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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 235, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/235\hfil Nonlinear singular second-order differential equations]
{Combined effects in nonlinear singular second-order differential equations on the
half-line}

\author[I. Bachar \hfil EJDE-2015/235\hfilneg]
{Imed Bachar}

\address{Imed Bachar \newline
King Saud University College of Science,
Mathematics Department, P.O. Box 2455,
Riyadh 11451, Saudi Arabia}
\email{abachar@ksu.edu.sa}

\thanks{Submitted  May 3, 2015. Published September 11, 2015.}
\subjclass[2010]{34B15, 34B18, 34B27}
\keywords{Green's function; Karamata regular variation theory;  
\hfill\break\indent positive solution; Schauder fixed point theorem}

\begin{abstract}
 We consider the existence, uniqueness and the asymptotic behavior
 of positive continuous solutions to the second-order boundary-value problem
 \begin{gather*}
 \frac{1}{A}(Au')'+a_1(t)u^{\sigma _1}+a_2(t)u^{\sigma _2}=0,\quad t\in (0,\infty ), \\
 \lim_{t\to 0^+} u(t)=0, \quad \lim_{t\to \infty } \frac{u(t)}{\rho (t)}=0,
 \end{gather*}
 where $\sigma _1,\sigma _2\in (-1,1)$, $A$ is a continuous
 function on $[0,\infty )$, positive and differentiable on
 $(0,\infty )$ such that $\int_0^1\frac{1}{A(t)}dt<\infty $ and
 $\int_0^{\infty }\frac{1}{A(t)}dt=\infty $.
 Here $\rho (t)=\int_0^{t}\frac{1}{A(s)}ds$ and for
 $i\in \{1,2\}$, $a_i$ is a nonnegative continuous function in
 $(0,\infty )$ such that there exists $c>0$ satisfying for $t>0$,
 \begin{equation*}
 \frac{1}{c}\frac{h_i(m(t))}{A^{2}(t)( 1+\rho (t)) ^{\mu _i}}
 \leq a_i(t) \leq c\frac{h_i(m(t))}{A^{2}(t)
 ( 1+\rho (t)) ^{\mu_i}},
 \end{equation*}
 where $m(t)=\frac{\rho (t)}{1+\rho (t)}$ and
 $h_i(t)=c_it^{- \lambda _i}\exp (\int_{t}^{\eta }\frac{z_i(s)}{s}ds)$,
 $c_i>0$, $\lambda _i\leq 2$, $\mu _i>2$ and $z_i$ is continuous on $[0,\eta ]$
 for some $\eta >1$ such that $z_i(0)=0$. The comparable asymptotic rate of
 $a_i(t)$ determines the asymptotic behavior of the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Boundary-value problems on the half-line, have been studied widely in the
literature (see, for example, \cite{AO,GR4,GY,MM,U,Z} and the references therein). 
The motivation for these studies stems from the fact that such problems arise 
naturally in the study of radially symmetric solutions of nonlinear elliptic 
equations (see, \cite{BS,BK,GR4,R,RR,U,Zhang} and also
many physical models, for example, the model of gas pressure in a
semi-infinite porous medium, the Thomas-Fermi model for determining the
electric potential in an isolated neutral atom (see, the Monographs,
\cite{AO,GR1} and the references therein). Therefore it is very important
to investigate the boundary-value problems for differential equations on
half-line.

Zhao \cite{Z} considered the problem
\begin{equation}
\begin{gathered}
u''+\varphi (.,u)=0, \quad \text{on }(0,\infty ),\\
u>0,\quad \text{on }(0,\infty ), \\
\lim_{t\to 0^+} u(t)=0, 
\end{gathered} \label{e1.1}
\end{equation}
where $\varphi $ is a measurable function on $(0,\infty )\times
(0,\infty )$, dominated by a convex positive function. Then he showed that
there exists $b>0$ such that for each $\mu \in (0,b]$, there exists a
positive continuous solution $u$ of \eqref{e1.1} satisfying 
$\lim_{t\to \infty }\frac{u(t)}{t}=\mu$.

 On the other hand, in \cite{B}, the author studied the singular problem
\begin{equation}
\begin{gathered}
\frac{1}{A}(Au')'+\varphi (\cdot,u)=0,\quad t\in (0,\infty ), \\
u>0,\quad \text{on }(0,\infty ), \\
\lim_{t\to 0^+} u(t)=0,\quad \lim_{t\to \infty } \frac{u(t)}{\rho (t)}=0,
\end{gathered} \label{e1.2}
\end{equation}
 where $A$ is a continuous function on $[0,\infty )$, positive and
differentiable on $(0,\infty )$ such that 
$\int_0^1\frac{1}{A(t)} dt<\infty $ and
$\int_0^{\infty }\frac{1}{A(t)}dt=\infty $.
Here $\rho (t)=\int_0^{t}\frac{1}{A(s)}ds$ and the function
$\varphi :(0,\infty )\times (0,\infty )\to \lbrack 0,\infty )$ 
is required to be continuous, non-increasing with respect to the second 
variable such that for each $c>0$, $\varphi (.,c)\neq 0$ and 
$\int_0^{\infty }A(s)\min (1,\rho (s))\varphi (s,c)ds<\infty $.
The author proved the existence of a unique positive solution $u$ in 
$C([0,\infty ))\cap C^{2}((0,\infty ))$ to problem \eqref{e1.2}.

 Recently, in \cite{BM}, the authors considered problem \eqref{e1.2}
 with $\varphi (t,u)=a(t)u^{\sigma }$, $\sigma <1$, (which include
the sublinear case) and $a$ is a nonnegative continuous function on 
$(0,\infty )$ satisfying some appropriate assumptions related to Karamata
regular variation theory. They have proved the existence, uniqueness and the
global asymptotic behavior of positive solutions to problem \eqref{e1.2}.

