\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2007(2007), No. 40, pp. 1--40.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/40\hfil Asymptotic solutions]
{Asymptotic solutions of forced nonlinear second order
differential equations and their extensions}

\author[A. B. Mingarelli and K. Sadarangani\hfil EJDE-2007/40\hfilneg]
{Angelo B. Mingarelli, Kishin Sadarangani}  % in alphabetical order

\address{Angelo B. Mingarelli \newline
School of Mathematics and Statistics \\
Carleton University, Ottawa, Ontario, Canada, K1S 5B6,
\newline
Departamento de Matem\'aticas, Universidad de Las Palmas de Gran
Canaria, Campus de Tafira Baja \\
 35017 Las Palmas de Gran Canaria, Spain}
\email{amingare@math.carleton.ca, amingarelli@dma.ulpgc.es}

\address{Kishin Sadarangani \newline
Departamento de Matem\'aticas,
 Universidad de Las Palmas de Gran Canaria,
Campus de Tafira Baja \\
 35017 Las Palmas de Gran Canaria, Spain}
\email{ksadaran@dma.ulpgc.es}

\thanks{Submitted February 15, 2007. Published March 9, 2007.}
\thanks{The first author is supported by a research grant from NSERC Canada}
\subjclass[2000]{39A11, 34E10, 34A30, 34C10, 45D05, 45G10, 45M05}
\keywords{Second order differential equations;
nonlinear; non-oscillation; \hfill\break\indent
integral inequalities; Atkinson's theorem; asymptotically linear;
asymptotically constant;
\hfill\break\indent oscillation; differential inequalities;
fixed point theorem; Volterra-Stieltjes; integral equations}

\begin{abstract}
 Using a modified version of Schauder's fixed point theorem,
 measures of non-compactness and classical techniques,
 we provide new general results on the asymptotic behavior
 and the non-oscillation of second order scalar nonlinear
 differential equations on a half-axis. In addition, we extend
 the methods and present new similar results for integral
 equations and Volterra-Stieltjes integral equations,
 a framework whose benefits include the unification of
 second order difference and differential equations.
 In so doing, we enlarge the class of nonlinearities and in
 some cases remove the distinction between superlinear,
 sublinear, and linear differential equations that is
 normally found in the literature. An update of papers,
 past and present, in the theory of Volterra-Stieltjes
 integral equations is also presented.
\end{abstract}

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

\section{Introduction}

We present in this paper results pertaining to the nonlinear differential
equation
\begin{equation} \label{non0}
y''(x) + F(x, y(x)) = g(x), \quad x \in I = [x_0, \infty),\; x_0 \geq 0,
\end{equation}
where $F: \mathbb{R}^+ \times \mathbb{R} \longrightarrow \mathbb{R}$
is a general nonlinearity on which we will impose mostly criteria of
integral type and $g(x)$ is given. Our main interest lies in the
formulation of results regarding the non-oscillation and asymptotic
behavior of its solutions. Some of the results will then be
formulated for pure integral equations and ultimately for
 Volterra-Stieltjes integral equations (see \eqref{nlvs1})
and Volterra-Stieltjes integro-differential equations, that is,
in the linear case, equations of the form
\begin{equation}\label{nlvs11}
y'(x) = y'(0) - \int_{0}^{x}y(t)\,d\sigma(t),
\end{equation}
and, in the nonlinear case, equations of the form
\begin{equation}\label{nlvs21}
y'(x) = y'(0) - \int_{0}^{x}F(t,y(t))\,d\sigma(t),
\end{equation}
where $\sigma$ is generally a function locally of bounded
variation on $I$ and the resulting integrals are understood in the
Riemann-Stieltjes sense.  An advantage of the more general
framework suggested by say, \eqref{nlvs11}, above is that one can
incorporate corresponding theorems for three term linear
recurrence relations such as
\begin{equation}
\label{31trr}c_ny_{n+1}+c_{n-1}y_{n-1} + b_ny_n = 0,\quad n\in \mathbb{N},
\end{equation}
and its nonlinear versions or, equivalently, second order linear
difference equations such as
\begin{equation} \label{2bdef1}
{{\Delta}^2}{y_{n-1}}+b_ny_n = 0, \quad n\in \mathbb{N}.
\end{equation}
and its nonlinear analogs, as  \emph{corollaries} so that no new
proof is  required to obtain the discrete analogs.

We recall that a solution of a real second order differential
equation is said to be  \emph{oscillatory} on $[x_0, \infty)$
provided it exists on a  semi-axis and it has arbitrarily large
zeros on that semi-axis. If the equation has at least one
non-trivial solution with a finite number of zeros it is termed
 \emph{non-oscillatory}. Recent work in asymptotics of \eqref{non0}
has dealt primarily with pointwise criteria on both $F$ and $g$
sufficient for the asymptotic linearity of at least one solution
(e.g., \cite{Con,mr,mr2,Yin,Zhao})  On the other hand,  \emph{
integral type} criteria cover by their very nature a wider
collection of nonlinearities and we strive to obtain such criteria
throughout. Thus, in Section~\ref{sec2} we give more general
integral type criteria on $F$ which are sufficient for the
existence of an uncoutable family of solutions of the unforced
equation \eqref{non0}. This extends the validity of the results
presented in Dub\'{e}-Mingarelli \cite{dm}. In addition, we note
that our criteria of integral type such as \eqref{tkt} and
\eqref{atk} below are extended over the whole half-line (that is
we obtain  \emph{global existence}, see \cite{mr}) rather than local
existence or existence for sufficiently large values of the
variable. In this regard, see \cite{mps} for an extensive complete
study of a specific nonlinear equation and \cite{mr} for a
bibliographical study of unforced equations of the form $y''(x) +
F(x, y(x), y'(x)) = 0$. For results which compare the
non-oscillatory behavior of forced equations of the form
\eqref{non0} with those of the associated unforced equation,
\eqref{non} below, and possible equations with delays, we refer
the reader to \cite{abu}, \cite{da} and \cite{ka}.

One should not forget that even though the literature is filled
with  sufficient criteria for oscillation/non-oscillation of
unforced equations like
\begin{equation} \label{non}
y''(x) + F(x, y(x)) = 0, \quad x \in I,
\end{equation}
in some cases, classical methods can actually be superior to the
use of such fixed point theorems for the determination of the
oscillatory character of an equation. For example, consider the
equation
$$
y'' + \frac{y\, \cos{2y^3}}{4(x+1)^2} =0, \quad x\in I,
$$
whose nonlinearity fails to comply with the conditions of Nehari's
theorem \cite{zn}, Atkinson's theorem \cite{1955}, the Coffman and
Wong results in \cite{jsww3,jsww4} and other more recent theorems.
However, every solution of this equation is non-oscillatory as can
be gathered by comparison with a non-oscillatory Euler equation
(and use of Sturm's comparison theorem \cite{jcf}).

Next, we note that the use of maximum principles allows for an
easy understanding of the oscillatory nature of an equation like
\eqref{non}. For example, if in some interval $[a,b)$ (finite or
not), we have $y\in C^2[a,b)$ and $y''(x) >0$ (or $y''(x) <0$)
then $y(x)$ can have at most two zeros there. Thus, whenever a
solution $y\in C^2$ of \eqref{non} satisfies $y''(x) \neq 0$, for
$x\in I$, or, more generally, for all sufficiently large $x$, then
we have non-oscillation on $I$. This explains the non-oscillatory
character of equations like the Painlev\'{e} I, where $F(x,y) =
-6y^2+x$, for $x > 0$ (see Hille~\cite{hille} or
Ince~\cite{ince}). It follows that if $F$ is continuous and
$F(x,y) < 0$ for all sufficiently large $x$ and all $y$, then we
always non-oscillatory solutions on $I$ (and only such solutions).
Thus, the only  interesting cases with regards to oscillations are
those for which ultimately either $F(x,y) >0$ on its domain or
$F(x,y)$ takes on both signs there. This motivates the main
assumptions we will be making throughout.

As can be expected, introduction of the forcing term $g$ and its
double primitive $f$, i.e.,  a function $f$ such that $g(x) =
f''(x)$, can alter the original asymptotics. Loosely speaking, the
case where $F$ dominates $g$ at infinity leads to solutions
asymptotic to a double primitive of $g$ (see Section~\ref{sec3}).
If $g$ is small in comparison to $F$, itself sufficiently small at
infinity, then asymptotically linear solutions persist (see
Section~\ref{ieq}). Motivated by Atkinson~\cite{fva} we introduce
a novel necessary and sufficient condition for the existence of a
solution of an integral equation of the form
\begin{equation*}
y(x) = f(x) - \int_{x}^{\infty}(t-x)F(t,y(t))\,dt, \quad x\geq x_0
\end{equation*}
in terms of associated solutions of differential inequalities
(Theorem~\ref{th4}). Ramifications of this result are noted and
classical methods are used to obtain criteria for every solution
of \eqref{non0} to be non-oscillatory. We also present an
extension of Nehari's necessary and sufficient condition for
non-oscillation [\cite{zn}, Theorem I], and Coffman and Wong's
version \cite{jsww4} of the same in terms of solution asymptotics.
We then proceed to a corresponding study of Volterra-Stieltjes
integro-differential equations in Section~\ref{vse} and give
conditions similar but more general than those in the previous
sections. Finally, we apply this theory to obtain results for
nonlinear three-term recurrence relations (or nonlinear second
order difference equations). In addition, we give a long needed
update of the theory of Volterra-Stieltjes integral equations in
our Introduction to Section~\ref{vse}. For the purpose of clarity
of exposition, we also proceed throughout the paper in order of
increasing generality and leave the proofs until the very last
section.

\section{Asymptotic results for nonlinear differential equations}

\subsection{Asymptotically linear solutions}
\label{sec2}
The present technique invokes a version of Schauder's fixed-point theorem
 and measures of non-compactness and is based, as in \cite{dm},
on the simple premise that in the variables separable case, the
nonlinearity in the dependent variable $y$ in \eqref{non} maps a
given compact interval back into (and not necessarily onto)
itself. For the rudiments of the notions of a measure of
non-compactness and their applications, see the book by Bana\'{s}
and Goebel \cite{bg}.

In the sequel, the space $ BC(\mathbb{R}^+)$ represents the space
of real bounded continuous functions defined on $\mathbb{R}^+$.
For given $a \geq 0, b> 0$, we consider the space
\[
Y =\{u\in C[0,\infty): {\sup_{t \geq 0} \frac{|u(t)|}{at+b}} <\infty\}.
\]
Obviously, $Y$ is a vector space over $\mathbb{R}$. Now, for
 $u\in Y$  the quantity
\[
\|u\|_Y =\sup_{t \geq 0} \frac{|u(t)|}{at+b}
\]
is a norm on $Y$. Consideration of the mapping
\[
\begin{array}{rlcl}
\Psi:& Y& \longrightarrow & BC(\mathbb{R}^+)\\
&u& \mapsto & \Psi (u)(t)= \frac{u(t)}{at+b}
\end{array}
\]
shows that $\Psi$ is a linear operator and, moreover, $\Psi$ is an onto isometry. Consequently, as $BC(\mathbb{R}^+)$ is complete, $Y$ is
a Banach space isometric to $BC(\mathbb{R}^+)$. More generally, for a given positive continuous function $p$, the space, $C_p$, of all tempered continuous functions (see [\cite{bg}, p.45]) consisting of all real-valued functions $ u \in C[0, \infty)$ such that $\sup_{t \geq 0} |u(t)|p(t) < \infty$, is a Banach space.

\begin{theorem} \label{th}
Let $a \geq 0, b>0$, and $X = \{ u \in Y :  0 \leq u(t) \leq {at+b},
\text{ for all }  t \geq 0\}$. Assume that
$F: \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+$ is continuous and
that for any $u \in X$,
\begin{equation} \label{atk}
\int_{0}^{\infty} t\, F(t, u(t))\, dt \leq b.
\end{equation}
In addition, we assume that there exists a function $k:\mathbb{R}^+ \to \mathbb{R}^+$ with
\begin{gather} \label{kt}
\int_{0}^{1}\, k(t)\, dt  < \infty,
\\ \label{tktt}
\int_{0}^{1} t\, k(t)\, dt  < \infty,
\\ \label{teek}
\int_{0}^{\infty} t^2\, k(t)\, dt  < \infty.
\end{gather}
and such that for any $u, v \in \mathbb{R}^+$,
\begin{equation} \label{lip}
| F(t, u) - F(t, v) |  \leq k(t) | u-v|, \quad t \geq 0.
\end{equation}
Then \eqref{non} has a positive (and so non-oscillatory) asymptotically
linear solution on $[0, \infty)$, i.e., $y(x) = ax+b + o(1)$,
as $x \to \infty$.
\end{theorem}

\begin{remark}\label{uno} \rm
Note that \eqref{teek} does not necessarily imply neither \eqref{tktt}
nor \eqref{kt}. However \eqref{kt}, \eqref{tktt}, and \eqref{teek}
together do imply that
$$
\int_{0}^{\infty} t\, k(t)\, dt < \infty, \quad \int_{0}^{\infty} k(t)\, dt
< \infty,
$$
conditions that are used in various places in the proof.
\end{remark}


\begin{remark} \rm
We note in passing that if $a, b$ are chosen so that
\begin{equation} \label{111}
\frac{1}{b}\, \max\{a,b\}\, \int_{0}^{\infty} t\,(t+1)\, k(t)\, dt < 1,
\end{equation}
in the inequality \eqref{cont}, then $T$ is a contraction on $X$ and
so the resulting fixed point is  \emph{unique} in $X$.
\end{remark}



\subsection{Asymptotic solutions in the forced nonlinear case}
\label{sec3}

In the sequel, the space $ BC([1,\infty))$ represents the space of all
 real bounded continuous functions defined on $[1,\infty)$.
For a given function $g$ in \eqref{non} we assume that it has a second
primitive $f : [1,\infty) \to \mathbb{R}$, such that for some $\delta > 0$
\begin{equation}\label{ft}
|f(x)| \geq \delta, \quad x \in [1,\infty)
\end{equation}
a condition that we will return to and discuss at various points.
Of course, since $f$ is continuous it is clear that \eqref{ft} implies
that $f$ is of one sign on $[1,\infty)$, but the sign itself is of no
concern to us here. Now, consider the vector space over $\mathbb{R}$
defined by
\begin{equation} \label{spaceY}
Y =\{u\in C[1,\infty): {\sup_{x \geq 1} \frac{|u(x)|}{|f(x)|}} <\infty\}.
\end{equation}
Now, for $u\in Y$ the quantity
\begin{equation} \label{normY}
\|u\|_Y =\sup_{x \geq 1} \frac{|u(x)|}{|f(x)|}
\end{equation}
is a norm on $Y$. Consideration of the mapping
\[
\begin{array}{rlcl}
\Psi:& Y& \longrightarrow & BC([1,\infty))\\
&u& \mapsto & \Psi (u)(x)= \frac{u(x)}{|f(x)|}
\end{array}
\]
shows that $\Psi$ is a linear operator and, moreover, $\Psi$ is an onto isometry. Consequently, as $BC([1,\infty))$ is complete, $Y$ is
a Banach space isometric to $BC([1,\infty))$.
More generally, for a given positive continuous function $p$,
 the space, $C_p$, of all tempered continuous functions
(see [\cite{bg}, p.45]) consisting of all real-valued functions
$ u \in C[x_0, \infty)$ such that $\sup_{x \geq x_0} |u(x)|p(x) < \infty$,
is a Banach space.

