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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 32, pp. 1--29.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/32\hfil Multi-point BVPs with general growth]
{Multiple positive solutions for singular multi-point 
 boundary-value problem with general growth on the positive half line}

\author[S. Djebali, K. Mebarki \hfil EJDE-2011/32\hfilneg]
{Sma\"il Djebali, Karima Mebarki}  % in alphabetical order

\address{Sma\"il Djebali \newline
Department of Mathematics, E.K.S. \\
PO Box 92, 16050 Kouba. Algiers, Algeria}
\email{djebali@ens-kouba.dz, djebali@hotmail.com}

\address{Karima Mebarki \newline
Department of Mathematics \\
A.E. Mira University, 06000. Bejaia, Algeria}
\email{mebarqi@hotmail.fr}

\thanks{Submitted August 26, 2010. Published February 23, 2011.}
\subjclass[2000]{34B15, 34B16, 34B18, 34B40}
\keywords{Fixed point; multiple solutions; multi-point; singularity;
\hfill\break\indent infinite interval; cone}

\begin{abstract}
This work is devoted to the existence of nontrivial positive
solutions for a class of second-order nonlinear multi-point
boundary-value problems on the positive half-line. The novelty of
this work is that the nonlinearity may exhibit a singularity at
the origin simultaneously with respect to the solution and its
derivative; moreover it satisfies quite general growth conditions
far from the origin, including polynomial growth. New existence
results of single, twin and triple solutions are proved using the
fixed point index theory on appropriate cones in weighted Banach
spaces together with two-functional and three-functional fixed
point theorems. The singularity is treated by means of
approximation and compactness arguments. The proofs of the
existence results rely heavily on several sharp estimates and
useful properties of the corresponding Green's function.
\end{abstract}

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


\section{Introduction}

This article concerns the existence of positive
solutions to the multi-point boundary value problem posed on the
positive half-line:
\begin{equation}\label{GP1}
\begin{gathered}
-y''+cy'+\lambda y=\Phi(t)f(t,y(t),e^{-ct}y'(t)),\quad t\in I\\
y(0)=\sum_{i=1}^{n}k_iy(\xi_i),\quad\lim_{t\to\infty}e^{-ct}y'(t)=0,
\end{gathered}
\end{equation}
where, for $i\in\{1,\dots,n\}$, $k_i\geq0$ and the multi-points
$0<\xi_1<\xi_2<\dots<\xi_n<\infty$ satisfy
\begin{equation} \label{H0}
\sum_{i=1}^{n}k_ie^{r_2\xi_i}<1,
\end{equation}
and where
$$
r_2=\frac{c-\sqrt{c^2+4\lambda}}{2}<0<r_1=
\frac{c+\sqrt{c^2+4\lambda}}{2}
$$
are the roots of the algebraic equation $-r^2+cr+\lambda=0$.
The parameters $c$ and $\lambda$ are real positive constants while the
function
$f=f(t,y,z): I^2\times\mathbb{R}^*\to\mathbb{R}^+$
is continuous and is allowed to have space singularities at $y=0$
and/or $z=0$, and $\Phi:I\to I$ is a continuous function.
Recall that $f$ is said to be singular at $y=0$ if
$\lim_{y\to\,0}f(t,y,z)=+\infty$ uniformly in
$(t,z)\in I\times\mathbb{R}^*$. Here and hereafter
$I:=(0,+\infty)$ denotes the set of positive real numbers,
$\mathbb{R}^+\colon=[0,+\infty)$, and
$\mathbb{R}^*\colon=\mathbb{R}\setminus\{0\}$.

Throughout this paper, by positive solution we mean $y\in
C^1([0,\infty))$ such that $y''$ exists and $y$ satisfies
\eqref{GP1} with $y(t)\ge0$ on $(0,\infty)$.

Many problems in physics, chemistry and biology are governed by
boundary value problems on the half-line, e.g., the flow of a
premixed mixture inducing the propagation of a nonadiabatic flame
in a long tube. For instance, the equation
$$
-y''(t)+cy'(t)+\lambda y(t)=f(t,y(t))
$$
subject to the boundary conditions
$$
y(0)=y(+\infty)=0
$$
extends the classical Fisher-Kolmogorov model equation (see
\cite{Fisher}) with no heat exchange, i.e. $\lambda=0$. The
positive nonlinear term is governed by classical physical laws. In
combustion theory, the source term in the energy equation obeys
Arrhenius' Law where $f=f(y)$ behaves as $y^ne^{-y}$ near positive
infinity (see e.g., \cite{Aris, Bai, Brit}). This motivates the
general growth of the nonlinearity considered in this work,
extending polynomials. In epidemiology, the propagation of
epidemics through given populations is governed by the generalized
Fisher autonomous equation $-y''+cy'+\lambda y=yh(y)$ (see
\cite{DKM, DMe3} for a mathematical investigation). Here the
positive constant $c$ is the velocity of the travelling wave and
the real parameter $\lambda$ is a removal rate \cite{Mur}. The
function $y$ represents a density of infectives. Thus, only
positive solutions corresponding to a density, a
temperature,\dots are useful from a physical point of view.

Moreover, various physiological processes in non-Newtonian fluid
theory, boundary layer theory and nonlinear phenomena (see e.g.,
\cite{ORegan}) are modelled by singular equations such that the
Emden-Fowler equation $y''=-\varphi(t)y^{-\gamma}\;(\gamma>0)$.
Also, the boundary value problem for the electrical potential in
an isolated neutral atom was derived in 1927 independently by
Thomas \cite{Thomas} and  Fermi \cite{Fermi}; it can be written
as
\begin{gather*}
y''=\sqrt{y^3/t}\\
y(0)=1,\quad y(+\infty)=0.
\end{gather*}
Another example is provided by the boundary layer equation for
steady flow over a semi-infinite plate (see \cite{CallegNach}):
\begin{gather*}
y''=-\frac{t}{2y^2}\\
y(0)=y(+\infty)=0.
\end{gather*}
These behaviors of the nonlinearities have motivated our
investigation of problem $\eqref{GP1}$ with a nonlinearity allowed
to have a singularity not only in $y$ but also in $y'$.

There have been recently so much work devoted to the investigation
of existence of positive solutions for boundary value problems on
infinite intervals of the real line and where the nonlinearity
satisfies either superlinear or sublinear growth assumptions (see
\cite{DKM, DMe1, DMe2, GuoGe, TianGe, TianGeShan} and the
references therein). A few methods have been employed to deal with
such problems which lack compactness; we cite upper and lower
solution techniques \cite{OregYanAgar}, fixed point theorems in
special Banach spaces and index fixed point theory on cones of
special Banach spaces \cite{BaiGe, LianGe, TianGe} as well as
diagonalization processes. Existence of single or multiple
solutions have been proved for two-point boundary value problems,
three-point and even multi-point BVPs in \cite{LianGe, Ma,
TianGe, TianGeShan}. We point out that several existence results
for general problems posed on unbounded intervals may be found in
the book by Agarwal and O'Regan \cite{AgaOR}.

In \cite{LiuLiuWu}, the authors have recently considered the
generalized Fisher equation $-y''+py'+qy=h(t)f(t,y)$ with $h$
singular in time while the nonlinearity $f$ may change sign. When
$f$ further depends on the first derivative, existence of multiple
solutions is given in \cite{DMe4} and the nonlinearity includes
sublinear and superlinear growth conditions; fixed point theory in cones of
special Banach spaces is employed. In \cite{DSaifi, DSY}, the
authors combine the fixed point index theory with the upper and
lower solution method to prove existence of solutions when the
nonlinearity satisfies various growth assumptions.

The second-order differential equation
$(p(t)y'(t))'+\lambda\phi(t)f(t,y(t))=0$ with
$\lim_{t\to+\infty}p(t)y'(t)=0$ as a boundary condition is
studied in \cite{LianGe, ZhangLiuWu} while the same equation where
$f$ also depends on $y'$ is considered in \cite{TianGeShan} with
Dirichlet condition at positive infinity; fixed point theorems in
cones are used to prove existence of positive solutions; the
condition $\int_0^{+\infty}\frac{dt}{p(t)}<\infty$ is assumed. A
discussion along with the smallness of the parameter $\lambda$ is also
given in \cite{WangLiuWu} for a nonlinearity of the form
$\lambda(f(t,y)-k^2y)$.

A three-point boundary value problem associated with the
Sturm-Liouville differential equation
\[
\big(\frac{1}{p(t)}(p(t)y'(t)\big)'+q(t)f(t,y(t),p(t)y'(t))=0
\]
is discussed in \cite{KangWei} and \cite{SunSunDebnath} with
$\lim_{t\to+\infty}p(t)y'(t)=b\ge0;$ the technique of upper
and lower solutions and the theory of fixed point theory are
employed to get existence of multiple solutions. The same
technique is employed in \cite{LianWangGe} when $f$ does not
depend on the first derivative. Notice that this equation is also
investigated in \cite{YanOregAgar} and existence of multiple
solutions is proved when $f$ may be singular at $y=0$ and $py'=0$.
We point out that in all of these works, the conditions
$\int_0^{+\infty}\frac{dt}{p(t)}<\infty$ is assumed which is not
the case in the present work since $p(t)=e^{-ct}$.

Our aim in this work is further to extend some of these works to
the case in which a positive nonlinearity does also depend on the
first derivative and is allowed to be singular at the origin in
both its second and third arguments; in addition it satisfies
general growth far from the singular origin, extending the
classical polynomial growth. We prove existence and multiplicity
of nontrivial positive solutions in a weighted Banach space. The
singularity of the nonlinearity is treated by approximating a
fixed point operator with the help of some compactness arguments.

The proofs of our existence theorems rely on recent fixed point
theorems of two or three functionals \cite{BaiGe, LiHan} together
with the fixed point index theory in cones of Banach spaces
\cite{GuoLak}. Some preliminaries needed to transform problem
\eqref{GP1} into a fixed point theorem are presented in Section 2
together with appropriate compactness criteria. In particular,
essential properties of the Green's function are given and the
main assumptions are enunciated. Then, we construct a special cone
in a weighted Banach space. The properties of a fixed point
operator denoted $T$ are studies in detail in Section 3. Section 4
is devoted to proving three existence results successively of a
single, twin and triple solutions. The existence theorems obtained
in this paper extend similar results available in the literature
in case the nonlinearity $f$ is either nonsingular or does not
depend on the first derivative (see e.g., \cite{DKM, DMe1, DMe2,
DMouss, LianGe, Ma, TianGe, TianGeShan}). We end the paper with an
example of application in Section 5 and some concluding remarks in
Section 6.

\section{Functional framework}

In this section, we present some definitions and lemmas which will
be needed in the proofs of the main results. Let
$$
C_l([0,\infty),\mathbb{R})=\{y\in C([0,\infty),\mathbb{R}):
\lim_{t\to\infty}y(t)\text{ exists}\}.
$$
It is easy to see that $C_l$ is a Banach space with the norm
$\|y\|_l=\sup_{t\in [0,\infty)}|y(t)|$. For a real
parameter $\theta>r_1$, consider the Banach space of Bielecki type
\cite{Biel} defined by
$$
X=C_{\infty}^1([0,\infty),\mathbb{R})=\big\{y\in
C^1([0,\infty),\mathbb{R}):
\lim_{t\to+\infty}\frac{y(t)}{e^{\theta t}}\,\text{ and
}\, \lim_{t\to+\infty}\frac{y'(t)}{e^{\theta t}}\,\text{
exist}\big\}
$$
with norm
$$
\|y\|_{\theta}=\max\{\Vert y\Vert_1,\Vert y\Vert_2\},
$$
where
$$
\Vert y\Vert_1=\sup_{t\in[0,\infty)}\frac{|y(t)|}{e^{\theta t}},\quad
\Vert y\Vert_2=\sup_{t\in
[0,\infty)}\frac{|y'(t)|}{e^{\theta t}}.
$$

\begin{lemma}[{\cite[Lemma 2.1]{DMe4}}] \label{lem2.1}
$X=C_{\infty}^1$ is a Banach space.
\end{lemma}

For some $0<\gamma<\delta$, let
\begin{equation}\label{Lambda}
0<\Lambda_0:=\min\{e^{r_2\delta},e^{r_1\gamma}-e^{r_2\gamma}\},\quad
 \Lambda=\Lambda_0 \max_{t\in[\gamma,\delta]}\sigma(t).
\end{equation}
Since $r_2<0$, we have $0<\Lambda_0<1$. Here
\[
\sigma(t)=
\begin{cases}
\min\Big(\frac{1-e^{(r_2-r_1)t}}{2r_1
(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})e^{(r_1-\theta)t}},
 \frac{1}{|r_2|}\Big),& t<\xi_1;\\[3pt]
\min\Big(\frac{1-\sum_{i=1}^{j}k_ie^{r_2\xi_i}-e^{(r_2-r_1)t}(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i})}
{2r_1(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})e^{(r_1-\theta)t}},
 \frac{1}{|r_2|}\Big),\\
\qquad  0<\xi_j\leq t\leq\xi_{j+1},\;j=1,2,\dots ,n-1;\\[3pt]
\min\Big(\frac{1-\sum_{i=1}^{j}k_ie^{r_2\xi_i}-e^{(r_2-r_1)t}(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})}
{2r_1(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})e^{(r_1-\theta)t}},
 \frac{1}{|r_2|}\Big),& t\ge\xi_n.
\end{cases}
\]
Then define the positive cone
\begin{equation}\label{cone}
\mathcal{P}=\big\{y\in X: y(t)\ge 0 \text{ on } \mathbb{R}^+,\;
y(t)\ge\Lambda\Vert y\Vert_2,\; \forall\,t\in[\gamma,\delta]\text{
and }\, y(0)=\sum_{i=1}^{n}k_iy(\xi_i) \big\}.
\end{equation}

\begin{lemma} \label{lem2.2}
Let $\rho=\frac{1}{\theta(1-\sum_{i=1}^{n}k_ie^{\theta
\xi_i})}$. Then
$\|y\|_{1}\le \rho\|y\|_2$ for all $y\in\mathcal{P}$.
\end{lemma}

\begin{proof}
Since $y(0)=\sum_{i=1}^{n}k_iy(\xi_i)$, then for every
$t\in\mathbb{R}^+$, we have
\begin{align*}
\frac{y(t)}{e^{\theta t}}
&= e^{-\theta t}\Big\{\int_0^ty'(s)ds+y(0)\Big\}\\
&= e^{-\theta t}\Big\{\int_0^ty'(s)ds+\sum_{i=1}^{n}k_iy(\xi_i)\Big\}\\
&= e^{-\theta t}\Big\{\int_0^te^{\theta s}\frac{y'(s)}{e^{\theta
s}}ds
+\sum_{i=1}^{n}k_ie^{\theta \xi_i}\frac{y(\xi_i)}{e^{\theta\xi_i}}\Big\}\\
&\leq e^{-\theta t}\Big\{\frac{1}{\theta}(e^{\theta t}-1)\|y\|_2
+\sum_{i=1}^{n}k_ie^{\theta\xi_i} \|y\|_1\Big\}\\
&\leq \frac{1}{\theta}(1-e^{-\theta t})\|y\|_2
+\sum_{i=1}^{n}k_ie^{\theta\xi_i} \|y\|_1.
\end{align*}
Passing to the supremum over $t\ge0$, we complete the proof.
\end{proof}