 In this article, we study the  boundary-value problem
\begin{equation}
\begin{gathered}
\frac{1}{A}(Au')'+a_1(t)u^{\sigma
_1}+a_2(t)u^{\sigma _2}=0,\quad t\in (0,\infty ), \\
u>0,\quad \text{on }(0,\infty ), \\
\lim_{t\to 0^+} u(t)=0,\quad \lim_{t\to \infty } \frac{u(t)}{\rho (t)}=0,
\end{gathered} \label{e1.4}
\end{equation}
where $\sigma _1,\sigma _2\in (-1,1)$, $A$ is a continuous
function on $[0,\infty )$, positive and differentiable on $(0,\infty )$ such
that $\int_0^1\frac{1}{A(t)}dt<\infty $,
$\int_0^{\infty }\frac{1}{A(t)}dt=\infty $ and
$\rho (t)=\int_0^{t}\frac{1}{A(s)}ds$, $t>0$.

Our goal is to study  \eqref{e1.4}, especially, when the
nonlinearity is the sum of a singular term and a sublinear term. Under
appropriate assumptions on $a_1$ and $a_2$ related to the Karamata class
$\mathcal{K}$ (see Definition \ref{def1.2}), we prove the existence,
uniqueness and the global asymptotic behavior of positive continuous
solution to problem \eqref{e1.4}.

 Throughout this paper and without loss of generality, we assume
that $\int_0^1\frac{1}{A(t)}dt=1$. To state our result, we need some
notation.

\begin{definition} \label{def1.2}\rm
The class $\mathcal{K}$ is the set of all Karamata functions $L $ defined 
on $(0,\eta ]$ by
\begin{equation*}
L(t):=c\exp \Big(\int_{t}^{\eta }\frac{z(s)}{s}ds\Big),
\end{equation*}
for some $\eta >1$, where $c>0$ and $z\in C([0,\eta ]\mathcal{)}$
such that $z(0)=0$.
\end{definition}

The theory of such functions was initiated by Karamata
in the fundamental paper \cite{K}. On the other hand, we emphasize that the
first use of the Karamata theory in the study of the growth rate of
solutions near the boundary was done in the paper of Cirstea and
 R\u{a}dulescu \cite{CR}.

\begin{remark} \label{rem1.3} \rm
 A function $L$ is in $\mathcal{K}$ if and only if $L$ is a
positive function in $C^1((0,\eta ])$, for some $\eta >1$, such that
 $\lim_{t\to 0^+} \frac{tL'(t)}{L(t)}=0$.
\end{remark}

 As a typical example of function belonging to the class 
$\mathcal{K }$ (see \cite{BGT,M,S}), we quote
\begin{equation*}
L(t)=2+\sin (\log _2(\frac{\omega }{t}))\quad\text{and}\quad
L(t)=\prod_{k=1}^m (\log _k(\frac{\omega }{t}))^{\xi _k},
\end{equation*}
where $\xi _k$ are real numbers, $\log _kx=\log \circ \log
\circ \dots\log x$ $(k$ times$)$ and $\omega $ is a sufficiently large
positive real number such that $L$ is defined and positive on 
$(0,\eta ]$, for some $\eta >1$.

 In the sequel, we denote by $B^{+}((0,\infty ))$ the set of
nonnegative Borel measurable functions in $(0,\infty )$ and by 
$C_0([0,\infty ))$ the set of continuous functions $v$ on $[0,\infty )$
such that $\lim_{t\to \infty} v(t)=0$. It is easy to see
that $C_0([0,\infty ))$ is a Banach space with the uniform norm 
$\| v\| _{\infty }=\sup_{t>0} |v(t)| $.

 For two nonnegative functions $f$ and $g$ defined on a set $S$,
the notation $f(t)\approx g(t)$, $t\in S$ means that there exists $c>0$ such
that $\frac{1}{c}f(t)\leq g(t)\leq cf(t)$, for all $t\in S$.

 Furthermore, let $G(t,s)=A(s)\min (\rho (t),\rho (s)))$, be the
Green's function of the operator $u\mapsto -\frac{1}{A}(Au')'$ on 
$(0,\infty )$ with the Dirichlet conditions 
$\lim_{t\to 0^{+}} u(t)=0$ and $\lim_{t\to \infty}\frac{u(t)}{\rho (t)}=0$.

 For $f\in B^{+}((0,\infty ))$, we put
\begin{equation*}
Vf(t)=\int_0^{\infty }G(t,s)f(s)dt,\quad \text{for }t>0.
\end{equation*}

 We point out that if the map $s\to A(s)\min (1,\rho(s))f(s)$ is continuous 
and integrable on $(0,\infty )$, then $Vf$ is the
solution of the boundary-value problem
\begin{equation}
\begin{gathered}
-\frac{1}{A}(Au')'=f,\quad \text{in }(0,\infty ), \\
\lim_{t\to 0^+} u(t)=0, \\
\lim_{t\to \infty} \frac{u(t)}{\rho (t)}=0.
\end{gathered} \label{e1.5}
\end{equation}

 For $\lambda \leq 2$, $\sigma \in (-1,1)$ and $L\in \mathcal{K}$
defined on $(0,\eta ]$ (for some $\eta >1$), we put for $t\in (0,\eta )$
\begin{equation}
\Psi _{L,\lambda ,\sigma }(t)
=\begin{cases}
\Big( \int_0^{t}\frac{L(s)}{s}ds\Big) ^{\frac{1}{1-\sigma }}, 
 & \text{if  }\lambda =2, \\
\big( L(t)\big) ^{\frac{1}{1-\sigma }}, & \text{if }1+\sigma <\lambda <2, \\
\Big( \int_{t}^{\eta }\frac{L(s)}{s}ds\Big) ^{\frac{1}{1-\sigma }}, &
\text{if }\lambda =1+\sigma , \\
1, & \text{if }\lambda <1+\sigma .
\end{cases}  \label{e1.6}
\end{equation}

