}, Rach\.{u}nkov\'{a} et al \cite{r1},
 Vasil'ev  and Klokov \cite{v1} for results, methods and references.

In this paper, we establish firstly an existence result for nonsingular 
problem \eqref{e1.1}. It is proved by a combination of the barrier strips
 technique with a global existence theorem which is due to Granas et al [5]. Next,
we apply the obtained existence result to construct a sequence 
$\{x_n\}$ of $C^2[n^{-1},1]$ solutions to the nonsingular problems
\begin{gather*}
 x''=f(t,x,x'),\quad t\in(0,1),\\
 x'(n^{-1})=A,\quad x(1)=Bx'(1)+C,\quad n\in N\setminus\{1\}.
\end{gather*}
Finally, using the Arzela-Ascoli theorem, we obtain a 
$C^1[0,1]\cap C^2(0,1]$ solution to singular problem \eqref{e1.1}
 as the limit of an uniformly convergent subsequence of $\{x_n\}$.

\section{Existence of a global solution}

Let $\Lambda x= x''+p(t)x'+q(t)x$, where the functions $p$ and $q$ are 
continuous on the interval $[a,b]$, and
$$
V_1(x)=a_1x(a)+b_1x'(a),\quad 
V_2(x)=a_2x(b)+b_2x'(b),
$$
where the constants are such that $a_i^2+b_i^2>0$ for $i=1,2$.
 Let $B_0$ be the set of functions satisfying the homogeneous boundary 
conditions $V_i(x)=0$, $i=1,2$, and $C_{B_0}^2[a,b]=C^2[a,b]\cap B_0$.

In this setting, we consider the boundary-value problem
\begin{equation}
\begin{gathered}
\Lambda x= f(t,x,x'),\quad t\in(a,b),\\
 V_1(x)=r_1,\quad V_2(x)=r_2,
\end{gathered}\label{e2.1}
\end{equation}
where $f:[a,b]\times D_x\times D_p\to R$, with
$D_x,D_p\subseteq \mathbb{R}$, and $r_i\in \mathbb{R}$, $i=1,2$.

The proof of our theorem guaranteeing the existence of 
nonsingular problems is based on the following global 
existence theorem which  is a slight modification of a well-known result.

\begin{theorem}[{\cite[Theorem 5.1]{g1}}] \label{thm2.1} 
 Assume that:
\begin{itemize}
\item[(i)]  The map $\Lambda:C_{B_0}^2[a,b]\to C[a,b]$ is one-to-one.
\item[(ii)]  Each solution $x\in C^2[a,b]$ to the family of problems
\begin{gather*}
\Lambda x=\lambda f(t,x,x'),\quad t\in(a,b),\\
V_1(x)=r_1,\;V_2(x)=r_2,
\end{gather*}
with $\lambda\in[0,1]$, satisfies the bounds
$$
m_i\leq x^{(i)}(t)\leq M_i,\quad i=0,1,2,\text{ for } t\in[a,b],
$$
where the constants $-\infty<m_i,M_i<\infty$, $i=0,1,2$, are independent of 
$\lambda$ and $x$.

\item[(iii)] There is a sufficiently small $\tau>0$ such that
$$
[m_0-\tau,M_0+\tau]\subseteq D_x,\quad [m_1-\tau,M_1+\tau]\subseteq D_p
$$
and $f(t,x,p)$ is continuous for 
$(t,x,p)\in[a,b]\times[m_0-\tau,M_0+\tau]\times[m_1-\tau,M_1+\tau]$.
\end{itemize}
Then \eqref{e2.1} has at least one solution in $C^2[a,b]$.
\end{theorem}


\section{Nonsingular problem}

Consider the nonsingular problem
\begin{equation}
\begin{gathered}
 x''= f(t,x,x'),\quad t\in(a,b), \\
 x'(a)=A,\;x(b)=B x'(b)+C,
\end{gathered}\label{e3.1}
\end{equation}
where $f:[a,b]\times D_x\times D_p\to \mathbb{R}$,
$D_x, D_p\subseteq {R}$.