Arguing as in \cite[Lemma 2.2]{DMe4}, we deduce the following
result.

\begin{lemma}\label{lemp}
Let $y\in \mathcal{P}$. Then, for any $t\in[\gamma,\delta]$, we
have
$y(t)\ge\Gamma\|y\|_{\theta}$,
 where $\Gamma=\Lambda/\max(1,\rho)$.
\end{lemma}

\subsection{Construction of the Green's function}

In the following lemma which generalizes \cite[Lemma 2.4]{DMe4},
we determine the Green's function for problem \eqref{GP1}.

\begin{lemma}\label{lemm5*}
Let $v$ be a continuous function such that
$\int_0^{\infty}e^{-r_1s}v(s)ds<\infty$ and
$\lim_{s\to+\infty}e^{-cs}v(s)=0$. Then $y\in C^1(I)$ is a
solution of
\begin{equation}\label{GP2}
\begin{gathered}
-y''+cy'+\lambda y=v(t),\quad t\in I\\
y(0)=\sum_{i=1}^{n}k_iy(\xi_i),\quad
\lim_{t\to\infty}\frac{y'(t)}{e^{ct}}=0,
\end{gathered}
\end{equation}
if and only if it may be expressed in the form
\begin{equation}\label{eq1}
y(t)=\int_0^{\infty}G(t,s)v(s)ds,\quad t\in I.
\end{equation}
Hereafter the positive Green's function $G$ is defined on
$I\times I$ by $G(t,s)=\frac{1}{\Delta}G^1(t,s)$ with
$\Delta=(r_1-r_2)(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})$ and
\[
G^1(t,s)=\begin{cases}
e^{r_2t}(e^{-r_2s}-e^{-r_1s}),\quad\text{if }
0<s\leq \min(t,\xi_1)<\infty;\\[3pt]
e^{r_1(t-s)}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})-e^{r_2t}
\Big(e^{-r_1s}-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)}\Big),\\
\quad\text{if }
0<t\leq s\leq\xi_1<\infty;\\[3pt]
e^{r_2t}\Big(e^{-r_2s}(1-\sum_{i=1}^{j}k_ie^{r_2\xi_i})
-e^{-r_1s}(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i})\Big),\\
\quad\text{if } 0<\xi_j\leq s\leq \xi_{j+1},\;s\le t,\; j=1,2,\dots ,
n-1;\\[3pt]
e^{-r_1s}\Big(e^{r_1t}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
-e^{r_2t}(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\\
- \sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s})\Big),\\
\quad\text{if } 0<\xi_j\leq s\leq \xi_{j+1},\;t\le s,\;
 j=1,2,\dots ,n-1;\\[3pt]
e^{r_2t}\Big(e^{-r_2s}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
-e^{-r_1s}(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})\Big),\\
\quad\text{if }0<\xi_n\leq s\leq t<\infty;\\[3pt]
e^{-r_1s}\Big(e^{r_1t}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})\\
 -e^{r_2t}(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})\big),\\
\quad\text{if }0<\max(\xi_n,t)\leq s<\infty.
\end{cases}
\]
\end{lemma}

\begin{proof}
(a) It is easy to show that the general solution of the
equation in the boundary value problem \eqref{GP2} reads
\begin{equation}\label{eq6}
y(t)=\frac{1}{r_1-r_2}\Big(Ae^{r_1t}+Be^{r_2t}
+\int_0^t\left(e^{r_2(t-s)}-e^{r_1(t-s)}\right)v(s)ds\Big)
\end{equation}
where
$A=y'(0)-r_2y(0)$ and $B=r_1y(0)-y'(0)$. Differentiating
\eqref{eq6} yields
\begin{equation}\label{eq7}
y'(t)=\frac{1}{r_1-r_2}\Big(Ar_1e^{r_1t}+Br_2e^{r_2t}
+\int_0^t(r_2e^{r_2(t-s)}-r_1e^{r_1(t-s)})v(s)ds\Big).
\end{equation}
 From \eqref{GP2} and \eqref{eq6}, we obtain
\begin{align*}
0&= y(0)-\sum_{i=1}^{n}k_iy(\xi_i)\\
&= \frac{1}{r_1-r_2}
\Big(A+B-\sum_{i=1}^{n}k_i(Ae^{r_1\xi_i}+Be^{r_2\xi_i})\\
&\quad + \int_0^{\xi_i}\left(e^{r_2(\xi_i-s)}
 -e^{r_1(\xi_i-s)}\right)v(s)ds\Big);
\end{align*}
that is,
\begin{equation}\label{eq8}
(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})A+(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})B=
\int_0^{\xi_i}\left(e^{r_2(\xi_i-s)}-e^{r_1(\xi_i-s)}\right)v(s)ds.
\end{equation}
Moreover, \eqref{eq7} yields
$$
\frac{y'(t)}{e^{ct}}=\frac{\Sigma(t)}{r_1-r_2}
$$
where
\begin{align*}
\Sigma(t)&=Ar_1e^{(r_1-c)t}+Br_2e^{(r_2-c)t}\\
&\quad +r_2e^{(r_2-c)t}\int_0^te^{-r_2s}v(s)ds
-r_1e^{(r_1-c)t}\int_0^te^{-r_1s}v(s)ds.
\end{align*}
We claim that
\begin{equation}\label{eq&1}
\lim_{t\to\infty}e^{(r_2-c)t}\int_0^te^{-r_2s}v(s)ds=0.
\end{equation}
Indeed, if $\int_0^{\infty}e^{-r_2s}v(s)ds<\infty$, then
\eqref{eq&1} holds. Now assume
$\int_0^{\infty}e^{-r_2s}v(s)ds=\infty$. Since
$\lim_{s\to\infty}e^{-cs}v(s)=0$, L'Hospital's rule
yields
\begin{align*}
\lim_{t\to\infty}e^{(r_2-c)t}\int_0^te^{-r_2s}v(s)ds
&= \lim_{t\to\infty}\frac{\int_0^te^{-r_2s}v(s)ds}
{e^{(c-r_2)t}} \\
&=  \lim_{t\to\infty}\frac{e^{-r_2t}v(t)}
{(c-r_2)e^{(c-r_2)t}}\\
&= \lim_{t\to\infty}\frac{e^{-ct}v(t)} {c-r_2}=0.
\end{align*}
From \eqref{eq8}, \eqref{eq&1} and the boundary conditions, we
find the values
\begin{gather*}
A=\int_0^{\infty}e^{-r_1s}v(s)ds,\\
\begin{split}
B&=(1-\sum_{i=1}^{n}k_ie^{r_1\xi_2})^{-1}
\Big\{\sum_{i=1}^{n}\int_0^{\xi_i}(e^{r_2(\xi_i-s)}
-e^{r_1(\xi_i-s)})v(s)ds\\
&\quad -(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})
\int_0^{\infty}e^{-r_1s}v(s)ds\Big\}.
\end{split}
\end{gather*}
By substitution in \eqref{eq6}, we obtain
\begin{align*}
y(t)&=
\frac{1}{r_1-r_2}\Big(\int_0^{\infty}e^{r_1(t-s)}v(s)ds\\
&\quad +
\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\big)^{-1}
\sum_{i=1}^{n}\int_0^{\xi_i}(e^{r_2(t+\xi_i-s)}
-e^{r_1(\xi_i-s)+r_2t})v(s)ds\Big)\\
&\quad -\Big(\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\big)^{-1}
\big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\big)
\int_0^{\infty}e^{r_2t-r_1s}v(s)ds\\
&\quad -\int_0^t(e^{r_2(t-s)}-e^{r_1(t-s)})v(s)ds\Big)
:=\frac{1}{\Delta}y_1(t),
\end{align*}
with
\[
y_1(t)=\begin{cases}
\int_0^te^{r_2t}(e^{-r_2s}-e^{-r_1s})v(s)ds
+\int_t^{\xi_1}\Big((1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})e^{r_1(t-s)}\\
- e^{r_2t}(e^{-r_1s}-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)})\Big)v(s)ds\\
+\sum_{i=1}^{n}k_i\int_{\xi_1}^{\xi_i}(e^{r_2(\xi_i-s)}-e^{r_1(\xi_i-s)})v(s)ds\\
+\int_{\xi_1}^{+\infty}e^{-r_1s}
\Big((1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
e^{r_1t}-(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})e^{r_2t}\Big)v(s)ds,\\
\qquad \text{if } t\leq\xi_1;
\\[3pt]
(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
\int_{\xi_j}^{\xi_{j+1}}e^{r_1(t-s)}v(s)\\
-(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})
 \int_{\xi_j}^{\xi_{j+1}}e^{-r_1s}v(s)ds\\
+e^{r_2t}\Big(\sum_{i=j+1}^{n}k_i\int_{0}^{\xi_i}(e^{r_2(\xi_i-s)}
 -e^{r_1(\xi_i-s)})v(s)\Big),\\
\qquad\text{if } \xi_{j}\leq t\leq\xi_{j+1},\,j=1,2,\dots ,n-1;
\\[3pt]
\int_0^{\xi_n}e^{r_2t}(e^{-r_2s}-e^{-r_1s})v(s)ds\\
+\int_{\xi_n}^te^{r_2t}\Big((1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})e^{-r_2s}-
(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})e^{-r_1s}\Big)v(s)ds\\
+\int_t^{+\infty}e^{-r_1s}\Big((1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})e^{r_1t}-
(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})e^{r_2t}\Big)v(s)ds,\\
\qquad\text{if } t\ge\xi_n
\end{cases}
\]
whence the form of the Green's function $G$.

(b) Conversely, let $y\in C^1(I)$ be as defined by \eqref{eq1}.
 A direct differentiation of \eqref{eq1} yields
\begin{equation}\label{eq9}
y'(t)=\int_0^{\infty}G_t(t,s)v(s)ds,\quad t\in I,
\end{equation}
where $G_t(t,s)=\frac{1}{\Delta}G^1_t(t,s)$ is the partial
derivative of $G(t,s)$ with respect to $t$ and
\[
G_t^1(t,s)= \begin{cases}
r_2e^{r_2t}(e^{-r_2s}-e^{-r_1s}),\quad\text{if }
0<s\leq\min(t,\xi_1)<\infty;
\\[3pt]
r_1e^{r_1(t-s)}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})-r_2e^{r_2t}
\Big(e^{-r_1s}-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)}\Big),\\
\qquad\text{if } 0<t\leq s\leq\xi_1<\infty;
\\[3pt]
r_2e^{r_2t}\Big(e^{-r_2s}(1-\sum_{i=1}^{j}k_ie^{r_2\xi_i})
-e^{-r_1s}(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i})\Big),\\
\qquad \text{if }0<\xi_j\leq s\leq \xi_{j+1},\;s\le t,
\; j=1,2,\dots ,n-1;
\\[3pt]
e^{-r_1s}\Big(r_1e^{r_1t}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
-r_2e^{r_2t}(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\\
- \sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s})\Big),\\
\qquad\text{if } 0<\xi_j\leq s\leq \xi_{j+1},\;t\le s,\;
j=1,2,\dots ,n-1;
\\[3pt]
r_2e^{r_2t}\Big(e^{-r_2s}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
-e^{-r_1s}(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})\Big),\\
\qquad\text{if }0<\xi_n\leq s\leq t<\infty;
\\[3pt]
e^{-r_1s}\Big(r_1e^{r_1t}(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i})
-r_2e^{r_2t}(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})\Big),\\
\qquad\text{if } 0<\max(\xi_n,t)\leq s<\infty.
\end{cases}
\]
Differentiating again \eqref{eq9} yields
\begin{align*}
y''(t) &=  -v(t)+c\int_0^{\infty}G_t(t,s)v(s)ds+
\lambda\int_0^{\infty}G(t,s)v(s)ds\\
&=  -v(t)+c y'(t)+ \lambda y(t),\quad t\in I.
\end{align*}
Hence $y\in C^1(I)$ and $y$ satisfies \eqref{GP2}.
\end{proof}

The following two lemmas are crucial; the proofs are lengthy; so
we only prove the second one.

\begin{lemma}\label{lem4}
The function $G(t,s)$ given by Lemma \ref{lemm5*} satisfies
\begin{itemize}
\item[(a)] $G(t,s)\geq0$ for all $t,s \in I$

\item[(b)] $e^{-\mu t}G(t,s)\leq e^{-r_1s}G(s,s)$,
 for all $t, s\in I$  and all $\mu\geq r_1$.

\item[(c)] $G(t,s)\geq\Lambda_0G(s,s)e^{-r_1s}$ for all
$t\in[\gamma, \delta]$ and all $s\in I$,

\end{itemize}
where $\Lambda_0$ is as defined by \eqref{Lambda}.
\end{lemma}

\begin{lemma}\label{lem5}
Assume that
\begin{equation} \label{(H00)}
\begin{gathered}
1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}>0,\\
1-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}>0, \quad 0<s\leq \xi_1\\
1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}-\sum_{i=j+1}^{n}k_i
 e^{r_2(\xi_i-s)+r_1s}>0,\quad
0<\xi_j\leq s\leq\xi_{j+1},\;1\le j\le n-1.
\end{gathered}
\end{equation}
Then, we have the estimates
\begin{equation} \label{(a)}
 e^{-\mu t}|G_t(t,s)|\leq e^{-r_1s}\overline G(s),
\quad\forall\,t, s\in I, \; \mu\geq r_1,
\end{equation}
where
\[
\overline G(s)=\begin{cases}
\max\Big(\vert r_2\vert G(s,s),
\frac{r_1}{\Delta}\Big(2-\sum_{i=1}^{n}k_ie^{r_2\xi_i}
-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big),\\
\qquad\text{if }  s\le\xi_1;\\[3pt]
\max\Big(\vert r_2\vert
G(s,s),\frac{r_1}{\Delta}\Big(2-\sum_{i=1}^{j}k_ie^{r_1\xi_i}
-\sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\\
-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)\Big),\\
\qquad \text{if }\xi_j\le s\le \xi_{j+1},\;1\le j\le n-1;\\[3pt]
\max\Big(\vert r_2\vert
G(s,s),\frac{r_1}{\Delta}\Big(2-\sum_{i=1}^{n}k_ie^{r_1\xi_i}
-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)\Big),
\quad\text{if } s\ge\xi_n.
\end{cases}
\]
and
\begin{equation} \label{(b)}
(b)\quad e^{-\theta t}\sigma(t)|G_t(t,s)|\leq
e^{-r_1s}G(s,s),\quad \forall\,t, s\in I.
\end{equation}
\end{lemma}

\begin{proof}
(a) For any $s\in I$, we have
$$
G^1(s,s)=\begin{cases}
1-e^{(r_2-r_1)s},& 0\le s\leq\xi_1\\
1-\sum_{i=1}^{j}k_ie^{r_2\xi_i}-e^{(r_2-r_1)s}
(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}),\\
\qquad\text{if } \xi_j\le s\le\xi_{j+1}, j=1,2,\dots,n-1\\[3pt]
1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}-e^{(r_2-r_1)s}(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}),&
s\ge\xi_n.
\end{cases}
$$
We  distinguish  four cases.