 Throughout this article we assume the following condition:
\begin{itemize}
\item[(H1)] For $i\in \{1,2\}$, $a_i$ is a nonnegative continuous
function on $(0,\infty )$ such that
\begin{equation}
a_i(t)\approx \frac{1}{( A(t)) ^{2}}( \rho (t))
^{-\lambda _i}( 1+\rho (t)) ^{\lambda _i-\mu _i}L_i(m(t)),
\quad t>0,  \label{e1.7}
\end{equation}
 where $\lambda _i\leq 2$, $\mu _i>2$, 
$m(t):=\frac{\rho (t)}{1+\rho (t)}$, for $t>0$ and 
$L_i\in \mathcal{K}$ defined on $(0,\eta ]$ $($
for some $\eta >1)$ such that
\begin{equation}
\int_0^{\eta }s^{1-\lambda _i}L_i(s)ds<\infty .  \label{e1.8}
\end{equation}
\end{itemize}
 As it will be seen, the numbers
\begin{equation}
\beta _1=\min (1,\frac{2-\lambda _1}{1-\sigma _1})\quad\text{and}\quad
\beta_2=\min (1,\frac{2-\lambda _2}{1-\sigma _2})  \label{e1.9}
\end{equation}
will play an important role in the study of asymptotic behavior of
solution. Without loss of generality, we may assume that 
\[
\frac{2-\lambda_1}{1-\sigma _1}\leq \frac{2-\lambda _2}{1-\sigma _2}
\]
and we define the function $\theta $ on $(0,\infty )$ by
\begin{equation}
\theta (t)=\begin{cases}
(m(t))^{\beta _1}\Psi _{L_1,\lambda _1,\sigma _1}(m(t)) 
& \text{if }\beta _1<\beta _2 \\
(m(t))^{\beta _1}(\Psi _{L_1,\lambda _1,\sigma _1}(m(t))+\Psi
_{L_2,\lambda _2,\sigma _2}(m(t))) & \text{ if \ }\beta _1=\beta_2,
\end{cases} \label{e1.10}
\end{equation}
where $m(t):=\frac{\rho (t)}{1+\rho (t)}$, for $t>0$.

For an explicit form of the function $\theta $ see \eqref{e3.01}.
Now, we are ready to state our main results.

\begin{theorem} \label{thm1.4}
Let $\sigma _1,\sigma _2\in (-1,1)$ and assume that {\rm (H1)}
is fulfilled. Then for $t\in (0,\infty )$,
\begin{equation}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx \theta
(t).  \label{e1.11}
\end{equation}
\end{theorem}

 By applying the above theorem and using the Schauder fixed point
theorem, we prove the following result.

\begin{theorem} \label{thm1.5}
Let $\sigma _1,\sigma _2\in (-1,1)$ and assume that
{\rm (H1)} is fulfilled. Then \eqref{e1.4} has a unique
positive continuous solution $u$ satisfying for $t\in (0,\infty )$
\begin{equation}
u(t)\approx \theta (t).  \label{e1.12}
\end{equation}
\end{theorem}

 The content of this paper is organized as follows. 
In Section 2, we present some fundamental properties of the Karamata 
class $\mathcal{K}$ including sharp estimates on some potential functions. 
In Section 3, exploiting the results of the previous section, we first 
prove Theorem \ref{thm1.4} which allow us to prove Theorem \ref{thm1.5} 
by means of the Schauder fixed point theorem.

\section{Properties of Karamata regular variation theory}

 We collect in this section some properties of functions belonging
to the Karamata class $\mathcal{K}$.

\begin{proposition}[\cite{M,S}] \label{prop2.1}
 The following hold:
\begin{itemize}
\item[(i)] Let $L_1,L_2\in \mathcal{K}$ and $p\in \mathbb{R}$. Then
$L_1+L_2\in \mathcal{K}$, $L_1L_2\in \mathcal{K}$ and 
$L_1^{p}\in \mathcal{K}$.

\item[(ii)] Let $L\in \mathcal{K}$ and $\varepsilon >0$. Then 
$\lim_{t\to 0^+} t^{\varepsilon }L(t)=0$.
\end{itemize}
\end{proposition}

 Applying Karamata's theorem (see \cite{M,S}), we obtain
the following result.

\begin{lemma} \label{lem2.2}
 Let $\nu \in \mathbb{R}$ and $L$ be a function in $\mathcal{K}$
defined on $(0,\eta ]$. We have
\begin{itemize}
\item[(i)] If $\nu <-1$, then $\int_0^{\eta }s^{\nu }L(s)ds$ diverges
and $\int_{t}^{\eta }s^{\nu }L(s)ds\underset{t\to 0^{+}}{\sim }-
\frac{t^{\nu +1}L(t)}{\nu +1}$.

\item[(ii)] If $\nu >-1$, then $\int_0^{\eta }s^{\nu }L(s)ds$
converges and $\int_0^{t}s^{\nu }L(s)ds\underset{t\to 0^{+}}{\sim }
\frac{t^{\nu +1}L(t)}{\nu +1}$.
\end{itemize}
\end{lemma}

The proof of the next lemmas can be found in \cite{CMMZ}.

\begin{lemma} \label{lem2.3} 
Let $L$ be a function in $\mathcal{K}$ defined on $(0,\eta ]$
$(\eta >1)$. Then 
\[
\lim_{t\to 0^+} \frac{L(t)}{\int_{t}^{\eta }\frac{L(s)}{s}ds}=0.
\]
 In particular  $t\to \int_{t}^{\eta }\frac{L(s)}{s}ds\in \mathcal{K}$.

If further $\int_0^{\eta }\frac{L(s)}{s}ds$ converges, then 
\begin{equation*}
\lim_{t\to 0^+} \frac{L(t)}{\int_0^{t}\frac{L(s)}{s}ds}=0.
\end{equation*}
 In particular $t\to \int_0^{t}\frac{L(s)}{s}ds\in \mathcal{K}$.
\end{lemma}

\begin{lemma}\label{lem2.4}
For $i\in \{1,2\}$, let $L_i\in \mathcal{K}$ be defined on 
$(0,\eta ]$ $(\eta >1)$ and put for $t\in (0,\eta )$,
\begin{equation*}
M(t)=\Big( \int_{t}^{\eta }\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\sigma_1}}
+\Big( \int_{t}^{\eta }\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _2}}.
\end{equation*}
Then for $t\in (0,\eta )$ we have
\begin{equation*}
\int_{t}^{\eta }\frac{(M^{\sigma _1}L_1+M^{\sigma _2}L_2)(s)}{s}
ds\approx M(t).
\end{equation*}
\end{lemma}

\begin{lemma}\label{lem2.5}
For $i\in \{1,2\}$, let $L_i\in \mathcal{K}$ be defined on 
$(0,\eta ]$ $(\eta >1)$ such that 
$\int_0^{\eta }\frac{L_i(s)}{s} ds<\infty $. Put for $t\in (0,\eta )$,
\begin{equation*}
N(t)=\Big( \int_0^{t}\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _1}}
+\Big( \int_0^{t}\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _2}}.
\end{equation*}
Then  for $t\in (0,\eta )$ we have
\begin{equation*}
\int_0^{t}\frac{(N^{\sigma _1}L_1+N^{\sigma _2}L_2)(s)}{s}
ds\approx N(t).
\end{equation*}
\end{lemma}