Let $F: [1, \infty) \times \mathbb{R} \to \mathbb{R}$ be continuous (not necessarily positive as in Section~\ref{sec2}) and assume that
\begin{equation} \label{sf0}
\int_{1}^{\infty} s\, | F(s, 0) |\, ds < \infty.
\end{equation}
With $f$ as defined as in \eqref{ft}, we assume that there exists a
function $k: [1,\infty) \to \mathbb{R}^+$ satisfying
\begin{equation} \label{sfk}
\int_{1}^{\infty}s\, |f(s)|\, k(s) \, ds  < \infty.
\end{equation}
An an additional restriction on both $F$ and $k$ we assume that for
any $u, v \in \mathbb{R}$,
\begin{equation} \label{lip01}
| F(x, u) - F(x, v) |  \leq k(x) | u-v|, \quad x \geq 1.
\end{equation}

Given such functions $F, k, f, g$ satisfying \eqref{ft}, \eqref{sf0},
\eqref{sfk} and \eqref{lip01} we consider the forced nonlinear
equation \eqref{non} on the interval $I = [x_0, \infty)$ where
$x_0$ is chosen so large that $x_0 \geq 1$ and for $x \geq x_0$,
\begin{equation} \label{t0}
 \max  \left \{ \int_{x}^{\infty} \,(s-x)\, |f(s)|\, k(s) \, ds ,
\int_{x}^{\infty} \,(s-x)\, |F(s,0)| \, ds \right \} \leq \frac{\delta}{4},
\end{equation}
the finiteness of the integrals in \eqref{t0} being ensured on account
of \eqref{sf0} and \eqref{sfk}.

\begin{theorem} \label{th1}
Let the terms in \eqref{non} satisfy the conditions
\eqref{ft}, \eqref{sf0}, \eqref{sfk}, \eqref{lip01} and \eqref{t0}.
Consider \eqref{non} on $I = [x_0, \infty)$ where $x_0$ is defined as
in \eqref{t0}.
Then \eqref{non} has a solution $y(x)$ satisfying
\begin{enumerate}
\item $y(x) \sim f(x)$ as $x \to \infty$ and actually,
$y(x) = f(x) + o(1)$, as $x \to \infty$, and
\item
\begin{equation}
\label{eqyf}
\sup_{x\in I} \frac{|y(x)|}{|f(x)|} \leq 2.
\end{equation}
\end{enumerate}
\end{theorem}



\subsection{Discussion}

Since the proof of Theorem~\ref{th1} uses the contraction mapping
principle, it follows that one can approximate the actual solution
in question arbitrarily closely using a standard iterative technique.

The upper bound appearing in \eqref{eqyf} is by no means precise but
will do for our purposes of obtaining global existence of solutions.
Indeed, it is easily seen that one can modify the proof a little in order
 to find a non-uniform bound that depends  on $x_0$.

The unforced case $g(x) \equiv 0$ is included in our theorem and is
reflected in the expression $f(x) =ax +b$ above, that is we obtain
the existence of asymptotically linear solutions for \eqref{non0}.
In this case Theorem~\ref{th1} extends the main results of
Hallam \cite{tgh} for $n=1$, Dub\'{e}-Mingarelli \cite{dm},
and Mustafa-Rogovchenko \cite{mr}. It should be emphasized here that
our conditions on the nonlinearity $F(x,y)$ and the forcing term $g(x)$
are essentially of  \emph{integral type} and not pointwise criteria as
in most papers in the area, e.g., \cite{mr} is a recent one. In addition,
Theorem~\ref{th1} provides an extension of some results in Atkinson \cite{fva}
where, in addition, it is assumed that $F$ is positive and non-decreasing
in its second variable (cf., also \cite{gp}), a condition we will return
to occasionally.

Of course, since $f$ is continuous, \eqref{ft} implies that $f(x)$ is of
 \emph{one sign} on the half-line $I$. Theorem~\ref{th1} then implies that
the forced equation \eqref{non} is  \emph{non-oscillatory}.
If \eqref{ft} is not satisfied then
\begin{equation}
\label{liminff}
\liminf_{x\to \infty} |f(x)| = 0,
\end{equation}
a condition used often in many papers in conjunction with the
questions under investigation here (\cite{fva}, \cite{ak},
\cite{ak2}, \dots). In this respect, the condition \eqref{liminff}
is known to furnish examples of oscillatory equations of the form
\eqref{non}, cf., \cite{fva}, \cite{ak2}. In addition, necessary
conditions for the existence of a positive solution of \eqref{non}
under the assumption $f(x)> 0$, yet more restrictive conditions on
the nonlinearity, may be found in [\cite{fva}, Section 4]. We note
that \eqref{ft} and \eqref{sfk} together imply that
\begin{equation}\label{sks}\int_{1}^{\infty}sk(s)\,ds < \infty,\end{equation}
 so that, as expected, one needs to ensure that the nonlinearity $F$
decreases quickly enough (see \eqref{lip01}) at infinity to ensure
nonoscillation. Our results apply to linear problems with small forcing
terms as well. The following example serves as illustration.

\begin{example} \label{ex2.3} \rm
Let $F(x,y) = (1+y)/x^5$, $g(x)=1$, $x \geq 1$,
in \eqref{non}. Choosing the double primitive $f(x) = x^2/2$,
we see that $\delta =1/2$ is a suitable lower bound for $f(x)$ in \eqref{ft}.
Note that \eqref{sf0}-\eqref{lip01} are all satisfied with the choice
$k(x) = 1/x^5$. In addition, \eqref{t0} holds for all $x \geq x_0$
where $x_0=3$. Theorem~\ref{th1} now applies to show that the equation
$$
y''+(1+y)/x^5 = 1, \quad x\geq 3,
$$
has a solution $y(x) \sim x^2/2$ as $x \to \infty$, defined by
solving \eqref{tux} for its fixed point. This solution can actually
be calculated using Bessel functions but its exact form is of no
particular interest here. Successive approximations to it show that
if we define $y_0(x)=1$, then $y_1(x) = x^2/2 - 1/{6x^3}$, and
$$
y_2(x) = \frac{x^2}{2} - \left (\frac{1}{4x} +\frac{1}{12x^3}-\frac{1}{252x^6}
\right), \quad etc.,
$$  the asymptotic nature of $y(x)$ can readily be ascertained.
\end{example}

A few more remarks on the case $g(x)=0$ in \eqref{non} are in order.
Our condition \eqref{sf0} is compatible with Nehari's \cite{zn} necessary
and sufficient condition for the existence of a bounded solution
(albeit under additional assumptions on $F(x,y)$ such as positivity
and monotonicity in its second variable). In this vein we can formulate
the following immediate corollary for asymptotically constant solutions
which does not assume neither the monotonicity nor the positivity of $F$.

\begin{corollary} \label{coro2.4}
Consider the equation \eqref{non} for $x \geq 1$. Let $F, k, \sigma $
satisfy \eqref{sf0}, \eqref{lip01}, \eqref{t0} and \eqref{sks}, for some
$\delta=M>0$ and for all $x \geq x_0$.
Then \eqref{non} has a solution satisfying $y(x) \to M $ as $x \to \infty$,
and $|y(x)| \leq 2M$ for all $x \geq x_0$.
\end{corollary}

Similar additional results may be formulated for the case of asymptotically
 linear solutions and so are left to the reader. For an excellent survey
up to the mid-seventies of nonlinear two term ordinary differential
equations of Emden-Fowler type, see \cite{jsww}.

Next, we consider
\begin{equation}
\label{pert} y''(x) + F(x, y(x)) = g(x), \quad x\geq 0.
\end{equation}
where $g(x) \equiv f''(x)$ and $f : \mathbb{R}^+ \to \mathbb{R}$
is not necessarily of one sign (as opposed to \eqref{ft}) but
$f''\in C(\mathbb{R}^+)$. Next, we seek to find asymptotic
theorems for equations of the form \eqref{pert} which may violate
\eqref{ft}. The trade-off here is that we need that the
nonlinearity be  \emph{positive}.

\begin{theorem}\label{th2}
Let $f \in L^{\infty}( \mathbb{R}^+) \bigcap C^2( \mathbb{R}^+) $.
Suppose that $F: \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+$
is continuous and such that for some $b > 0$, \eqref{atk} is satisfied
for any $u \in X$ where
$X=\{ u \in BC(\mathbb{R}^+) : |u(x)| \leq \|f\|_{\infty} + b, x \geq 0\}$.
Let $F$ satisfy
\begin{equation} \label{lip02}
| F(x, u) - F(x, v) |  \leq k(x) | u-v|, \quad x \geq 0.
\end{equation}
for any $u, v \in \mathbb{R}$, where $k:\mathbb{R}^+ \to \mathbb{R}^+$
is such that
\begin{equation}
\label{tkt} \int_{0}^{\infty} t\, k(t)\, dt < 1.
\end{equation}
Then \eqref{pert} has a solution $y(x)$ defined on $\mathbb{R}^+$ with
$|y(x) - f(x)| \to 0$ as $x \to \infty$ and
$\|y\|_{\infty} \leq \|f\|_{\infty} + b, x \geq 0$.
\end{theorem}

\begin{remark} \label{rmk3} \rm
 Uniqueness of the solution in Theorem~\ref{th2} may be lost in case
we relax the requirement on $k$ as given by \eqref{tkt} to the integral
being merely finite. In this case, a proof using measures of non-compactness
such as the one in Theorem~\ref{th1} may be used to prove
\end{remark}

\begin{theorem}\label{th3}
Let $f \in L^{\infty}( \mathbb{R}^+) \bigcap C^2( \mathbb{R}^+) $ and
suppose that $F: \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+$
is continuous and such that for some $b > 0$, \eqref{atk} is satisfied
for any $u \in X$ where
$X=\{ u \in BC(\mathbb{R}^+) : |u(x)| \leq \|f\|_{\infty} + b, x \geq 0\}$.
Let $k:\mathbb{R}^+ \to \mathbb{R}^+$  satisfy \eqref{atk}, \eqref{lip} and
\begin{equation}\label{ttt}
 \int_{0}^{\infty} t\, k(t)\, dt < \infty.
\end{equation}
Then \eqref{pert} has at least one solution $y(x)$ defined on
$\mathbb{R}^+$ with $|y(x) - f(x)| \to 0$ as $x \to \infty$
and $\|y\|_{\infty} \leq \|f\|_{\infty} + b, x \geq 0$.
\end{theorem}

\subsection{Discussion}

We make no claim as to  \emph{positivity} of the solution in question in
either of Theorems~\ref{th2},\,\ref{th3}, since $f(x)$ may be of both signs,
 only that $y(x)=f(x) + o(1)$ as $x \to \infty$.  The following example
illustrates this.

\begin{example} \label{ex2.7}\rm
Consider the equation
$$
y'' + F(x,y)= - \sin x, \quad x\geq 0,
$$
with $F(x,y)= \lambda (x+1)^{-4}$ where $\lambda > 0$ is arbitrary but
fixed and $b \geq \lambda/6$, where $b$ is defined in \eqref{atk}.
For any constants $c_1, c_2$, we can choose the double primitive
$f(x) = \sin x + c_1x+c_2$. The assumptions of Theorem~\ref{th2}
are readily verified for our choice of $b$ and $F$. It follows that
there is a solution of this equation such that $y(x) = f(x) + o(1)$
as $x \to \infty$. In fact the solution is given by
$$
y(x) = f(x) - \frac{\lambda}{6(x+1)^2},
$$
for every $x$, from which the asymptotic estimate follows, as well as
the  \emph{a priori} bound on the solution, namely, that
$\|y\|_{\infty} \leq \|f\|_{\infty} + b$, valid for every $x \geq 0$.
Thus, choosing $c_1 =0, c_2=0$, we see that for large $x$ the solution
will generally have both signs.
\end{example}

\section{Asymptotics for solutions of integral equations}
\label{ieq}

Motivated by Atkinson's paper \cite{fva} we produce a sharpening of
the results in [\cite{fva}, Section 3] by studying  \emph{integral} inequalities.
Our purpose is now to provide a formulations of some of the results of the
previous sections to a wider framework, namely, that of integral equations,
and ultimately to Volterra-Stieltjes integral equations with a view at
obtaining discrete analogs for three-term recurrence relations.

Instead of beginning this study with a differential equation of the
form \eqref{non0} we pass immediately to its integral equation counterpart,
that is,
\begin{equation}
\label{ie1}
y(x) = f(x) - \int_{x}^{\infty}(t-x)F(t,y(t))\,dt, \quad x\geq x_0
\end{equation}
under various assumptions on the terms involved (after all, all our
 preceding proofs were of this nature). Once again we strive to minimize
the requirements on the forcing term, here, $f(x)$. In \cite{fva} this
term is assumed to be small at infinity in the differential equation and
differential inequality formulation. If $f\in C^2(I)$ we can recover
results for the nonlinear equation \eqref{non0} by setting $g=f''$.