(1) If either $0<s\leq\min(t,\xi_1)<\infty$ or $0<\xi_j\leq
s\leq\xi_{j+1}$, $s\le t$, $j=1,2,\dots,n-1$ or
 $\xi_n\le s\le t$, then for any $\mu\geq r_1$,
$$
e^{-\mu t}|G_t(t,s)|=e^{-\mu t}|r_2G(t,s)|\le
|r_2|e^{-r_1s}G(s,s),\quad\forall \mu\geq r_1.
$$
(2)  If $0<t<s\leq\xi_1<\infty$, then for any $\mu\geq r_1$,
\begin{align*}
&e^{-\mu t}|G^1_t(t,s)|\\
&=\Big|r_1e^{-\mu
t}e^{r_1(t-s)}\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)
-r_2e^{(r_2-\mu)t} \Big(e^{-r_1s} -\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)}
\Big)\Big|\\
&\le r_1e^{-r_1s}\Big(e^{(r_1-\mu)t}\Big(1-\sum_{i=1}^{n}
k_ie^{r_2\xi_i}\Big)+
\frac{|r_2|}{r_1}e^{(r_2-\mu )t}\Big(1-\sum_{i=1}^{n}k_i
e^{r_2(\xi_i-s)+r_1s}\Big)\Big)\\
&\le r_1e^{-r_1s}\Big(\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)
+\Big(1-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big)\\
&=r_1e^{-r_1s}\Big(2-\sum_{i=1}^{n}k_ie^{r_2\xi_i}
-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big).
\end{align*}

(3) If $0<\xi_j\leq s\leq\xi_{j+1}$, $t\le s$,
$j=1,2,\dots,n-1$, then for any $\mu\geq r_1$,
\begin{align*}
&e^{-\mu t}|G^1_t(t,s)|\\
&=e^{-r_1s}\Big|r_1e^{(r_1-\mu)t}\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}
\Big)\\
&\quad -r_2e^{(r_2-\mu)t}\Big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}-
\sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big|\\
&\le r_1e^{-r_1s}\Big(e^{(r_1-\mu)t}\Big(1-\sum_{i=1}^{n}
k_ie^{r_2\xi_i}\Big)+\frac{|r_2|}{r_1}e^{(r_2-\mu)t}
\Big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\\
&\quad -\sum_{i=j+1}^{n} k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big)\\
&\le r_1e^{-r_1s}\Big(\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)+
\Big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}
-\sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big)\\
&=r_1e^{-r_1s}\Big(2-\sum_{i=1}^{j}k_ie^{r_1\xi_i}-\sum_{i=j+1}^{n}
k_ie^{r_2(\xi_i-s)
+r_1s}-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big).
\end{align*}

(4) If $0<\max(\xi_n,t)\leq s<\infty$, then for any $\mu\geq r_1$,
\begin{align*}
&e^{-\mu t}|G^1_t(t,s)|\\
&= e^{-r_1s}\Big|r_1e^{(r_1-\mu)t}
\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)-
r_2e^{(r_2-\mu )t}\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)\Big|\\
&\le r_1e^{-r_1s}\Big(e^{(r_1-\mu)t}\Big(1-\sum_{i=1}^{n}k_i
e^{r_2\xi_i}\Big)+\frac{|r_2|}{r_1}e^{(r_2-\mu
)t}\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)\Big)\\
&\le r_1e^{-r_1s}\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}+
(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i})\Big)\\
&=r_1e^{-r_1s}\Big(2-\sum_{i=1}^{n}k_ie^{r_1\xi_i}
-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big).
\end{align*}
Hence
$$
e^{-\mu t}|G_t(t,s)|\leq e^{-r_1s}\overline G(s), \quad\forall
t, s\in I;\;\forall\,\mu\geq r_1.
$$

(b) For any $s\in I$, we have the discussion

(1) If either $0<s\leq\min(t,\xi_1)<\infty$ or
$0<\xi_j\leq s\leq\xi_{j+1}$, $s\le t$, $j=1,2,\dots,n-1$ or
$\xi_n\le s\le t$,
for any $\mu\geq r_1$, then we have
$$
e^{-\mu t}|G_t(t,s)|=e^{-\mu t}|r_2G(t,s)|\le|r_2|e^{-r_1s}G(s,s).
$$
Hence
$$
\frac{e^{-r_1s}G(s,s)}{e^{-\mu t}|r_2G(t,s)|}
\ge\frac{1}{|r_2|},\quad\forall\,\mu\geq r_1.
$$

(2) If $0<t<s\leq\xi_1<\infty$, then for any $\mu\geq r_1$,
\begin{align*}
&\frac{e^{-r_1s}G(s,s)}{e^{-\mu t}|G_t(t,s)|}\\
&=\frac{e^{-r_1s}\big(1-e^{(r_2-r_1)s}\big)}{\big|r_1e^{-\mu
t}e^{r_1(t-s)}\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\big)
-r_2e^{(r_2-\mu)t} \big(e^{-r_1s}-\sum_{i=1}^{n}k_ie^{r_2(\xi_i-s)}
\big)\big|}\\
&\ge\frac{e^{-r_1s}(1-e^{(r_2-r_1)s})}
{r_1e^{-r_1s}\Big(e^{(r_1-\mu)t}\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}
\big)+\frac{|r_2|}{r_1}e^{(r_2-\mu )t}\big(1-\sum_{i=1}^{n}
k_ie^{r_2(\xi_i-s)+r_1s}\big)\Big)}\\
&\ge\frac{1-e^{(r_2-r_1)t}}{2r_1(1-\sum_{i=1}^{n}k_i
e^{r_2\xi_i})e^{(r_1-\mu)t}}\,.
\end{align*}

(3) If $0<\xi_j\leq s\leq\xi_{j+1}$, $t\le s$,
$j=1,2,\dots,n-1$, then for any $\mu\geq r_1$,
\begin{align*}
\frac{e^{-r_1s}G(s,s)}{e^{-\mu t}|G_t(t,s)|}
&=\frac{e^{-r_1s}\Big(1-\sum_{i=1}^{j}k_ie^{r_2\xi_i}-e^{(r_2-r_1)s}
\big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\big)\Big)} {e^{-r_1s}}\\
&\quad\times
\Big|r_1e^{(r_1-\mu)t}\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}
\big)- r_2e^{(r_2-\mu)t}\Big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\\
&\quad -\sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big|^{-1}
\\
&\ge\frac{\Big(1-\sum_{i=1}^{j}k_ie^{r_2\xi_i}-e^{(r_2-r_1)s}
\big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\big)\Big)}
{r_1}\\
&\quad\times
\Big(e^{(r_1-\mu)t}\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\big)
+\frac{|r_2|}{r_1}e^{(r_2-\mu)t}
\Big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\\
&\quad -\sum_{i=j+1}^{n}k_ie^{r_2(\xi_i-s)+r_1s}\Big)\Big)^{-1}
\\
&\ge\frac{1-\sum_{i=1}^{j}k_ie^{r_2\xi_i}-e^{(r_2-r_1)t}
\big(1-\sum_{i=1}^{j}k_ie^{r_1\xi_i}\big)}
{2r_1\big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\big)e^{(r_1-\mu)t}}\,.
\end{align*}

(4) If $0<\max(\xi_n,t)\leq s<\infty$, then for any $\mu\geq
r_1$,
\begin{align*}
&\frac{e^{-r_1s}G(s,s)}{e^{-\mu t}|G_t(t,s)|}\\
&=\frac{e^{-r_1s}\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}
-e^{(r_2-r_1)s}\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)\Big)}
{e^{-r_1s}\Big|r_1e^{(r_1-\mu)t}\Big(1-\sum_{i=1}^{n}
k_ie^{r_2\xi_i}\Big)-
r_2e^{(r_2-\mu )t}\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)\Big|}\\
&\ge\frac{1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}-e^{(r_2-r_1)s}
\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)}
{r_1\Big(e^{(r_1-\mu)t}\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\big)
+\frac{|r_2|}{r_1}e^{(r_2-\mu)t}
\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)\Big)}\\
&\ge\frac{1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}-e^{(r_2-r_1)t}
\Big(1-\sum_{i=1}^{n}k_ie^{r_1\xi_i}\Big)}
{2r_1\Big(1-\sum_{i=1}^{n}k_ie^{r_2\xi_i}\Big)e^{(r_1-\mu)t}}\,.
\end{align*}
Hence
$$
e^{-\mu t}\sigma(t)|G_t(t,s)|\leq e^{-r_1s}G(s,s),\;\forall\,t,
s\in I,\;\forall\,\mu\geq r_1.
$$
\end{proof}

\subsection{A compact fixed point operator}

On the space $X$, define the mapping $T$ by
\begin{equation}\label{operatorF}
Ty(t)=\int_0^{\infty}G(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds,\quad
t\in I.
\end{equation}

\begin{remark}\label{rmk2.1} \rm
Let $y\in X$ be a fixed point of $T$ in $X$. Then it is a solution
of problem \eqref{GP1} provided the integral in \eqref{operatorF}
converges.
\end{remark}

Recall that an operator is called completely continuous if it
is continuous and maps bounded sets into relatively compact sets.

\begin{lemma}[{\cite[p. 62]{Cord}}] \label{lem1}
Let $M\subseteq C_l(\mathbb{R}^+,\mathbb{R})$. Then $M$ is
relatively compact in $C_l(\mathbb{R}^+,\mathbb{R})$  if the
following conditions hold:
\begin{itemize}
\item[(a)] $M$ is uniformly bounded in
$C_l(\mathbb{R}^+,\mathbb{R})$;

\item[(b)] Functions belonging
to $M$ are almost equicontinuous on $\mathbb{R}^+$; i.e.,
equicontinuous on every compact interval of $\mathbb{R}^+$.

\item[(c)] The functions from $M$ are equiconvergent; that is,
given $\varepsilon>0$, there corresponds $T(\varepsilon)>0$ such
that $|x(t)-l|<\varepsilon$ for any $t\geq T(\varepsilon)$ and
$x\in M$,
\end{itemize}
\end{lemma}

From the above lemma we easily deduce the following result
(see e.g., \cite{DMe4}).

\begin{lemma}\label{lem2}
Let $M\subseteq C^1_{\infty}(\mathbb{R}^+,\mathbb{R})$. Then $M$
is relatively compact in $C^1_{\infty}(\mathbb{R}^+,\mathbb{R})$
if the following conditions hold:
\begin{itemize}
\item[(a)] $M$ is uniformly
bounded in $ C^1_{\infty}(\mathbb{R}^+,\mathbb{R})$.

\item[(b)] The functions belonging to the sets
$\{y: y(t)= x(t)/e^{\theta t}, x \in M\}$  and
$\{z: z(t)= x'(t)/e^{\theta t}, x\in M\}$
are locally equicontinuous on $\mathbb{R}^+$.

\item[(c)] The functions from the sets
$\{y: y(t)= x(t)/e^{\theta t}, x\in M\}$  and
$\{z| z(t)= x'(t)/e^{\theta t}, x\in M\}$
are equiconvergent at $+\infty$.
\end{itemize}
\end{lemma}

\subsection{General assumptions}

Regarding the growth of the function
$F(t,u,v)=f(t,ue^{\theta t},ve^{\theta t})$,
we first enunciate the main
assumptions to be considered in this paper:
\begin{itemize}
\item[(H1)] $F:I^2\times\mathbb{R}^*\to\mathbb{R}^+$
is a continuous function and there exist functions $g,\,w\in C(I,I)$
and $h,\,k\in C(\mathbb{R}^*,I)$ such that
$$
0\le F(t,u,v)\leq(g(u)+w(u))(h(v)+k(v)),\quad\forall\,(t,u,v)\in
I^2\times\mathbb{R}^*
$$
where $g,h$ are non-increasing functions, $w/g$, $k/h$
are nondecreasing functions and for all positive number $R$
$$
\Pi(R)=\int_0^{+\infty} e^{-r_1s}\max\{G(s,s),\overline
G(s)\}\Phi(s)g(e^{-\theta s}\Gamma R)h(-e^{-cs}R)ds<\infty.
$$

\item[(H2)] There exits $R_0>0$ such that
\begin{equation}\label{Rhypothesis}
\Big(1+\frac{w(R_0)}{g(R_0)}\Big)
\Big(1+\frac{k(R_0)}{h(R_0)}\Big)\Pi(R_0)< R_0.
\end{equation}
\end{itemize}

\section{Properties of the operator $T$}

In the subsequent two lemmas, we study the properties of the
operator $T$ including its compactness when the nonlinearity $f$
is assumed to have no singularities.