 Next, we have the following fundamental sharp estimates on the
potential function $Vb$, for
\begin{equation*}
b(t)=\frac{1}{( A(t)) ^{2}}( \rho (t)) ^{-\beta
}( 1+\rho (t)) ^{\beta -\gamma }\widetilde{L}(m(t)),
\end{equation*}
 where $\beta \leq 2$, $\gamma >2$, $\widetilde{L}\in \mathcal{K}$
and $m(t)=\frac{\rho (t)}{1+\rho (t)}$ for $t>0$.

\begin{proposition}[\cite{BM}] \label{prop2.6}
Let $\beta \leq 2$, $\gamma >2$ and $\widetilde{L}\in \mathcal{K}$ be 
defined on $(0,\eta ]$ $(\eta >1)$ such that 
$\int_0^{\eta }s^{1-\beta }\widetilde{L}(s)ds<\infty $. 
Then for $t>0$,
\begin{equation*}
Vb(t)\approx \phi _{\beta }(m(t)),
\end{equation*}
where for $r\in (0,1]$,
\begin{equation*}
\phi _{\beta }(r)=\begin{cases}
\int_0^{r}\frac{\widetilde{L}(s)}{s}ds & \text{if  }\beta =2,\\
r^{2-\beta }\widetilde{L}(r) & \text{if  }1<\beta <2, \\
r\int_{r}^{\eta }\frac{\widetilde{L}(s)}{s}ds & \text{if  }\beta=1, \\
r & \text{if }\beta <1.
\end{cases}
\end{equation*}
\end{proposition}

\section{Proof of main results}

 Let $\sigma _1,\sigma _2\in (-1,1)$, assume (H1) and for 
$i\in \{1,2\}$, let $L_i\in \mathcal{K}$ defined on $(0,\eta ]$ 
(for some$\eta >1$) satisfying \eqref{e1.7} and \eqref{e1.8}. 
Let $b$, $L$, $M$ and $N$ be the nonnegative functions defined in
 $(0,\eta )$ by
\begin{gather*}
b(t):=\Big(\int_{t}^{\eta }\frac{L_1(s)}{s}ds\Big)^{\frac{1}{1-\sigma _1}}, \\
L(t):=(L_1(t))^{\frac{1}{1-\sigma _1}}+(L_2(t))^{\frac{1}{1-\sigma _2
}}, \\
M(t):=\Big(\int_{t}^{\eta }\frac{L_1(s)}{s}ds\Big)^{\frac{1}{1-\sigma _1}}
+\Big(\int_{t}^{\eta }\frac{L_2(s)}{s}ds\Big)^{\frac{1}{1-\sigma _2}}, \\
N(t):=\Big( \int_0^{t}\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _1}}
+\Big( \int_0^{t}\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _2}},\quad
\text{if }\int_0^{\eta }\frac{L_i(s)}{s}ds<\infty .
\end{gather*}

 First, we give an explicit form of the function $\theta $ defined
by \eqref{e1.10}. We recall that for $i\in \{1,2\}$, $\lambda _i\leq 2$
and $\beta _i=\min (1,\frac{2-\lambda _i}{1-\sigma _i})$. 
Since $\beta_1<\beta _2$ is equivalent to 
$\frac{2-\lambda _1}{1-\sigma _1}<\frac{2-\lambda _2}{1-\sigma _2}$ and 
$1+\sigma _1<\lambda _1$, we deduce that for $t\in (0,\infty )$,
\begin{equation}
\theta (t)=\begin{cases}
\Big( \int_0^{m(t)}\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _1}}, 
& \text{if }\lambda _1=2\text{ and }\lambda _2<2, \\
(m(t))^{\frac{2-\lambda _1}{1-\sigma _1}}( L_1(m(t))) ^{
\frac{1}{1-\sigma _1}}, 
& \text{if }\frac{2-\lambda _1}{1-\sigma _1}<
\frac{2-\lambda _2}{1-\sigma _2}\text{ and }1+\sigma _1<\lambda
_1<2, \\
(m(t))^{\frac{2-\lambda _1}{1-\sigma _1}}L(m(t)), 
& \text{if }\frac{2-\lambda _1}{1-\sigma _1}=\frac{2-\lambda _2}{1-\sigma _2}\text{
and }1+\sigma _1<\lambda _1<2, \\
m(t)M(m(t)), 
& \text{if }\lambda _1=1+\sigma _1\text{ and }\lambda_2=1+\sigma _2, \\
m(t)( 1+b(m(t))) , 
& \text{if }\lambda _1=1+\sigma _1\text{ and }\lambda _2<1+\sigma _2, \\
2m(t), & \text{if }\lambda _1<1+\sigma _1, \\
N(m(t)), & \text{if }\lambda _1=\lambda _2=2,
\end{cases}  \label{e3.01}
\end{equation}
 where $m(t)=\frac{\rho (t)}{1+\rho (t)}$.

