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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 273, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/273\hfil Singular second-order IVP]
{Solvability of singular second-order initial-value problems}

\author[P. Kelevedjiev \hfil EJDE-2016/273\hfilneg]
{Petio Kelevedjiev}

\address{Petio Kelevedjiev \newline
Technical University of Sofia,
Branch Sliven, Bulgaria}
\email{keleved@mailcity.com}

\thanks{Submitted April 8, 2016. Published October 12, 2016.}
\subjclass[2010]{34A12, 34A36}
\keywords{ Initial value problem; second order differential equation;
 singularity; 
\hfill\break\indent existence; barrier conditions}

\begin{abstract}
 This article concerns the solvability of the initial-value problem
 $x''=f(t,x,x')$, $x(0)=A$, $x'(0)=B$, where the scalar function $f$
 may be unbounded as $t\to 0$. Using barrier strip type arguments,
 we establish the existence of monotone and/or positive solutions
 in $C^1[0,T]\cap C^2(0,T]$.
\end{abstract}

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

\section{Introduction}

 In this article we study the solvability of the initial value problem (IVP)
\begin{equation}
\begin{gathered}
  x''=f(t,x,x'),\\
x(0)=A,\quad x'(0)=B,
\end{gathered}\label{e1.1}
\end{equation}
where the scalar function $f(t,x,p)$ is defined for
$(t,x,p)\in D_t\times D_x\times D_p$, and $D_t,D_x,D_p\subseteq R$,
but there may be sets $X\subseteq D_x$ and $P\subseteq D_p$ such that
$f$ is unbounded as $t\to 0$ and $(x,p)\in X\times P$.

The solvability of various nonsingular and singular second order IVPs has 
been studied by  Aslanov \cite{a3}, Agarwal and O'Regan \cite{a1,a2}, 
 Bobisud and  O'Regan \cite{b1},  Bobisud and  Lee \cite{b2},  
Cabada et al. \cite{c1,c2,c3},   Cid \cite{c4},  Maagli and  Masmoudi\cite{m1},
  Rach\.{u}nkov\'{a} and   Tome\v{c}ek \cite{r1,r2,r3},  
Yang \cite{y1,y2} and Zhao \cite{z1}. 
Yang \cite{y1,y2}, for example, established the solvability in $C^1[0,1]$ and 
$C[0,1]\times C^2(0,1)$ in the case $A=B=0$. 
In these works the function $f(t,x,p)\in C((0,1),(0,\infty)^2)$ is allowed 
to be singular at ${t=0,t=1}, x=0$ or $p=0$ and is such that
$$
0<f(t,x,p)\leq k(t)F(x)G(p)\quad \text{for }(t,x,p)\in(0,1)\times(0,\infty)^2,
$$
where $k,F$ and $G$ are suitable functions.

Here we present sufficient conditions guaranteeing monotone and/or 
positive solutions to \eqref{e1.1} in $C^1[0,T]\times C^2(0,T]$.
They are established by adapting ideas from  Kelevedjiev and  Popivanov \cite{k1} 
and  Kelevedjiev et al. \cite{k2} (sse also Kelevedjiev \cite{k3}),
 where \eqref{e1.1} may be singular at $x=A$ and/or $p=B$. 
The results in these works rely on 
a combination of a barrier type condition with the assumption that there 
is a constant $k<0$ such that
\begin{equation}
f(t,x,p)\leq k \label{e1.2}
\end{equation}
on a suitable bounded subset of the domain of $f$. It turned out,
 however, that \eqref{e1.2} is not necessary when  \eqref{e1.1}
is singular only at $t=0$, that is why we pay a special attention to this case.