\begin{lemma}\label{lemma1}
Under Assumptions {\rm (H1), (H2)}, the operator $T$
maps $\mathcal{P}$ into itself, where the cone $\mathcal{P}$ is as
defined by \eqref{cone}.
\end{lemma}

\begin{proof}
{\bf Claim 1.} $T(\mathcal{P})\subset X$. Indeed, from
Assumptions (H1) and (H2) and with Lemma
\ref{lem4}(a), (b) and Lemma \ref{lem5}(a) with $\mu=\theta$,
we obtain, for any $y\in \mathcal{P}$, and $t\in \mathbb{R}^+$ the
following estimates:
\begin{align*}
|Ty(t)|e^{-\theta t}
&=\int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds\\
&= \int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)
f\Big(s,\frac{e^{\theta s}}{e^{\theta s}}\, y(s),
 \frac{e^{\theta s}}{e^{\theta s}}\,e^{-cs}y'(s)\Big)ds\\
&\leq \int_0^{+\infty}e^{-r_1s}G(s,s)\Phi(s)F(s,e^{-\theta s}y(s),e^{-(c+\theta )s}y'(s))ds\\
&\leq \int_0^{+\infty}e^{-r_1s}G(s,s)\Phi(s)\left(g(e^{-\theta
s}y(s))+w(e^{-\theta s}y(s))\right)\\
&\quad \times \Big(h(e^{-(c+\theta)s}y'(s))+k(e^{-(c+\theta)s}y'(s))
\Big)ds\\
&= \int_0^{+\infty}e^{-r_1s}G(s,s)\Phi(s)\Big(1+\frac{w(e^{-\theta
s}y(s))}{g(e^{-\theta s}y(s))}\Big)\\
&\quad\times \Big(1+\frac{k(e^{-(c+\theta )s}y'(s))}{h(e^{-(c+\theta
)s}y'(s))}\Big)g(e^{-\theta
s}y(s))h(e^{-(c+\theta)s}y'(s))ds\\
&\leq \Big(1+\frac{w(\|y\|_{\theta})}{g(\|y\|_{\theta})}\Big)
\Big(1+\frac{k(\|y\|_{\theta})}{h(\|y\|_{\theta})}\Big)
\Pi(\|y\|_{\theta})<\infty,
\end{align*}
and
\begin{align*}
|(Ty)'(t)|e^{-\theta t}
&=\int_0^{+\infty}e^{-\theta t}G_t(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds\\
&\leq \int_0^{+\infty}e^{-r_1s}\overline{G}(s)\Phi(s)
\left(g(e^{-\theta s}y(s))+w(e^{-\theta s}y(s))\right)\\
&\quad\times \Big(h(e^{-(c+\theta)s}y'(s))
+k(e^{-(c+\theta)s}y'(s))\Big)ds\\
&\leq \Big(1+\frac{w(\|y\|_{\theta})}{g(\|y\|_{\theta})}\Big)
\Big(1+\frac{k(\|y\|_{\theta})}{h(\|y\|_{\theta})}\Big)
\Pi(\|y\|_{\theta})<\infty.
\end{align*}

{\bf Claim 2.} $T(\mathcal{P})\subset \mathcal{P}$. Let
$y\in \mathcal{P}$. Clearly, $Ty(t)\ge 0$ for all $t\in I$.
Moreover, by Lemma \ref{lem4}(c) and Lemma \ref{lem5}(b), for
$t\in [\gamma,\delta]$, we have
\begin{align*}
Ty(t)
&= \int_0^{+\infty}G(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds,\\
&\geq \int_0^{\infty}\min_{t\in[\gamma,\delta]}G(t,s)\Phi(s)f(s,y(s),
 e^{-cs}y'(s))ds\\
&\geq \int_0^{\infty}e^{-r_1s}\Lambda_0 G(s,s)\Phi(s)f(s,y(s),
 e^{-cs}y'(s))ds \\
&\geq \int_0^{\infty}\Lambda_0\sigma(\tau)
e^{-\theta\tau}G_t(\tau,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds.
\end{align*}
Passing to the supremum over $\tau\in\mathbb{R}^+$, we obtain
\begin{align*}
Ty(t)&\geq \Lambda_0 \sup_{\tau\in\mathbb{R}^+}
\Big(\sigma(\tau)\frac{(Ty)'(\tau)}{e^{\theta \tau}}\Big)\\
&= \Lambda_0 \sup_{\tau\in\mathbb{R}^+}
\sigma(\tau)\sup_{\tau\in\mathbb{R}^+}\frac{(Ty)'(\tau)}{e^{\theta
\tau}}\\
&\geq \Lambda_0 \sup_{\tau\in[\gamma,\delta]}
\sigma(\tau)\sup_{\tau\in\mathbb{R}^+}\frac{(Ty)'(\tau)}{e^{\theta
\tau}}.
\end{align*}
Hence
$$
Ty(t)\ge\Lambda\|Ty\|_2, \quad \forall\;t\in[\gamma,\delta].
$$
Finally, by the property of the Green's function
$$
Ty(0)=\sum_{i=1}^{n}k_iTy(\xi_i).
$$
\end{proof}

\begin{lemma}\label{lemma2}
Under Assumptions {\rm (H1), (H2)}, the
mapping $T: \mathcal{P}\to \mathcal{P}$ is
completely continuous.
\end{lemma}

\begin{proof}
{\bf Claim 1.} $T: \mathcal{P}\to \mathcal{P}$ is
continuous. Let a sequence $\{y_n\}_{n\geq1}\subseteq \mathcal{P}$
and $y_0\in \mathcal{P}$ with
$\lim_{n\to+\infty}y_n\to y_0$ in
$\mathcal{P}$. Then, there exists an $M>0$ such that
$\max\{\|y_n\|_{\theta},\|y_0\|_{\theta}\}\le M$ for all
$n\in\{1,2,\dots \}$. Thus, arguing as in Lemma \ref{lemma1}, Claim 1
and using Assumptions (H1) and (H2), we
arrive at the estimates
\begin{align*}
&\int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)f(s,y_n(s),
 e^{-cs}y_n'(s))ds\\
&\le\Big(1+\frac{w(M)}{g(M)}\Big)\Big(1+\frac{k(M)}{h(M)}\Big)
\Pi(\|y_n\|_{\theta}) <\infty
\end{align*}
and
\begin{align*}
&\int_0^{+\infty}e^{-\theta t}|G_t(t,s)\Phi(s)|f(s,y_n(s),
 e^{-cs}y_n'(s))ds\\
&\le\Big(1+\frac{w(M)}{g(M)}\Big)
\Big(1+\frac{k(M)}{h(M)}\Big)\Pi(\|y_n\|_{\theta})<\infty.
\end{align*}
By continuity of $f$, we obtain
$$
\lim_{n\to+\infty}
f(t,y_n(t),e^{-ct}y_n'(t))=f(t,y_0(t),e^{-ct}y'_0(t)), \quad t\in
I.
$$
Then the Lebesgue Dominated Convergence Theorem implies
\begin{align*}
&\sup_{t\in \mathbb{R}^+}\{|Ty_n(t)-Ty_0(t)|e^{-\theta t}\}\\
&= \sup_{t\in \mathbb{R}^+}\Big|\int_0^{\infty}e^{-\theta
t}G(t,s)\Phi(s)\left(f(s,y_n(s),e^{-cs}y_n'(s))-f(s,y_0(s),
e^{-cs}y_0'(s)\right)ds\Big|\\
&\leq \sup_{t\in \mathbb{R}^+}\int_0^{\infty}
G(s,s)\Phi(s)e^{-r_1 s}
\left|f(s,y_n(s),e^{-cs}y_n'(s))-f(s,y_0(s),e^{-cs}y_0'(s))\right|ds\\
&\to0, \quad \text{as }\, n\to +\infty
\end{align*}
and
\begin{align*}
&\sup_{t\in \mathbb{R}^+}\{|(Ty_n)'(t)-(Ty_0)'(t)|e^{-\theta t}\}\\
&=  \sup_{t\in \mathbb{R}^+}\Big|\int_0^{\infty}e^{-\theta
t}G_t(t,s)\Phi(s)\left(f(s,y_n(s),e^{-cs}y_n'(s))
-f(s,y_0(s),e^{-cs}y_0'(s)\right)ds\Big|\\
&\leq  \sup_{t\in \mathbb{R}^+}\int_0^{\infty}\overline
G(s)\Phi(s)e^{-r_1 s}
\left|f(s,y_n(s),e^{-cs}y_n'(s))-f(s,y_0(s),e^{-cs}y_0'(s))\right|ds\\
&\to0, \quad \text{as }\,n\to +\infty.
\end{align*}
As a result
$$
\|Ty_n-Ty_0\|_{\theta }\to0, \quad n\to+\infty.
$$

{\bf Claim 2.} Let $\Omega\subset X$ be a bounded subset,
say $\Omega=\{y\in X: \|y\|_{\theta}\le r\}$. We prove that
$T(\Omega\cap \mathcal{P})$ is relatively compact.

(a) For some $y\in \Omega\cap \mathcal{P}$, we have
$$
\|Ty\|_{\theta}\le\Big(1+\frac{w(r)}{g(r)}\Big)
\Big(1+\frac{k(r)}{h(r)}\Big)\Pi(\|y\|_{\theta}),
$$
yielding that $T(\Omega\cap \mathcal{P})$ is uniformly bounded.

(b) $T(\Omega\cap \mathcal{P})$ is locally equicontinuous on $I$.
The functions in $\{Ty(t)/e^{\theta t},y\in
\Omega\cap \mathcal{P}\}$ and the functions belonging to
$\{(Ty)'(t)/e^{\theta t},y\in \Omega\cap \mathcal{P}\}$
are locally equicontinuous on $I$. Indeed, $G(t,s)$ is
continuously differentiable in $t$ on $[0,\infty)$ except for
$t=s;$ so the Lebesgue dominated convergence theorem yields
\begin{align*}
|Ty(t_1)-Ty(t_2)|e^{-\theta t}
&\leq \int_0^{\infty}e^{-\theta
t}|G(t_1,s)-G(t_2,s)|\Phi(s)
f(s,y(s),e^{-cs}y'(s))ds\\
&\to 0, \quad \text{ as}\; t_1\to t_2,
\end{align*}
as well as
\begin{align*}
&|(Ty)'(t_1)-(Ty)'(t_2)|e^{-\theta t}\\
&\leq \int_0^{\infty}e^{-\theta t}|G_t(t_1,s)-G_t(t_2,s)|\Phi(s)
f(s,y(s),e^{-cs}y'(s))ds\\
&\to 0, \quad \text{ as}\;t_1\to t_2,
\end{align*}

(c) $T(\Omega\cap \mathcal{P})$ is locally equiconvergent at
$+\infty$. Let $y\in\Omega\cap \mathcal{P}$. From the expression
of the Green's function $G$ in Lemmas \ref{lem4}, \ref{lem5}, we infer
that
\begin{equation}\label{eq10}
\lim_{t\to+\infty}\frac{G(t,s)}{e^{\theta t}}=0,\quad
\lim_{t\to+\infty}\frac{G_t(t,s)}{e^{\theta t}}=0,
\quad s\in [0,+\infty).
\end{equation}
With the estimates in Lemma \ref{lemma1}, Claim 1 and the Lebesgue
dominated convergence theorem, we finally obtain
\begin{align*}
&\lim_{t\to+\infty}\big|e^{-\theta
t}Ty(t)-\lim_{s\to+\infty}e^{-\theta s}Ty(s)\big|\\
&= \lim_{t\to+\infty}\Big|\int_0^{\infty}e^{-\theta
t}G(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds\Big|\\
&\leq \int_0^{\infty}\lim_{t\to+\infty}|e^{-\theta
t}G(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds|=0
\end{align*}
and
\begin{align*}
&\lim_{t\to+\infty}\big|e^{-\theta
t}(Ty)'(t)-\lim_{t\to+\infty}e^{-\theta
s}(Ty)'(s)\big|\\
&= \lim_{t\to+\infty}\Big|\int_0^{\infty}e^{-\theta
t}G_t(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds\Big|=0.
\end{align*}
By Lemma \ref{lem2}, $T(\Omega\cap P)$ is relatively compact.
\end{proof}

\section{Main existence results}

\subsection{Single solution}

The following Lemmas are needed in this section. The proofs and
more details on the index fixed point theory in cones can be found
in \cite{AMO, Deim, GuoLak, Kras, Zeid}.

\begin{lemma}\label{lemA}
Let $\Omega$ be a bounded open set in a real Banach space $E,
\mathcal{P}$ be a cone of $E,\theta\in \Omega$ and $A:
\overline{\Omega}\cap \mathcal{P}\to\mathcal{P}$ be a
completely continuous operator. Assume that
$$
Ax\neq\lambda x, \quad \forall\,x\in\partial\Omega\cap \mathcal{P},\;
\lambda\ge1.
$$
Then $i(A,\Omega\cap \mathcal{P},\mathcal{P})=1$.
\end{lemma}

\begin{lemma}\label{lemB}
Let $\Omega$ be a bounded open set in a real Banach space $E,
\mathcal{P}$ be a cone of $E,\theta\in \Omega$ and $A:
\overline{\Omega}\cap \mathcal{P}\to \mathcal{P}$ be a
completely continuous operator. Assume that
$$
Ax\not\leq x, \quad \forall\,x\in\partial\Omega\cap \mathcal{P}.
$$
Then $i(A,\Omega\cap \mathcal{P},\mathcal{P})=0$.
\end{lemma}

We are now in position to prove our first existence result. Let
$$
\ell:=\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)ds.
$$

\begin{theorem}\label{thm1}
Assume {\rm (H1), (H2)} hold together with
\begin{itemize}
\item[(H3)]
$ f(t,y,z)\ge\varphi(t,y)$ for all $t\in[\gamma,\delta]$ and all
$ (y,z)\in(0,+\infty)\times \mathbb{R}^*$,
where $\varphi\in C([\gamma,\delta]\times(0,+\infty))$
 satisfies
\[
\liminf_{y\to0}\min_{t\in[\gamma,\delta]}
\frac{\varphi(t,y)}{y}>\frac{1}{\Lambda_0\ell}\,.
\]
\end{itemize}
Then problem \eqref{GP1} has at least one positive solution $y$
such that
$$
\|y\|_{\theta}\le R_0,\quad
y(t)\ge\Gamma\|y\|_{\theta},\quad \forall\,t\in[\gamma,\delta].
$$
\end{theorem}

\begin{proof}
For each $n\in\{1,2,\dots\}$, define a sequence of functions by
\begin{equation}\label{approximatingfunction}
f_n(t,y,z)=f\left(t,\max\{e^{\theta t}/n,y(t)\},\max\{e^{\theta
t}/n,z(t)\}\right).
\end{equation}
Then, for $y\in\mathcal{P}$, define a sequence of operators by
\begin{equation}\label{approximatingoperator}
T_ny(t)=\int_0^{+\infty}G(t,s)\Phi(s)
f_n(s,y(s),e^{-cs}y'(s))ds,\; t\in I.
\end{equation}
Lemma \ref{lemma2} guarantees that $T_n: \mathcal{P}\to
\mathcal{P}$ is a completely continuous operator. By the
inequality of (H3), there exist an $r>0$ and
$\varepsilon>0$ such that
\begin{equation}\label{eq14}
\varphi(t,y)\geq\Big(\frac{1}{\Lambda_0\ell}+\varepsilon\Big)y,
\quad\text{ for each } y\in[0,r]\text{ and } t\in[\gamma,\delta].
\end{equation}
 Let $R_0$ be as defined by Assumption (H2) and
$\widetilde{R}=\min(R_0/2,r/e^{\theta\delta})$ and
consider the open sets
$$
\Omega_1:=\{y\in X:\|y\|_\theta<R_0\},\quad
\Omega_2:=\{y\in X:\|y\|_\theta<\widetilde{R}\}.
$$