We will always assume that the ``forcing term" $f\in C(I)$ in \eqref{ie1}
where $I$ as usual is of the form $I=[x_0,\infty)$, $x_0 \geq 0$, and
uniformly bounded there. This last requirement will be denoted by the
relation $f \in L^{\infty}(I)$, a minor abuse of notation.
This is the only requirement we will impose upon $f$. The main result
in this section follows:

\begin{theorem}
\label{th4}
Let $f \in L^{\infty}(I)$ and suppose that the nonlinearity $F$ in
$\eqref{ie1}$ satisfies
\begin{enumerate}
\item $F:I \times \mathbb{R}\to \mathbb{R}^+$ is continuous on this domain
\item $F(x, \cdot)$ is nondecreasing for every $x \in I$
\item For every $M > 0$, $$\int_{0}^{\infty}t\,F(t,M)\,dt < \infty$$
\item For every $y, z \in \mathbb{R}$ and every $x \in I$,
$$ | F(x,y)-F(x,z)| \leq k(x)|y-z|
$$
where
\item $\int_{x_0}^{\infty}tk(t)\,dt < 1$.
\end{enumerate}
Then \eqref{ie1} has a (continuous) solution $y \in L^{\infty}(I)$ if and
only if there are two (continuous) functions $u, v \in L^{\infty}(I)$
such that $u(x) \leq v(x)$, $x \in I$,
\begin{equation}
\label{uf0}
u(x) \leq f(x) - \int_{x}^{\infty}(t-x)F(t,v(t))\,dt
\end{equation}
for $x\geq x_0$, and
\begin{equation}
\label{vf0}
v(x) \geq f(x) - \int_{x}^{\infty}(t-x)F(t,u(t))\,dt
\end{equation}
for $x\geq x_0$.
\end{theorem}

\begin{remark} \label{rem4} \rm
A corresponding result is valid for  \emph{positive} solutions $y$
of \eqref{ie1}. In this case $u(x) >0$ in Theorem~\ref{th4} and
the positivity assumption on $F$ can be restated as $F(x,y) \geq 0$
for every $y \geq 0$, the remaining assumptions being the same.
\end{remark}

As a consequence we obtain a differential equations counterpart.
Under the basic assumptions (1)-(5) of Theorem~\ref{th4}, we obtain
the following two theorems.

\begin{theorem} \label{cor01}
Equation~\eqref{non0} with $g=f''$ has a solution $y$ with
$y(x)=f(x) + o(1)$, $y'(x)=f'(x)+o(1)$, as $x \to \infty$ if and only
if there exists two functions $u, v \in L^{\infty}(I)$ such that
$u(x) \leq v(x)$, $x \in I$, satisfying \eqref{uf0} and \eqref{vf0},
\end{theorem}


\begin{theorem}\label{thx}
Let $u, v, f$, $u(x) \leq v(x)$, for $x\in I$, be three twice continuously
differentiable functions satisfying the differential inequalities
$$
u''(x) + F(x,v(x)) \leq g(x) \leq v''(x) + F(x,u(x)), \quad x \in I,
$$
where $g=f''$. If, in addition, $u, u', v, v'$ have vanishing limits
at infinity, then \eqref{pert} has a positive solution
$y \in L^{\infty}(I)$, with $y(x) \sim f(x)$ as $x \to \infty$.
\end{theorem}

\begin{example} \label{ex3.4} \rm
We consider the integral equation \eqref{ie1} with $F(x,y)=y/x^4$ and
$f(x)=1+1/(6x^2)$ on $I=[1,\infty)$. Note that $y=1$ is a solution
of \eqref{ie1} on $I$ and that for such $x$ all the conditions of
Theorem~\ref{th4} and Remark~\ref{rem4} are satisfied with the choice
$k(x)=1/x^4$. The functions $u, v$, whose existence is guaranteed
 by this result, are given by $u(x)=1/2$ and $v(x) = 2$ for $x \in I$.
\end{example}


\subsection{Discussion}

Our result gives, for a given (akin to `superlinear')
nonlinearity, a necessary and sufficient condition for the
existence of asymptotic  solutions of a given type in terms of
solutions to specific integral inequalities (according to
\cite{jsww3} a superlinear $F$ is one which satisfies condition
`2' in Theorem~\ref{th4}). The stated theorem gives uniqueness as
well as uniform bounds for the required solution. It is likely
that hypotheses `4' and `5' in the theorem may be relaxed albeit
at the possible loss of uniqueness. Theorem~\ref{th4} appears to
be new even when considered from the viewpoint of  second order
nonlinear differential equations (as in, e.g.,
Theorem~\ref{cor01}).

As mentioned earlier, Nehari \cite{zn} gives a necessary and
sufficient  condition for the existence of a non-oscillatory
solution of an equation akin to \eqref{non0} in terms of the
nonlinearity. Under a  \emph{strong superlinear} condition that is,
for some $\varepsilon > 0$ there holds
\begin{equation}
\label{sts}
y_2^{-\varepsilon}F(x,y_2) >y_1^{-\varepsilon}F(x,y_1) ,
\end{equation}
for $0< y_1<y_2<\infty$, he shows in \cite{zn} that if the
nonlinear  equation has the special form
\begin{equation}
\label{zn11} y''+yF(x, y^2) = 0, \quad x\geq 0,
\end{equation}
then it has a bounded non-oscillatory solution if and only if, for some $M >0$,
\begin{equation}
\label{zn2} \int_{}^{\infty}tF(t,M)\, dt < \infty,
\end{equation}
i.e., condition `3' in Theorem~\ref{th4} holds for  \emph{some}
$M>0$  (and $x=0$).

His result was extended later by Wong \cite{jsww2} by relaxing the
monotonicity condition on $F$ somewhat and taken up again by
Coffman-Wong \cite{jsww3}, \cite{jsww4},  where further
developments in a sublinear case were given (in particular, see
the Table on p.123 in \cite{jsww2} for a useful visual display of
known necessary and sufficient conditions for non-oscillation). A
more precise version of Nehari's result can be found in the strong
superlinear case in \cite{jsww4}, that is, \eqref{zn11} has a
bounded  \emph{asymptotically linear} solution (the special case
$f(x)=ax+b$ in our set up) if and only if \eqref{zn2} holds for
some $M >0$.

Although we require that \eqref{ie1} be `superlinear' (i.e.,
condition `2'  in Theorem~\ref{th4}) it need not be strongly so.
On the other hand, we require that \eqref{zn2} hold for every $M
>0$, but then we are also strengthening the conclusion. Indeed,
our result is also valid for a wider class of equations, not only
second order nonlinear differential equations.

In a recent paper \cite{mr2}, the authors implicitly assume an
integrability condition on $f$ and that this double primitive
$f(x) \to 0$ as $x \to \infty$, in the spirit of \cite{fva} and
\cite{ak}. The nature of such a decay condition on the forcing
term is that the basic tenet  underlying the asymptotic behavior
of a given nonlinear differential equation \eqref{non0} appears to
be the interplay between the rate of decay of the nonlinearity as
opposed to the rate of decay of the forcing term. For example, if
the nonlinearity is small in the sense of the applicability of
conditions (3-5) in Theorem~\ref{th4} and the forcing term is not
( e.g., perhaps not integrable on I) then solutions may be
expected to be asymptotic to a double primitive of the forcing
term. On the other hand, if both forcing term and nonlinearity are
``small" in some suitable sense then the solutions of \eqref{non0}
may be expected to be asymptotically linear (or asymptotic to the
solutions of the same equation with nonlinearity and forcing term
omitted). This philosophy may be used in our basic understanding
of nonlinear equation asymptotics on a half-line on account of the
following unpublished result, reproduced here for completeness.

\begin{theorem}[Atkinson-Mingarelli, 1976, unpublished] \label{atm}
Let $g:I\to (0,\infty)$ be continuous and satisfy
\begin{equation}
\label{tgt}\int_{x_0}^{\infty}t^i\, g(t)\, dt < \infty
\end{equation}
for $i=0,1$. We assume that $G:I \times \mathbb{R}\to \mathbb{R}^+$ verifies assumptions (1-2) in Theorem~\ref{th4} (i.e., $G$ is continuous on this domain and $G(x, \cdot)$ is nondecreasing for every $x \in I$). In addition, let $G_{x}(x,y) \equiv \partial{G}/ \partial{x}$ exist, be continuous and non-positive on the same domain. If
\begin{equation}
\label{tgKt} \int_{x_0}^{\infty}t\, G(t, Kt)\, dt < \infty
\end{equation}
for every $K > 0$, then every solution $y$ of
\begin{equation}
\label{non2} y'' + yG(x,y) = g(x)
\end{equation}
is either asymptotically linear or $ y(x) < 0$, $y'(x) \leq 0$,
$y'' \geq 0$ for all sufficiently large $x$. Either way, every
solution is nonoscillatory.
\end{theorem}

\begin{remark} \label{rmk5} \rm
The double primitive of $g$ does not enter the picture here as in
Theorem~\ref{th4} since $g$ is already itself small, as evidenced
by \eqref{tgt}.  By a solution $y$ that is ``asymptotically
linear" we mean that the solution has the property that $y(x)=
Ax+B+o(1)$ as $x \to \infty$, for some constants $A, B$, where the
cases $A=0, B> 0$, $A > 0, B=0$, $A=B=0$ can all occur. This
result is in the same spirit as Theorem~\ref{th4} above except for
clear differences in the behavior of the nonlinearities involved.
Despite these differences, these nonlinearities are each small at
infinity thereby leading to the stated  \emph{linear} asymptotics.
This theorem extends a theorem of [Nehari \cite{zn}, Theorem III]
\end{remark}

The following complementary result to Theorem~\ref{atm} is for the
case where $g$ is not integrable at infinity. In this case the
nonlinearity is still small in comparison to the growth of $g$ in
\eqref{fr} but now the forcing term is integrably large, so the
solutions are asymptotic to a double primitive of the forcing
term.

\begin{theorem}[Atkinson-Mingarelli, 1976, unpublished] \label{atm2}
Let $g:I\to (0,\infty)$ be continuous and satisfy
\begin{equation}
\label{tgt2}\int_{x_0}^{\infty} g(t)\, dt = + \infty
\end{equation}
for $i=0,1$. We assume that $G:I \times \mathbb{R}\to
\mathbb{R}^+$ verifies assumptions (1-2) in Theorem~\ref{th4}
(i.e., $G$ is continuous on this domain and $G(x, \cdot)$ is
nondecreasing for every $x \in I$). In addition, let $G_{x}(x,y)
\equiv \partial{G}/ \partial{x}$ exist, be continuous and
non-positive on the same domain. If for some double primitive $f$
(i.e., $g = f''$) and some $\varepsilon > 0$
\begin{equation}
\label{fr} \int_{x_0}^{x}\sup_{|u| \leq (1+\varepsilon)f} |F(t,u)|\, dt = o\left \{\int_{x_0}^{x}g(t)\, dt\right\},
\end{equation}
as $x \to \infty$, then every solution $y$ of
\begin{equation}
\label{non21} y'' + yG(x,y) = g(x)
\end{equation}
is asymptotic to $f(x)$ as $x \to \infty$.
\end{theorem}

\subsection{Discussion}
\label{dis4}

One of the advantages in using classical methods over fixed point
theorems is exhibited in Theorems~\ref{atm} and~\ref{atm2} above.
For example, it is difficult to obtain  \emph{a priori} bounds such
as \eqref{ener3} on the derivative or \eqref{yKx} on the solution
$y(x)$ using fixed point theorems. Both techniques can be used
interchangeably, preference being only a function of the
conclusion desired, and on the nature of the hypotheses, nothing
more.

Note, however, the absence of conditions such as \eqref{lip} or
\eqref{tkt}  in these two results, hypotheses that were deemed
necessary in the proofs of most of the results in this section.
Conditions such as \eqref{lip} and \eqref{tkt} could also be
interpreted as being conditions on the rate of growth of
$\partial{G}/\partial{y}$ in the domain under consideration.

\section{Asymptotic theory of nonlinear Volterra-Stieltjes integral equations}
\label{vse}

We begin this section with a non-exhaustive review and update of
the  results over the past 20 years in this fascinating area which
can be used to unify discrete and continuous phenomena. The
unification allows for the simultaneous study of both differential
and difference equations of the second order and even includes
equations that are, in some sense,  \emph{between} these two as we
gather from the discussion that follows (and from the references).
A   \emph{Volterra-Stieltjes integral equation} is basically a
Volterra integral operator on a space $X$ of suitable functions in
which the integral appearing therein is a Stieltjes integral (in
whatever sense it can be defined, more on this below). The
prototype (linear) Volterra-Stieltjes integral equation that we
use in this work is of the form $I \equiv 0 \leq x < b
\leq\infty$,
\begin{equation}\label{vs1}
y(x) = y(0)+xy'(0) - \int_{0}^{x}(x-t)y(t)d\sigma(t)\quad x \in I,
\end{equation}
where $\sigma: I \to \mathbb{R}$ is a function that is locally of bounded
variation on $I$. A  \emph{solution} of \eqref{vs1} is an absolutely continuous
 function with a right-derivative that exists for each $x \in I$ and is
locally of bounded variation on $I$. In this case, the integral in
\eqref{vs1} can be understood in the \emph{Riemann-Stieltjes}
sense and we take this for granted throughout this section. It is
known that this formulation, which includes the use of a simple
Riemann-Stieltjes integral, is adequate (and sufficient) for the
unification purposes referred to above (see e.g., \cite{abmb}
among other possible references).  Other frameworks that can be
used as a unification tool for discrete and continuous nonlinear
equations include the theory of  \emph{time scales}. However, we
do not entertain these studies here (unless results overlap) and
so refer the interested reader to e.g.,
 \cite{mbo,lhe,erk,billur,shs}, and the references therein for further
information.

Associated with \eqref{vs1} is the Volterra-Stieltjes
integro-differential  equation
\begin{equation}\label{vs2}
y'(x) = y'(0) - \int_{0}^{x}y(t)d\sigma(t)
\end{equation}
obtained by differentiating the equation \eqref{vs1}. The
derivative appearing in \eqref{vs2} is now understood generally as
a right-derivative and this function is locally of bounded
variation on $I$. Local existence and uniqueness of solutions of
initial value problems associated with either \eqref{vs1} or
\eqref{vs2}  and the basic theory of such equations, as we define
them, was developed by Atkinson \cite{fvab}, and also continued by
Mingarelli \cite{abmb}, Mingarelli and Halvorsen \cite{mhb} among
others. The reader may also wish to consult the monographs of
Corduneanu \cite{cc}, H\"{o}nig \cite{csh}, Schwabik  \emph{et al.}
\cite{ssc3} and \cite{ssc1} for different approaches and
generalizations of both the method and the context. We also refer
the reader to Groh \cite{jg0} for an extensive list of references
to this subject, some not included here.

