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

\AtBeginDocument{{\noindent\small
Tenth MSU Conference on Differential Equations and Computational
Simulations,\newline
\emph{Electronic Journal of Differential Equations},
Conference 23 (2016),  pp. 131--138.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document} \setcounter{page}{131}
\title[\hfilneg EJDE-2016/Conf/23 \hfil Existence of three solutions]
{Existence of three solutions for a two-point singular boundary-value
problem with an unbounded weight}

\author[D. Rajendran, E. Ko, R. Shivaji \hfil EJDE-2016/conf/23 \hfilneg]
{Dhanya Rajendran, Eunkyung Ko, Ratnasingham Shivaji}

\address{Dhanya Rajendran\newline
Departamento de Matem\'atica, 
Universidad de Concepci\'on, Chile}
\email{dhanya.tr@gmail.com}

\address{Eunkyung Ko \newline
PDE and Functional Analysis Research Center,
Department of Mathematical Sciences, Seoul National University,
Seoul, 151-747, South Korea}
\email{ekko1115@snu.ac.kr}

\address{Ratnasingham Shivaji \newline
Department of Mathematics and Statistics,
University of North Carolina at Greensboro, Greensboro, NC 27412, USA}
\email{shivaji@uncg.edu}


\thanks{Published March 21, 2016}
\subjclass[2010]{35J25, 35J55}
\keywords{Three solutions theorem; singular boundary value problems}

\begin{abstract}
 We show the existence of three solution for the singular 
 boundary-value problem
\begin{gather*}
 -z'' = h(t) \frac{f(z)}{z^\beta} \quad \text{in } (0,1) ,\\
 z(t)> 0 \quad \text{in } (0,1),\\
 z(0)= z(1)=0 ,
 \end{gather*}
 where $ 0 < \beta <1$, $f\in C^1([0,\infty), (0,\infty))$ and
 $ h\in C((0,1], (0,\infty))$ is such that $h(t)\leq C/t^\alpha$
 on $(0,1]$ for some $C>0$ and $0<\alpha<1-\beta$. When there exist
 two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where
 $\psi_1\leq \psi_2\leq \phi_1, \psi_1\leq \phi_2\leq \phi_1 $
 with $\psi_2 \not \leq \phi_2$, and $\psi_2 ,\phi_2$ are strict sub
 and super solutions. The establish the existence of at least  three solutions 
 $z_1, z_2, z_3$ satisfying  $z_1\in [\psi_1,\phi_2]$, $z_2\in [\psi_2,\phi_1]$ 
 and $z_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.
\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}
\allowdisplaybreaks

\section{Introduction}

In this article we consider the two point boundary-value problem
\begin{equation} \label{eI}
 \begin{gathered}
 -z'' = h(t) \frac{f(z)}{z^\beta} \quad \text{in } (0,1),\\
 z(t)> 0 \quad \text{in } (0,1),\\
 {z(0)} = z(1) =0 , \\
 \end{gathered}
\end{equation}
 where $ 0 < \beta <1$, $f\in C^1([0,\infty),(0,\infty))$
is nondecreasing, $ h\in C((0,1], (0,\infty))$ is such that
$h(t)\leq C/t^\alpha$ on $(0,1]$ for some $C>0$ and $0< \alpha<1-\beta$.
 In particular, we are interested in weights $h$ which are unbounded
at the origin.
This makes the reaction term in \eqref{eI} singular at $t=0$ not only due
to the term $z^{\beta}$ in the denominator but also due to this unbounded weight
$h$.