{\bf Claim 1.} $T_ny\neq\lambda y$ for any
$y\in\partial\Omega_1\cap P,\,\lambda\geq1$  and $n\ge
n_0>\frac{1}{R_0}$. Let $y\in\partial\Omega_1\cap \mathcal{P}$. By
Assumptions (H1) and (H2), we obtain
successively the following estimates
\begin{align*}
&e^{-\theta t}|T_ny(t)|\\
&= \int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)f_n(s,y(s),
 e^{-cs}y'(s))ds\\
&= \int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)
f\left(s,\max\{e^{\theta s}/n,y(s)\},\max\{e^{\theta s}/n,
 e^{-cs}y'(s)\}\right)ds\\
&= \int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)
F(s,\max\{1/n,e^{-\theta s}y(s)\},\max\{1/n,e^{-(c+\theta )s}y'(s)\})ds\\
&\leq \int_0^{+\infty}e^{-r_1s}G(s,s)\Phi(s)\left(g(\max\{1/n,e^{-\theta
s}y(s)\})+w(\max\{1/n,e^{-\theta s}y(s)\})\right)\\
&\quad \times\left(h(\max\{1/n,e^{-(c+\theta )s}y'(s)\})
+k(\max\{1/n,e^{-(c+\theta )s}y'(s)\})\right)ds\\
&= \int_0^{+\infty}\left(1+\frac{w(\max\{1/n,e^{-\theta
s}y(s)\})}{g(\max\{1/n,e^{-\theta s}y(s)\})}\right)\left(1
+\frac{k(\max\{1/n,e^{-(c+\theta
)s}y'(s)\})}{h(\max\{1/n,e^{-(c+\theta)s}y'(s)\})}\right)\\
&\quad \times e^{-r_1s}\max\{G(s,s),\overline G(s)\}\Phi(s)g(e^{-\theta
s}y(s))h(e^{-(c+\theta)s}y'(s))ds\\
&\leq \Big(1+\frac{w(\max\{1/n,\|y\|_{\theta}\})}
 {g(\max\{1/n,\|y\|_{\theta}\})}\Big)
 \Big(1+\frac{k(\max\{1/n,\|y\|_{\theta}\})}
 {h(\max\{1/n,\|y\|_{\theta}\})}\Big)\\
&\quad \times\int_0^{+\infty} e^{-r_1s}\max\{G(s,s),\overline
G(s)\}\Phi(s)g(e^{-\theta s}\Gamma
\|y\|_{\theta})h(-e^{-cs}\|y\|_{\theta})ds\\
&\leq \Big(1+\frac{w(R_0)}{g(R_0)}\Big)
\Big(1+\frac{k(R_0)}{h(R_0)}\Big)\Pi(R_0)
< R_0
\end{align*}
and
\begin{align*}
&e^{-\theta t}|(T_ny)'(t)|\\
&= \int_0^{+\infty}e^{-\theta t}G_t(t,s)\Phi(s)
F(s,\max\{1/n,e^{-\theta s}y(s)\},\max\{1/n,e^{-(c+\theta )s}y'(s)\})ds\\
&\leq \int_0^{+\infty}e^{-r_1s}\overline
G(s)\Phi(s)\left(g(\max\{1/n,e^{-\theta s}y(s)\})
+w(\max\{1/n,e^{-\theta s}y(s)\})\right)\\
&\quad\times\left(h(\max\{1/n,e^{-(c+\theta )s}y'(s)\})
+k(\max\{1/n,e^{-(c+\theta )s}y'(s)\})\right)ds\\
&= \int_0^{+\infty}\Big(1+\frac{w(\max\{1/n,e^{-\theta
s}y(s)\})}{g(\max\{1/n,e^{-\theta s}y(s)\})}\Big)
\Big(1 +\frac{k(\max\{1/n,e^{-(c+\theta
)s}y'(s)\})}{h(\max\{1/n,e^{-(c+\theta
)s}y'(s)\})}\Big)\\
&\quad\times e^{-r_1s}\max\{G(s,s),\overline G(s)\}\Phi(s)g(e^{-\theta
s}y(s))h(e^{-(c+\theta
)s}y'(s))ds\\
&\leq \Big(1+\frac{w(\max\{1/n,\|y\|_{\theta}\})}{g(\max\{1/n,
\|y\|_{\theta}\})}\Big)
\Big(1+\frac{k(\max\{1/n,\|y\|_{\theta}\})}{h(\max\{1/n,
\|y\|_{\theta}\})}\Big)\Pi(\|y\|_{\theta})\\
&\leq \Big(1+\frac{w(R_0)}{g(R_0)}\Big)
\Big(1+\frac{k(R_0)}{h(R_0)}\Big)
\Pi(R_0)
< R_0.
\end{align*}
Passing to the supremum over $t$, we infer that
\begin{equation}\label{eq11}
\|T_ny\|_\theta<R_0=\| y\|_\theta,\quad
\forall\,y\in\partial\Omega_1\cap\mathcal{P}.
\end{equation}
As a consequence, we may conclude that
\begin{equation}\label{eq12}
T_ny\neq\lambda y,\quad \forall\,y\in\partial\Omega_1\cap
\mathcal{P},\;\forall\,\lambda\geq1,\; n\ge n_0.
\end{equation}
Otherwise, for some $n_1\ge n_0$, there would exist $y_1\in
\partial\Omega_1\cap \mathcal{P}$ and $\lambda_1\geq1$ such that
$T_{n_1}y_1=\lambda_1y_1$. Thus
$$
\|T_{n_1}y_1\|_\theta=\lambda_1\|y_1\|_\theta\geq\|y_1\|_\theta=R_0,
$$
contradicting \eqref{eq11}. This implies that \eqref{eq12} holds.
Therefore, Lemma \ref{lemA} and \eqref{eq12} imply
\begin{equation}\label{eq13}
i (T_n,\Omega_1\cap \mathcal{P},\mathcal{P})=1,\quad
 \forall\,n\in \{n_0,n_0+1,\dots\}.
\end{equation}

{\bf Claim 2.} $T_ny\not\le y$ for any
$y\in\partial\Omega_2\cap \mathcal{P}$. Otherwise, let
$y_2\in\partial\Omega_2\cap P$ and ${n_2}\ge n_0$ with
\begin{equation}\label{eq15}
T_{n_2}y_2\le y_2.
\end{equation}
 From \eqref{eq14} and the fact that
$\frac{|y_2(t)|}{e^{\theta t}}\le\|y_2\|_{\theta}=\widetilde{R}\le
\frac{r}{e^{\theta\delta}}$, we infer that $y_2(t)\le r$,
for each $t\in[\gamma,\delta]$. Then
\begin{equation}\label{eq16}
\varphi(t,y_2(t))\geq\Big(\frac{1}{\Lambda_0\ell}+\varepsilon\Big)
y_2(t),\quad \forall\,t\in[\gamma,\delta].
\end{equation}
By  \eqref{eq15}, \eqref{eq16} and Lemma
\ref{lem4}, the following estimates are straightforward:
\begin{align*}
y_2(t)
&\geq \int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)
f\left(s,\max\{e^{\theta s}/n_2,y_2(s)\},\max\{e^{\theta s}/n_2,e^{-cs}y_2'(s)\}\right)ds\\
&\geq \Lambda_0\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)\varphi\left(s,\max\{e^{\theta s}/n_2,y_2(s)\}\right)ds\\
&\geq \Lambda_0\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)
\Big(\frac{1}{\Lambda_0\ell}+\varepsilon\Big)
\max\{e^{\theta s}/n_2,y_2(s)\}ds\\
&\geq \Lambda_0\left(\frac{1}{\Lambda_0\ell}+\varepsilon\right)
\min_{t\in[\gamma,\delta]}y_2(t)\int_{\gamma}^{\delta}
e^{-r_1s}G(s,s)\Phi(s)ds\\
&= \Lambda_0\ell\left(\frac{1}{\Lambda_0\ell}+\varepsilon\right)
\min_{t\in[\gamma,\delta]}y_2(t)\\
&> \min_{t\in[\gamma,\delta]}y_2(t),
\quad \forall\,t\in[\gamma,\delta],
\end{align*}
contradicting the continuity of the function $y_2$ on the compact
interval $[\gamma,\delta];$ this implies that Claim 2 holds.
Then, Lemma \ref{lemB} yields
\begin{equation}\label{eq17}
i (T_n,\Omega_1\cap \mathcal{P},\mathcal{P})=0,\quad \forall\, n\in
\{1,2,\dots\}.
\end{equation}
Consequently, from \eqref{eq13}, \eqref{eq17} and the fact that
$\overline{\Omega}_1\subset\Omega_2$, we find
\begin{equation}\label{eq18}
i (T_n,(\Omega_1\setminus\overline{\Omega}_2)\cap
\mathcal{P},\mathcal{P})=-1,\quad \forall\, n\in \{n_0,n_0+1,\dots\}.
\end{equation}
This equality and the solution property of the fixed point index
imply that, for each $n\ge n_0$, there exists some
$y_n\in(\Omega_1\setminus\overline{\Omega}_2)\cap \mathcal{P}$
such that
$T_ny_n=y_n$ with $0<\widetilde{R}<\|y_n\|_{\theta}<R_0$.
Consider the sequence of functions $\{y_n\}_{n\ge n_0}$.
Clearly, the functions belonging to
$\{\frac{y_n(t)}{e^{\theta t}},\;n\ge n_0\}$ and the functions
belonging to $\{\frac{y_n'(t)}{e^{\theta t}},\;n\ge n_0\}$
are uniformly bounded on $\mathbb{R}^+$. Since
$\widetilde{R}<\|y_n\|_{\theta}<R_0$, (H1) and
(H2) imply that, for each $n\ge n_0$,
\begin{align*}
&\int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)
f\left(s,\max\{e^{\theta s}/n,y_n(s)\},\max\{e^{\theta s}/n,e^{-cs}y_n'(s)\}\right)ds\\
&\leq \int_0^{+\infty}e^{-r_1s}G(s,s)\Phi(s)\left(g(\max\{1/n,e^{-\theta
s}y_n(s)\})
+w(\max\{1/n,e^{-\theta s}y_n(s)\})\right)\\
&\quad \times\left(h(\max\{1/n,e^{-(c+\theta )s}y_n'(s)\})
+k(\max\{1/n,e^{-(c+\theta )s}y_n'(s)\})\right)ds\\
&= \int_0^{+\infty}\Big(1 +\frac{w(\max\{1/n,e^{-\theta
s}y_n(s)\})}{g(\max\{1/n,e^{-\theta s}y_n(s)\})}\Big)
\Big(1+\frac{k(\max\{1/n,e^{-(c+\theta
)s}y_n'(s)\})}{h(\max\{1/n,e^{-(c+\theta)s}y_n'(s)\})}\Big)\\
&\quad \times e^{-r_1s}\max\{G(s,s),\overline G(s)\}\Phi(s)g(e^{-\theta
s}y_n(s))h(e^{-(c+\theta
)s}y_n'(s))ds\\
&\leq \Big(1+\frac{w(\max\{1/n,\|y_n\|_{\theta}\})}
{g(\max\{1/n,\|y_n\|_{\theta}\})}\Big)
\Big(1+\frac{k(\max\{1/n,\|y_n\|_{\theta}\})}
{h(\max\{1/n,\|y_n\|_{\theta}\})}\Big)\Pi(\|y_n\|_{\theta})\\
&\leq \Big(1+\frac{k(R_0)}{h(R_0)}\Big)
\Big(1+\frac{k(R_0)}{h(R_0)}\Big)\\
&\quad\times \int_0^{+\infty} e^{-r_1s}\max\{G(s,s),\overline
G(s)\}\Phi(s)g(e^{-\theta s}\Gamma \widetilde{R})h(-e^{-cs}R_0)ds
\end{align*}
and
\begin{align*}
&\int_0^{+\infty}e^{-\theta t}G_t(t,s)\Phi(s)
f\left(s,\max\{e^{\theta s}/n,y_n(s)\},\max\{e^{\theta s}/n,e^{-cs}y_n'(s)\}\right)ds\\
&\leq \Big(1+\frac{w(\max\{1/n,\|y_n\|_{\theta}\})}
 {g(\max\{1/n,\|y_n\|_{\theta}\})}\Big)
\Big(1+\frac{k(\max\{1/n,\|y_n\|_{\theta}\})}
{h(\max\{1/n,\|y_n\|_{\theta}\})}\Big) \Pi(\|y_n\|_{\theta})\\
&\leq \Big(1+\frac{w(R_0)}{g(R_0)}\Big)
\Big(1+\frac{k(R_0)}{h(R_0)}\Big)\\
&\quad\times\int_0^{+\infty} e^{-r_1s}\max\{G(s,s),\overline
G(s)\}\Phi(s)g(e^{-\theta s}\Gamma \widetilde{R})h(-e^{-cs}R_0)ds.
\end{align*}
Then, for some $a>0$ and $t_1,t_2\in[0,a]$, we have for $n\in
\{n_0,n_0+1,\dots\}$,
\begin{align*}
|y_n(t_1)-y_n(t_2)|e^{-\theta t}
&\leq \int_0^{\infty}e^{-\theta t}|G(t_1,s)-G(t_2,s)|\Phi(s)\\
&\quad\times f\left(s,\max\{e^{\theta s}/n,y_n(s)\},\max\{e^{\theta
s}/n,e^{-cs}y_n'(s)\}\right)ds
\end{align*}
and
\begin{align*}
|y_n'(t_1)-y_n'(t_2)|e^{-\theta t}
&\leq\int_0^{\infty} e^{-\theta t}|G_t(t_1,s)-G_t(t_2,s)| \Phi(s)\\
&\quad\times f\left(s,\max\{e^{\theta s}/n,y_n(s)\},\max\{e^{(\theta-c)
s}/n,e^{-cs}y_n'(s)\}\right)ds.
\end{align*}
Consequently, the functions belonging to
$\{\frac{y_n(t)}{e^{\theta t}},\;n\ge n_0\}$ and the functions
belonging to $\{\frac{y_n'(t)}{e^{\theta t}},\;n\ge n_0\}$ are
locally equicontinuous on $\mathbb{R}^+$. Similarly, we have
\begin{align*}
&\lim_{t\to+\infty}\sup_{n\ge
n_0}\big|e^{-\theta t}y_n(t)-\lim_{s\to+\infty}e^{-\theta
s}y_n(s)\big|\\
&= \lim_{t\to+\infty}\sup_{n\ge
n_0}\Big|\int_0^{\infty}e^{-\theta
t}G(t,s)\Phi(s)f_n(s,y_n(s),e^{-cs}y_n'(s))ds\Big|\\
&\leq \int_0^{\infty}\lim_{t\to+\infty}\left|e^{-\theta
t}G(t,s)\Phi(s)f_n(s,y_n(s),e^{-cs}y_n'(s))\right|ds=0
\end{align*}
and
\begin{align*}
&\lim_{t\to+\infty}\sup_{n\ge n_0}\big|e^{-\theta
t}y'_n(t)-\lim_{s\to+\infty}e^{-\theta s}y'_n(s)\big|\\
&\leq \lim_{t\to+\infty}\int_0^{\infty}e^{-\theta
t}\left|G_t(t,s)\Phi(s)f(s,y_n(s),e^{-cs}y_n'(s))\right|ds=0.
\end{align*}
Thus, the functions functions belonging to
$\{\frac{y_n(t)}{e^{\theta t}},\;n\ge n_0\}$ and the functions
belonging to $\{\frac{y'_n(t)}{e^{\theta t}},\;n\ge n_0\}$ are
locally equiconvergent on $+\infty$. Consequently, Lemma
\ref{lem2} guarantees that there is a convergent subsequence
$\{y_{n_j}\}_{j\ge 1}$ of $\{y_n\}_{n\ge n_0}$ such that
$\lim_{j\to+\infty}y_{n_j}=y$ strongly $X$.
Moreover the continuity of $f$ yields
\begin{align*}
\lim_{j\to+\infty}f_{n_j}(t,y_{n_j},y'_{n_j})&= \lim_{j\to+\infty}
f\left(t,\max\{e^{\theta t}/_{n_j},y_{n_j}\},\max\{e^{\theta t}/_{n_j},e^{-ct}y'_{n_j}\}\right)\\
&= f(t,y(t),e^{-ct}y'(t)).
\end{align*}
Then the dominated convergence theorem guarantees that
\begin{align*}
y(t)&= \lim_{j\to+\infty}y_{n_j}(t)\\
&= \lim_{j\to+\infty}
\int_0^{+\infty}G(t,s)\Phi(s)f\Big(t,\max\{e^{\theta
t}/_{n_j},y_{n_j}\},
\max\{e^{\theta t}/_{n_j},e^{-ct}y'_{n_j}\}\Big)\\
&= \int_0^{+\infty}G(t,s)\Phi(s)f(s,y(s),e^{-cs}y'(s))ds,\quad
t\in I.
\end{align*}
Finally, $\widetilde{R}<|y_{n_j}\|_{\theta}<R_0$,
for all $j\ge1$ implies $\widetilde{R}\le\|y\|_{\theta}\le R_0$.
Hence
$$
0<\widetilde{R}\le\|y\|_{\theta}\le R_0,\quad
y(t)\ge\Gamma\|y\|_\theta,\quad \forall\, t\in[\gamma,\delta],
$$
as claimed.
\end{proof}