\subsection*{Proof of Theorem \ref{thm1.4}}

\begin{lemma} \label{lem3.1} 
For $r,s>0$, we have
\begin{equation}
2^{-\max (1-\sigma _1,1-\sigma _2)}(r+s)\leq r^{1-\sigma
_1}(r+s)^{\sigma _1}+s^{1-\sigma _2}(r+s)^{\sigma _2}\leq 2(r+s).
\label{e3.02}
\end{equation}
\end{lemma}

\begin{proof}
Let $r,s>0$ and put $t=\frac{r}{r+s}$. Since $0\leq t\leq 1$, then we obtain
\begin{equation*}
2^{-\max (1-\sigma _1,1-\sigma _2)}\leq t^{1-\sigma
_1}+(1-t)^{1-\sigma _2}\leq 2.
\end{equation*}
Which implies the result.
\end{proof}

 Now we are ready to prove Theorem \ref{thm1.4}.
We recall that for $i\in \{1,2\}$, $a_i$ is a nonnegative
continuous function on $(0,\infty )$ such that
\begin{equation*}
a_i(t)\approx \frac{1}{( A(t)) ^{2}}( \rho (t))
^{-\lambda _i}( 1+\rho (t)) ^{\lambda _i-\mu _i}L_i(m(t)),
\text{ }t>0,
\end{equation*}
where $\lambda _i\leq 2$ and $\mu _i>2$.

 Note that throughout the proof, we use Proposition \ref{prop2.1}
and Lemma \ref{lem2.3} to verify that some functions are in $\mathcal{K}$.
We distinguish the following cases.
\smallskip

\noindent\textbf{Case 1:} $\lambda _1=2$ and $\lambda _2<2$.
We have 
\[
\theta (t)=\Big( \int_0^{m(t)}\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\sigma _1}}. 
\]
Therefore
\begin{align*}
&a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t) \\
&\approx 
\frac{( \rho (t)) ^{-2}( 1+\rho (t)) ^{2-\mu }}{(
A(t)) ^{2}}L_1(m(t))\Big( \int_0^{m(t)}\frac{L_1(s)}{s}
ds\Big) ^{\frac{\sigma _1}{1-\sigma _1}} \\
&\qquad+\frac{( \rho (t)) ^{-\lambda _2}( 1+\rho (t))
^{\lambda _2-\mu }}{( A(t)) ^{2}}L_2(m(t))\Big(
\int_0^{m(t)}\frac{L_1(s)}{s}ds\Big) ^{\frac{\sigma _2}{1-\sigma_1}}.
\end{align*}

 Since for $i\in \{1,2\}$, the function 
$t\to \widetilde{L}_i(t):=L_i(t)( \int_0^{t}\frac{L_1(s)}{s}ds) ^{\frac{
\sigma _i}{1-\sigma _1}}\in \mathcal{K}$ and $\lambda _2<2$, we deduce
by Proposition \ref{prop2.1} that
\begin{equation*}
a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t)
\approx \frac{( \rho (t)) ^{-2}( 1+\rho (t)) ^{2-\mu }}{(
A(t)) ^{2}}\widetilde{L}_1(m(t)).
\end{equation*}
Moreover, since $\lambda _1=2$, we have
\begin{equation*}
\int_0^{\eta }\frac{\widetilde{L}_1(s)}{s}ds
\leq c\Big( \int_0^{\eta
}\frac{L_1(r)}{r}dr\Big) ^{\frac{1}{1-\sigma _1}}<\infty ,
\end{equation*}
it follows by applying Proposition \ref{prop2.6} with 
$\beta =\lambda _1=2$, $\gamma =\mu $, we obtain
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx
\int_0^{m(t)}\frac{L_1(s)}{s}\Big( \int_0^{s}\frac{L_1(r)}{r}
dr\Big) ^{\frac{\sigma _1}{1-\sigma _1}}ds\approx \theta (t).
\end{equation*}
\smallskip

\noindent \textbf{Case 2:} $\frac{2-\lambda _1}{1-\sigma _1}<\frac{
2-\lambda _2}{1-\sigma _2}$ and $1+\sigma _1<\lambda _1<2$.
 Since 
\[
\theta (t)=(m(t))^{\frac{2-\lambda _1}{1-\sigma _1}
}( L_1(m(t))) ^{\frac{1}{1-\sigma _1}},
\]
we obtain
\begin{align*}
&a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t)\\
&\approx 
\frac{( \rho (t)) ^{\frac{(2-\lambda _1)\sigma _1}{1-\sigma
_1}-\lambda _1}( 1+\rho (t)) ^{\lambda _1-\frac{(2-\lambda
_1)\sigma _1}{1-\sigma _1}-\mu }}{( A(t)) ^{2}}(L_1L_1^{
\frac{\sigma _1}{1-\sigma _1}})(m(t)) \\
&\quad +\frac{( \rho (t)) ^{\frac{(2-\lambda _1)\sigma _2}{
1-\sigma _1}-\lambda _2}( 1+\rho (t)) ^{\lambda _2-\frac{
(2-\lambda _1)\sigma _2}{1-\sigma _1}-\mu }}{( A(t)) ^{2}}
(L_2L_1^{\frac{\sigma _2}{1-\sigma _1}})(m(t)).
\end{align*}
Since in this case $\frac{(2-\lambda _1)\sigma _2}{1-\sigma
_1}-\lambda _2>\frac{(2-\lambda _1)\sigma _1}{1-\sigma _1}-\lambda
_1$, we deduce by Proposition \ref{prop2.1} that
\begin{align*}
&a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t)\\
&\approx \frac{( \rho (t)) ^{\frac{(2-\lambda _1)\sigma _1}{1-\sigma
_1}-\lambda _1}( 1+\rho (t)) ^{\lambda _1-\frac{(2-\lambda
_1)\sigma _1}{1-\sigma _1}-\mu }}{( A(t)) ^{2}}(L_1L_1^{
\frac{\sigma _1}{1-\sigma _1}})(m(t)).
\end{align*}
Therefore applying Proposition \ref{prop2.6} with 
$\beta =\lambda _1-\frac{(2-\lambda _1)\sigma _1}{1-\sigma _1}\in (1,2)$, $\gamma
=\mu $ and $\widetilde{L}=L_1L_1^{\frac{\sigma _1}{1-\sigma _1}
}=L_1^{\frac{1}{1-\sigma _1}}\in \mathcal{K}$, we obtain
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx
(m(t))^{2-\beta }( L_1(m(t))) ^{\frac{1}{1-\sigma _1}}=\theta
(t).
\end{equation*}
\smallskip