Regarding the possible different interpretations of the
nomenclature ``Volterra-Stieltjes integral equation" in the
literature, the main ideas and developments depend strongly on the
definition of the particular Stieltjes integral being used. Once
this is in place, one can define a solution, develop a basic
theory (ask about existence and uniqueness of solutions,
continuous dependence on initial conditions, etc) and suggest
additional applications. Thus, studies which have depended upon
the use of the \emph{Kurzweil-Henstock integral} in \eqref{vs1}
and equations like it include, and are not restricted to, Kurzweil
\cite{jk}, Schwabik \cite{ssc4}, Tvrd\'y \cite{mtu1},
Federson-Bianconi \cite{feb}. One of the very first  researchers
in this area, Martin~\cite{rhm} uses the \emph{Cauchy
right-integral} while the \emph{Dushnik integral} appears in  the
papers by H\"{o}nig~\cite{csh}, \cite{csh3} and
Dzhgarkava~\cite{ddt2}. The Stieltjes integral, when viewed as a
\emph{Lebesgue-Stieltjes} integral, makes its appearance in
Ding-Wang \cite{xqd1}, \cite{xqd2} and Caizhong~\cite{lcz}.
Finally, but not exhaustively, Dressel~\cite{fgd} uses the Young
integral. There may be overlap between some of these definitions
as they have developed in the past century, but the aim is to show
that different integral definitions may produce different
applications and may account for the large literature on this
subject.

Various abstract formulations of such equations can be found in
the works of Helton~\cite{bwh} who considered such equations over
rings, Ashordia~\cite{mash}, \cite{mash1}, Dzhgarkava~\cite{ddt},
H\"{o}nig~\cite{csh4},\cite{csh5}, Ryu~\cite{ksr},
Schwabik~\cite{ssc0}, Travis~\cite{cct} and Young~\cite{dfy}.
Controllability of equations of this general type has been studied
by Barbanti~\cite{lb}, Dzhgarkava \cite{ddt2}, Groh~\cite{jg2},
Yong~\cite{jmy}, and Young~\cite{dfy2}. Investigations related to
 systems of equations include those of Gopalsamy
 \emph{et al.} \cite{kgo}, Herod~\cite{jvh}, Hildebrandt~\cite{thh},
 Hinton~\cite{dbh}, H\"{o}nig~\cite{csh2}, Schwabik~\cite{ssc},
and Wheeler~\cite{rlw}. For the relationship between
Volterra-Stieltjes integral equations and their applications to
difference equations (or recurrence relations see
Atkinson~\cite{fvab},
 Mingarelli~\cite{abmb}, Mingarelli-Halvorsen \cite{mhb},
Petrovanu~\cite{dp}, and Schwabik~\cite{ssc4}.

The study of  scalar Volterra-Stieltjes integral equations and
subsequent qualitative, quantitative, and spectral theory can be
found in the works by Banas  \emph{et
al}~\cite{bg2,bg3,bg4,bg5,bg6,bg7,bg8}, Caballero  \emph{et
al}~\cite{crs}, Cao~\cite{zjc}, Cerone-Dragomir~\cite{pce},
Chen~\cite{hyc,hyc1}, El-Sayed~\cite{wes}, Gibson~\cite{wlg}, Gil'
and Kloeden~\cite{mig,mig2}, Hu~\cite{qyh}, Jiang~\cite{zmj,zmj0},
Lou~\cite{lou}, Marrah and Proctor~\cite{marr},
Mingarelli~\cite{abm0,abmb}, Mingarelli and Halvorsen~\cite{mhb},
Parhi~\cite{np}, Randels~\cite{wcr}, Schwabik~\cite{ssc2}, Spigler
and Vianello \cite{rsv1,rsv}, Tritjinsky~\cite{tri},
Wang~\cite{zwa}, Wong and Yeh~\cite{wy0,wy}.

Finally, there is a relationship which has seen little follow-through in
the past 70 years or so since its beginnings. Basically, one asks
about a relationship between the notion of a generalized derivative
( \emph{\`{a} la Feller})  of the form
$$
- \frac{dy'}{d\sigma} = f(x), \quad x\in [0,b],
$$
and the integro-differential equation (cf., \eqref{vs2} above),
 $$
y'(x) = c - \int_{0}^{x}f(t)\,d\sigma(t),
$$
where $c$ is a constant and $\sigma$ is an non-decreasing
(actually increasing) function defined on $[0,b]$ with an appropriate
 Stieltjes integral. This approach was pioneered by the probabilist
 Feller~\cite{wfe} (see the references in \cite[ p.316]{abmb}) and
the resulting theory, found in many papers in probability
(not all quoted here) now includes the keywords: Feller derivatives,
 Krein-Feller operators, Generalized differential operators, etc.,
see Jiang~\cite{zmj2}, Albeverio-Nizhnik~\cite{sal}, Fleige~\cite{anf},
 Mingarelli~\cite{abmb}.

That there is an equivalence between these last two displays should not
be surprising yet the lines of development of the resulting theories
seem to have diverged over the years with each equation taking on a
life of its own, so to speak. For papers dealing with generalized
differential expressions see Groh~\cite{jg1}, Jiang~\cite{zmj2},
Volkmer~\cite{hv}, and Mingarelli~\cite{abmb} among others.
We emphasize here the importance of the contributions of
 Ka\v{c}, Kre\v{i}n and  Langer to the study of the spectral
theory of the operators associated with the generalized
differential expressions above. Although these references are not
included here specifically for reasons of length, we refer the
interested reader to the more than 80 historical references in
Mingarelli~\cite{abmb}, in addition to those in Atkinson~\cite[pp.
529-533]{fvab}, , the list of references in Fleige~\cite{anf} and
the references contained within each of the articles mentioned in
the bibliography below. Altogether these should give the reader an
essentially complete view of this vast field as of today.

\subsection{Asymptotically linear solutions of nonlinear equations}
\label{alis}

The asymptotic theory of solutions of equations of the form \eqref{vs1}
or \eqref{vs2} is still in its infancy with few basic results in existence
in the literature. In the linear case \eqref{vs1} we can cite
 Atkinson~\cite[Theorem~12.5.2]{fvab}. Fewer are specific results
dealing with the nonlinear case \eqref{nlvs1} or \eqref{nlvs2}.
One such result may be found in \cite[Theorem 2.3.1]{abmb},  in
the case where $F(x,y) = p(x)q(y)$, a result which extends
Butler's necessary and sufficient condition for non-oscillation
\cite{Bu}. In the remaining sections we produce extensions of the
results in the previous sections to this framework along with some
possible refinements.
\begin{gather}\label{nlvs1}
y(x) = y(0)+xy'(0) - \int_{0}^{x}(x-t)F(t,y(t))\,d\sigma(t)
\\ \label{nlvs2}
y'(x) = y'(0) - \int_{0}^{x}F(t,y(t))\,d\sigma(t)
\end{gather}
Using the methods in Atkinson~\cite[Chapter 12]{fvab},  one can
readily prove the existence and uniqueness of solutions of initial
value problems for equations of the form \eqref{nlvs1} or
\eqref{nlvs2} under a locally Lipschitz condition on the
continuous nonlinearity $F$. Recall that a solution of
\eqref{nlvs1} (resp.\eqref{nlvs2}) is an absolutely continuous
function such that its right derivative exists at every point of
$I$ and $y(x)$ (resp.$y'(x)$) satisfies the equation \eqref{nlvs1}
(resp.\eqref{nlvs2}) at every point in $I$. Unless otherwise
specified we always assume the minimum requirement that such
solutions exist and are unique. In some cases below we actually
get existence, uniqueness and asymptotic limits as a by-product of
the techniques used.

For simplicity of notation we will assume hereafter, unless
otherwise  specified, that the interval $I$ in question is
$I=[0,\infty)$, but it could well be any half-line, of the form
$I=[x_0, \infty)$, with minor changes throughout (obtained by a
change of independent variable). Our first general result is a
counterpart of Theorem~\ref{atm} for asymptotically linear
solutions of \eqref{nlvs1}.

\begin{theorem} \label{th5}
Let $\sigma : I \to \mathbb{R}$ be right continuous and locally of bounded variation on $I$. Suppose that the nonlinearity $F$ in $\eqref{nlvs1}$ satisfies
\begin{enumerate}
\item $F:I \times \mathbb{R}\to \mathbb{R}^+$ is continuous on this domain
\item $F(x, \cdot)$ is nondecreasing for every $x \in I$
\item For some $M > 1$, $$\int_{0}^{\infty}F(t,Mt)\,|d\sigma(t)| < \infty$$
\end{enumerate}
Then \eqref{nlvs1} has an asymptotically linear solution, viz.,
a solution $y$ with $y(x) = Ax+B +o(1)$ as $x \to \infty$ for some
appropriate choice of real numbers $A, B$.
\end{theorem}


\begin{remark} \label{rmk6} \rm
The assumption that $F$ is nondecreasing in its second variable may be
 weakened at the expense of additional smoothness as a function of that
 variable (e.g., a Lipschitz condition of type \eqref{lip} and \eqref{ttt}
as we have seen above) and use of a fixed point theorem as the next result
shows.
\end{remark}

\begin{theorem}\label{th6}
Let $\sigma:I\longrightarrow\mathbb{R}$ be a
non-decreasing right-continuous function,
$F: I \times\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ be continuous
such that for some $M > 0$,
\begin{itemize}
\item[(a)] $\int_{0}^{\infty}t\,F(t,y(t))\,d\sigma(t)\leq M$, for $y\in X$,
\end{itemize}
where $X=\{y\in \mathcal{C}(I): 0\leq y(x)\leq M,\; x\in I\}$,
\begin{itemize}
\item[(b)] $|F(x,u)-F(x,v)|\leq k(x)|u-v|$, $x \in I$, $u, v \in
\mathbb{R^+}$
\end{itemize}
where $k: I \to \mathbb{R}^{+}$ is continuous and
\begin{itemize}
\item[(c)] $\int_{0}^{\infty}t\,k(t)\,d\sigma(t)<\infty$.
\end{itemize}
Then the Volterra-Stieltjes integro-differential equation \eqref{nlvs2}
has a monotone increasing solution $y(x)$ with $0 \leq y(x) \leq M$
for $x \in I$ and $y(x) \to M$ as $x \to \infty$.
\end{theorem}

It appears at first sight as if condition (a) in Theorem~\ref{th6}
may be  difficult to verify. However, the following simple
corollary shows that pointwise estimates on $F(x,y)$ can be used
to imply the same conclusion.

\begin{corollary} \label{coro}
Assume that $F, \sigma$ are as in Theorem~\ref{th6}. Let $M > 0$ and let
\begin{equation}\label{atko}
 F(x, y)  \leq  p(x) q(y),\quad x \geq 0, y \in \mathbb{R}^+,
\end{equation}
for some function $q$, where $q : [0, M] \to [0, M] $ is continuous on
$[0,M]$. Let $p\in C[0,\infty)$ and suppose that
\begin{equation}\label{effo}
 \int_{0}^{\infty}t\, p(t)\, d\sigma(t)\, \leq 1.
\end{equation}
Assume further that there exists a function
$k:\mathbb{R}^+ \to \mathbb{R}^+$ such that $k$ is continuous and
\begin{equation*}%\label{teek}
 \int_{0}^{\infty}t\, k(t)\, d\sigma(t)\, < 1
\end{equation*}
such that for any $u, v \in \mathbb{R}^+$, we also have
\begin{equation*} %\label{lip}
| F(x, u) - F(x, v) | \leq  k(x) | u-v|, \quad x \geq 0\,.
\end{equation*}
Then \eqref{nlvs2} has a positive (and so non-oscillatory) monotone solution
on $I$ such that $y(x) \to M$ as $x \to \infty$.
\end{corollary}


\subsection{Discussion}

Note that if $\int_{0}^{\infty}t\,F(t,0)\,d\sigma(t)<\infty$ then
this condition, along with assumptions (b) and (c) in the theorem
together imply (a). In particular, (a) is satisfied if $F(x,0)=0$
for every $x \in I$.