The following result can be proved in an analogous manner.
The proof is omitted.

\begin{theorem}\label{thm1b}
Assume  {\rm (H1)--(H2)} hold and
\begin{itemize}
\item[(H3')] $f(t,y,z)\ge \varphi'(t,y)$
for all $t\in[\gamma,\delta]$ and all
$(y,z)\in(0,+\infty)\times\ \mathbb{R}^*$,
where $\varphi'\in C([\gamma,\delta]\times(0,+\infty))$
satisfies
$$
\liminf_{y\to+\infty}\min
_{t\in[\gamma,\delta]}
\frac{\varphi'(t,y)}{y}>\frac{1}{\Lambda_0 \ell}\,.
$$
\end{itemize}
Then problem \eqref{GP1} has at least one nontrivial positive
solution.
\end{theorem}

\subsection{Twin solutions}

Let $\mathcal{P}$ be a cone of a real Banach space $E$. Let
$0<c<d$ be constants and $\beta$, $\alpha$ two continuous
functionals on $\mathcal{P}$ convex
and concave, respectively. Define the
convex sets:
\begin{gather*}
P_{d}=\{y\in\mathcal{P}:\Vert y\Vert<d\},\\
P(\beta,d)=\{x\in \mathcal{P}: \beta(x)\leq d\},\\
P(\beta,\alpha,c,d)=\{x\in \mathcal{P}: \alpha(x)\ge c,\;
\beta(x)\leq d\}.
\end{gather*}
We will apply the following fixed point theorem to prove the
existence of two positive fixed points for the operator $T$.

\begin{lemma}[\cite{LiHan}] \label{theomA}
Let $A: \mathcal{P}\to \mathcal{P}$ be a completely
continuous operator. Let $\beta$ and $\alpha$ be continuous convex
and concave functionals on $\mathcal{P}$, respectively. Let $d$
and $c$ be real numbers. Assume that
\begin{itemize}
\item[(i)] $0\in\{x\in\mathcal{P}: \beta(x)<d\}$ and the
 set $\{x\in\mathcal{P}:
\beta(x)<d\}$ is bounded;

 \item[(ii)] $\{x\in P(\beta,\alpha,c,d): \beta(x)<d\}\neq\emptyset$
and $\beta(Ax)<d$ for all $x\in P(\beta,\alpha,c,d)$;

 \item[(iii)] $\beta(Ax)<d$ for all $x\in P(\beta,d)$ with
$\alpha(Ax)<c$;

\item[(iv)] $i(A,P_r,\mathcal{P})=0$ for sufficiently
small positive number $r$, $i(A,P_R,\mathcal{P})=0$
for sufficiently large positive
number $L$.
\end{itemize}
Then $A$ has at least two fixed points $x_1$, $x_2$ in
$\mathcal{P}$ such that $\|x_1\|>r$ with $\beta(x_1)<d$, and
$\|x_2\|<L$ with $\beta(x_2)>d$.
\end{lemma}

Our main result in this section is as follows.

\begin{theorem}\label{thm2}
Assume {\rm  (H1)--(H2)} and
\begin{itemize}
\item[(H4)] $f(t,y,z)\ge\varrho(t,y)$ for all
$t\in[\gamma,\delta]$ and all $(y,z)\in(0,+\infty)\times\mathbb{R}^*$,
where the function $\varrho\in C([\gamma,\delta]\times(0,+\infty))$
satisfies
\[
\liminf_{y\to0}\min_{t\in[\gamma,\delta]}
\frac{\varrho(t,y)}{y}>\frac{1}{\Lambda_0\ell},\quad
\liminf_{y\to+\infty}\min_{t\in[\gamma,\delta]}
\frac{\varrho(t,y)}{y}>\frac{1}{\Lambda_0\ell}\,.
\]
\end{itemize}
Then, problem \eqref{GP1} has at least two positive solutions
$y_1$, $y_2$ such that
$$
0<\|y_1\|_{\theta}\le R_0\le \|y_2\|_{\theta}.
$$
\end{theorem}

\begin{proof}
 Define a sequence of operators $T_n$ by \eqref{approximatingoperator}
and then consider the nonnegative,
continuous concave and convex functionals $\alpha$, $\beta$
defined respectively by
$$
\alpha(y)=\min_{y\in[\gamma,\delta]}\frac{y(t)}{e^{\theta t}},\quad
\beta(y)=\Vert y\Vert_\theta.
$$
Lemmas \ref{lemma1} and \ref{lemma2} guarantee that $T_n:
\mathcal{P}\to\mathcal{P}$ is completely continuous. So,
we only have to verify the conditions of Lemma \ref{theomA}.

{\bf Claim 1.} $\beta(y)=\|y\|_{\theta}$. For $R_0$
given by the inequality \eqref{Rhypothesis} in Assumption
(H2), it is clear that $0\in\{\;y\in \mathcal{P}:
\beta(y)<R_0\}$ and the set $\{y\in \mathcal{P}: \beta(y)<R_0\}$
is bounded.

{\bf Claim 2.} The set $\{y\in
P(\beta,\alpha,\frac{R_0}{2}e^{-\theta\delta},R_0):
\beta(y)<R_0\}$ is nonempty since it contains the constant function
$y_0\equiv \frac{R_0}{2}$. Indeed,
$\beta(y_0)=\frac{R_0}{2}\sup_{t\in\mathbb{R}^+}\,e^{-\theta
t}<R_0$ and $\alpha(y_0)=\frac{R_0}{2}e^{-\theta\delta}$. Let
$y\in P(\beta,\alpha,\frac{R_0}{2}e^{-\theta\delta},R_0);$ then
$\beta(y)=\|y\|_{\theta}\leq R_0$. As in the proof of Theorem
\ref{thm1}, Claim (a), for $n\ge n_0>\frac{1}{R_0}$, we can check
that $\beta(T_ny)=\|T_ny\|_\theta<R_0$. So the condition (ii) of
Lemma \ref{theomA} is satisfied.

{\bf Claim 3.} Arguing as in Claim 2, we obtain
$$
\beta(T_ny)=\|T_ny\|_\theta<R_0,\quad \forall\,y\in
P(\beta,R_0),\; \forall\,n\in\{n_0,n_0+1,\dots\}.
$$
So the condition (iii) of Lemma \ref{theomA} is satisfied.

{\bf Claim 4.} Since
$\liminf_{y\to0}\min_{t\in[\gamma,\delta]}
\frac{\varrho(t,y)}{y}>\frac{1}{\Lambda_0 \ell}$,
there exist $\varepsilon_0$ and $r_0>0$ such that
$$
\varrho(t,y)\geq\big(\frac{1}{\Lambda_0\ell}+\varepsilon_0\big)y,\quad
\forall y\in[0,r_0]\,\text{ and }\,\forall\,t\in[\gamma,\delta].
$$
We choose a sufficiently small
$r=\min(R_0/2,r_0/e^{\theta\delta})$. Proceeding
as in the proof of Theorem \ref{thm1}, Claim (b), we can prove that
$$
T_ny\not\le y,\quad \text{for any }\,y\in\partial P_{r}.
$$
According to Lemma \ref{lemB}, we infer that
$$
i(T_n,P_{r},\mathcal{P})=0.
$$

{\bf Claim 5.} Since
$\liminf_{y\to+\infty}\min_{t\in[\gamma,\delta]}
\frac{\varrho(t,y)}{y}>\frac{1}{\Lambda_0
\ell}$, there exist $\varepsilon_1$ and $\sigma>0$ such that
\begin{equation}\label{eqs}
\varrho(t,y)\geq\big(\frac{1}{\Lambda_0\ell}+\varepsilon_1\big)y,\quad
\text{for each }\;y\geq \sigma\,\text{ and }\,t\in[\gamma,\delta].
\end{equation}
Choose sufficiently large $L=\max(2R_0,\frac{\sigma}{\Gamma})$. So
$y\in\partial P_{L}$ implies
$$
y(t)\ge \Gamma \|y\|_{\theta}\ge L\Gamma\ge
\frac{\sigma}{\Gamma}\Gamma=\sigma,\quad t\in[\gamma,\delta].
$$
Then, using the inequality
$$
\varrho(t,y(t))\geq\big(\frac{1}{\Lambda_0\ell}+\varepsilon\big)
y(t),\quad\text{for any }\;t\in[\gamma,\delta]
$$
and arguing as in the proof of Theorem \ref{thm1}, Claim (b),
we can prove that
$$
T_ny\not\le y,\;\text{ for any }\,y\in\partial P_{L}.
$$
By Lemma \ref{lemB}, we deduce that
$$
i (T_n,P_{L},\mathcal{P})=0.
$$
Thus, the condition (vi) of Lemma \ref{theomA} is satisfied.
According to this lemma with $c=\frac{R_0}{2}e^{-\theta\delta}$
and $d=R_0$, we infer that, for each $n\in\{n_0,n_0+1,\dots \}$,
$T_n$ has at least two positive fixed points $y_{n,1}, y_{n,2}\in
\mathcal{P}$ such that
$r<\|y_{n,1}\|_{\theta}<R_0<\|y_{n,2}\|_{\theta}<L$. Now consider
the sequence of functions $\{y_{n,i}\}_{n\ge n_0},\; i=1,2$.
Essentially the same argument used for $\{y_n\}_{n\ge n_0}$ in
Theorem \ref{thm1} guarantees that $\{y_{n,i}\}_{n\ge n_0},\;
i=1,2$ has a convergent subsequence $\{y_{n_j,i}\}_{j\ge 1}$  such
that $\lim_{j\to+\infty}y_{n_j,i}=y_i,\ i=1,2$ for
the norm topology of $X$. Consequently, $y_1$ and $ y_2$ are two
positive solutions of problem \eqref{GP1} with
$$
r\le\|y_1\|_{\theta}\le R_0\le\|y_2\|_{\theta}\le L.
$$
\end{proof}

\subsection{Triple nonnegative solutions}

Let $r>a>0$, $L>0$ be constants, $\psi$ a nonnegative continuous
concave functional and $\alpha, \beta$ nonnegative continuous
convex functionals on a cone $\mathcal{P}$ of a Banach space
$(E,\Vert\cdot\Vert)$. Define the convex sets:
\begin{gather*}
P(\alpha,r;\beta,L)=\{x\in \mathcal{P}:
\alpha(x)<r, \;\beta(x)<L\},
\\
\overline{P}(\alpha,r;\beta,L)=\left\{x\in \mathcal{P}:
\alpha(x)\leq r,\;\beta(x)\leq L\right\},
\\
P(\alpha,r;\beta,L;\psi,a)=\left\{x\in \mathcal{P}:
\alpha(x)<r,\;\beta(x)<L,\;\psi(x)>a\right\},
\\
\overline{P}(\alpha,r;\beta,L;\psi,a)=\left\{x\in \mathcal{P}:
\alpha(x)\leq r,\;\beta(x)\leq L,\;\psi(x)\geq a\right\}.
\end{gather*}
The following assumptions about the nonnegative continuous convex
functionals $\alpha, \beta$ will be considered:
\begin{itemize}
\item[(A1)] there exists $M>0$ such that
$\|x\|\leq M \max\{ \alpha(x),\beta(x)\}$, for all $x\in
\mathcal{P}$;

\item[(A2)] $P(\alpha,r;\beta,L)\neq\emptyset\;$ for all $r>0$, $L>0$.
\end{itemize}

\begin{lemma}[\cite{BaiGe}]\label{theomB}
Let $E$ be a Banach space $\mathcal{P}\subset E$ a cone and
$r_2\ge d>c>r_1>0$, $L_2\ge L_1>0$ be constants. Assume that
$\alpha, \beta$ are nonnegative continuous convex functionals
satisfying (A1) and (A2). Let $\psi$ be
a nonnegative continuous concave functional on $\mathcal{P}$ such that
$\psi(x)\leq\alpha(x)$ for all
$x\in\overline{P}(\alpha,r_2;\beta,L_2)$ and let
 $A: \overline{P}(\alpha,r_2;\beta,L_2)\to\overline{P}(\alpha,r_2;\beta,L_2)$
be a completely continuous operator. Assume
\begin{itemize}
\item[(B1)]
$\{x\in\overline{P}(\alpha,d;\beta,L_2;\psi,c):
\psi(x)>c\}\neq\emptyset$ and $\psi(Ax)>c$, for all
$x$ in $\overline{P}(\alpha,d;\beta,L_2;\psi,c)$;

\item[(B2)] $\alpha(Ax)<r_1$, $\beta(Ax)<L_1$,
for all $x\in\overline{P}(\alpha,r_1;\beta,L_1)$;

\item[(B3)] $\psi(Ax)>c$ for all
$x\in\overline{P}(\alpha,r_2;\beta,L_2;\psi,c)$ with
$\alpha(Ax)>d$.