Our main focus in this paper is to establish the existence of three positive
solutions in $C^1[0,1]\cap C^2(0,1)$ when certain pair of sub-super solutions
can be constructed for \eqref{eI}. By a sub solution $\psi$ of \eqref{eI}
we mean a function $\psi\in C^2(0,1)\cap C[0,1]$ such that
 \begin{gather*}
 -\psi'' \leq h(t) \frac{f(\psi)}{\psi^\beta} \quad \text{in } (0,1),\\
 \psi(t)> 0 \quad \text{in } (0,1),\\
 \psi(0) = \psi(1)=0 ,
 \end{gather*}
and by a super solution of \eqref{eI} we mean a function
$\phi\in C^2(0,1)\cap C[0,1]$ such that
 \begin{gather*}
 -\phi'' \geq h(t) \frac{f(\phi)}{\phi^\beta} \quad \text{in } (0,1),\\
 \phi(t)> 0 \quad \text{in } (0,1),\\
 \phi(0) = \phi(1)=0.
 \end{gather*}
Let $g(z)=(f(z)-f(0))/z^\beta$, then problem \eqref{eI} can be
equivalently re-formulated as
\begin{equation} \label{eP}
 \begin{gathered}
 -z''- \frac{h(t)f(0)}{z^\beta} = h(t)g(z) \quad\text{in } (0,1),\\
z(0) =z(1) =0 .
\end{gathered}
\end{equation}


 By the mean value theorem $ g(t)=f'(s)t^{1-\beta}$ for some
$s\in (0,t)$. Since $0<\beta<1$ and $\lim_{s\to 0}|f'(s)|<\infty $,
we have $g(0)=0$. Thus we can treat $g$ as a H\"older
continuous function on $[0,\infty)$ with $g(0)=0$ and extend $g$ to be
identically zero on the negative $x$-axis. We assume that
\begin{itemize}
 \item[(G1)] There exists a non-negative constant $\tilde{k}$ such that
$\tilde{g}(t) = g(t)+\tilde{k} t $ is strictly increasing in
 $[0,\infty)$.
\end{itemize}

\begin{remark}\label{rem1} \rm
Without lost of generality, we assume throughout this article that $g$
is strictly increasing in $\mathbb{R}^+ $(i.e $\tilde{k}=0$ in (G1)).
If not, we can study
\begin{equation} \label{ePtilde}
\begin{gathered}
-z''-\frac{h(t)f(0)}{z^\beta} +\tilde{k}z = h(t) \tilde{g}(z)\quad \text{in }(0,1),\\
{z(0)} = z(1) =0
\end{gathered}
\end{equation}
instead of \eqref{eP} and establish our results.
 \end{remark}

In this article, we prove the following results:

 \begin{theorem}[Minimal and maximal solutions] \label{thm1.1}
 Let $\psi, \phi$ be positive sub and super solution of \eqref{eI} satisfying
$\psi\leq \phi$. Then there exists a minimal as well as a maximal solution for
\eqref{eI} in the ordered interval $[\psi,\phi]$.
 \end{theorem}

 \begin{theorem}[Three solution theorem] \label{thm1.2} 
 Suppose there exists two pairs of ordered sub and super solutions
$(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ of \eqref{eI} with the property that
 $\psi_1 \leq \psi_2 \leq \phi_1$, $\psi_1 \leq \phi_2 \leq \phi_1$ and
$\psi_2\not \leq \phi_2$. Additionally assume that $\psi_2 , \phi_2$ are
not solutions of \eqref{eI}. Then there exists at least three solutions
$z_1,z_2,z_3$ for \eqref{eI} where $z_1\in [\psi_1,\phi_2]$,
$z_2\in [\psi_2,\phi_1]$ and
$z_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.
\end{theorem}

We first note that when a sub solution $\psi$ and a super solution $\phi$
exists such that $\psi\leq \phi$ there are results in the history which
establish a solution $z$ for \eqref{eI} such that $z\in [\psi,\phi]$
(see \cite{Cui}). Here the author establishes the result by first studying the non
singular boundary value problem
\begin{equation} \label{eIepsilon}
\begin{gathered}
 -z'' = h(t) \frac{f(z)}{z^\beta} \quad \text{in } [\epsilon,1-\epsilon],\\
 z(\epsilon) = \psi(\epsilon),\quad z(1-\epsilon)=\psi(1-\epsilon) \\
 \end{gathered}
\end{equation}
for $\epsilon>0$ and then by analyzing the limit of the solution
$u_\epsilon\in [\psi, { \phi}]$ of (I$_\epsilon$) as $\epsilon\to 0$. In this paper we will
 provide a direct method and in Theorem \ref{thm1.1} we establish also
the existence of maximal and minimal solutions of \eqref{eI} in $[\psi, {\phi}]$.