In the next lemma, we use the assumption
\begin{itemize}
\item[(R1)]   There are constants $L_i,F_i,i=1,2$, and a sufficiently small 
$\tau>0$ such that
\begin{gather*}
L_2-\tau\geq L_1\geq A\geq F_1\geq F_2+\tau,\\
[m_0-\tau,M_0+\tau]\subseteq D_x,\quad
[F_2,L_2]\subseteq D_p,
\end{gather*}
where
\begin{gather*}
m_0=-\max\{|L_1|, |F_1|\}(|B|+b-a)+C,\\ 
M_0=\max\{|L_1|, |F_1|\}(|B|+b-a)+C,\\
f(t,x,p)\in C\bigl([a,b]\times[m_0-\tau,M_0+\tau]\times[F_1-\tau,L_1+\tau]\bigr),\\
f(t,x,p)\leq0\quad \text{for }(t,x,p)\in[a,b]\times D_x\times[L_1,L_2],\\
f(t,x,p)\geq0\quad \text{for }(t,x,p)\in[a,b]\times D_x\times[F_2,F_1].
\end{gather*}
\end{itemize}
Our first result shows that the strips $[a,b]\times[L_1,L_2]$ and 
$[a,b]\times[F_2,F_1]$ are barriers to the values of all $C^2[a,b]$ solutions 
to the family
\begin{equation}
\begin{gathered}
\quad x''=\lambda f(t,x,x'),\quad t\in(a,b),\\
x'(a)=A,\;x(b)=B x'(b)+C,
\end{gathered}\label{e3.2}
\end{equation}
where $\lambda\in[0,1]$.

\begin{lemma} \label{lem3.1} 
 Let {\rm (R1)} hold and let $x\in C^2[a,b]$ be a solution to family 
\eqref{e3.2} with $\lambda\in[0,1]$. 
Then
$$
m_0\leq x(t)\leq M_0,\quad 
F_1\leq x'(t)\leq L_1, \quad 
m_2\leq x''(t)\leq M_2
$$
for $t\in[a,b]$, where $m_2=\min f(t,x,p)$ and $M_2=\max f(t,x,p)$ on
 $[a,b]\times[m_0,M_0]\times[F_1,L_1]$.
\end{lemma}

\begin{proof}
Suppose that the set
$$
S_-=\{t\in[a,b]:L_1<x'(t)\leq L_2\}
$$
is not empty. Then  $x'(a)=A\leq L_1$ and $x'\in C[a,b]$ imply that 
there exists a $\gamma\in S_-$ such that
$x''(\gamma)>0$.
Since $x(t)$ is a solution of the differential equation, 
$(t,x(t),x'(t))\in(a,b)\times D_x\times D_p$. 
In particular, we have 
$(\gamma,x(\gamma),x'(\gamma))\in S_-\times D_x\times(L_1,L_2]$. 
So, we can use (R1) to obtain
$$
x''(\gamma)=\lambda f(\gamma,x(\gamma),x'(\gamma))\leq0,
$$
a contradiction. Thus, $S_-$ is empty which means
$$
x'(t)\leq L_1\quad text{for } t\in[a,b].
$$
Further, assuming that the set
$$
S_+=\{t\in[a,b]:F_2\leq x'(t)<F_1\}
$$
is not empty and arguing as in the above part of the proof, we obtain
$$
F_1\leq x'(t)\quad \text{for } t\in[a,b].
$$
Now, by the mean value theorem, for each $t\in[a,b)$ there exists
 $\xi\in(t,b)$ such that $x(b)-x(t)=x'(\xi)(b-t)$,
from where, using the proved bounds for $x'(t)$, we obtain
$$
m_0\leq x(t)\leq M_0\quad \text{for } t\in[a,b].
$$
Finally, the bounds for $x''(t)$ are an elementary consequence of the 
continuity of $f(t,x,p)$ on the compact set 
$[a,b]\times[m_0,M_0]\times[F_1,L_1]$.
\end{proof}

We are ready to formulate an  existence result.

\begin{theorem} \label{thm3.2} 
 Let {\rm (R1)} hold. Then nonsingular problem \eqref{e3.1} has at 
least one solution in $C^2[a,b]$.
\end{theorem}

\begin{proof}
 A combination of the a priori bounds of Lemma \ref{lem3.1}
 with Theorem \ref{thm2.1} 
gives the assertion at once. Notice only that (i) of Theorem \ref{thm2.1} 
follows from the fact that for each $y\in C[a,b]$ the homogeneous BVP
$$
x''=y,\quad x'(a)=0,\quad x(b)-Bx'(b)=0
$$
has a unique solution. 
\end{proof}