In our considerations we use two results from \cite{k2} for the nonsingular problem
\begin{equation}
\begin{gathered}
x''=f(t,x,x'),\\
x(a)=A,\;x'(a)=B,
\end{gathered} \label{e1.3}
\end{equation}
where $f:D_t\times D_x\times D_p\to R,\;D_t,D_x,D_p\subseteq R$.
They are based on the  assumption
\begin{itemize}
\item[(A1)]   There are constants $T>a$, $m_1,\overline{m}_1,M_1,\overline{M}_1$
and a sufficiently small $\tau>0$ such that
\begin{gather*}
\overline{M}_1-\tau\geq M_1\geq B\geq m_1\geq \overline{m}_1+\tau, \\
[a,T]\subseteq D_t,[m_0-\tau,M_0+\tau]\subseteq D_x,\quad
 [\overline{m}_1,\overline{M}_1]\subseteq D_p,
\end{gather*}
where $M_0=\max\{|m_1|, |M_1|\}(T-a)+|A|$, and $m_0=-M_0$,
\begin{gather*}
f(t,x,p)\in C\bigl([a,T]\times[m_0-\tau,M_0+\tau]\times[m_1-\tau,M_1+\tau]\bigr),\\
f(t,x,p)\leq0\quad \text{for } (t,x,p)\in[a,T]\times D_x\times[M_1,\overline{M}_1],\\
f(t,x,p)\geq0\quad \text{for }(t,x,p)\in[a,T]\times D_{M_0}\times[\overline{m}_1,m_1],
\end{gather*}
where $D_{M_0}=D_x\cap(-\infty,M_0]$.
\end{itemize}

So, we need the following result.

\begin{lemma}[\cite{k2}] \label{lem1.1}
Let {\rm (A1)} hold and $x\in C^2[a,T]$ be a solution to \eqref{e1.3}.
 Then
$$
m_0\leq x(t)\leq M_0,\quad
m_1\leq x'(t)\leq M_1,\quad
m_2\leq x''(t)\leq M_2\quad \text{for }t\in[a,T],
$$
where $m_2=\min f(t,x,p)$ and $M_2=\max f(t,x,p)$ for
$(t,x,p)\in[a,T]\times[m_0,M_0]\times[m_1,M_1]$.
\end{lemma}

This lemma was used in the proof of the following theorem.

\begin{theorem}[\cite{k2}] \label{thm1.1}
Let {\rm (A1)} hold. Then nonsingular IVP \eqref{e1.3}  has at least
one solution in $C^2[a,T]$.
\end{theorem}


\section{Existence results}

Returning our attention to singular problem \eqref{e1.1}, we assume that
\begin{itemize}
\item[(A2)]   There are constants $T>0$, $m_1,\overline{m}_1,M_1,\overline{M}_1$
and a sufficiently small $\tau>0$ such that
\begin{gather*}
\overline{M}_1-\tau\geq M_1\geq B\geq m_1\geq \overline{m}_1+\tau, \\
(0,T]\subseteq D_t,[\tilde{m}_0-\tau,\tilde{M}_0+\tau]\subseteq D_x,\quad
 [\overline{m}_1,\overline{M}_1]\subseteq D_p,
\end{gather*}
where $\tilde{M}_0=\max\{|m_1|, |M_1|\}T+|A|$, and $\tilde{m}_0=-\tilde{M}_0$,
\begin{gather}
f(t,x,p)\in C\bigl((0,T]\times[\tilde{m}_0-\tau,\tilde{M}_0+\tau]
\times[m_1-\tau,M_1+\tau]\bigr),\label{e2.1} \\
f(t,x,p)\leq0\quad\text{for }(t,x,p)\in(0,T]\times D_x\times[M_1,\overline{M}_1],
\nonumber \\
f(t,x,p)\geq0\quad \text{for }(t,x,p)\in(0,T]\times D_{\tilde{M}_0}
\times[\overline{m}_1,m_1],\nonumber
\end{gather}
where $D_{\tilde{M}_0}=D_x\cap(-\infty,\tilde{M}_0]$.
\end{itemize}
We are now in a position to state our first existence theorem.

\begin{theorem} \label{thm2.1}
Let {\rm (A2)} hold.  Then  \eqref{e1.1}  has at least one solution in
 $C^1[0,T]\cap C^2(0,T]$ such that
\begin{gather*}
m_1t+A\leq x(t)\leq M_1t+A\quad\text{for }t\in[0,T], \\
m_1\leq x'(t)\leq M_1\quad\text{for }t\in[0,T].
\end{gather*}
\end{theorem}

\begin{proof}
 We will do the proof in several steps considering the family of 
nonsingular problems
\begin{equation}
\begin{gathered}
x''=f(t,x,x'),\\
x(n^{-1})=A,\quad x'(n^{-1})=B,
\end{gathered}\label{e2.2}
\end{equation}
where $n\in N_T=\{n\in N:n^{-1}<T\}$.
\smallskip