Next, in \cite{EES} the authors study positive radial solutions on exterior
domains to problem of the form
\begin{equation} \label{eIE}
 \begin{gathered}
 -\Delta u = \lambda K(|x|) \frac{f(u)}{u^\beta} \quad \text{in } \Omega_E,\\
 u(x)=0 \text{ on }|x|=r_0,\\
 u(x)\to 0 \quad\text{as } |x| \to \infty,
 \end{gathered}
\end{equation}
where $\lambda>0$, $\Omega_E= \{x\in \mathbb{R}^N :|x|>r_0, {r_0>0}, N>2\}$
and $K\in C((r_0,\infty),(0,\infty))$ such that
$K(|x|)\to 0$ as $|x|\to \infty$.
Assume $0\leq K(r)\leq \frac{\tilde{C}}{r^{N+\mu_1}}$ where
$\beta (N-2)<\mu_1<N-2$.
 Then the change of variables $r=|x|$ and $t=(\frac{r}{r_0})^{2-N}$
transform \eqref{eIE} into
\begin{equation} \label{eIEtilde}
 \begin{gathered}
 -u'' = \lambda\tilde{h}(t) \frac{f(u)}{u^\beta} \quad \text{in } (0,1),\\
 u(0) = u(1)=0 , 
 \end{gathered}
\end{equation}
 where
\[
\tilde{h}(t)\leq \frac{\tilde{C}}{(N-2)^2 r_0^{N-2+\mu_1}}t^{-1+\frac{\mu_1}{N-2}}
\]
and hence this study reduces to study of positive solutions to the boundary-value
problem of the form \eqref{eI}. In \cite{EES}, the authors study
 classes of nonlinearities $f$ where for certain range of $\lambda$ they are able
to construct the two pairs of sub-super solutions as in
 Theorem \ref{thm1.2}. Using the result by Cui \cite{Cui} they were able to conclude
the existence of two positive solutions for these ranges of $\lambda$.
However, now using Theorem \ref{thm1.2} we conclude that there are in fact three
positive solutions in this range of $\lambda$.
 This three solutions theorem (Theorem \ref{thm1.2}) is motivated by earlier work for
non-singular problems by Amann\cite{amann}, Shivaji\cite{shivaji}, and by our
recent work in \cite{DES} for singular problems on bounded domain without
the burden of an unbounded weight in the reaction term.
We will establish some preliminaries and then prove Theorems 
.

\section{Proofs of main theorems}
 Now onwards instead of looking for a positive solution of \eqref{eI}
we work with the equivalent formulation \eqref{eP}.

\begin{lemma} \label{lem2.1}
 There exists a unique positive weak solution $\theta\in H^1_0(0,1)\cap C^2(0,1)$
to $-\theta''-\frac{h(t)}{\theta^\beta}=0$ in
 $(0,1)$ and $\theta(0)=\theta(1)=0$.
\end{lemma}