\end{itemize}
Then $A$ has at least three fixed points $x_1$, $x_2$ and $x_3$ in
$\overline{P}(\alpha,r_2;\beta,L_2)$ with
\begin{gather*}
x_1\in P(\alpha,r_1;\beta,L_1),\\
x_2\in\{x\in\overline{P}(\alpha,r_2;\beta,L_2;\psi,c): \psi(x)>c\},\\
x_3\in\overline{P}(\alpha,r_2;\beta,L_2)\setminus
\overline{P}(\alpha,r_2;\beta,L_2;\psi,c)\cup\overline{P}
(\alpha,r_1;\beta,L_1).
\end{gather*}
\end{lemma}

Now we arrive at our final existence result in this paper.

\begin{theorem}\label{thm3}
 Assume the following assumptions hold:
\begin{itemize}
\item[(H1')] $F:I^2\times\mathbb{R}^*\to\mathbb{R}^+$ is a continuous
function and there exist functions 
$g,w\in C((1,\infty),I)$ and $h,k\in C(\mathbb{R}^*,I)$ such
that
$$
0\le F(t,u,v)\leq(g(u+1)+w(u+1))(h(v)+k(v)),\quad\forall\,(t,u,v)\in
I^2\times\mathbb{R}^*
$$
where $g,h$ are non-increasing functions,
$w/g$ and $k/h$ are nondecreasing functions.

\item[(H2')] For all $\Re>0$,
$$
\widetilde{\Pi}(\Re)=\int_0^{+\infty}
e^{-r_1s}\max\{G(s,s),\overline G(s)\}\Phi(s)g(e^{-\theta
s})h(-e^{-cs}\Re)ds<\infty
$$
and there exists constants $R_1,\,R_2$ with
$R_2<\frac{\Lambda_0}{2e^{\theta\delta}}R_1 $ such that for
$i=1,2$
\begin{equation}\label{eq3solu}
\Big(1+\frac{w(R_i+1+\frac{1}{\Gamma})}
{g(R_i+1+\frac{1}{\Gamma})}\Big)
\Big(1+\frac{k(R_i)}{h(R_i)}\Big)\widetilde{\Pi}(R_i)<R_i.
\end{equation}

\item[(H5)] $f(t,y,z)\ge \zeta(t,y)$ for all
$t\in[\gamma,\delta]$ and all
$(y,z)\in(0,+\infty)\times\ \mathbb{R}^*$,
where $\zeta\in C([\gamma,\delta]\times(0,+\infty))$  satisfies
$$
\liminf_{y\to0}\min_{t\in[\gamma,\delta]}
\frac{\zeta(t,y)}{y}=+\infty.
$$
\end{itemize}
Then \eqref{GP1} has at least three nonnegative solutions
(two of which are positive)  $y_1, y_2$ and $y_3$ in
$\overline{P}(\alpha,R_1;\beta,R_1)$ such that for $t\in[0,\infty)$,
\begin{gather*}
e^{-\theta t}|y_1(t)|\le R_2,\quad e^{-\theta t}|y'_1(t)|\le R_2,
\\
e^{-\theta t}|y_2(t)|\le R_1,\quad e^{-\theta t}|y'_2(t)|\le R_1,
\\
R_2\le e^{-\theta t}|y_3(t)|\le R_1,\quad
R_2 \le e^{-\theta t}|y'_3(t)|\le R_1,
\end{gather*}
and for $t\in[\gamma,\delta]$,
\[
|y_2(t)|\ge\frac{\Lambda_0}{2e^{\theta \delta}}R_1,\quad
|y_3(t)|\le\frac{\Lambda_0}{2e^{\theta \delta}}R_1.
\]
\end{theorem}

\begin{proof}
 Define an operator sequence by \eqref{approximatingoperator}
and consider the functionals
$$
\alpha(y)=\frac{1}{\Gamma}+\sup_{t\in\mathbb{R}^+}\frac{|y(t)|}{e^{\theta
t}},\quad
\beta(y)=\sup_{t\in\mathbb{R}^+}\frac{|y'(t)|}{e^{\theta t}},\quad
\psi(y)=\frac{1}{\Gamma}+\min_{t\in[\gamma,\delta]}
\frac{|y(t)|}{e^{\theta t}}.
$$
Then $\alpha, \beta$ are nonnegative continuous convex functionals
satisfying (A1) and (A2); $\psi$ is a
nonnegative continuous concave functional with
$\psi(y)\leq\alpha(y)$ for all $y\in\mathcal{P}$.
Here $\mathcal{P}$ is the cone defined in \eqref{cone}.
For this, we will apply Theorem
\ref{thm1b} to verify the existence of fixed points for the
operator $T_n$. Lemmas \ref{lemma1} and \ref{lemma2} guarantee
that $T_n: \mathcal{P}\to \mathcal{P}$, is completely continuous.

{\bf Claim 1.}
$T_n:\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1)\to
\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1)$, for
$n\ge n_0>\frac{1}{R_1+\frac{1}{\Gamma}}$. Indeed, if
$y\in\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1)$, then
$\alpha(y)\leq R_1+\frac{1}{\Gamma}$ and $\beta(y)\leq R_1$.
Arguing as in the proof of Theorem \ref{thm1}, Claim 1, we obtain,
using Assumptions (H1') and (H2'), the
following estimates valid for $t\in\mathbb{R}^+$:
\begin{align*}
&\frac{1}{\Gamma}+e^{-\theta t}|T_ny(t)|\\
&= \frac{1}{\Gamma}+\int_0^{+\infty}e^{-\theta t}G(t,s)
 \Phi(s)f_n(s,y(s),e^{-cs}y'(s))ds\\
&= \frac{1}{\Gamma}+\int_0^{+\infty}e^{-\theta t}G(t,s)\Phi(s)
f\left(s,\max\{e^{\theta s}/n,y(s)\},
\max\{e^{\theta s}/n,e^{-cs}y'(s)\}\right)ds\\
&\leq \frac{1}{\Gamma}+\int_0^{+\infty}e^{-r_1s}G(s,s)\Phi(s)
\Big(g(\max\{1/n,e^{-\theta s}y(s)\}+1)\\
&\quad + w(\max\{1/n,e^{-\theta s}y(s)\}+1\Big)
\Big(h(\max\{1/n,e^{-(c+\theta )s}y'(s)\})\\
&\quad +k(\max\{1/n,e^{-(c+\theta )s}y'(s)\})\Big)ds\\
&\leq \frac{1}{\Gamma}+\int_0^{+\infty}
\left(1+\frac{w(\max\{1/n,e^{-\theta
s}y(s)\}+1)}{g(\max\{1/n,e^{-\theta s}y(s)\}+1)}\right)\\
&\quad\times  \left(1 +\frac{k(\max\{1/n,e^{-(c+\theta
)s}y'(s)\})}{h(\max\{1/n,e^{-(c+\theta )s}y'(s)\})}\right)\\
&\quad \times e^{-r_1s}\max\{G(s,s),\overline G(s)\}\Phi(s)g(e^{-\theta
s}y(s)+1)h(e^{-(c+\theta )s}y'(s))ds.
\end{align*}
Hence
\begin{align*}
&\frac{1}{\Gamma}+e^{-\theta t}|T_ny(t)|\\
&\leq \frac{1}{\Gamma}+\left(1+\frac{w(\max\{1/n,\alpha(y)\}+1)}{g(\max\{1/n,
\alpha(y)\}+1)}\right)
\left(1+\frac{k(\max\{1/n,\beta(y)\})}{h(\max\{1/n,\beta(y)\})}\right)\\
&\quad \times\int_0^{+\infty} e^{-r_1s}\max\{G(s,s),\overline
G(s)\}\Phi(s)g(e^{-\theta
s}\Gamma\alpha(y))h(-e^{-cs}\beta(y))ds\\
&\leq \frac{1}{\Gamma}+
\Big(1+\frac{w\left(R_1+1+\frac{1}{\Gamma}\right)}
 {g\left(R_1+1+\frac{1}{\Gamma}\right)}\Big)
\Big(1+\frac{k(R_1)}{h(R_1)}\Big)
\widetilde{\Pi}(R_1)\\
&<R_1<R_1+\frac{1}{\Gamma}.
\end{align*}
Therefore,  $\alpha(T_ny)\le R_1+\frac{1}{\Gamma}$, and
\begin{align*}
&e^{-\theta t}|(T_ny)'(t)|\\
&=  \int_0^{+\infty}e^{-\theta
t}G_t(t,s)\Phi(s)F(s,\max\{1/n,e^{-\theta s}y(s)\},
\max\{1/n,e^{-(c+\theta )s}y'(s)\})ds\\
&\leq \int_0^{+\infty}e^{-r_1s}\overline{G}(s)\Phi(s)\left(g(\max\{e^{\theta
s}/n,y(s)\}+1)+
w(\max\{1/n,e^{-\theta s}y(s)\}+1\right)\\
&\quad\times\left(h(\max\{1/n,e^{-(c+\theta )s}y'(s)\})
+k(\max\{1/n,e^{-(c+\theta )s}y'(s)\})\right)ds\\
&\leq \int_0^{+\infty}\left(1 +\frac{w(\max\{1/n,e^{-\theta
s}y(s)\}+1)}{g(\max\{1/n,e^{-\theta s}y(s)\}+1)}\right) \left(1
+\frac{k(\max\{1/n,e^{-(c+\theta
)s}y'(s)\})}{h(\max\{1/n,e^{-(c+\theta
)s}y'(s)\})}\right)\\
&\quad\times e^{-r_1s}\max\{G(s,s),\overline G(s)\}\Phi(s)g(e^{-\theta
s}y(s)+1)h(e^{-(c+\theta
)s}y'(s))ds\\
&\leq \Big(1+\frac{w\left(R_1+1+\frac{1}{\Gamma}\right)}
 {g\left(R_1+1+\frac{1}{\Gamma}\right)}\Big)
\Big(1+\frac{k(R_1)}{h(R_1)}\Big) \widetilde{\Pi}(R_1)<R_1.
\end{align*}
Consequently, $\beta(T_ny)\le R_1$.

{\bf Claim 2.} Condition \eqref{eq3solu} implies that
$T_n: \overline{P}(\alpha,R_2+\frac{1}{\Gamma};\beta,R_2)\to
\overline{P}(\alpha,R_2+\frac{1}{\Gamma};\beta,R_2)$ for
$n\ge n_1>\frac{1}{R_2+\frac{1}{\Gamma}}$. The proof is identical to
that in Claim 1. So the condition $(\mathfrak{B}2)$ of Lemma
\ref{theomB} is satisfied.

{\bf Claim 3.} The set
$\{y\in \overline{P}(\alpha,\frac{R_1}{2}
 +\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1):\psi(y)>\frac{\Lambda_0}{2e^{\theta\delta}}R_0)\}$
is nonempty. Notice that the constant function $y_0\equiv
\frac{R_1}{2}$ lies in the set
$\overline{P}(\alpha,\frac{R_1}{2}+\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1)$ and $\psi(y_0)>\frac{\Lambda_0}{2e^{\theta
\delta}}R_1$. Indeed,
$\alpha(y_0)=\frac{R_1}{2}\sup_{t\in[0,\infty)}\,e^{-\theta
t}+\frac{1}{\Gamma}\leq \frac{R_1}{2}+\frac{1}{\Gamma},\;$
$\beta(y_0)=0$ and $\psi(y_0)=e^{-\theta
\delta}\frac{R_1}{2}>\frac{\Lambda_0}{2e^{\theta \delta}}R_1$ for
$\Lambda_0<1$.