\noindent \textbf{Case 3:} 
$\frac{2-\lambda _1}{1-\sigma _1}=\frac{2-\lambda _2}{1-\sigma _2}$ and 
$1+\sigma _1<\lambda _1<2$.
We have
\begin{equation*}
\theta (t)=(m(t))^{\frac{2-\lambda _1}{1-\sigma _1}}L(m(t)).
\end{equation*}
So
\begin{equation*}
a_1(t)\theta ^{\sigma _1}(t)\approx \frac{( \rho (t)) ^{\frac{
(2-\lambda _1)\sigma _1}{1-\sigma _1}-\lambda _1}( 1+\rho
(t)) ^{\lambda _1-\frac{(2-\lambda _1)\sigma _1}{1-\sigma _1}
-\mu }}{( A(t)) ^{2}}(L_1L^{\sigma _1})(m(t))
\end{equation*}
 Hence using again Proposition \ref{prop2.6} with 
$\beta =\lambda _1-\frac{(2-\lambda _1)\sigma _1}{1-\sigma _1}\in (1,2)$, 
$\gamma =\mu $ and $\widetilde{L}=L_1L^{\sigma _1}\in \mathcal{K}$, we obtain
\begin{equation*}
V(a_1\theta ^{\sigma _1})(t)\approx (m(t))^{\frac{2-\lambda _1}{
1-\sigma _1}}(L_1L^{\sigma _1})(m(t)).
\end{equation*}
 On the other hand, since $1+\sigma _2<\lambda _2<2$, we
similarly obtain
\begin{equation*}
V(a_2\theta ^{\sigma _2})(t)\approx (m(t))^{\frac{2-\lambda _1}{
1-\sigma _1}}(L_2L^{\sigma _2})(m(t)).
\end{equation*}
Using Lemma \ref{lem3.1}, we deduce that
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx (m(t))^{
\frac{2-\lambda _1}{1-\sigma _1}}L(m(t))=\theta (t).
\end{equation*}
\smallskip

\noindent \textbf{Case 4:} $\lambda _1=1+\sigma _1$ and 
$\lambda_2=1+\sigma _2$.
In this case we have
$\theta (t)=m(t)M(m(t))$
 By calculations, we obtain
\begin{equation*}
a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t)\approx
\frac{( \rho (t)) ^{-1}( 1+\rho (t)) ^{1-\mu }}{(
A(t)) ^{2}}(M^{\sigma _1}L_1+M^{\sigma _2}L_2)(m(t))
\end{equation*}
Using Proposition \ref{prop2.6} with $\beta =1$, $\gamma =\mu $
and $\widetilde{L}=M^{\sigma _1}L_1+M^{\sigma _2}L_2$, we deduce that
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx
m(t)\int_{m(t)}^{\eta }\frac{\widetilde{L}(s)}{s}ds.
\end{equation*}
Hence the results follows from Lemma \ref{lem2.4}.
\smallskip

\noindent \textbf{Case 5:} $\lambda _1=1+\sigma _1$ and $\lambda
_2<1+\sigma _2$.
 Since $\lim_{t\to 0^+} b(t)\in (0,\infty ]$,
it follows that $\theta (t)\approx m(t)b(m(t))$. So, we obtain that
\begin{align*}
a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t) 
&\approx  \frac{( \rho (t)) ^{-1}( 1+\rho (t)) ^{1-\mu }}{(
A(t)) ^{2}}(L_1b^{\sigma _1})(m(t)) \\
&\quad +\frac{( \rho (t)) ^{-\lambda _2+\sigma _2}( 1+\rho
(t)) ^{\lambda _2-\sigma _2-\mu }}{( A(t)) ^{2}}
(L_2b^{\sigma _2})(m(t)).
\end{align*}
 Using the fact that for $i\in \{1,2\}$, the function 
$t\to \widetilde{L}_i(t):=L_i(t)b^{\sigma _i}(t)\in \mathcal{K}$ and that 
$\lambda _2-\sigma _2<1$, we deduce by Proposition \ref{prop2.1} that
\begin{equation*}
a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t)\approx
\frac{( \rho (t)) ^{-1}( 1+\rho (t)) ^{1-\mu }}{(
A(t)) ^{2}}(L_1b^{\sigma _1})(m(t)).
\end{equation*}
 Hence applying Proposition \ref{prop2.6} with $\beta =1$, $\gamma
=\mu $, we obtain that
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx
m(t)\int_{m(t)}^{\eta }\frac{\widetilde{L}_1(s)}{s}ds\approx \theta (t).
\end{equation*}
\smallskip

\noindent\textbf{Case 6:} $\lambda _1<1+\sigma _1$.
Since $\theta (t)=2m(t)$, we obtain
\begin{equation*}
a_1(t)\theta ^{\sigma _1}(t)\approx \frac{( \rho (t))
^{-\lambda _1+\sigma _1}( 1+\rho (t)) ^{\lambda _1-\sigma
_1-\mu }}{( A(t)) ^{2}}L_1(m(t)).
\end{equation*}
Applying Proposition \ref{prop2.6} with $\beta =\lambda
_1-\sigma _1<1$, $\gamma =\mu $ and $\widetilde{L}=L_1$, we obtain
\begin{equation*}
V(a_1\theta ^{\sigma _1})(t)\approx m(t).
\end{equation*}
On the other hand, since also $\lambda _2<1+\sigma _2$, we
similarly obtain
\begin{equation*}
V(a_2\theta ^{\sigma _2})(t)\approx m(t).
\end{equation*}
Hence
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx
m(t)\approx \theta (t).
\end{equation*}
\smallskip

\noindent\textbf{Case 7:} $\lambda _1=\lambda _2=2$.
We have $\theta (t)=N(m(t))$. By calculus, we conclude that
\begin{equation*}
a_1(t)\theta ^{\sigma _1}(t)+a_2(t)\theta ^{\sigma _2}(t)\approx
\frac{( \rho (t)) ^{-2}(1+\rho (t))^{2-\mu }}{( A(t))
^{2}}(N^{\sigma _1}L_1+N^{\sigma _2}L_2)(m(t))
\end{equation*}
On the other hand, since $s\to ( N^{\sigma
_1}L_1+N^{\sigma _2}L_2) (s)\in \mathcal{K}$ and from Lemma
\ref{lem2.5},
\begin{equation*}
\int_0^1\frac{( N^{\sigma _1}L_1+N^{\sigma _2}L_2) (s)
}{s}ds\approx N(1)<\infty ,
\end{equation*}
then by Proposition \ref{prop2.6} with $\beta =2$, $\gamma =\mu $
and $\widetilde{L}=N^{\sigma _1}L_1+N^{\sigma _2}L_2$, we deduce that
\begin{equation*}
V(a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2})(t)\approx
\int_0^{m(t)}\frac{( N^{\sigma _1}L_1+N^{\sigma _2}L_2)
(s)}{s}ds.
\end{equation*}
Hence the results follows from Lemma \ref{lem2.5}.