As in the differential equation case before, if
$\int_{0}^{\infty}t\,k(t)\,d\sigma(t)<1$ then relation
\eqref{ayxaz} in its proof gives us
$$
\|Ay-Az\|_{\infty}\leq\|x-y\|_{\infty}\int_{0}^{\infty}t\,k(t)\,d\sigma(t),
$$
and the Banach contraction mapping theorem applies immediately to
gives us existence and uniqueness of the solution of our
integro-differential equation. Note the similarity between
hypothesis (3) in Theorem~\ref{th5}  and assumption (a) in
Theorem~\ref{th6}: In condition (a) the integrand involves a class
of functions all bounded by the constant $M$, whereas in
hypothesis (3) the ``class of functions" is replaced by the class
of linear functions of the form $Mt$. In the former case there are
asymptotically constant solutions while, in the latter case, there
are asymptotically linear solutions. This is reflected in the form
of the respective assumptions. Indeed, since the constant function
$y(x)=M$ is in $X$, assumption (a)  \emph{includes} the condition
\begin{equation}\label{tftm}
\int_{0}^{\infty}t\,F(t,M)\, d\sigma(t) \leq M.
\end{equation}
Next, Theorem~\ref{th6} gives the existence of an asymptotically
constant  solution whenever there exists a constant $M >0$
satisfying condition (a) (the other two assumptions being
independent of $M$ we assume as implicitly verified). Thus, if (a)
is assumed for  \emph{each} $M > 0$ then there is an asymptotically
constant solution tending to that limit, $M$. A similar
observation appplies to Theorem~\ref{th5}. A moment's reflection
shows that if, in addition, we assume that $F(x, \cdot)$ is
non-decreasing for each $x \in I$, then the existence of a
solution can be obtained satisfying the improved estimate $$ M -
\int_{0}^{\infty}t\,F(t,M)\, d\sigma(t) \leq y(x) \leq M$$ in
Theorem~\ref{th6}. Since \eqref{tftm} holds for that $M$, the left
hand side is non-negative. Since \eqref{tftm} is reminiscent of
Nehari's criterion \cite{zn} for the existence of bounded
nonoscillatory solutions, it is of interest to investigate the
validity of this criterion in this more general setting and this
is the subject of the next result.

\begin{lemma}
\label{lem1} Let $\sigma:I\longrightarrow\mathbb{R}$ be a
non-decreasing right-continuous function,  $F: I
\times\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ be continuous
and such that for some $M > 0$,
\begin{equation}
\label{zn1}
\int_{0}^{\infty}t\,F(t,M)\,d\sigma(t)< \infty.
\end{equation}
Then every eventually positive solution of the Volterra-Stieltjes
integro  differential equation \eqref{nlvs2} is either of the form
$y(x)\sim Ax$ as $x \to \infty$  for some constant $A\neq 0$ or
$y(x)/x \to 0$ as $x \to \infty$.
\end{lemma}


We now formulate an analog of Nehari's necessary and sufficient
criterion \cite{zn} for the existence of a bounded nonoscillatory
solution of our equation (recall that a solution $y$ of
\eqref{nlvs1} or \eqref{nlvs2} is said to be  \emph{nonoscillatory}
provided $y(x) \neq 0$ for all sufficiently large $x$).

\begin{theorem}
\label{th7} Let $\sigma:I\longrightarrow\mathbb{R}$ be a
non-decreasing right-continuous function,  $F: I
\times\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ be continuous
and non-decreasing in its second variable (i.e., $F(x, y)$ is
nondecreasing in $y$ for $y>0$, for each $x\in I$). Then
\eqref{nlvs2} has bounded eventually positive solutions if and
only if \eqref{zn1} holds for some $M>0$.
\end{theorem}


\begin{corollary}\label{cor001}
Let $\sigma$ be as in Theorem~\ref{th7},  $G: I\times
\mathbb{R}^+\to \mathbb{R^+}$ be continuous and positive in
$I\times \mathbb{R}^+$. In addition, let $G(x,y)$ be nondecreasing
for every $y>0$, $x\in I$. Then
\begin{equation}\label{nlvs2p}
y'(x) = y'(0) - \int_{0}^{x}y(t)G(t,y^2(t))\,d\sigma(t)
\end{equation}
has bounded nonoscillatory solutions if and only if there holds
\begin{equation}
\label{zn18}
\int_{0}^{\infty}t\,G(t,c)\, d\sigma(t) < \infty,
\end{equation}
for some $c>0$.
\end{corollary}

The proof of the next result is an immediate consequence of the
theorem.

\begin{corollary}\label{cor02}
Let $\sigma, F$ be as in Theorem~\ref{th7}. Then \eqref{nlvs1}
has asymptotically constant positive solutions if and only if \eqref{zn1}
holds for some $M>0$.
\end{corollary}

Of course, Corollary~\ref{cor02} deals with  \emph{bounded} solutions
of \eqref{nlvs1}. An analogous result for possibly unbounded solutions
follows (although strong superlinearity \eqref{sts} is to be imposed).

\begin{theorem}\label{th8}
Let $\sigma$ be as in Theorem~\ref{th7}. Assume that  $F: I\times
\mathbb{R}^+\to \mathbb{R^+}$ is continuous and positive in
$I\times \mathbb{R}^+$. In addition, let $F$ satisfy the strong
superlinearity condition \eqref{sts} for $\varepsilon =0$, as well
as for some $\varepsilon > 1$. Then \eqref{nlvs1} has an
eventually positive solution if and only if  \eqref{zn1} holds for
some $M>0$.
\end{theorem}



\subsection{Discussion} Theorem~\ref{th7} is an improvement of
 Nehari's theorem \cite{zn} to the framework of Volterra-Stieltjes
integral equations \eqref{nlvs1}, or Volterra-Stieltjes integro-differential
 equations \eqref{nlvs2}.  Although Nehari's theorem \cite[Theorem I]{zn}
was stated for equations of the form \eqref{zn11}, we choose the
more general form stated here, with an arbitrary nonlinearity
(this explains the apparently odd restriction on $\varepsilon > 1$
rather than $\varepsilon > 0$ as in the original Nehari result).
As pointed out in the proof of Corollary~\ref{cor001} the form
\eqref{zn11} is actually guided by the wish that both $y$ and $-y$
be solutions of the same equation. Nehari's theorem as such is
actually a special case of Corollary~\ref{cor001} with
$\sigma(t)=t$ throughout. The integral equation \eqref{nlvs2p}
then produces a differential equation of the form \eqref{zn11}
(since the indefinite integral is continuously differentiable).
Indeed, Corollary~\ref{cor02} (via the techniques in the proof
Corollary~\ref{cor001}) also includes an extension of Nehari's
theorem by Coffman and Wong \cite{jsww4}, \cite[Theorem E]{jsww3}.

The Volterra-Stieltjes framework provides for recurrence relation
(discrete) analog or even intermediate mixed type integro
differential equations as a direct consequence (see the next
Section for applications). In addition, Corollary~\ref{cor02}
shows that the sufficiency of the proof of Theorem~\ref{th7}
actually provides a criterion for the existence of asymptotically
constant solutions of either \eqref{nlvs1} or \eqref{nlvs2}. As we
gather from the proof of said theorem, we can choose the
asymptotic limit $A$ appearing in \eqref{ii0} to be any number
between $(0, M)$, where the $M$ appears in \eqref{zn1}. It follows
that if \eqref{zn1} is valid for every $M>0$ then \eqref{nlvs1}
has solutions whose limits can be any prescribed positive number.



Theorem~\ref{th8} includes a slight modification of an additional
result of Coffman and Wong [\cite{jsww4}, Section 6]. Observe
that, if the solution in the necessity of Theorem~\ref{th8} is
unbounded, then \eqref{zn1} must hold for  \emph{every} $M>0$, just
as in the case of ordinary differential equations, cf.,
\cite{jsww4}. That is, the existence of at least one unbounded
eventually positive solution of \eqref{nlvs1} implies the
convergence of the integral \eqref{zn1}, not only for the $M$ in
question, but for every $M>0$ (see also Lemma~\ref{lem2} below in
this regard).

In order not to restrict ourselves only to the study of
asymptotically constant solutions of either \eqref{nlvs1} or
\eqref{nlvs2}, we now present further results relating to
asymptotically linear solutions. Lemma~\ref{lem2} below
complements Theorem~\ref{th5} above.

\begin{lemma} \label{lem2}
Let $\sigma$ be right-continuous and nondecreasing on $I$,
$F: I\times \mathbb{R}^+\to \mathbb{R^+}$ be continuous and positive
in $I\times \mathbb{R}^+$. In addition, let $F(x,y)$ be nondecreasing
for every $y>0$, $x\in I$. If either \eqref{nlvs1} or \eqref{nlvs2}
has a solution $y(x) \sim Ax+B$ as $x \to \infty$, where $A>0$, $B$
are constants, then
\begin{equation} \label{tmt}
\int_{0}^{\infty} F(t, Mt)\, d\sigma(t) < \infty
\end{equation}
for some $M > 0$.
\end{lemma}

\begin{remark} \label{rmk7} \rm
Incidentally, this proof also shows that the existence of at least one
asymptotically linear solution with asymptotic slope $A$ implies that
\eqref{tmt} is satisfied for every $M$, with $0 < M < A$.
\end{remark}

\begin{theorem} \label{th9}
Let $\sigma$ be right-continuous and nondecreasing on $I$,
$F: I\times \mathbb{R}^+\to \mathbb{R^+}$ be continuous and positive in
$I\times \mathbb{R}^+$. In addition, let $F(x,y)$ be nondecreasing for
every $y>0$, $x\in I$. Then \eqref{nlvs2} has an asymptotically linear
 solution if and only if \eqref{tmt} holds for some $M>0$.
\end{theorem}

\subsection{Discussion}
The previous result extends another result of Nehari \cite[Theorem
II]{zn} to this more general setting. Although we did not exhibit
``Stieltjes analogs" (i.e., for equations of the form
\eqref{nlvs1} or \eqref{nlvs2}) of the results in the first few
sections for reasons of length, we do not foresee any difficulties
in their respective formulations and proofs. In this vein a
Stieltjes analog of Theorem~\ref{th6} is readily available, the
only major difference being the definition of the space which in
this case is $L^{\infty}(I)$. The result is stated next and we
leave the proof to the reader.

\begin{theorem}
\label{th10}
Let $f \in L^{\infty}(I)$, $\sigma$ be right-continuous and non-decreasing
on $I$, and suppose that the nonlinearity
$F:I \times \mathbb{R}\to \mathbb{R}^+$ in
\begin{equation}
\label{vvv}
y(x) = f(x) - \int_{x}^{\infty} (t-x)\,F(t,y(t))\, d\sigma(t), \quad x\geq x_0
\end{equation}
is continuous on this domain, that $F(x, y)$ is nondecreasing in $y$ for
every $x \in I$, $y>0$ and for every $M > 0$,
$$
\int_{0}^{\infty}t\,F(t, M)\,d\sigma(t) < \infty.
$$
In addition, we assume that for every $y, z \in \mathbb{R}$ and every
$x \in I$,
$$
| F(x,y)-F(x,z)| \leq k(x)|y-z|
$$
where
$$
\int_{x_0}^{\infty}tk(t)\,d\sigma(t) < 1.
$$
Then \eqref{vvv} has a solution $y \in L^{\infty}(I)$ if and only if there
are two functions $u, v \in L^{\infty}(I)$ such that $u(x) \leq v(x)$,
$x \in I$, and for $x\geq x_0$,
\begin{equation} \label{uf}
u(x) \leq f(x) - \int_{x}^{\infty}(t-x)F(t,v(t))\,d\sigma(t)
\end{equation}
and
\begin{equation} \label{vf}
v(x) \geq f(x) - \int_{x}^{\infty}(t-x)F(t,u(t))\,d\sigma(t)
\end{equation}
\end{theorem}

Finally, we give a result that completely parallels
Theorem~\ref{th1} above in this wider setting.


Let $f\in L^{\infty}[1,\infty)$ with the usual essential supremum
norm, $\|\cdot \|$, satisfy \eqref{ft} for some $\delta > 0$.
Define $Y = \{ u\in L^{\infty}[1,\infty): \|u(x)/f(x)\| <
\infty\}$.

The subset $X = \{ u\in Y: \|u(x)/f(x)\| \leq 2\}$, is a closed subset of $Y$.
  Let $F: [1, \infty) \times \mathbb{R} \to \mathbb{R}$ be continuous
(and not necessarily positive), and let $\sigma$ be a
right-continuous non-decreasing function defined on $[1, \infty)$.
In addition, let
\begin{equation} \label{sf00}
\int_{1}^{\infty} s\, | F(s, 0) |\, d\sigma(s) < \infty.
\end{equation}
With $f$ as above let there exist a function $k: [1,\infty) \to \mathbb{R}^+$
satisfying
\begin{equation} \label{sfk0}
\int_{1}^{\infty}s\, |f(s)|\, k(s) \, d\sigma(s)  < \infty.
\end{equation}
We assume the usual Lipschitz condition on $F$ as before, that is, for any
$u, v \in \mathbb{R}$,
\begin{equation} \label{lip010}
| F(x, u) - F(x, v) |  \leq k(x) | u-v|, \quad x \geq 1.
\end{equation}
For such functions $F, k, f, \sigma$ satisfying \eqref{ft},
\eqref{sf00}, \eqref{sfk0} and \eqref{lip010} we consider the
``forced" nonlinear equation defined by, for $y \in X$,
\begin{equation}
\label{fnon}
Ty(x) = f(x) - \int_{x}^{\infty} F(t,y(t))\, d\sigma(t), \quad x\geq a.
\end{equation}
on the interval $I = [a, \infty)$ where $a$ is chosen so large that
$a \geq 1$ and for $x \geq a$,
\begin{equation} \label{t000}
 \max  \Big\{ \int_{x}^{\infty} \,(s-x)\, |f(s)|\, k(s) \, d\sigma(s) ,
 \int_{x}^{\infty} \,(s-x)\, |F(s,0)| \, d\sigma(s) \Big\}
\leq \frac{\delta}{4}.
\end{equation}
 Fix such an $a$ for the next result.

\begin{theorem} \label{th15}
Let $f, F, k, \sigma$ defined above satisfy
\eqref{ft}, \eqref{sf00}, \eqref{sfk0}, \eqref{lip010} and \eqref{t000}.
Then the operator $T$ has a unique fixed point in $X$, and this point
corresponds to a solution of the integral equation
$$
y(x) = f(x) - \int_{x}^{\infty} F(t,y(t))\, d\sigma(t), \quad x\geq a.
$$
such that $y \in X$ and $y(x) \sim f(x)$ as $x \to \infty$.
\end{theorem}

\begin{remark} \label{rmk8} \rm
If $f $ is, in addition, absolutely continuous on $[a,\infty)$, then so is
$y$, in which case its right derivative satisfies \eqref{nlvs2} for every
$x \geq a$.
\end{remark}


\section{Applications to differential and difference equations}

The main reason for the developments of the previous sections to
Volterra-Stieltjes integral and integro-differential equations of
the form \eqref{nlvs1}, \eqref{nlvs2} is that this wider framework
can be used as a tool for unifying discrete and continuous
phenomena such as differential equations and difference equations
(or recurrence relations). This approach was emphasized by
Atkinson \cite{fvab}, H\"{o}nig \cite{csh}, Mingarelli \cite{abmb}
and Mingarelli-Halvorsen \cite{mhb} among the earliest such
textual sources. See these texts for basic terminology and other
examples of theorems in this wider framework along with their
developments to discrete phenomena. Although such generalizations
seem to be academic at best, their main thrust lies in their
applicability to cases that are not ``continuous" as we see below.

The simplest of all applications of the results in
Section~\ref{vse} is to differential equations of the second
order, linear or not. This is accomplished by choosing
$\sigma(t)=t$ throughout that section. The correponding results
for ordinary differential equations then arise as corollaries of
the results therein. Thus, as pointed out in that section the
various theorems therein, some even new for the case of ordinary
differential equations, extend essential results in nonlinear
theory due to Atkinson, Nehari, Coffman and Wong, etc. to this
wider framework.