\begin{proof}
Let us define the functional
 \begin{equation}
 E_1(u)=\frac{1}{2}\int_0^1 |u'|^2-\frac{1}{1-\beta}\int_0^1 h(t)(u^{+})^{1-\beta}.
 \end{equation}
By the Sobolev embedding $H^1_0(0,1)\hookrightarrow C^{\gamma}[0,1]$ for some
$\gamma\in (0,1)$ and the fact that $\alpha < 1-\beta$
we have $E_1$ is a well-defined map in the entire space $H^1_0(0,1)$.
The functional $E_1$ restricted to the set
 $H^+= \{u\in H^1_0(0,1): u\geq 0\}$ is convex.
Let $\varphi_1$ be a positive eigenfunction corresponding to
the first eigenvalue $\lambda_1$ of $-u''=\lambda u$ with $u(0)=u(1)=0$.
Then, we note that for all $\epsilon$ small enough
$$
E_1(\epsilon \varphi_1)=\frac{\epsilon^2\lambda_1}{{2}} \int_0^1{\varphi_1}^2
- \frac{\epsilon^{1-\beta}}{{1-\beta}}\int_0^1 h(t)\varphi_1^{1-\beta}< 0 = E_1(0).
$$
Hence coercive and weakly lower semicontinuous functional $E_1$ admits a
non-zero global minimizer $\theta$ in $H^1_0(0,1)$.
We observe that $E_1(|\theta|)\leq E_1(\theta)$.
Unless $\theta^-\equiv 0$, this would contradict the fact that $\theta$ is
a global minimizer of $E_1$. Hence $\theta(t)\geq 0$ in $[0,1]$.
One can repeat the arguments in \cite[Lemma A.2]{GTI} and infer that the
functional $E_1$ is Gateaux differentiable at any $u\geq \epsilon \varphi_1(t)$,
and for $w\in H^1_0(0,1)$ we have
$$
\langle E_1'(u),w\rangle =\int_0^1 u'w'-\int_0^1 h(t)u^{-\beta} w.
$$
Thus to prove that global minimizer $\theta$ is a weak solution,
it suffices to prove the following claim.
\begin{quote}
Claim: $\theta(t)\geq \epsilon_0 \varphi_1(t)$ for some positive constant $\epsilon_0$.
\end{quote}
 For a given $\epsilon>0$, let $v=(\epsilon \varphi_1-\theta)^+$ and
$\Omega_+=\{x\in (0,1): v(x)>0\}$. On the contrary, suppose that $v$ does not
vanish in $\Omega$
for $\epsilon$ small enough, and then we derive a contradiction. Clearly,
$\theta+v \geq \epsilon \varphi_1$ and
\begin{equation} \label{eqn4.2}
\begin{aligned}
 \langle E_1'(\theta+v),v\rangle
&= \int_{\Omega_+} \epsilon \varphi_1'(\epsilon\varphi_1'-\theta')
 -\int_{\Omega_+} h(t)(\epsilon \varphi_1)^{-\beta}(\epsilon \varphi_1-\theta)\\
&= \lambda_1\int_{\Omega_{+}} (\epsilon \varphi_1)(\epsilon \varphi_1-\theta)
 -\int_{\Omega_+} h(t)(\epsilon \varphi_1)^{-\beta}(\epsilon \varphi_1-\theta)\\
&= \int_{\Omega_{+}}(\epsilon \varphi_1)^{-\beta} [\lambda_1(\epsilon \varphi_1)^{1+\beta}-h(t)](\epsilon\varphi_1-\theta).
\end{aligned}
\end{equation}
If we choose $\epsilon_0$ small enough so that
$\lambda_1(\epsilon \varphi_1)^{1+\beta}-\inf_{t\in (0,1)} h(t)<0$, then we have
 \begin{equation}\label{En}
 (E_1'(\theta+v),v)<0 \quad \text{for all } \epsilon\leq \epsilon_0.
 \end{equation}

To complete the claim we need to show that $\Omega_+$ is empty or
$v\equiv 0$ for $\epsilon$ small enough.
 Let $\xi(s)=E_1(\theta+s v)$ for $s\in [0,\infty)$.
The function $\xi : [0,\infty) \to \mathbb{R}$ is convex, since it is a
composition of the convex function $E_1$ restricted to $H^+$ with a linear
function. We already know that $\theta$
is a global minimizer of $E_1$ and hence we have
$\xi(s)\geq \xi(0)$. Also $\theta+sv \geq \text{max}\{\theta,\epsilon s \varphi_1\}
\geq s \epsilon \varphi_1 $ whenever $0<s\leq 1$.
Thus $\xi$ is differentiable for all $s\in (0,1]$. Also we note that
$\xi'$ is nondecreasing and is non-negative since $\xi(s)\geq \xi(0)$. Thus,
\begin{equation}\label{Ep}
 0\leq \xi'(1)-\xi'(s) = (E_1'(\theta+v),v)- \xi'(s) \leq (E_1'(\theta+v),v).
\end{equation}
From \eqref{En} and \eqref{Ep} we have a contradiction for all
$\epsilon \leq \epsilon_0$. Hence $v=0$, or in
other words $\theta \geq \epsilon_0 \varphi_1$. Finally, we conclude that $\theta(t)$
is a weak solution of $-\theta''-\frac{h(t)}{\theta^\beta}=0$ and by the interior
regularity $\theta \in C^2(0,1)$.
Since $h(t)>0$ we can prove the uniqueness of weak solution
by a standard test function approach(for e.g. see \cite[Lemma 3.2]{DES}).
\end{proof}