\noindent\textbf{Step 1}  Construction of a sequence $\{x_n\}$ of
 $C^2[n^{-1},T]$-solutions to \eqref{e2.2}. 
It is not hard to check that each problem of \eqref{e2.2} satisfies  (A1)
 for $a=n^{-1}$, $M_0=\max\{|m_1|, |M_1|\}(T-n^{-1})+|A|<\tilde{M}_0$, and 
$m_0=-M_0$. Thus, according to Theorem \ref{thm1.1}, \eqref{e2.2} has a solution
$$
x_n\in C^2[n^{-1},T]\quad \text{for each }n\in N_T.
$$
In addition, for each $n\in N_T$ Lemma \ref{lem1.1} guarantees the bounds
\begin{gather*}
\tilde{m}_0<m_0\leq x_n(t)\leq M_0<\tilde{M}_0\quad\text{for }t\in[n^{-1},T],\\
m_1\leq x'_n(t)\leq M_1\quad \text{for }t\in[n^{-1},T].
\end{gather*}
\smallskip

\noindent\textbf{Step 2}  Construction of a $C^2(0,T]$-solution to the differential 
equation. Now, we introduce a numerical sequence 
$\{\theta_i\},i\in N$, having the properties
$$
\theta_i\in(0,T),\quad \theta_{i+1}<\theta_i\quad \text{for } 
 i\in N\text{ and }\lim_{t\to\infty}\theta_i=0,
$$
and consider the sequence $\{x_n\}$ of $C^2[n^{-1},T]$-solutions of 
family \eqref{e2.2} only for $n\in N_1=\{n\in N_T:n^{-1}<\theta_1\}$.
 Clearly, the bounds
\begin{gather}
\tilde{m}_0< x_n(t)<\tilde{M}_0\quad \text{for }t\in[\theta_1,T],\label{e2.3}\\
m_1\leq x'_n(t)\leq M_1\quad\text{for }t\in[\theta_1,T],\label{e2.4}
\end{gather}
independent of $n\in N_1$. In view of \eqref{e2.1}, $f(t,x,p)$ is continuous 
on the set $[\theta_1,T]\times[\tilde{m}_0,\tilde{M}_0]\times[m_1,M_1]$
 and so there is a constant $M_{1,2}$, independent on $n$, such that
$$
|x''_n(t)|\leq M_{1,2}\quad\text{for }t\in[\theta_1,T].
$$
The obtained bounds for $x_n(t),x'_n(t)$ and $x''_n(t)$ on the interval 
$[\theta_1,T]$ allows 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 
$x_{\theta_1}\in C^2[\theta_1,T]$ such that
$$
\|x_{1,n_{k}}-x_{\theta_1}\|_1\to 0\quad\text{on }t\in[\theta_1,T];
$$
that is, the sequences $\{x_{1,n_{k}}\}$ and $\{x'_{1,n_{k}}\}$ converge 
uniformly on $[\theta_1,T]$ to $x_{\theta_1}$ and $x'_{\theta_1}$, respectively.
Since \eqref{e2.3} and \eqref{e2.4} are valid in particular for the elements
 of $\{x_{1,n_{k}}\}$ and $\{x'_{1,n_{k}}\}$, letting $k\to\infty$, we obtain
\begin{gather}
\tilde{m}_0\leq x_{\theta_1}(t)\leq\tilde{M}_0\quad\text{for }t\in[\theta_1,T],
\label{e2.5}\\
m_1\leq x'_{\theta_1}(t)\leq M_1\quad\text{for }t\in[\theta_1,T].\label{e2.6}
\end{gather}
On the other hand, on using that the functions $x_{1,n_{k}}(t),n_{k}\in N_1$, 