In order to derive results for equations other than ordinary
differential equations we can choose $\sigma(t)$ to be a function
that is part step-function and part absolutely continuous, or even
all step-function or by the same token, all absolutely continuous.
The three different choices lead to three intrinsically different
kinds of equations.

\subsection{The case of three-term recurrence relations}

In order to derive the special results in this case, we appeal to
the methods described in [\cite{abmb}, Chapter 1]. Thus, starting
from  any infinite sequence of real numbers
$\{b_n\}_{n=0}^{\infty}$ we produce an absolutely continuous
function $b: \mathbb{N}\to \mathbb{R}$ by simply joining the
various points $(n, b_n)$, $n =0,1,2,\dots$ in the plane by a line
segment. The resulting polygonal curve is clearly locally
absolutely continuous on its domain (we call this curve the  \emph{
polygonal extension} of the the sequence of points to a curve).
Next, we define a right-continuous step-function (or simple
function)  by defining its jumps to be at the integers (or any
other suitable countable set, [\cite{abmb}, xi]) of magnitude
$\sigma(n)-\sigma(n-0) = - b_n$, for $n\geq 0$ (so $\sigma(t) =
{\rm constant}$ in between any two consecutive integers). Defining
$F(x,y):=y$ for simplicity of exposition, we can show that (see
[\cite{abmb}, pp.12-15]) the solution $y(x)$ of the equation
\eqref{nlvs2} with right-derivatives has the property that
\begin{equation*}
{{\Delta}^2}{y_{n-1}}+b_n\,y_n = 0, \quad n\in \mathbb{N},
\end{equation*}
where $y(n)=y_n$ for every $n$, and $\Delta$ is the forward
difference  operator defined here classically by $\Delta y_{n-1}=
y_n - y_{n-1}$. No more generality is gained by looking at the
three-term recurrence relation in standard form, that is,
\begin{equation}\label{3trr}
c_ny_{n+1}+c_{n-1}y_{n-1} + b_ny_n = 0,\quad n\in \mathbb{N},
\end{equation}
where $c_n \neq 0$ for every $n$. The change of dependent variable
$y_n = \alpha_nz_n$ where the $\alpha_n$ satisfy the recurrence relation
$\alpha_{n+1} = \{c_{n-1}/c_n\}\alpha_{n-1}$, $n\in \mathbb{N}$,
brings \eqref{3trr} into the form
$$
{{\Delta}^2}{z_{n-1}}+\beta_n\,z_n = 0, \quad n\in \mathbb{N},
$$
 for some appropriately defined sequence $\beta_n$. Conversely,
every such second order linear difference equation  is equivalent
to a three term recurrence relation of the form \eqref{3trr} with
$y_n=z_n$, $c_n=1$ and $b_n = \beta_n-2$.

If $F$ is defined generically as in Section~\ref{vse} then the same
choice of the step-function $\sigma$ in \eqref{nlvs2} produces the the
second order difference equation
\begin{equation}
\label{2bdef}
{{\Delta}^2}{y_{n-1}}+b_n\,F(n,y_n) = 0, \quad n\in \mathbb{N}.
\end{equation}
The pure nonlinear difference equation
\begin{equation}
\label{2odef}
{{\Delta}^2}{y_{n-1}}+F(n,y_n) = 0, \quad n\in \mathbb{N}.
\end{equation}
is obtained by setting the $b_n=1$ and defining the resulting
step-function $\sigma$ as above.

Conversely, starting with any nonlinear difference equation of the
form \eqref{2odef} we can produce a Volterra-Stieltjes
integro-differential equation of the form \eqref{nlvs2} by
``extending" the domain of this discrete solution $y_n$ to a half
axis by joining the points $(n,y_n)$ by line segments. Call this
new function $y(x)$. Define the step-function $\sigma$ by jumps of
magnitude $\sigma(n)-\sigma(n-0) = - 1$ and right-continuity, and
$F(x,y)$, the polygonal extension of the sequence  $F(n, y_n)$ to
an absolutely continuous function $F(x,y)$ (obtained by joining
the points $(n,y_n,F(n,y_n))$, $(n+1,y_{n+1},F(n+1,y_{n+1}))$,
$n\in \mathbb{N}$,  by a line segment). In this case, the
Riemann-Stieltjes integral appearing in \eqref{nlvs2} exists for
each $x$. The resulting function $y(x)$ is locally absolutely
continuous and its right-derivative exists at every point and is
locally of bounded variation on the half-axis.  It can be shown
that this new function $y(x)$, now satisfies \eqref{nlvs2} with
right-derivatives. If more smoothness is required on the function
$F$ we can use interpolating polynomials in $\mathbb{R}^3$  \emph{in
lieu} of the polygonal extension$\dots$. This duality between
equations of the form \eqref{nlvs2} and \eqref{2odef} underlines
the importance of this approach.

With these facts in hand we formulate the recurrence relation
corollary of Theorem~\ref{th5} above.

\begin{theorem} \label{th11}
Let $F:I\times \mathbb{R}\to \mathbb{R}^+$ with values $F(x,y)$,
 be continuous on this domain, nondecreasing in its second variable
for every $x\in I$ and assume that for some $M>1$ and for some real
sequence $\{b_n\}_{n=0}^{\infty}$, we have
$$
\sum_{n=0}^{\infty}F(n, Mn)\, |b_n| < \infty.
$$
Then \eqref{2bdef} has asymptotically linear solutions, that is solutions
of the form $y_n \sim An+B$ as $n \to \infty$ for some constants $A,B$.
\end{theorem}

Another such consequence is a discrete analog of
Theorem~\ref{th6}.

\begin{theorem} \label{th12}
Let $X=\{y\in \mathcal{C}(I): 0\leq y(x)\leq M,\; x\in I\}$, where
$M>0$ is given and fixed. Let $F: I
\times\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ be continuous
on this domain, and $\{b_n\}_{n=0}^{\infty}$ a given non-negative
sequence such that
\begin{itemize}
\item[(a)] $\sum_{n=0}^{\infty}n\,b_n\,F(n,y(n))\leq M$, \quad{for all $y\in X$,}
\end{itemize}
\begin{itemize}
\item[(b)] $|F(x,u)-F(x,v)|\leq k(x)|u-v|$, \quad{$x \in I$, $u, v \in
\mathbb{R^+}$}
\end{itemize}
where $k: I \to \mathbb{R}^{+}$ is continuous and for $k(n):=k_n$,
\begin{itemize}
\item[(c)] $\sum_{n=0}^{\infty}n\,k_n\,b_n<\infty$.
\end{itemize}
Then the difference equation \eqref{2bdef} has a monotone increasing
solution $y_n$ satisfying $0 \leq y_n \leq M$ for each $n$, and $y_n \to M$
as $n \to \infty$.
\end{theorem}

Finally, we formulate a version of Nehari's theorem [\cite{zn},
Theorem I] for second order difference equations as a result of
our investigations. We leave the proof to the reader (note that we
use $b_n=1$ in this case).

\begin{theorem} \label{th13}
Let $F: I \times\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ be
 continuous on this domain and non-decreasing in its second variable
(i.e., $F(x, y)$ is nondecreasing in $y$ for $y>0$, for each $x\in I$).
Then \eqref{2odef} has bounded eventually positive solutions if and only if
$$
\sum_{n=0}^{\infty}n\,F(n, M) < \infty
$$
holds for some $M>0$.
\end{theorem}

This should convince the reader that difference equation analogs
of Lemma~\ref{lem1}, Corollary~\ref{cor001},
Corollary~\ref{cor02},Theorem~\ref{th8}, Theorem~\ref{th15} can be
formulated without undue difficulty and their proof is simply a
consequence of the results in the previous section with the
necessary choices of functions as detailed above.

Next, we note that equations  \emph{intermediate} between difference
and differential equations are also included in our framework of
equations of the form \eqref{nlvs2}. That is, we can assume that
our function $\sigma$ consists of a discrete part and a part that
is possibly continuous and of bounded variation (but not
necessarily absolutely continuous). Indeed, on $I=[0,\infty)$ for
a given $p>0$ we define $\sigma (t)$ by its jumps on $(0,p]$, so
that $\sigma (n) -\sigma(n-1) = -b_n$, for $n=0,1,2,\dots,p$ where
$b_n$ is a given arbitrary sequence and $\sigma$ is
right-continuous at its jumps. Let $\sigma (t) := h(t)$ where $h$
is a fixed function, right-continuous and locally of bounded
variation on $[p, \infty)$. In the framework of these equations,
Nehari's theorem takes the following form:

\begin{theorem} \label{th131}
Let $F: I \times\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ be
continuous on this domain and non-decreasing in its second variable
(i.e., $F(x, y)$ is nondecreasing in $y$ for $y>0$, for each $x\in I$).
Then the integro-differential-difference equation of Stieltjes type,
\begin{equation}
\label{iddi}
y'(x) = y'(0) - \sum_{n=0}^{p}F(n, y(n))b_n -\int_{p}^{x}F(t,y(t))\,dh(t)
\end{equation}
for $x > p$, has bounded eventually positive solutions if and only if
$$
\int_{p}^{\infty}t\,F(t, M)\,dh(t) < \infty
$$
holds for some $M>0$.
\end{theorem}

\subsection{Discussion}
A solution $y$ of our equation \eqref{iddi} above is a polygonal curve
whenever $0<x<p$ (since the integral term is absent in \eqref{iddi})
while for $x> p$ it is an absolutely continuous curve locally of bounded
variation. Thus the values $y(n) := y_n$ actually satisfy a second order
difference equation for small $x$ ($x<p$) while for large $x$ ($x>p$)
this $y(x)$ is the solution of a pure integral equation of Volterra-Stieltjes type along with some discrete parts (as seen in \eqref{iddi}). The special case $h(t)=t$ is clearly included in this discussion. For this choice, \eqref{iddi} takes the form
\begin{equation} \label{iddie}
y'(x) = y'(0) - \sum_{n=0}^{p}F(n, y(n))b_n -\int_{p}^{x}F(t,y(t))\,dt,
\end{equation}
``almost" a second order differential equation except for the interface
conditions at a prescribed set of points in $[0,p]$. Under the usual
conditions on $F$ as required by Theorem~\ref{th7}, \eqref{iddie} will
have eventually positive solutions if and only if
$$
\int_{p}^{\infty}t\,F(t, M)\,dt < \infty
$$
holds for some $M>0$ (which is precisely Nehari's necessary and
sufficient  criterion for second order nonlinear differential
equations). For this choice of $\sigma$ this result is to be
expected, in some sense, since we are dealing with large $x$
anyhow and so the equation \eqref{iddie} behaves very much like a
differential equation. However, we could  \emph{spread} the discrete
part all over the interval $I$ in which case this argument is no
longer tenable, as it is  \emph{a priori} conceivable that
oscillations may occur therein (but cannot by Theorem~\ref{th7}).

\section{Proofs}

\begin{proof}[Proof of Theorem~\ref{th}]
We note that $X$ is a closed subset of  the Banach space $Y$
above.  This is most readily seen by writing the space $X$ as $X =
\{ u \in Y |:  0 \leq \frac{u(t)}{at+b} \leq 1,\text{ for all } t
\geq 0\}$ and applying standard arguments. In addition, it is easy
to see that $X$ is convex. Now we define a map $T$ on $X$ by
setting
\begin{equation} \label{map}
(Tu)(x) = ax+b - \int_{x}^{\infty} (t-x) \, F(t, u(t))\, dt
\end{equation}
for $u \in X$. Note that the right-side of \eqref{map} converges
for each $x \geq 0$, because of \eqref{atk}. Indeed, for $u \in X$,
$x \geq 0$,
\begin{equation} \label{bound}
0 \leq  \int_{x}^{\infty}(t-x)F(t, u(t))\, dt \leq \int_{0}^{\infty} t\,
F(t, u(t))\, dt \leq b,
\end{equation}
as $F(t, u(t)) \geq 0$ for such $u$ (which implies that $(Tu)(x) \leq ax+b$)
and the indefinite integral is a non-increasing function of $x$ on
$[0, \infty)$. Since $a \geq 0$, we get that $(Tu)(x) \geq 0$ for any
$x \geq 0$. On the other hand, it is easy to see that for
$u \in X$, $Tu$ is a continuous function on $[0, \infty)$. So,
$TX \subseteq X$.

Next, we prove that $T$ is a continuous map on $X$. For $u, v \in X$,
\begin{align*}
|(Tu)(x) - (Tv)(x) |
& \leq   \int_{x}^{\infty} (t-x) | F(t, u(t)) - F(t, v(t)) | \, dt  \\
& \leq   \int_{x}^{\infty} (t-x)  k(t) | u(t) - v(t) |\, dt  \\
& \leq  \| u - v \|_{Y} \, \int_{0}^{\infty} t\, k(t)\,(at+b)\, dt,
\end{align*}
where we have used \eqref{lip} and the fact that
$\int_{x}^{\infty}(t-x)k(t)(at+b)\, dt$ is a non-increasing function of $x$ for
$x \in [0, \infty)$, since $k(t)(at+b) \geq 0$. It follows that for $x \geq 0$,
\begin{equation}\label{cont}
| \Psi (Tu)(x) - \Psi (Tv)(x) |  \leq  \frac{1}{b}\, \| u - v
\|_{Y} \, \max\{a,b\}\, \int_{0}^{\infty} t\,(t+1)\, k(t)\, dt,
\end{equation}
from which we conclude that
$$
\| Tu - Tv \|_{Y} \leq \alpha \, \| u - v \|_{Y},
$$
where $\alpha < \infty$ on account of \eqref{teek} and \eqref{tktt}.
It follows that $T$ is continuous on $X$.