\section{Singular problem}

Now, we turn our attention to problem \eqref{e1.1} by considering the  case
\begin{equation}
\begin{gathered}
f(t,x,p) \text{ is defined for } (t,x,p)\in([0,1]\times D_x\times D_p)\setminus S,
\text{ where }\\
D_x,D_p\subseteq \mathbb{R}\text{  and
$S=\{0\}\times X\times P $ for some sets
$X \subseteq D_x$ and $P\subseteq D_p$}
\end{gathered}\label{e4.1}
\end{equation}
which allows $f(t,x,p)$ to be unbounded at $t=0$ if
 $(x,p)\in X\times P$.

Now, assume that
\begin{itemize}
\item[(S1)]   There are constants $L_i,F_i$, $i=1,2$, and a sufficiently small 
$\tau>0$  such that
\begin{gather*}
 L_2-\tau\geq L_1\geq A\geq F_1\geq F_2+\tau,\\
 [m_0-\tau,M_0+\tau]\subseteq D_x,\quad [F_2,L_2]\subseteq D_p,
\end{gather*}
where 
\begin{gather*}
m_0=-K(|B|+1)+C,\;M_0=K(|B|+1)+C,\\
K=\max\{|L_1|, |F_1|\},\\
f(t,x,p)\in C\bigl((0,1]\times[m_0-\tau,M_0+\tau]\times[F_1-\tau,L_1+\tau]\bigr),\\
f(t,x,p)\leq0\quad\text{for }(t,x,p)\in(0,1]\times D_x\times[L_1,L_2],\\
f(t,x,p)\geq0\quad \text{for }(t,x,p)\in(0,1]\times D_x\times[F_2,F_1].
\end{gather*}
\item[(S2)]  The functions $f_t(t,x,p)$, $f_x(t,x,p)$ and $f_p(t,x,p)$ are 
continuous for $(t,x,p)$ in $(0,1]\times [m_0,M_0]\times[F_1,L_1]$, where
 $m_0,M_0,F_1$ and $L_1$ are as in  (S1).
\end{itemize}

We are now in position to state the main existence theorem of this paper.

\begin{theorem} \label{thm4.1} 
 Assume \eqref{e4.1} is satisfied. Assume also {\rm (S1)} and {\rm (S2)} hold. 
Then singular BVP \eqref{e1.1} has at least one $C^1[0,1]\cap C^2(0,1]$ solution.
\end{theorem}

\begin{proof} 
It is easy to see that for each fixed $n\in N\setminus\{1\}$,
 (R1) holds for the corresponding nonsingular BVP
\begin{equation}
\begin{gathered}
 x''=f(t,x,x'),\\
x'(n^{-1})=A,\quad x(1)=B x'(1)+C.
\end{gathered}\label{e4.2}
\end{equation}
Consequently, for each  $n\in N\setminus\{1\}$ we can apply
Theorem \ref{thm3.2} to construct a sequence $\{x_n\}$ of $C^2[n^{-1},1]$ solutions
to family \eqref{e4.2}.

Further, we introduce a numerical sequence 
$\{\theta_i\}$, $i\in N$, such that $\theta_i\in(0,1)$,
$\theta_{i+1}<\theta_i$ for $i\in N$ and $\lim_{i\to\infty}\theta_i=0$.

In view of Lemma \ref{lem3.1}, for each $n\in N_1$ we have the bounds
\begin{gather}
m_0\leq x_n(t)\leq M_0\quad\text{for } t\in[\theta_1,1],\label{e4.3}\\
F_1\leq x'_n(t)\leq L_1\quad\text{for } t\in[\theta_1,1],\label{e4.4}
\end{gather}
independent of $n$. Also, the continuity of $f(t,x,p)$ on 
$[\theta_1,1]\times[m_0,M_0]\times[F_1,L_1]$ yields the bound
\begin{equation}
m_{\theta_1}\leq x''_n(t)\leq M_{\theta_1}\quad \text{for } t\in[\theta_1,1],
\label{e4.5}
\end{equation}
independent of $n$. Thus, we can use  (S2) to conclude that $x'''_n(t)$ exists,
 $x'''_n\in C[\theta_1,1]$ and
$$
x'''_n(t)=f_t(t,x_n(t),x'_n(t))+f_x(t,x_n(t),x'_n(t))x'_n(t)
+f_p(t,x_n(t),x'_n(t))x''_n(t),
$$
from where it follows that there is a constant $\overline M_{\theta_1}$,
independent of $n$, such that
\begin{equation}
|x'''_n(t)|\leq \overline M_{\theta_1}\quad \text{for }
 t\in[\theta_1,1]\text{ and all }n\in N_1.\label{e4.6}
\end{equation}