are solutions of the differential equation \eqref{e2.2}, we have
$$
x'_{1,n_{k}}(t)=x'_{1,n_{k}}(\theta_1)
+\int^t_{\theta_1}f(s,x_{1,n_{k}}(s),x'_{1,n_{k}}(s))ds,\;t\in(\theta_1,T].
$$
Next, we need to show that the sequence $\{f(s,x_{1,n_{k}}(s),x'_{1,n_{k}}(s))\}$, 
${n_{k}\in N_1,}$ converges uniformly on the interval $[\theta_1,T]$. 
To this aim we observe at first that since $f(t,x,p)$ is uniformly continuous 
on the compact set $[\theta_1,T]\times[\tilde{m}_0,\tilde{M}_0]\times[m_1,M_1]$, 
for each $\varepsilon>0$ there is a $\delta>0$ such that
\begin{equation}
|f(t_0,x_0,p_0)-f(t_1,x_1,p_1)|<\varepsilon\label{e2.7}
\end{equation}
if $(t_0,x_0,p_0),(t_1,x_1,p_1)\in[\theta_1,T]
\times[\tilde{m}_0,\tilde{M}_0]\times[m_1,M_1]$ and
$$
\sqrt{(t_0-t_1)^2+(x_0-x_1)^2+(p_0-p_1)^2}<\delta.
$$
Now, from the uniform convergence of $\{x_{1,n_{k}}\}$ and
$\{x'_{1,n_{k}}\}$ on $[\theta_1,T]$ it follows that there is a
$N_{\delta(\varepsilon)}$ with the properties
$$
|x_{1,n_{k}}-x_{\theta_1}|<\frac{\delta}{\sqrt{2}}\quad\text{and}\quad
|x'_{1,n_{k}}-x'_{\theta_1}|<\frac{\delta}{\sqrt{2}}\quad\text{for }t\in[\theta_1,T]
$$
and each $n_k>N_{\delta(\varepsilon)}$. As a result, for $t\in[\theta_1,T]$ we obtain
\begin{equation}
\sqrt{(t-t)^2+(x_{1,n_{k}}-x_{\theta_1})^2+(x'_{1,n_{k}}-x'_{\theta_1})^2}<\delta.
\label{e2.8}
\end{equation}
Finally, for $t\in[\theta_1,T]$ and $n_k>N_{\delta(\varepsilon)}$
from \eqref{e2.3}-\eqref{e2.6} we obtain
\begin{equation}
(t,x_{1,n_{k}}(t),x'_{1,n_{k}}(t)),(t,x_{\theta_1}(t),x'_{\theta_1}(t))\in
[\theta_1,T]\times[\tilde{m}_0,\tilde{M}_0]\times[m_1,M_1].\label{e2.9}
\end{equation}
On combining \eqref{e2.8} and \eqref{e2.9} with \eqref{e2.7},
we establish that for an arbitrary $\varepsilon>0$ there exists
$N_{\delta(\varepsilon)}$ such that for $n_k>N_{\delta(\varepsilon)}$ we have
$$
|f(s,x_{1,n_{k}}(s),x'_{1,n_{k}}(s))-f(s,x_{\theta_1}(s),x'_{\theta_1}(s))|
<\varepsilon\quad\text{for }t\in[\theta_1,T],
$$
i.e. the sequence $\{f(s,x_{1,n_{k}}(s),x'_{1,n_{k}}(s))\},n_{k}\in N_1$,
converges uniformly on the interval $[\theta_1,T]$ to
$f(s,x_{\theta_1}(s),x'_{\theta_1}(s))$. Then, returning to the integral
equation and letting $k\to\infty$ yield
$$
x'_{\theta_1}(t)=x'_{\theta_1}(t)+\int^t_{\theta_1}f(s,x_{\theta_1}(s),
x'_{\theta_1}(s))ds,\quad t\in(\theta_1,T],
$$
from where it follows that $x_{\theta_1}(t)$ is a $C^2[\theta_1,T]$-solution
to the differential equation $x''=f(t,x,x')$ on $[\theta_1,T]$.