Next, we show that $TX$ is compact, that is, $T$ sends bounded
subsets of $X$  onto relatively compact subsets. For $M$ a subset
of $X$ we have that to prove that $TM$ is relatively compact. By
virtue of the isometry $\Psi$, this is equivalent to proving that
$\Psi (T(M))$ is relatively compact. To this end, we use the
measure of noncompactness on $BC(\mathbb{R}^+)$ defined for $A \in
BC(\mathbb{R}^+)$ by
\[
\mu (A)= \lim_{L\to \infty}\left( \lim_{\varepsilon \to 0}
 w^L (A,\varepsilon)\right) + \limsup_{t\to\infty}\, {\rm diam}\, A(t),
\]
see [\cite{bg}, Theorem 9.1.1(d), p.46], where
$${\rm diam}\, A(t) = {\rm sup} \{|x(t)-y(t)| : x, y \in A\},$$
and
$$
w^L (A,\varepsilon) = {\rm sup} \{ w^L(x,\varepsilon) : x \in A\},
$$
with
$$
w^L(x,\varepsilon) = {\rm sup} \{|x(t)-x(s)| : t, s \in [0,L], |t-s|
 \leq \varepsilon\}.
$$
We fix $\varepsilon >0$, $L>0$, $u\in M\subset X$ and
$t_1,t_2\in\mathbb{R}^+$ with $t_2-t_1\leq \varepsilon$ and,
without loss of generality $t_2>t_1$. Then
\begin{align*}
&|\Psi(Tu)(t_2)-\Psi(Tu)(t_1)|\\
&= \Big|\int_{t_2}^{\infty}\frac{ (s-t_2)F(s,u(s))}{at_2 +b}ds-
\ \int_{t_1}^{\infty}\frac{ (s-t_1)F(s,u(s))}{at_1 +b}ds \Big|  \\
&\leq \Big|\ \int_{t_2}^{\infty}\left[\frac{(s-t_2)}{at_2+b}-\
\frac{(s-t_1)}{at_1 +b}\right]F(s,u(s))ds -
 \int_{t_1}^{t_2} \frac{(s-t_1)}{at_1 +b}F(s,u(s))ds\Big| \\
& \leq  \int_{t_2}^{\infty}\frac{(as
+b)(t_2-t_1)}{(at_2+b)(at_1+b)}F(s,u(s))ds +
\int_{t_1}^{t_2}\frac{(s-t_1)}{at_1 +b}F(s,u(s))ds
\\
&\leq  \frac{\varepsilon}{b^2}\ \int_{t_2}^{\infty}(as +b)
F(s,u(s))ds +
\frac{\varepsilon}{b}\int_{t_1}^{t_2}F(s,u(s))ds  \\
&\leq   \frac{a \varepsilon}{b} + \
\frac{\varepsilon}{b}\int_{t_1}^{\infty}F(s,u(s))ds,
\end{align*}
by \eqref{atk}, since $F \geq 0$ for $u \in M$. Combining these
estimates we deduce that
\begin{equation} \label{28}
\begin{aligned}
|\Psi(Tu)(t_2)-\Psi(Tu)(t_1)|
&\leq  \frac{a \varepsilon}{b} +
\frac{\varepsilon}{b}\int_{t_1}^{\infty}F(s,u(s))ds,  \\
& \leq   \frac{a \varepsilon}{b} + \frac{\varepsilon}{b}\left \{ \int_{t_1}^{t_2}F(s,u(s))ds + \int_{t_2}^{\infty}F(s,u(s))ds \right \}, \\
& \equiv  \frac{a \varepsilon}{b} +
\frac{\varepsilon}{b}\left \{ I_1 +I_2 \right \}.
\end{aligned}
\end{equation}
We estimate the two integral quantities in \eqref{28} in turn.
This said, use of  \eqref{lip} and \eqref{atk}  for $u \in M$ gives
\begin{equation} \label{28a} %\label{28b}
\begin{aligned}
I_2&\leq  \int_{t_2}^{\infty} |F(s,u(s))-F(s,0)|ds
  + \int_{t_2}^{\infty} F(s,0)ds  \\
&\leq   \int_{0}^{\infty} k(s)u(s)ds + \int_{0}^{1}F(s,0)ds
  +  \int_{1}^{\infty}sF(s,0)ds  \\
&\leq  \|u\|_Y \int_0^{\infty}k(s)(as +b)ds
  +  \sup_{s \in [0,1]}F(s,0) +\int_{0}^{\infty}sF(s,0)ds   \\
&\leq  C_2 \equiv \int_0^{\infty} k(s)(as +b)ds +   \sup_{s \in
[0,1]}F(s,0) + b
\end{aligned}
\end{equation}
(since, for $u\in M$, $\|u\|_Y \leq 1$), where $C_2$ is finite and
independent of $\varepsilon$, because of \eqref{tktt} and \eqref{kt}.
On the other hand, arguing as in \eqref{28a}, we obtain
\begin{equation}
\begin{aligned}
I_1&\leq   \int_{t_1}^{t_2} |F(s,u(s))-F(s,0)|ds+ \int_{t_1}^{t_2} F(s,0)ds  \\
&\leq   \int_{0}^{\infty} k(s)u(s)ds + \int_{0}^{\infty}F(s,0)ds  \\
&\leq  C_2
\end{aligned}\label{29a}.
\end{equation}
Combining \eqref{29a}, \eqref{28a} and \eqref{28} we get finally,
\begin{equation}
|\Psi(Tu)(t_2)-\Psi(Tu)(t_1)|  \leq  \ \frac{a \varepsilon}{b} +
\frac{2C_2 \varepsilon}{b}. \label{30}
\end{equation}
Passing to the supremum over $u \in M \subset X$ we find that
\[
\lim_{\varepsilon \to 0} w^{L}(\Psi (TM),\varepsilon)=0
\]
Since $L>0$ is arbitrary we deduce that
\[
\lim_{L\to \infty} \left(\lim_{\varepsilon \to 0}w^L
(\Psi (TM),\varepsilon)\right)=0.
\]
To complete the proof we need to analyze the term related to the diameter.
Taking $u, v\in M$ and $t\in  \mathbb{R}^+$ then, proceeding as in
the continuity argument above leading to \eqref{cont}, we see that
\begin{align*}
|\Psi (Tu)(t)-\Psi (Tv)(t)|
&\leq   \frac{\|u-v\|_{Y}}{b} \int_t^{\infty} s(as+b)k(s)ds \\
& \leq   \frac{\|u-v\|_Y}{b}\left[a  \int_t^{\infty} s^2 k(s)ds
  +b \int_t^{\infty}sk(s)ds\right] \\
& \leq  \frac{2}{b} \left[a  \int_t^{\infty} s^2 k(s)ds +b
\int_t^{\infty} sk(s)ds\right],
\end{align*}
since $\|u-v\|_Y \leq 2$ for $u, v \in M$. Consequently
\[
\limsup_{t\to \infty}\mathop{\rm diam} \Psi (TM)(t)
 \leq \frac{2}{b}\limsup_{t\to \infty}\left[ a\int_t^{\infty} s^2 k(s)ds
+b\int_t^{\infty} sk(s)ds \right]=0,
\]
on account of \eqref{teek} and Remark~\ref{uno}. Thus, $\mu (\Psi (TM))=0$.
This fact tells us that $\Psi (TM)$ is relatively compact in
$BC(\mathbb{R}^+)$ and, since $\Psi$ is an isometry, $TM$ is relatively
compact in $Y$. Hence, $T$ is compact, and so Schauder's theorem
gives the existence of a fixed point $u \in X$ for $T$. This fixed
point is necessarily a solution of \eqref{non0} asymptotic to the
line $ax+b$ as $x \to \infty$. This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th1}]
  Let $Y$ be the Banach space defined in \eqref{spaceY} with the norm
\eqref{normY}, where we replace the interval $[1,\infty)$ by $x_0,\infty)$.
Let $X$ be the closed subset defined by
$X =\{u\in Y: {\sup_{x \geq x_0}\{|u(x)|/|f(x)|\}\leq 2} \}$.
Define a map $T$ on $X$ by $u \in X$,
\begin{equation}
\label{tux}
Tu(x) = f(x) - \int_{x}^{\infty}{(s-x)F(s,u(s))\,ds},\;  x\geq x_0.
\end{equation}
Clearly, for $u \in C(I)$ we have, because of our assumptions on $F$,
$Tu \in C(I)$. In addition, \eqref{lip01} gives that for
$u \in X$, $|F(s,u(s))| \leq k(s)|u(s)| + |F(s,0)|$.
Combining this with \eqref{tux}, dividing \eqref{tux} throughout
 by $f(x)$ a simple estimation gives that for $ x \geq x_0$,
\[
\Big | \frac{Tu(x)}{f(x)}\Big |
\leq 1 + \frac{1}{|f(x)|}\int_{x}^{\infty}(s-x)|k(s)f(s)|
\Big |\frac{u(s)}{f(s)}\Big | \, ds
+ \frac{1}{|f(x)|}\int_{x}^{\infty}(s-x)|F(s,0)|\, ds.
\]
Now, the use of \eqref{ft} shows that
\begin{equation}
\label{Tuf} \Big | \frac{Tu(x)}{f(x)}\Big | \leq 1 + \Big \|
\frac{u}{f}\Big \|
\frac{1}{\delta}\int_{x}^{\infty}(s-x)|k(s)f(s)|\, ds
+ \frac{1}{\delta}\int_{x}^{\infty}(s-x)|F(s,0)|\, ds,
\end{equation}
which, since $u \in X$ and \eqref{t0} is enforced, furnishes the bound
\[
\Big \| \frac{Tu(x)}{f(x)}\Big \| \leq
1 + 2 \frac{1}{\delta}\frac{\delta}{4}\,+ \frac{1}{\delta}\frac{\delta}{4}
= \frac{7}{4} < 2.
\]
Thus $T$ is a self-map on $X$.  In order to show that $T$ is a
contraction on $X$, consider the simple estimate derived from
\eqref{tux}, namely, for $x \geq x_0$,
\begin{equation} \eqref{lip01}
\begin{aligned}
\Big | \frac{Tu(x) - Tv(x)}{f(x)} \Big |
&\leq \frac{1}{|f(x)|} \int_{x}^{\infty}{(s-x)|F(s,u(s))-F(s,v(s))|\, ds},\\
&\leq \frac{1}{|f(x)|} \int_{x}^{\infty}{(s-x)k(s)|u(s)-v(s)|\, ds},\quad
({\rm by}\,  ) \\
&\leq \frac{1}{\delta}\|u-v\| \int_{x}^{\infty}{(s-x)k(s)|f(s)| \, ds},
\quad ({\rm by}\, \eqref{ft} ) .
\end{aligned}
\end{equation}
Since the last display is valid for every $x \geq x_0$ it follows
from \eqref{t0} that,
$$
\| Tu - Tv\| \leq (1/4)\, \|u-v\|,
$$
so that $T$ is a contraction on $X$. It is easily seen that the subsequent
fixed point, say $u(x)$, obtained by applying the classical fixed point
theorem of Banach, is a solution of \eqref{non0} satisfying the
 conclusion (2) stated in the theorem, since $u \in X$. On the other
hand, since our fixed point $u$ satisfies \eqref{tux}, we have
$$
\frac{u(x)}{f(x)} = 1 - \frac{1}{f(x)}\int_{x}^{\infty} (s-x)F(s,u(s))\, ds.
$$
An estimation of this integral similar to the one leading to the
right-side of \eqref{Tuf} gives that
$$
\lim_{x \to \infty} \frac{1}{f(x)}\int_{x}^{\infty} (s-x)F(s,u(s))\, ds = 0,
$$
on account of the finiteness of all the integrals involved.
This shows that $u(x) \sim f(x)$ as $x \to \infty$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th2}]
 Note that $X$ is a closed subset of the Banach space $BC(\mathbb{R}^+)$.
For $u \in X$ we define a map $T$ by setting
$$
Tu(x) = f(x) - \int_{x}^{\infty}(t-x)F(t, u(t))\, dt, \quad x \geq 0.
$$
Then for $u \in X$ it is clear that $Tu \in C(\mathbb{R}^+)$ and
since $F(t,u(t)) \geq 0$ for such $u$ and all $t \geq 0$, we have
$|Tu(x)| \leq \|f\|_{\infty} + b$, for every $x \geq 0$, where we have
used the fact the integral in question is a non-increasing function of $x$
for all $x \geq 0$. Hence $T$ is a self-map on the ball $X$. Finally,
 an argument similar to the corresponding one in Theorem~\ref{th}
gives that $T$ is a contraction on $X$ provided there holds \eqref{tkt}.
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th4}]
 The necessity is simple. If $y$ is such a solution then set $u=v=y$ throughout.
 For the sufficiency we appeal, as usual, to a fixed point theorem.
Consider the space $BC(I)$ of (uniformly) bounded continuous functions
on $I$ with the uniform norm. Since $u, v$ are uniformly bounded by
hypothesis, the subset $X$ defined by
$$
X=\{ y \in BC(I) : u(x) \leq y(x) \leq v(x), x \in I\}
$$
with the induced metric, is complete. Define a map $T$ on $X$ by the usual
\[
Ty(x) = f(x) - \int_{x}^{\infty}(t-x)F(t,y(t))\,dt, \quad x \in I
\]
for $y \in X$. Since $y \in X$, then $y \in L^{\infty}(I)$; it follows
from hypotheses (2) and (3) that the integral on the right is finite
for every $x \in I$ and this defines a continuous function that is
uniformly bounded on $I$. Thus, $Ty$ is continuous and uniformly
bounded on $I$, since $f$ is. Thus, $T$ is well-defined. On the other hand,
by hypothesis (2), $F(t,u(t)) \leq F(t,y(t)) \leq F(t,v(t))$ for
$t \in I$; it follows that, for $x \in I$,
\begin{align*}
Ty(x) &\leq f(x) - \int_{x}^{\infty}(t-x)F(t,u(t))\,dt \leq v(x)\\
& \geq  f(x) - \int_{x}^{\infty}(t-x)F(t,v(t))\,dt \geq u(x)
\end{align*}
 where we have used assumptions (b) and (c) in order to estimate the integrals.
 Thus $T$ is self-map on $X$. That $T$ is a contraction on $X$ follows
the usual route. Briefly, for $x \in I$, $y, z \in X$,
\begin{align*}
|Ty(x) - Tz(x)|
&\leq \int_{x}^{\infty}(t-x)|F(t,y(t))-F(t,z(t))|\, dt\\
&\leq  \int_{x}^{\infty}(t-x)k(t)|y(t)-z(t)|\, dt\\
& \leq  \left (\int_{x_0}^{\infty}tk(t)\,dt\right ) \; \|y-z\|_{\infty}
\end{align*}
and so,
$$
\|Ty - Tz\|_{\infty} \leq  \alpha \|y-z\|_{\infty},
$$
where $\alpha < 1$ is the integral in question (cf., assumption (5)).
\end{proof}