Bounds \eqref{e4.3}-\eqref{e4.6} allow us to apply the Arzela-Ascoli theorem 
on the sequence $\{x_n\}$ to conclude that there are a subsequence 
$\{x_{1,n_{k}}\}$, $k\in N$, $n_{k}\in N_1$, and a function 
$x_{\theta_1}\in C^2[\theta_1,1]$ such that
 $\{x_{1,n_{k}}\}$, $\{x'_{1,n_{k}}\}$ and $\{x''_{1,n_{k}}\}$ converge uniformly on
$[\theta_1,1]$ to $x_{\theta_1}$, $x'_{\theta_1}$ and $x''_{\theta_1}$, respectively.

As a solution of \eqref{e4.2}, each function $x_{1,n_{k}},\,n_{k}\in N_1$, 
is such that
\begin{gather*}
x''_{1,n_{k}}(t)=f(t,x_{1,n_{k}}(t),x'_{1,n_{k}}(t))\quad\text{for }
 t\in[\theta_1,1),\\
x_{1,n_{k}}(1)=Bx'_{1,n_{k}}(1)+C,
\end{gather*}
from where, keeping in mind \eqref{e4.3}, \eqref{e4.4} and the continuity 
of $f$ on the compact set $[\theta_1,1]\times[m_0,M_0]\times[F_1,L_1]$, we obtain
\begin{gather*}
x''_{\theta_1}(t)=f(t,x_{\theta_1}(t),x'_{\theta_1}(t))\quad\text{for }
 t\in[\theta_1,1),\\
x_{\theta_1}(1)=Bx'_{\theta_1}(1)+C.
\end{gather*}
Now consider the sequence $\{x_{1,n_{k}}\}$ on the interval $[\theta_2,1]$.
Arguing as above, we extract a subsequence 
$\{x_{2,n_{k}}\}$, $n_k\in N_2=\{n\in N:n^{-1}<\theta_2\}$, of
 $\{x_{1,n_{k}}\}$ converging uniformly on the new interval $[\theta_2,1]$ 
to a new function $x_{\theta_2}(t)$, which is a $C^2[\theta_2,1]$ solution
 to the differential equation $x''=f(t,x,x')$ on the interval $[\theta_2,1)$ 
with the property $x_{\theta_2}(1)=Bx'_{\theta_2}(1)+C$ and
$$
x_{\theta_2}(t)=x_{\theta_1}(t)\quad\text{for } t\in[\theta_1,1].
$$
Continuing this process, we establish that for each $i\in N$ there is a 
function $x_{\theta_i}(t)$ which is a $C^2[\theta_i,1]$ solution of 
$x''=f(t,x,x')$ on the interval $[\theta_i,1)$, 
$x_{\theta_i}(1)=Bx'_{\theta_i}(1)+C$ and
$$
x_{\theta_{i+1}}(t)=x_{\theta_i}(t)\quad\text{for } t\in[\theta_i,1].
$$
Moreover, for each $i\in N$ there is a subsequence 
$\{x_{i,n_{k}}\}$, $n_k\in N_i=\{n\in N:n^{-1}<\theta_i\}$, such that
\begin{equation}
\|x_{i,n_k}-x_{\theta_i}\|_2\to0\quad\text{on the interval }
[\theta_i,1],\label{e4.7}
\end{equation}
where 
$$
\|x\|_2=\max\big\{
 \max_{t\in[\theta_i,1]}|x(t)|, 
 \max_{t\in[\theta_i,1]}|x'(t)|,
 \max_{t\in[\theta_i,1]}|x''(t)| \big\}
$$
is the norm in the Banach space $C^2 [\theta_i,1]$.