 \begin{lemma}\label{lem4.2}
 For a given nonnegative function $v\in C[0,1]$ there exists a unique weak
solution $w\in C^{1,\epsilon}[0,1]$ of
$-w''-\frac{h(t)}{w^\beta}=v(t)$ in $(0,1)$ and $w(0)=w(1)=0$.
Also there exists a constant $C=C(\|v\|_{\infty},\beta,\alpha)$ such that
 $$
\|w\|_{C^{1,\epsilon}[0,1]}\leq C \quad \text{where } \epsilon = 1-\beta-\alpha.
$$
 \end{lemma}

\begin{proof}
 The existence of a unique solution $w\in H^1_0(0,1)$ follows exactly as in
\cite[Lemma 4.1]{DGPS} or \cite[Lemma 3.2]{DES}.
 Note that
$-w''-\frac{h(t)}{w^\beta}=v(t) \geq 0 = -\theta''-\frac{h(t)}{\theta^\beta}$
and hence by comparison principle
 $w(t)\geq \theta(t)\geq \epsilon_0 \varphi_1$.
 By the Greens representation formula:
 \begin{equation}
 w(t)=\int_0^1 G(t,\xi)\Big(\frac{h(\xi)}{w^\beta}+v(\xi)\Big)d\xi,
 \end{equation}
where
$$
G(t,\xi)=\begin{cases}
 (1-t)\xi & 0\leq \xi\leq t\\
 (1-\xi)t & t\leq \xi\leq 1.
\end{cases}
$$
If we write ${h_1}(\xi)= \frac{h(\xi)}{w^\beta}+v(\xi)$,
then from the lower estimate on $w$ there exists
$C_0$ such that
$$
h_1(\xi) \leq \frac{C_0 \xi^{ {-\alpha} }}{ \varphi_1^{\beta}(\xi) }
\leq {C_0}\xi^{{-\alpha}}d(\xi)^{-\beta},
$$
where the distance function $d(\xi)$ is
 $$
d(\xi)=\begin{cases}
 \xi & 0\leq \xi\leq \frac{1}{2}\\
 (1-\xi) & \frac{1}{2}\leq \xi\leq 1.
 \end{cases}
$$
For $0<t<s<1$ we have
\[
w'(t)-w'(s)={ \int_t^s} h_1(\xi) d\xi.
\]
 We can write
$h_1(\xi)\leq \tilde{C} d(\xi)^{{-\alpha}-\beta}$ where $\tilde{C}$ depends
on $\|v\|_{\infty}$ and thus there exists a constant
$C=C(\|v\|_{\infty},\beta,\alpha)$ such that
\begin{equation}
 |w'(t)-w'(s)| \leq C |t-s|^{{1-\beta -\alpha}}
\end{equation}
which completes the proof.
\end{proof}

We now recall some results from Amann\cite{amann}.
Let $e\in C^2[0,1]$ denote the unique positive solution of
\begin{gather*}
-e''(s) = 1\quad \text{in }(0,1),\\
e(0)=e(1)=0.
\end{gather*}
Then $e(s)=\frac{1}{2}s(1-s)$ and $e'(1)=-e'(0)=-1$.
Also $e(s)\geq k\, d(s)$ for some constant $k>0$.
Let $C_e[0,1]$ be the set of functions in $u \in C_0[0,1]$ such that
 $-s e \leq u\leq s e$ for some $s>0$. $C_e[0,1]$ equipped with
$ \|u\|_e = \inf\{ s > 0 : -s e \leq u \leq s e\} $ is a Banach space.
Also the following continuous embedding holds
 \begin{equation}\label{emb}
 C_0^1[0,1] \hookrightarrow C_e[0,1] \hookrightarrow C_0[0,1].
 \end{equation}