Further, we consider the sequence $\{x_{1,n_{k}}\}$ on the new interval 
$[\theta_2,T]$ and for $n_k\in N_2=\{n_k\in N_T,k\in N:n_k^{-1}<\theta_2\}$.
 Obviously, for $n_k\in N_2$ we have
\begin{gather*}
\tilde{m}_0\leq x_{1,n_{k}}(t)\leq\tilde{M}_0\quad\text{for }t\in[\theta_2,T],\\
m_1\leq x'_{1,n_{k}}(t)\leq M_1\quad\text{for }t\in[\theta_2,T].
\end{gather*}
Besides, there is a constant $M_{2,2}$, independent on $n_k$, such that
$$
|x''_{1,n_{k}}(t)|\leq M_{2,2}\quad\text{for }t\in[\theta_2,T].
$$
Having obtained bounds, we apply the Arzela-Ascoli theorem on the 
sequence $\{x_{1,n_{k}}\}$ to conclude that there exist a subsequence 
$\{x_{2,n_{k}}\}, k\in N, n_k\in N_2$, and a function 
$x_{\theta_2}\in C^2[\theta_2,T]$ such that
$$
\|x_{2,n_{k}}-x_{\theta_2}\|_1\to 0\quad \text{on the new interval }[\theta_2,T].
$$
As above we establish also that $x_{\theta_2}(t)$ is a $C^2[\theta_2,T]$-solution 
to the differential equation $x''=f(t,x,x')$ on $[\theta_2,T]$ and
\begin{gather*}
\tilde{m}_0\leq x_{\theta_2}(t)\leq\tilde{M}_0\quad\text{for }t\in[\theta_2,T],\\
m_1\leq x'_{\theta_2}(t)\leq M_1\quad\text{for }t\in[\theta_2,T].
\end{gather*}
In addition, since $\{x_{2,n_{k}}\}$ is a subsequence of $\{x_{1,n_{k}}\}$, 
then $\{x_{2,n_{k}}\}$ converges  uniformly to $x_{\theta_1}$ on the interval 
$[\theta_1,T]$ which means
$$
x_{\theta_2}(t)\equiv x_{\theta_1}(t)\quad\text{for }t\in[\theta_1,T].
$$
Applying the same procedure repeatedly for $\theta_i\to0$, we establish 
that for each $i\in N$ there exists a function $x_{\theta_i}(t)$ which 
is a $C^2[\theta_i,T]$-solution to the equation $x''=f(t,x,x')$ 
on the interval $[\theta_i,T]$,
\begin{equation}
\|x_{i,n_{k}}-x_{\theta_i}\|_1\to 0\quad \text{on the interval }
[\theta_i,T]\label{e2.10}
\end{equation}
as $k\to\infty$ and $n_k\in N_i=\{n_k\in N_T,k\in N:n_k^{-1}<\theta_i\}$,
\begin{gather*}
\tilde{m}_0\leq x_{\theta_i}(t)\leq\tilde{M}_0\quad\text{for }t\in[\theta_i,T],\\
m_1\leq x'_{\theta_i}(t)\leq M_1\quad\text{for }t\in[\theta_i,T], \\
x_{\theta_{i+1}}(t)\equiv x_{\theta_i}(t)\quad\text{for }t\in[\theta_i,T].
\end{gather*}
Thanks to the properties of the functions of $\{x_{\theta_i}\}$, we conclude
that there is some function $x_0(t)$ which is a $C^2(0,T]$-solution to the
equation $x''=f(t,x,x')$ on the interval $(0,T]$,
\begin{gather}
\tilde{m}_0\leq x_0(t)\leq\tilde{M}_0\quad\text{for }t\in(0,T],\nonumber \\
m_1\leq x'_0(t)\leq M_1\quad\text{for }t\in(0,T], \label{e2.11} \\
x_0(t)\equiv x_{\theta_{i}}(t)\quad\text{for }t\in[\theta_i,T].\label{e2.12}
\end{gather}
\smallskip