The existence of the sequence $\{x_{\theta_i}\}$ allows us to conclude that 
there is a function $x_0(t)$, which is a $C^2(0,1]$ solution of 
 $x''=f(t,x,x')$ on the interval $(0,1)$, $x_0(1)=Bx'_0(1)+C$,
\begin{equation}
x_0(t)=x_{\theta_i}(t)\quad \text{for } t\in[\theta_i,1].\label{e4.8}
\end{equation}
In what follows we will show in addition that
\begin{equation}
\lim_{t\to0^+}x'_0(t)=A.\label{e4.9}
\end{equation}
Reasoning by contradiction, assume that there are sufficiently small
$\varepsilon>0$ and $\delta_0>0$ such that
\begin{equation}
x'_0(t)\notin(A-\varepsilon,A+\varepsilon)\quad\text{for }
 t\in(0,\delta_0).\label{e4.10}
\end{equation}

Now, from $x'_n\in C[n^{-1},1]$ and $x'_n(n^{-1})=A$ it follows that for each 
sufficiently large $n$ and the chosen $\varepsilon$  there exists a sufficiently 
small $\delta_n>0$, depending on $n$ and $\varepsilon$, such that
 $(n^{-1},\delta_n)\subset(0,\delta_0)$
and
$$
x'_n(t)\in(A-\varepsilon/2,A+\varepsilon/2)\quad\text{for } t\in(n^{-1},\delta_n).
$$
Besides, for each sufficiently large $n$ there exists any $i\in N$ such that 
$\theta_i>n^{-1}$ and
$$
[\theta_i, \theta_{i-1}]\subset(n^{-1},\delta_n)\subset(0,\delta_0);
$$
the assumption that the interval $[\theta_i, \theta_{i-1}]$ does not exist
 contradicts to the fact that $t=0$ is an accumulation point of the sequence 
$\{\theta_i\}$. In summary, for each sufficiently large $n$ there exists  
$i\in N$ such that
\begin{equation}
x'_n(t)\in(A-\varepsilon/2,A+\varepsilon/2)\quad\text{for }
 t\in[\theta_i, \theta_{i-1}]\subset(0,\delta_0).\label{e4.11}
\end{equation}
But, for each sufficiently large $n$ and its $i$ from \eqref{e4.7}
 ($\theta_i>n^{-1}$ means that $n\equiv n_k\in N_i$ for any
$k\in N$), \eqref{e4.8} and \eqref{e4.10} we obtain
$$
x'_n(t)\notin(A-\varepsilon,A+\varepsilon)\quad\text{for }
t\in[\theta_i,\theta_{i-1}],
$$
which contradicts \eqref{e4.11}. Thus, \eqref{e4.9} is true.

Now, we introduce a function $x(t)$ such that
$$
x(t)=x_0(t)\quad\text{for } t\in(0,1]
$$
and
$$
x'(t)=\begin{cases}
x'_0(t) &\text{for } t\in(0,1]\\
A       &\text{for } t=0\,.
\end{cases}
$$
Because of \eqref{e4.9}, $x'(t)$ is continuous on $[0,1]$, which means
 $x(t)$ is also continuous on $[0,1]$, from where it follows that $x(t)$ 
is a $C^1[0,1]\cap C^2(0,1]$ solution to singular BVP \eqref{e1.1}.
\end{proof}

As an application of the above theorem, we will establish existence results 
for singular problems \eqref{e1.2} and \eqref{e1.3}. 
The first result concerns \eqref{e1.2}.

\begin{corollary} \label{coro4.2} Assume that:
\begin{itemize}
\item[(i)] $g(t,x)$ is bounded; i.e., there is a constant $M>0$ such that
$$
|g(t,x)|\leq M\quad \text{for }(t,x)\in[0,1]\times D_x.
$$

\item[(ii)]  There is a $\tau>0$ such that $[m_0-\tau,M_0+\tau]\subseteq D_x$ and
$$
f(t,x)\in C([0,1]\times[m_0-\tau,M_0+\tau]),
$$
where $m_0=-M(|B|+1)/k+C$ and $M_0=M(|B|+1)/k+C$.