Further $C_e[0,1]$ is an ordered Banach space(OBS) whose positive cone
$P_e = \{u \in C_e[0,1] : u(s)\geq 0\}$
is normal and has non empty interior.
In particular the interior $P_e^0$ consists of all those functions
$u\in {C_e[0,1]}$ with $s_1 e\leq u\leq s_2 e$ for some $s_1,s_2 >0$.

For a given $v\in C_e[0,1]$, let $\tilde{v}(t)=h(t)g(v(t))$.
Using the assumptions on $g$ and the blow up estimate on
$h$, we have $|\tilde{v}(t)|\leq C t^{{-\alpha}} |v(t)|^{1-\beta}$.
This upper bound implies that $\tilde{v}\in
C_0[0,1]$ and by Lemma \ref{lem4.2} there exists a unique solution
$w\in C^{1,\epsilon}[0,1]\cap H^1_0(0,1)$ solving
$-w''-\frac{f(0)h(t)}{w^{\beta}}=h(t)g(v(t))$. We make the following definition.

\begin{definition} \label{def2.1} \rm
 We define the operator $A_g: C_e[0,1]\to C^{1,\epsilon}_0[0,1]$ as $A_g(v)=w$,
where $w$ is the unique positive weak solution of
 $-w''-\frac{h(t)f(0)}{w^\beta}=h(t)g(v(t))$ in $(0,1)$ and $w(0)=w(1)=0$.
\end{definition}

 We first establish the following result.

\begin{proposition} \label{prop2.1}
 The map $A_g: C_e[0,1]\to C_e[0,1]$ is completely continuous and strictly
increasing.
\end{proposition}

\begin{proof}
 Clearly $A_g$ maps $C_e[0,1]$ into $C^{1,\epsilon}_0[0,1]$ and we need to prove
that the mapping is continuous.
For a proof consider $v, v_0 \in C_e[0,1]$ and $\|v-v_0\|_{C_e}<\epsilon$.
Let $A_g(v_0)=w_0$ and $A_g(v)=w$. Then from the weak formulation
of the solution
\begin{align*}
&\int_0^1 |w'-w_0'|^2 \\
&= \int_0^1 h(t)f(0)\left(\frac{1}{w^\beta}-\frac{1}{w_0^\beta} \right)(w-w_0)
+\int_0^1 h(t) (g(v)-g(v_0))(w-w_0)\\
&\leq \int_0^1 h(t) (g(v)-g(v_0))(w-w_0).
\end{align*}
Since $g$ is assumed to be H\"older continuous of exponent $1-\beta$ and by
the continuous embedding \eqref{emb} we have the  estimate
\begin{align*}
|g(v(t))-g(v_0(t))|
&\leq C_0|v(t)-v_0(t)|^{1-\beta} \\
&\leq C_0 \|v-v_0\|_{C_0(\overline{\Omega})}^{1-\beta}\\
&\leq C_1 \|v-v_0\|_{C_e(\Omega)}^{1-\beta}<C_1 \epsilon^{1-\beta}.
\end{align*}
Hence,
$$
 \int_0^1 |w'-w_0'|^2  \leq  C_2\, \epsilon^{1-\beta}\int_0^1 \frac{ |w-w_0|}{t}t^{{1-\alpha}}.
$$
Now by Hardy's inequality we have
$$
\int_0^1 |w'-w_0'|^2 \leq C \epsilon^{1-\beta}\|w-w_0\|_{H^1_0(0,1)} .
$$
Therefore if $v_n\to v_0$ in $C_e[0,1]$ then the corresponding solutions
$A_g(v_n)=w_n\to w_0=A_g(v_0) $ in $H^1_0(0,1)$. Next we note that if
$v_n\to v_0$ in $C_e[0,1]$ then $\tilde{v_n}=h(t)g(v_n)$
is uniformly bounded in $C_0[0,1]$ and hence by Lemma \ref{lem4.2},
$\|w_n\|_{C^{1,\epsilon}[0,1]}$
is bounded. By Ascoli-Arzela theorem $w_n$ has a convergent subsequence in
$C^{1,\epsilon'}[0,1]$ for every $\epsilon'<\epsilon$ and the from the previous discussion
the limit has to be $w_0$. Hence $A_g: C_e[0,1] \to C_0^{1,\epsilon'}[0,1]$
is continuous. Since $C^{1,\epsilon}_0[0,1] \subset \subset C^1_0[0,1]$
(compact imbedding), we have $A_g: C_e[0,1] \to C_0^1[0,1]$
is completely continuous.
Therefore, $A_g: C_e[0,1]\to C_e[0,1]$ is completely continuous.