\noindent\textbf{Step 3} Construction of a $C^1[0,T]\cap C^2(0,T]$-solution to
 \eqref{e1.1}. To define a $C[0,T]$-solution to \eqref{e1.1} we need to show that
\begin{equation}
\lim_{t\to0^+}x_0(t)=A.\label{e2.13}
\end{equation}
To this aim, we assume firstly on the contrary that for some $\varepsilon>0$
there exists $\delta>0$ such that $(0,\delta)\subset[0,T]$ and
\begin{equation}
x_0(t)\notin(A-\varepsilon,A+\varepsilon)\quad\text{for }t\in(0,\delta).\label{e2.14}
\end{equation}
Returning our attention to the sequence $\{x_n\}$, from $x_n\in C[0,T]$
and $x_n(n^{-1})=A$ deduce that there is a number $n_\delta$ such that
for each $n\geq n_\delta, n\in N$, there exists a sufficiently small
$\delta_n>n^{-1}$  with the properties $(n^{-1},\delta_n)\subset(0,\delta)$ and
$$
x_n(t)\in(A-\varepsilon/2,A+\varepsilon/2)\quad\text{for }t\in(n^{-1},\delta_n).
$$
On the other hand, there exists a number $n^*$ such that for each
$n\geq n^*, n\in N$, there exists some $i^*\in N$ for which
$$
[\theta_{i^*},\theta_{i^{*}-1}]\subset(n^{-1},\delta_n)\subset(0,\delta);
$$
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 $\{\theta_i\}$. As a result, for each $n\geq\max\{n_\delta,n^*\}$
there exists $i^*\in N$ such that
\begin{equation}
A-\varepsilon/2<x_n(t)<A+\varepsilon/2\quad\text{for }
t\in[\theta_{i^*},\theta_{i^{*}-1}]\subset(0,\delta).\label{e2.15}
\end{equation}
It is easy to see, for every $i^*$ there is a number $n_{i^*}$  such that
\eqref{e2.15} holds for each $n_k\in N_{i^*}, k\in N$, with
$n_k\geq\max\{ n_{i^*}, n_\delta,n^*\}$, that is,
\begin{equation}
A-\varepsilon/2<x_{i^*,n_k}(t)<A+\varepsilon/2\quad\text{for }t\in[\theta_{i^*},
\theta_{i^{*}-1}]\subset(0,\delta).\label{e2.16}
\end{equation}
Further, from \eqref{e2.10} and \eqref{e2.12} for each $i\in N$ we obtain
\begin{equation}
\|x_{i,n_{k}}-x_0\|_1\to 0\quad \text{on $[\theta_i,T]$ when
$k\to\infty$ and }n_{k}\in N_i,\label{e2.17}
\end{equation}
which means that for each $i\in N$ there is a number $\overline{n}_i$ such
that for each $n_{k}\in N_i$ with $n_{k}\geq \overline{n}_i$ we have
$$
-\varepsilon/2<x_{i,n_{k}}(t)-x_0(t)<\varepsilon/2\quad\text{for }t\in[\theta_i,T]
$$
or
$$
x_{i,n_{k}}(t)-\varepsilon/2<x_0(t)<x_{i,n_{k}}(t)+\varepsilon/2
\quad\text{for }t\in[\theta_i,T].
$$
In particular, for each $n_k\in N_{i^*}$ with
$n_k\geq\max\{ n_{i^*},\overline{n}_{i^*},n_\delta,n^*\}$, $k\in N$, we obtain
$$
x_{i^*,n_{k}}(t)-\varepsilon/2<x_0(t)<x_{i^*,n_{k}}(t)+\varepsilon/2
\quad\text{for }t\in[\theta_i^*,T].
$$
This combined with \eqref{e2.16} yields
$$
A-\varepsilon<x_0(t)<A+\varepsilon\quad\text{for }t\in[\theta_{i^*},
\theta_{i^{*}-1}]\subset(0,\delta),
$$
which contradicts to \eqref{e2.15} and so \eqref{e2.13} holds.

By exactly the same reasoning applied on the sequence $\{x'_n\}$ we establish
$$\lim_{t\to0^+}x'_0(t)=B.$$
Moreover, now we use that for each $i\in N$ and sufficiently large $n_k\in N_{i}, k\in N$, \eqref{e2.17} yields
$$-\varepsilon/2<x'_{i,n_{k}}(t)-x'_0(t)<\varepsilon/2\quad\text{for }t\in[\theta_i,T].$$
Next, introduce the function
$$
x(t)=\begin{cases}
A &\text{for }t=0,\\
x_0(t) &\text{for }t\in(0,T].
\end{cases}
$$
Clearly, $x'(t)=x'_0(t)$ for $t\in(0,T]$. Besides,
$$
x'(0)=\lim_{t\to0^+}\frac{x(t)-x(0)}{t-0}
=\lim_{t\to 0^+}x'(t)=\lim_{t\to0^+}x'_0(t)=B.
$$
Thus, $x'\in C[0,T]$ and so $x(t)$ is a $C^1[0,T]\cap C^2(0,T]$-solution 
to \eqref{e1.1}.