{\bf Claim 4.} We prove that
$\psi(T_ny)>\frac{\Lambda_0}{2e^{\theta \delta}}R_1$,
$\forall\,y\in
\overline{P}(\alpha,\frac{R_1}{2}+\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1)$. If $y\in
\overline{P}(\alpha,\frac{R_1}{2}+\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1)$, then $\alpha(y)\leq\frac{R_1}{2}+\frac{1}{\Gamma}$.
Moreover, the condition (H5) tells us that, if
$M_4=\frac{2e^{(\theta+r_1)\delta}}{\Lambda_0\Gamma\ell}$ then
there exists some $\mu>\frac{R_1}{2}e^{\theta \delta}$ such that
\begin{equation}\label{eqkx}
\zeta(t,y)\ge M_4y,\quad \forall\,y\in(0,\mu),
\forall\,t\in[\gamma,\delta].
\end{equation}
We can see that, for any
$y\in\overline{P}(\alpha,\frac{R_1}{2};\beta,R_1;
\psi,\frac{\Lambda}{2e^{\theta
\delta}}R_1)$ and $t\in[\gamma,\delta]$, we have
$$
\alpha(y)\leq\frac{R_1}{2}+\frac{1}{\Gamma}\,\Rightarrow\,y(t)\leq
\frac{R_1}{2}e^{\theta \delta}<\mu,\quad \forall\,t\in[\gamma,\delta].
$$
With Lemma \ref{lem4} (c) and \eqref{eqkx}, we obtain the
estimates:
\begin{align*}
\psi(T_ny)
&>\min_{t\in[\gamma,\delta]}\int_0^{+\infty}e^{-\theta t}G(t,s)\\
&\quad \times \Phi(s)f\left(s,\max\{e^{\theta s}/n,y(s)\},\max\{e^{\theta s}/n,e^{-cs}y'(s)\}\right)ds\\
&\geq \Lambda_0 e^{-\theta
\delta}\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)\zeta(s,\max\{e^{\theta s}/n,y(s)\})ds\\
&\geq \Lambda _0e^{-\theta
\delta}\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)
M_4\max\{e^{\theta s}/n,y(s)\}ds\\
&\geq \Lambda_0 e^{-\theta
\delta}\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)
M_4y(s)ds\\
&\geq \Lambda_0 M_4e^{-\theta
\delta}\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)
\Gamma\|y\|_{\theta}ds\\
&>\frac{1}{2}M_4\Lambda_0\Gamma \,e^{-(\theta+r_1)
\delta}\|y\|_{\theta}\int_{\gamma}^{\delta}e^{-r_1s}G(s,s)\Phi(s)ds\\
&= \|y\|_{\theta}\ge \psi(y)\ge\frac{\Lambda_0}{2e^{\theta
\delta}}R_1.
\end{align*}

{\bf Claim 5.}
$\psi(T_ny)>\frac{\Lambda_0}{2e^{\theta\delta}}R_1$ for all
$y\in\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1)$ with
$\alpha(T_ny)>\frac{R_1}{2}+\frac{1}{\Gamma}$. Let
$y\in\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1)$ be such that
$\alpha(T_ny)>\frac{R_1}{2}+\frac{1}{\Gamma}$. For any $\sigma
\in\mathbb{R}^+$, we know by Lemma \ref{lem4}(b),(c) that
\begin{align*}
\psi(T_ny)
&= \frac{1}{\Gamma}+e^{-\theta
\delta}\min_{t\in[\gamma,\delta]}\int_0^{+\infty}G(t,s)\Phi(s)f_n(s,y(s),e^{-cs}y'(s))ds\\
&\geq \frac{1}{\Gamma}+e^{-\theta
\delta}\int_0^{+\infty}\Lambda_0 e^{-r_1s}G(s,s)\Phi(s)f_n(s,y(s),e^{-cs}y'(s))ds\\
&\geq \frac{1}{\Gamma}+\Lambda_0 e^{-\theta
\delta}\int_0^{+\infty} e^{-\theta\sigma}G(\sigma,s)\Phi(s)f_n(s,y(s),e^{-cs}y'(s))ds\\
&= \frac{1}{\Gamma}+\Lambda_0 e^{-\theta
\delta}e^{-\theta\sigma}\int_{0}^{+\infty}
G(\sigma,s)\Phi(s)f_n(s,y(s),e^{-cs}y'(s))ds\\
&= \Lambda_0 e^{-\theta \delta}(\frac{1}{\Gamma}\frac{e^{\theta
\delta}}{\Lambda_0}+e^{-\theta\sigma}T_ny(\sigma))\\
&\geq \Lambda_0 e^{-\theta
\delta}(\frac{1}{\Gamma}+e^{-\theta\sigma}T_ny(\sigma)).
\end{align*}
Passing to the supremum over $\sigma$, we obtain that $y\in
\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1;\psi,
\frac{\Lambda_0}{2e^{\theta \delta}}R_1)$,
$$
\psi(T_ny)\ge\Lambda_0 e^{-\theta \delta}\alpha(T_ny)\ge\Lambda_0
e^{-\theta \delta}(R_1+\frac{1}{\Gamma})>
\frac{\Lambda_0}{2e^{\theta \delta}}R_1.
$$
To sum up, all of the hypotheses of Lemma \ref{theomB} are met if
we take $L_2=R_1,\;r_2=R_1+\frac{1}{\Gamma}$,
$L_1=R_2,\;r_1=R_2+\frac{1}{\Gamma}$
$d=\frac{R_1}{2}+\frac{1}{\Gamma}$ and
$c=\frac{\Lambda_0}{2e^{\theta \delta}}R_1$. Hence, for each
$n\in\{n_1,n_1+1,\dots \}$, $T_n$ has at least three nonnegative
fixed points
$y_{n,i}\in\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1)$,
$i=1,2,3$, with
\begin{gather*}
y_{n,1}\in P(\alpha,R_2+\frac{1}{\Gamma};\beta,R_2),\\
y_{n,2}\in \{\,y\in\overline{P}(\alpha,R_1+\frac{1}{\Gamma};
\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1): \psi(y)>\frac{\Lambda_0}{2e^{\theta \delta}}R_1\},\\
y_{n,3}\in\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1)\setminus
\overline{P}(\alpha,R_1+\frac{1}{\Gamma};\beta,R_1;\psi,\frac{\Lambda_0}{2e^{\theta
\delta}}R_1)\cup\overline{P}(\alpha,R_2;\beta,R_2).
\end{gather*}
Consider the sequence of functions $\{y_{n,i}\}_{n\ge n_1},\;
i=1,2,3$. Arguing as in the proof as in Theorem \ref{thm1}, we can
show that $\{y_{n,i}\}_{n\ge n_1},\; i=1,2,3$ has a convergent
subsequence $\{y_{n_j,i}\}_{j\ge 1}$, such that
$\lim_{j\to+\infty}y_{n_j,i}=y_i,\ i=1,2,3$ for the
strong topology of $X$. Consequently, $y_1,y_2$ and $y_3$ are
three different nonnegative solutions of problem \eqref{GP1} and
satisfy
\begin{gather*}
e^{-\theta t}|y_1(t)|\le R_2,\quad e^{-\theta t}|y'_1(t)|
\le R_2,\quad t\in[0,\infty),\\
e^{-\theta t}|y_2(t)|\le R_1,\quad e^{-\theta t}|y'_2(t)|
\le R_1,\quad t\in[0,\infty), \\
R_2< e^{-\theta t}|y_3(t)|\le R_1,\quad R_2< e^{-\theta t}|y'_3(t)
|\le R_1,\quad t\in[0,\infty), \\
|y_2(t)|\ge\frac{\Lambda_0}{2e^{\theta \delta}}R_1,\quad
|y_3(t)|\le\frac{\Lambda_0}{2e^{\theta \delta}}R_1,\quad t\in
[\gamma,\delta].
\end{gather*}
\end{proof}

\section{Examples}

Let $\Phi(t)=e^{-\mu t}$ and consider the nonlinearity
$$
f(t,y,z)=\left(g(ye^{-\theta t})+w(ye^{-\theta t})\right)
\left(h(ze^{-\theta t})+k(ze^{-\theta t})\right),\quad
(t,y,z)\in I^2\times \mathbb{R}^*
$$
where $g(u)=1/u$, $w(u)=u^2$ and the functions $h$ and $k$
are defined by
$$
h(v)=\begin{cases}
-v,&v\le -1; \\
\frac{1}{\sqrt{-v}},\; &-1\le v< 0; \\
\frac{1}{\sqrt{v}}, & v> 0.
\end{cases}
\quad k(v)=\begin{cases}
-v, &v\le -1; \\
\frac{1}{\sqrt{-v}}, &-1\le v< 0; \\
1+v,& v\ge 0.
\end{cases}
$$
To check the inequality \eqref{Rhypothesis} in
(H2), take $\gamma=1/3$, $\delta=1/2$, $c=1/2$,
$\lambda=1/3$, $\eta=2$, $\alpha=1/8$ and $\mu=100;$ so we can
choose $\theta=1$ and $R_0=5$. In addition, we have
$$
G(s,s)= \begin{cases}
\frac{1}{\Delta}\left(1-e^{(r_2-r_1)s}\right),
 & \text{if } s\le\eta ;\\
\frac{1}{\Delta}\left(1-\alpha e^{r_2\eta}-e^{(r_2-r_1)s}(1-\alpha
e^{r_1\eta})\right), & \text{if } s\ge\eta
\end{cases}
$$
and
$$
\overline G(s)= \begin{cases}
\frac{r_1}{\Delta}\left(2-\alpha e^{r_2\eta}
 -\alpha e^{r_2(\eta-s)+r_1s)}\right), & \text{if } s\le\eta; \\
\frac{r_1}{\Delta}\left(2-\alpha e^{r_2\eta}-\alpha
e^{r_1\eta}\right), & \text{if } s\ge\eta.
\end{cases}
$$
Using Matlab 7, we have found $\Pi(5)=6.9589.10^{-4}$, whence
$$
\Big(1+\frac{w(R_0)}{g(R_0)}\Big)\Big(1+\frac{k(R_0)}{h(R_0)}\Big)
\Pi(R_0)=1.2641.
$$
Therefore Assumptions (H1) and (H2) are met.
Also, Assumption (H3) in Theorem \ref{thm1} is
clearly satisfied. As a consequence, if
$$
f(t,y,y'e^{-t/2})=\left(g(ye^{-t})+
w(ye^{-t})\right)\left(h(y'e^{-3t/2})
+k(y'e^{-3t/2})\right)
$$
then the singular boundary value problem
\begin{equation}\label{example2}
\begin{gathered}
-y''+1/2y'+1/3y=e^{-10^2t}f(t,y,y'e^{-t/2}),\quad t>0\\
y(0)=\alpha y(\eta),\quad
\lim_{t\to\infty}e^{-t/2}y'(t)=0,
\end{gathered}
\end{equation}
has at least one nontrivial positive solution. Moreover, we can
check that Assumption $(\mathcal{H}_4)$ in Theorem \ref{thm2} is
fulfilled. Therefore, this problem has also two nontrivial
positive solutions.

Let $\hat{g}, \hat{w}$ the functions defined by
$$
\hat{g}(u)=\begin{cases}
1/(u-1),& u>1; \\
1/4, & 0\le u<1.
\end{cases}
 \quad \hat{w}(u)=(u-1)^2
$$
The inequality \eqref{eq3solu} in Theorem \ref{thm3}
holds true for $R_1=3$ and $R_2=4/10$.
Indeed $\widetilde\Pi(3)= 0.0020, \widetilde\Pi(4/10)=0.0055$ and
\begin{gather*}
\Big(1+\frac{\hat{w}(R_1+1+\frac{1}{\Gamma})}{\hat{g}(R_1+1
+\frac{1}{\Gamma})}\Big)
\Big(1+\frac{k(R_1)}{h(R_1)}\Big)\widetilde\Pi(R_1)= 0.6000< 3,\\
\Big(1+\frac{\hat{w}(R_2+1+\frac{1}{\Gamma})}{\hat{g}(R_2+1
+\frac{1}{\Gamma})}\Big)
\Big(1+\frac{k(R_2)}{h(R_2)}\Big)\widetilde\Pi(R_2)=  0.1875< 0.4.
\end{gather*}
Therefore, the singular boundary value problem
\begin{equation}\label{example3}
\begin{gathered}
-y''+1/2y'+1/3y=e^{-10^2t}\hat{f}(t,y,y'e^{-t/2}),\quad t>0\\
y(0)=\alpha
y(\eta),\quad\lim_{t\to\infty}e^{-t/2}y'(t)=0,
\end{gathered}
\end{equation}
where
$$
\hat{f}(t,y,y'e^{-t/2})
=\left(\hat{g}(ye^{-t}+1)+\hat{w}(ye^{-t}+1)\right)
\left(h(y'e^{-3t/2})+k(y'e^{-3t/2})\right)
$$
 has in fact three nonnegative solutions, at least two of
which are positive.

\section{Concluding remarks}

In this work, we have considered problem \eqref{GP1} when the
nonlinearity may not only possess space-singularities in $y$ and $y'$
at the origin, but also takes quite general asymptotic behaviors
near positive infinity, including polynomial growth as a special
case. Indeed, we can consider the special cases in which $F$
behaves in the first argument as $g(u)+w(u)$ with
$g(u)=u^{-\sigma}$, $w(u)=u^m$ ($\sigma>0$, $m\in\mathbb{N}^*$) and
in the second argument as $h(v)+k(v)$ with
$h(v)=v^{-\mu}$, $k(v)=v^n$ ($\mu>0$, $n\in\mathbb{N}^*$). In this
respect, the main assumptions are (H1) and
(H2).

The existence results obtained in this paper have the advantages
to allow working in a special cone of a Banach space such that
most of solutions are positive hence nontrivial. With
(H3) (or (H3'), we have proved in
Theorem \ref{thm1} and \ref{thm1b} existence of at least one positive solution
with $y(t)\ge\Gamma\Vert y\Vert_\theta$ for
$t\in[\gamma,\delta]$; that is $y\in\mathcal{P}$. At this step,
notice that $[\gamma,\delta]$ is an arbitrary chosen interval
which helps to get nontrivial solutions; this does not always hold
true when one applies the Schauder fixed point theorem which rather
provides solutions in a ball. In addition (H3) covers
nonlinearities which are bounded below by sublinear functions near
the origin while in (H3'), $f$ may be superlinear at
positive infinity.

Using a recent fixed point theorem of two functionals, we have
obtained existence of a second solution in Theorem \ref{thm2}
 satisfying $0<\Vert y_1\Vert_\theta\le R\le\Vert y_2\Vert_\theta$.
However, we may notice that (H4) combines Assumptions
(H3) and (H3') and in counterpart yields
precise information about solutions.

Finally, Assumption (H5) is of the form of
(H2). However with a stronger assumption than
(H3), we have even proved existence of three
solutions by means of a three-functional fixed point theorem;
notice however that one of them lies in a ball and thus could be a
trivial solution.

The multi-point condition at $0$ has given rise to a new and
elaborated Green's function; its properties have enabled us to
choose an appropriate cone to contain the desired solutions. The
space singularities have been treated by approximation through the
nonlinearity \eqref{approximatingfunction} and the operator
\eqref{approximatingoperator} for which existence of fixed points
has been proved under sharp estimates of the Green's function.
Then, the solutions have been obtained as limits as $n\to+\infty$
via compactness sequential arguments.

The example of application shows that all of these hypotheses can
be satisfied for quite simple and general nonlinearities. One of
the novelty of this work is that we have considered a class of
space-singular nonlinearities at the origin with general growth at
positive infinity. We hope this work can provide improvements of
the rich literature developed for multi-point boundary value
problems on the positive half line.

\subsection*{Acknowledgments} The authors are thankful to
the anonymous referees for their careful reading of the manuscript,
 which led to a substantial improvement of the original
manuscript.

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\end{document}