To prove the map $A_g$ is strictly increasing we assume that $g$ is strictly
increasing (otherwise see Remark \ref{rem1}). We need to show
that if $v_1\leq v_2$, $v_1\not = v_2$ then $A_g(v_1)<A_g(v_2)$.
 By the test function approach one can easily show that
$0<w_1=A_g(v_1)\leq A_g(v_2)=w_2$.
Let $\rho:(0,1)\to \mathbb{R}$ be such that
$w_1(t)\leq \rho(t) \leq w_2(t)$ is defined by mean value theorem.
Then we have
\begin{equation}\label{eqn4.5}
 -(w_2-w_1)''+\frac{h(t)f(0)}{\rho(t)^{1+\beta}}(w_2-w_1)
=h(t)\left(g(v_2)-g(v_1)\right) \geq 0.
\end{equation}
Since $(w_1-w_2)|_{\{t=0,1\}}=0$, we have by
Theorem 3 in Chapter 1 of Protter and Weinberger \cite{PW},
$w_1<w_2$, or in other words $A_g$ is strictly increasing.
\end{proof}


\begin{lemma} \label{lem2.3}
 The map $A_g: C_e[0,1] \to C_e[0,1]$ is strongly increasing, i.e.
 $A_g(v_2)-A_g(v_1)\in P_e^0$.
 \end{lemma}

\begin{proof}
The idea of the proof is same as in \cite[Theorem 3.6]{DES} 
with a slight change in the singular exponent $\beta$. Let us write
$\tilde{w}=(w_2-w_1)$, then from equation \eqref{eqn4.5} and by the upper 
bound on $\frac{h(t)}{\rho^{\beta+1}}$ we have
$-\tilde{w}''+{ c \,\tilde{w}}\, {d(t)^{ {-1-\beta-\alpha} }}\geq 0$. I
f we denote $\beta'={\alpha}+\beta$, then $\beta'\in (0,1)$.
Thus we obtain
\begin{gather*}
 -\tilde{w}''+\frac{c \tilde{w}}{d(t)^{1+\beta'}} \geq  0\quad \text{in } (0,1),\\
\tilde{w} > 0 \text{ in } (0,1), \\[3mm]
 \tilde{w}(0)= \tilde{w}(1)=0.
\end{gather*}
Rest of the proof is exactly as in \cite[Theorem 3.6]{DES} 
and hence we skip the details.
\end{proof}

Now Proposition \ref{prop2.1} combined with \cite[Corollary 6.2]{amann} 
easily establishes the proof of Theorem \ref{thm1.1}. 
Further, since Lemma \ref{lem2.3} holds, repeating
the arguments in \cite{DES} the proof of Theorem \ref{thm1.2} follows.


\subsection*{Acknowledgements} 
The authors want to thank the anonymous referee for suggesting a number 
of valuable improvements and corrections.
This work was partially supported by a grant from the Simons 
Foundation (\#317872) to Ratnasingham Shivaji.


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