The inequalities \eqref{e2.11} give immediately
$$
m_1\leq x'(t)\leq M_1\quad \text{for }t\in[0,T],
$$
from where by integration from $0$ to $t\in(0,T]$ we obtain the bounds for $x(t)$.
\end{proof}

As an elementary consequence of Theorem \ref{thm2.1} we obtain results 
guaranteeing important properties of the solutions.

\begin{theorem} \label{thm2.2} 
 Let $B\geq0$ and let {\rm (A2)} hold for $m_1=0$. Then problem \eqref{e1.1} 
 has at least one nondecreasing solution in $C^1[0,T]\cap C^2(0,T]$.
\end{theorem}

\begin{theorem} \label{thm2.3} 
 Let $B>0$ and let {\rm (A2)} hold for $m_1>0$. Then problem \eqref{e1.1}
 has at least one strictly increasing solution in $C^1[0,T]\cap C^2(0,T]$.
\end{theorem}

\begin{theorem} \label{thm2.4} 
 Let $A>0$ $(A=0)$, $B\geq0$ and let {\rm (A2)} hold for $m_1=0$. 
Then problem \eqref{e1.1}  has at least one positive (nonnegative) 
nondecreasing solution in $C^1[0,T]\cap C^2(0,T]$.
\end{theorem}

\begin{theorem} \label{thm2.5} 
 Let $A\geq0, B>0$ and let {\rm (A2)} hold for $m_1>0$. Then problem
 \eqref{e1.1} has at least one strictly increasing solution in 
$C^1[0,T]\cap C^2(0,T]$ having positive values for $t\in(0,T]$.
\end{theorem}


\section{Example}

Consider the IVP
\begin{gather*}
x''=t^{-\frac{m}{n}}P_k(x'), \\
x(0)=A,\;x'(0)=B,
\end{gather*}
where $A\geq0$, $B>0$, $m,n\in N$, and the polynomial $P_k(p),k\geq2$, 
has simple zeros $p_1$ and $p_2$ such that $P'_k(p_1)<0$ and $0<p_1< B<p_2$.

Let $\theta>0$ be so small that $p_1-\theta>0$, $p_1+\theta<B<p_2-\theta$ and
$$
P_k(p)\ne0\quad \text{for }p\in[p_1-\theta,p_1)\cup(p_1,p_1+\theta)
\cup[p_2-\theta,p_2)\cup(p_2,p_2+\theta].
$$
Then $P'_k(p_1)<0$ implies
$$
P_k(p)>0\quad\text{for } p\in[p_1-\theta,p_1)\quad\text{and}\quad
P_k(p)<0\quad\text{for } p\in(p_1,p_1+\theta].
$$
Besides, we see easily that if
$$
P_k(p)<0\quad \text{for } p\in[p_2-\theta,p_2),
$$
then {\rm (A2)} holds for an arbitrary $T>0$,
$$
\overline{m}_1=p_1-\theta, \quad m_1=p_1,M_1=p_2-\theta,\quad
\overline{M_1}=p_2, \quad \tau=\theta/2,
$$
moreover $\tilde{M_0}=(p_2-\theta)T+A$, and if
$$
P_k(p)<0\quad\text{for } p\in(p_2,p_2+\theta],
$$
it is satisfied for an arbitrary $T>0$,
$$
\overline{m}_1=p_1-\theta,\quad 
m_1=p_1, \quad M_1=p_2, \quad \overline{M_1}=p_2+\theta, \quad
\tau=\theta/2,
$$
moreover $\tilde{M_0}=p_2T+A$. So, it follows from Theorem \ref{thm2.5} that 
for each $T>0$ the considered problem has a strictly increasing 
solution in $C^1[0,T]\cap C^2(0,T]$ which is positive on $(0,T]$. 


\subsection*{Acknowledgements}
The author is very grateful to the anonymous referee for his valuable suggestions.

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