\subsection*{Proof of Theorem \ref{thm1.5}}

The next Lemma will be useful to prove the uniqueness.

\begin{lemma}[\cite{B}] \label{lem3.2}
 Let $a\geq 0$ and $u\in C^1((a,\infty)) $ be a function satisfying
\begin{equation}
\begin{gathered}
-\frac{1}{A}(Au')'\geq 0,\quad \text{in }(a,\infty ), \\
\lim_{t\to a^{+}} u(t)=0, \quad \lim_{t\to \infty} \frac{u(t)}{\rho (t)}=0.
\end{gathered}\label{e3.0}
\end{equation}
 Then $u$ is nondecreasing and nonnegative function on $(a,\infty)$. 
\end{lemma}

Now we are ready to prove Theorem \ref{thm1.5}.
Let $\sigma _1,\sigma _2\in (-1,1)$, assume (H1) and put 
$\widetilde{\omega }:=a_1\theta ^{\sigma _1}+a_2\theta ^{\sigma _2}$
and $v:=V(\widetilde{\omega })$.
From Theorem \ref{thm1.4}, there exists $M>1$ such that for each 
$t>0$,
\begin{equation}
\frac{1}{M}\theta (t)\leq v(t)\leq M\theta (t).  \label{e3.3}
\end{equation}
On the other hand, using hypothesis (H1), \eqref{e3.01},
 Lemma\ref{lem2.2} and Lemma \ref{lem2.5}, we verify that
\begin{equation}
\int_0^{\infty }A(s)\min (1,\rho (s))\widetilde{\omega }(s)ds<\infty .
\label{e3.4}
\end{equation}
Put $\sigma =\max (| \sigma _1| ,|
\sigma _2| )$, $c=M^{\frac{1+\sigma }{1-\sigma }}$, where the
constant $M$ is given in \eqref{e3.3} and let
\begin{equation*}
\Lambda =\{\omega \in C_0([0,\infty )):\frac{\theta (t)}{c( 1+\rho
(t)) }\leq \omega (t)\leq \frac{c\theta (t)}{1+\rho (t)},\text{ \ }
t>0\}.
\end{equation*}
Using \eqref{e3.01}, Proposition \ref{prop2.1} and Lemma \ref{lem2.3}, 
we verify that the function
$t\mapsto \frac{\theta (t)}{1+\rho (t)}\in C_0([0,\infty ))$
and so $\Lambda $ is not empty.

 We define the operator $T$ on $\Lambda $ by
\begin{equation}
\begin{aligned}
T\omega (t)
&=\frac{1}{1+\rho (t)}\int_0^{\infty }G(t,s)\big[a_1(s)(1+\rho
(s))^{\sigma _1}\omega ^{\sigma _1}(s)\\
&\quad +a_2(s)(1+\rho (s))^{\sigma
_2}\omega ^{\sigma _2}(s)\big]ds.  
\end{aligned}\label{e3.6}
\end{equation}
We shall prove that $T$ has a fixed point in $\Lambda $.

 First observe that for this choice of $c$, by using \eqref{e3.3},
we have for all $\omega \in \Lambda $ and $t>0$
\[
T\omega (t) 
\leq \frac{1}{1+\rho (t)}V(a_1c^{\sigma }M^{\sigma }\theta
^{\sigma _1}+a_2c^{\sigma }M^{\sigma }\theta ^{\sigma _2})(t)=\frac{
c^{\sigma }M^{\sigma }}{1+\rho (t)}v(t)\leq \frac{c\theta (t)}{1+\rho (t)}
\]
and 
\[
T\omega (t) \geq V(a_1c^{-\sigma }M^{-\sigma }\theta
^{\sigma _1}+a_2c^{-\sigma }M^{-\sigma }\theta ^{\sigma _2})(t)=\frac{
c^{-\sigma }M^{-\sigma }}{1+\rho (t)}v(t)\geq \frac{\theta (t)}{c(
1+\rho (t)) }.
\]
On the other hand, for all $\omega \in \Lambda $ we have
\begin{equation}
| a_1(t)(1+\rho (t))^{\sigma _1}\omega ^{\sigma
_1}(t)+a_2(t)(1+\rho (t))^{\sigma _2}\omega ^{\sigma
_2}(t)| \leq c^{\sigma }M^{\sigma }\widetilde{\omega }(t),
\label{e3.7}
\end{equation}
and for all $t,s>0$, we have
\begin{equation}
\frac{G(t,s)}{1+\rho (t)}\leq A(s)\min (1,\rho (s)).  \label{e3.8}
\end{equation}
Since for each $s>0$, the function 
$t\to \frac{G(t,s)}{1+\rho (t)}$ is in $C_0([0,\infty ))$, we deduce by 
using \eqref{e3.7}, 
\eqref{e3.8} and \eqref{e3.4} that the family 
$\{t\to T\omega (t)$, $\omega \in \Lambda \}$ is relatively compact in 
$C_0([0,\infty ))$. Therefore, $T(\Lambda )\subset \Lambda $.