\item[(iii)]  $g_t,g_x$ are continuous for $(t,x)\in(0,1]\times[-M/k-\tau,M/k+\tau]$.
\end{itemize}
Then \eqref{e1.2} with $k=1,2$  has at least one solution in
 $C^1[0,1]\cap C^2(0,1]$.
\end{corollary}

\begin{proof} 
It is easy to check that (S1) and  (S2) hold for $F_1=-M/k$,
$F_2=-M/k-1$, $L_1=M/k$ and $L_2=M/k+1$ and $\tau=0.5$. 
So we can apply Theorem \ref{thm4.1} to conclude that the assertion is true.
\end{proof}

The following two results concern problem \eqref{e1.3}.

\begin{corollary} \label{coro4.3}
 Assume $g(t,x,p)$ satisfies {\rm (S1)} and {\rm (S2)}. 
Then \eqref{e1.3} has at least one solution in $C^1[0,1]\cap C^2(0,1]$.
\end{corollary}

\begin{proof} 
Since $a(t)\geq0$ for $t\in(0,1]$, the function $f(t,x,p)=-a(t)p+g(t,x,p)$ 
satisfies {\rm (S1)} and {\rm (S2)}  for the same constants 
$L_i,F_i,i=1,2$, for which $g$ satisfies them. Thus, the assertion 
follows from Theorem \ref{thm4.1}.
\end{proof}

\begin{corollary} \label{coro4.4} 
 Assume that
\begin{itemize}
\item[(i)] $g(t,x,p)$ is bounded; i.e., there is a constant $M>0$ such that
$$
|g(t,x,p)|\leq M\quad \text{for }(t,x,p)\in[0,1]\times D_x\times D_p
$$
and $a(t)\geq h>0$ for $t\in(0,1]$.

\item[(ii)]  There is a $\tau>0$ such that
\begin{gather*}
[m_0-\tau,M_0+\tau]\subseteq D_x,\quad [-M/h-\tau,M/h+\tau]\subseteq D_p,\\
g(t,x,p)\in C([0,1]\times[m_0-\tau,M_0+\tau]\times[-M/h-\tau,M/h+\tau]),
\end{gather*}
where $m_0=-M(|B|+1)/h+C$ and $M_0=M(|B|+1)/h+C$.

\item[(iii)] $g_t(t,x,p),\;g_x(t,x,p)$ and $g_p(t,x,p)$ are continuous for 
$(t,x,p)\in[0,1]\times[m_0-\tau,M_0+\tau]\times[-M/h-\tau,M/h+\tau]$.
\end{itemize}
Then \eqref{e1.3} has at least one solution in $C^1[0,1]\cap C^2(0,1]$.
\end{corollary}

\begin{proof} We can choose, for example, $F_1=-M/h$,
$F_2=-M/h-\tau,\,L_1=M/h$ and $L_2=M/h+\tau$ to see that  (S1) and  (S2)
 hold and so the assertion is true by Theorem \ref{thm4.1}.
\end{proof}

\section{Examples}

As a first example, we have the boundary-value problem
\begin{gather*}
x''=x'ln(t)+\frac{(x'+3)(x'-4)}{\sqrt{100-x^2}},\quad t\in(0,1),\\
x'(0)=2,\quad x(1)=x'(1),
\end{gather*}
which is solvable in $C^1[0,1]\cap C^2(0,1]$, by Theorem \ref{thm4.1}, 
since  (S1) and  (S2) hold for $F_2=-4$, $F_1=-3$, $L_1=3$, $L_2=4$ and 
$\tau=0.1$.

As a second example, we have the  boundary-value problem
\begin{gather*}
x''=-2t^{-1}x'+\sqrt{25-x^2}+1,\quad t\in(0,1),\\
x'(0)=0,\quad x(1)=0.5x'(1)-1,
\end{gather*}
which has a $C^1[0,1]\cap C^2(0,1]$ solution by Corollary 4.2.

As a third example, we have  boundary-value problem
\begin{gather*}
x''=-15t^{-1}x'+\sqrt{25-x^2}\sqrt{9-x'^2}+\cos{5t},\quad t\in(0,1),\\
x'(0)=0,\;x(1)=-2x'(1)+1,
\end{gather*}
which has a $C^1[0,1]\cap C^2(0,1]$ solution by Corollary 4.4.


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\end{document}