 Now, we shall prove the continuity of the operator $T$ in $\Lambda
$ in the supremum norm. Let $( \omega _k) _{k\in \mathbb{N}}$
be a sequence in $\Lambda $ which converges uniformly to a function $\omega $
in $\Lambda $. Then, for each $t>0$, we have
\begin{equation*}
| T\omega _k(t)-T\omega (t)|
 \leq \frac{1}{1+\rho (t)}
V[a_1(1+\rho (.))^{\sigma _1}| \omega _k^{\sigma _1}-\omega
^{\sigma _1}| +a_2(1+\rho (.))^{\sigma _2}| \omega
_k^{\sigma _2}-\omega ^{\sigma _2}| ](t).
\end{equation*}

On the other hand, by similar arguments as above, we have
\begin{equation*}
a_1(1+\rho (.))^{\sigma _1}| \omega _k^{\sigma _1}-\omega
^{\sigma _1}| +a_2(1+\rho (.))^{\sigma _2}| \omega
_k^{\sigma _2}-\omega ^{\sigma _2}| \leq \widetilde{c}
\widetilde{\omega }(s).
\end{equation*}
We conclude by \eqref{e3.4} and the dominated convergence theorem
that for all $t>0$,
\begin{equation*}
T\omega _k( t) \to T\omega ( t) \quad \text{as }k\to +\infty .
\end{equation*}
 Consequently, as $T( \Lambda ) $ is relatively compact
in $C_0([0,\infty ))$, we deduce that the pointwise convergence implies
the uniform convergence, namely,
\begin{equation*}
\| T\omega _k-T\omega \| _{\infty }\to 0\quad \text{as }k\to +\infty .
\end{equation*}
 Therefore, $T$ is a continuous mapping from $\Lambda $ into
itself. So the Schauder fixed point theorem implies the existence of 
$\omega \in \Lambda $ such that
\begin{equation*}
\omega (t)=\frac{1}{1+\rho (t)}V(a_1(1+\rho (.))^{\sigma _1}\omega
^{\sigma _1}+a_2(1+\rho (.))^{\sigma _2}\omega ^{\sigma _2})(t).
\end{equation*}
 Put $u(t)=1+\rho (t)\omega (t)$. Then $u$ is continuous and
satisfies
\begin{equation*}
u(t)=V(a_1u^{\sigma _1}+a_2u^{\sigma _2})(t).
\end{equation*}
Since the function $s\to A(s)\min (1,\rho
(s))[a_1(s)u^{\sigma _1}(s)+a_2(s)u^{\sigma _2}(s)]$ is continuous
and integrable on $(0,\infty )$, then it follows that $u$ is a solution of
problem \eqref{e1.4}.

 Finally, it remains to prove that $u$ is the unique positive
continuous solution satisfying \eqref{e1.12}. To this end, assume that
problem \eqref{e1.4} has two positive continuous solutions $u,v$
satisfying \eqref{e1.12}. Then there exists a constant $m>1$ such that
\begin{equation*}
\frac{1}{m}\leq \frac{u}{v}\leq m.
\end{equation*}
This implies that the set
\begin{equation*}
J=\{m\geq 1:\text{ }\frac{1}{m}\leq \frac{u}{v}\leq m\}
\end{equation*}
is not empty. Let $\sigma :=\max (| \sigma_1| ,| \sigma _2| )$ and put 
$c_0:=\inf J$. Then $c_0\geq 1$ and we have $\frac{1}{c_0}v\leq u\leq c_0v$. It
follows that for $i\in \{1,2\}$, 
$u^{\sigma _i}\leq c_0^{\sigma}v^{\sigma _i}$ and that the function 
$w:=c_0^{\sigma }v-u$ satisfies
\begin{gather*}
-\frac{1}{A}(Aw')'=a_1(c_0^{\sigma }v^{\sigma
_1}-u^{\sigma _1})+a_2(c_0^{\sigma }v^{\sigma _2}-u^{\sigma
_2})\geq 0, \\
\lim_{t\to 0^+} w(t)=0, \\
\lim_{t\to \infty} \frac{w(t)}{\rho (t)}=0.
\end{gather*}
 By Lemma \ref{lem3.2}, this implies that the function 
$w=c_0^{\sigma }v-u$ is nonnegative. 
By symmetry, we also have $v\leq c_0^{\sigma }u$. 
Hence $c_0^{\sigma }\in J$ and $c_0\leq c_0^{\sigma}$. 
Since $0\leq \sigma <1$, then $c_0=1$ and therefore $u=v$. 
\hfill\qed

\begin{example} \rm
Let $\sigma _1\in (-1,0)$, $\sigma _2\in (0,1)$ and $\lambda _1$, 
$\lambda _2<2$, such that $\frac{2-\lambda _1}{1-\sigma _1}\leq \frac{
2-\lambda _2}{1-\sigma _2}$. Let $\mu _1,\mu _2>2$ and $a_1,a_2$
be a positive continuous function on $(0,\infty )$ such that
\begin{equation*}
a_i(t)\approx \frac{1}{( A(t)) ^{2}}( \rho (t))
^{-\lambda _i}( 1+\rho (t)) ^{\lambda _i-\mu _i},\quad
\text{ for }i\in \{1,2\}.
\end{equation*}
 Then by Theorem \ref{thm1.5}, problem \eqref{e1.4} has a unique
positive continuous solution $u$ satisfying for $t>0$,
\begin{equation*}
u(t)\approx \begin{cases}
(\frac{\rho (t)}{1+\rho (t)})^{\frac{2-\lambda _1}{1-\sigma _1}}, &
\text{if }1+\sigma _1<\lambda _1<2,  \\
\frac{\rho (t)}{1+\rho (t)}(\log (\frac{2+2\rho (t)}{\rho (t)}))^{\frac{1}{
1-\sigma _2}}, & \text{if }\lambda _1=1+\sigma _1\text{ and }\lambda
_2=1+\sigma _2,  \\
\frac{\rho (t)}{1+\rho (t)}(\log (\frac{2+2\rho (t)}{\rho (t)}))^{\frac{1}{
1-\sigma _1}}, & \text{if }\lambda _1=1+\sigma _1\text{ and }\lambda
_2<1+\sigma _2,  \\
\frac{\rho (t)}{1+\rho (t)}, & \text{if }\lambda _1<1+\sigma _1.
\end{cases}
\end{equation*}
\end{example}

\subsection*{Acknowledgments}
This Project was funded by the National
Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City
for Science and Technology, Kingdom of Saudi Arabia, Award Number
(13-MAT1813-02).

 The author wants to thank the anonymous referee for the careful reading of the
manuscript,  and the useful suggestions.

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\end{document}