\begin{proof}[Proof of Theorem~\ref{cor01}]
 This is clear since we can integrate the differential equation \eqref{non0}
twice to obtain \eqref{ie1} and conversely, if we know that its solution
is $L^{\infty}$, we can differentiate \eqref{ie1} twice to recover
 \eqref{non0}. The result follows from an application of the theorem.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thx}]
 We integrate the inequalities twice over the half line to obtain both
\eqref{uf} and \eqref{vf}. An application of Theorem~\ref{th4} gives
that \eqref{ie1} has a solution $y(x) \sim f(x)$, as $x \to \infty$.
But the right side of \eqref{ie1} is twice differentiable, consequently
so is $y(x)$, that is \eqref{pert} is satisfied.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{atm}]
First, we show that solutions of \eqref{non2} exist on the half-line, $I$.
Introduce the usual energy functional $E(x)$ on solutions of \eqref{non2} by
\begin{equation}\label{ener}
E(x) = \frac{1}{2}{y'}^2 +  \int_{0}^{y}\eta G(x, \eta)\, d\eta
\equiv \frac{1}{2}{y'}^2 + \mathcal{I}(x,y),
\end{equation}
where $\mathcal{I}_{x}(x,y) \leq 0$ by hypothesis. A glance at \eqref{non2}
shows that
$$
E'  = y'g + \mathcal{I}_{x} \leq y'g \leq g \sqrt{2E}.
$$
So, $E'E^{-1/2} \leq \sqrt{2}g$ whenever $E>0$. It follows that if the
solution $y(x)$ exists for $x \in [a,b]$, $a \geq x_0$ and, at the same time,
$E(x) > 0$ for such $x$,  then
\begin{equation}
\label{ener2} \sqrt{E(b)} \leq \sqrt{E(a)}
+ \frac{\sqrt{2}}{2}\int_{a}^{b}g(t)\,dt.
\end{equation}
Of course, \eqref{ener2} is also true for any interval $[a,b]$ in which
the solution exists. For such an interval we have from \eqref{ener2}
\begin{equation}
\label{ener3} |y'(b)| \leq \sqrt{2E(a)} + \int_{a}^{b}g(t)\,dt,
\end{equation}
so, if the solution exists on an interval $[a,b)$ then it can be continued
to $x=b$ and thus to a right-neighborhood of $b$. Thus, we see that for
any $x \geq x_0$ a solution can be continued throughout $I$.

We now claim that for a given solution $y$ of \eqref{non2} there is an $X$
(depending on $y$) such that we cannot have for
\begin{equation}
\label{yayb}b > a \geq X, y(a)=y(b)=0, y(x) > 0, x \in (a,b).
\end{equation}
Note that by \eqref{tgt} and \eqref{ener3} we can suppose that $X$
is such that $X > 0$ and such that for some $K>0$ we have
\begin{equation}
\label{yKx} |y(x)| < Kx, \quad x \geq X.
\end{equation}
This already implies that all solutions are ``sublinear" or cannot
grow faster than a linear function. We fix this $K$ and consider
the differential equation $$z'' + G(x, Kx)z =0, \quad x \geq X.$$
Since this is a linear equation it is well known that (e.g.,
\cite{rb}),  assumption \eqref{tgKt} implies that this equation
has a solution $z(x) \to 1$ as $x \to \infty$. We choose $X_0> X$
so that $z(x) > 0$ for all $x \geq X_0$ and let $b>a \geq X_0$.
Now, writing $y = wz$ and making use of the equation for $z$, we
obtain the second order linear differential equation
\begin{equation}
\label{wz} w''z+2w'z' = g+wz\{G(x,Kx)-G(x,y)\}.
\end{equation}
However, \eqref{yayb}, \eqref{yKx} and $G$ non-decreasing in its
second variable,  would imply that the right of \eqref{wz} is positive
in $(a,b)$ so that $(w'z^2)' \geq 0$ on $(a,b)$. Indeed, \eqref{yayb}
would also force $w(a)=w(b)=0$ and $w(x)>0$ in $(a,b)$. On the other hand,
this leads to $w'(a) \geq 0$, $w'(b) \leq 0$. Since $w'z^2$ is non-decreasing,
this implies that $w'=0$, i.e., $w=0$ in $(a,b)$ resulting in a contradiction.
Thus, $y(x)$ is either ultimately positive or it is ultimately non-positive.


Now consider the case where $y(x)$ is ultimately positive. We may suppose
(see \eqref{yKx}) that
\begin{equation}
\label{0yKt}
0 < y(x) < Kx, \quad x \geq X.
\end{equation}
We modify the argument following \eqref{yKx} as follows: Consider
the differential equation $$z_1''+G(x,y(x))z_1 =0 , \quad x\geq
X.$$ As before, the integrability condition \eqref{tgKt} gives
that this will have a solution $z_1(x)$ with $z_1(x) \to 1$ as $x
\to \infty$. We can define $w_1$ as before by $y=w_1z_1$ and find,
as before, that $(w_1'{z_1}^2)' =gz_1$. But \eqref{tgt} along with
the fact that $w_1'{z_1}^2$ is non-decreasing implies that
${w_1}'$ tends to a non-negative finite limit at infinity. The
possibility that ${w_1}'(\infty)>K$ is excluded on account of
\eqref{0yKt}. Hence $ 0 \leq {w_1}'(\infty) \leq K$. If $A\equiv
{w_1}'(\infty)>0$  then necessarily $y(x) \sim Ax$ as $x \to
\infty$.

The other possibility is that \eqref{0yKt} holds but that
${w_1}'(\infty)=0$. In this case, the differential equation for $w_1$ yields
$$
{w_1}'(x) = - \{z_1(x)\}^{-2}\,\int_{x}^{\infty}g(t)z_1(t)\,dt,
$$
and since $z_1(x) \to 1$ as $x \to \infty$ we see that
\begin{equation}
\label{w1}
{w_1}'(x) \sim - \int_{x}^{\infty}g(t)\,dt
\end{equation}
as $x \to \infty$. Note that if \eqref{tgt} were false (for $i=1$)
it would follow that (since $g(x) \geq 0$), $w_1(x) \to -\infty$
as $x \to \infty$ and this contradicts the positivity of $y(x)$
for all large $x$. Thus, \eqref{tgt} is actually a necessary
condition. On the other hand, the hypothesis \eqref{tgt} implies
that $B\equiv w_1(\infty)$ is finite and necessarily non-negative,
because of the positivity of $y$, i.e., $y(x) \sim B$ as $x \to
\infty$, where $B \geq 0$.   If $y(x)$ is ultimately
non-positive, we can take it that $y(x) \leq 0$  for $x \geq X$,
and that $y(x) < 0$ on some unbounded subset of $x \geq X$. Since
$yG(x,y) \leq 0$ for $y \leq 0$, we get from \eqref{non2} that
$y'' \geq 0$. Applying Lemma 0 in \cite{fva} with $z(x)\equiv
-y(x)$ we see that $y'(x) \leq 0$ for $x \geq X$. This, in
conjunction with the fact that $y(x) < 0$ on some unbounded subset
implies that $y(x) < 0$ for all sufficiently large $x$. So, $ y(x)
< 0$, $y'(x) \leq 0$, $y'' \geq 0$ for all large $x$ which leads
to a counterpart of the positive solutions result.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{atm2}]
 As before we write $F(x,y)=yG(x,y)$ and we proceed as in the proof
 of Theorem~\ref{atm} up to \eqref{ener3}. The same argument therein
gives that solutions all exist on some half-axis. Indeed, since
$f'(\infty) = \infty$, by assumption, we can use \eqref{ener3}
 to derive that $|y'(x)| \leq (1+\varepsilon)f'(x)$ for all sufficiently
large $x$. In addition, another integration gives us a similar bound for
$y$ in the form $|y(x)| \leq (1+\varepsilon)f(x)$ for all sufficiently
large $x$, say $x \geq X_0$. Next, an integration of \eqref{non2} over
$[X_0, x)$ and use of \eqref{tgt2} and \eqref{fr} shows that
$$
y'(x) \sim \int_{x_0}^{x}g(t)\,dt, \quad x\to \infty.
$$
Finally, one last integration of the preceding equation gives the
desired  asymptotic estimate.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th5}]
Consider the solution $y$ whose initial conditions are $y(a)=a$,
$y'(a)=M$  with right-derivatives, where $a$ is to be chosen
later. Since $y'$ is right-continuous on $I$ there is a $b > a$
such that
\begin{equation} \label{m1}
\frac{M}{2} < |y'(x)| < 2M, \quad{x \in [a,b)}
\end{equation}
Since $y$ is absolutely continuous on $[a,b)$ it follows that
\begin{equation*}
|y(x)| \leq |y(a)| + |\int_{a}^{x}y'(t)\, dt | \leq a+ M(x-a)
\end{equation*}
that is,
\begin{equation} \label{m2}
|y(x)| < Mx, \quad{x \in [a,b)}
\end{equation}
since $M > 1$. Since $y$ is continuous, \eqref{m2} also holds at
$x=b$.  It follows from \eqref{nlvs1}, \eqref{nlvs2} that
\begin{equation} \label{m3}
y'(b) = y'(a) - \int_{a}^{b}F(t,y(t))\,d\sigma(t)
\end{equation}
i.e.,
\begin{equation} \label{m4}
| y'(b) - M|\leq  \int_{a}^{b}F(t,y(t))\,|d\sigma(t)|.
\end{equation}
On the other hand, $F$ is nondecreasing in its second variable by hypothesis,
so \eqref{m2} and assumption (3) together yield
\begin{equation} \label{m5}
| y'(b) - M|\leq  \int_{a}^{\infty}F(t,Mt)\,|d\sigma(t)|.
\end{equation}
Now, by assumption (3) again we can choose (and fix) $a$ so large that
$$
\int_{a}^{\infty}F(t,Mt)\,|d\sigma(t)| < M/4.
$$
Then \eqref{m1} holds for all $b > a$ and, as a result, \eqref{m2}
holds for all $x > a$. From this we see that for given $\varepsilon > 0$,
we can choose $X$ so large that for any $c> b > X$ we have
$$
\int_{b}^{c}F(t,Mt)\,|d\sigma(t)| < \varepsilon,
$$
and a double application of \eqref{m3} and the usual estimates, shows that
$$
|y'(c)-y'(b)| <\int_{b}^{c}F(t,Mt)\,|d\sigma(t)| < \varepsilon,
$$
for $c > b > X$. Hence $y'(x)$ tends to a limit $L$, say, as
$x \to \infty$, and $L \neq 0$ on account of \eqref{m1}.
From this it follows that $y(x)/x = L + o(1)$, i.e., $y$ is
asymptotically linear as $x \to \infty$.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{th6}]
 It suffices to show that the
 integral equation
\begin{equation}\label{eq1}
y(x)=M-\int_{x}^{\infty}(t-x)\,F(t,y(t))\,
d\sigma(t),
\end{equation}
has a fixed point in $X$ (since the
resulting solution will be absolutely continuous, with a
derivative that is locally of bounded variation and satisfying
\eqref{nlvs2}).  So, we define the operator $A$ on $X$ by
$$
(Ay)(x)=M-\int_{x}^{\infty}(t-x)F(t,y(t))\,d\sigma(t).
$$
Since $\sigma$ is non-decreasing and $F \geq 0$, for $y\in X$,
the function defined by
$$
\int_{x}^{\infty}(t-x)\,F(t,y(t))\,d\sigma(t)
$$
is nonincreasing, hence
$$
0\leq\int_{x}^{\infty}(t-x)\,F(t,y(t))\,d\sigma(t)\leq
\int_{0}^{\infty}t\,F(t,y(t))\,d\sigma(t)\leq M,
$$
by hypothesis (a). Consequently, for $y\in X$ we have y
\begin{equation}\label{0ay}
0\leq (Ay)(x)\leq M,\quad {for}x\in I.
\end{equation}
In order to show that for $y\in X$ then
$Ay\in\mathcal{C}(I)$, we note by Fubini's theorem that since
$$
\int_{x}^{\infty}(t-x)F(t,y(t))\, d\sigma(t)
= \int_{x}^{\infty}\int_{t}^{\infty}F(s,y(s))\, d\sigma(s)\, dt,
$$
for every $x \in I$ and the integral of a function that is locally
of bounded variation is locally absolutely continuous, it is in
particular continuous and so, for $y\in X$, we have
$Ay\in\mathcal{C}(I)$. This, in combination with \eqref{0ay} shows
that $Ay \in X$. Hence, the operator $A$ applies $X$ into itself.

 Now, we prove that $A$ is continuous on $X$. Indeed,
\begin{align*}
|(Ay_{n})(x)-(Ay)(x)|
&=  \Big|\int_{x}^{\infty}(t-x)[F(t,y_{n}(t))-F(t,y(t))]\, d\sigma(t)\Big| \\
&\leq \int_{x}^{\infty}(t-x)|F(t,y_{n}(t))-F(t,y(t))|\, d\sigma(t) \\
&\leq \|y_{n}-y\|_{\infty}\int_{x}^{\infty}(t-x)k(t)\, d\sigma(t) \\
&\leq \|y_{n}-y\|_{\infty}\int_{0}^{\infty}t\,k(t)\,d\sigma(t).
\end{align*}
It follows that $A$ is continuous on $X$ on account of assumption
$(c)$.

The proof that $A$ is compact uses ideas from the theory of
measures on non-compactness. First, we introduce some terminology.
Let us fix a nonempty bounded subset $X$ of $C[0,a]$. For
$\varepsilon >0$ and $y \in X$ denote by $w(y,\varepsilon)$ the
 \emph{modulus of continuity} of $y$ defined by
\[
w(y,\varepsilon)= \sup  \{| y(t)-y(s)|  : t,s \in [0,a],\, | t-s|
\leq \varepsilon\}
\]
Further, let us put
\begin{gather*}
w(X,\varepsilon)=\sup \{w(y,\varepsilon) : y\in X\}\\
w_0 (X)= \lim_{\varepsilon \rightarrow 0} w(X,\varepsilon),
\end{gather*}
It can be shown (see \cite{11}) that the function $\mu(X)=w_0 (X)$
is a regular measure of noncompactness in the space $C[0,a]$.
 Now, let $x_{1},x_{2}\in [0,\infty)$ be such that
$x_{2}-x_{1}\leq \varepsilon$ and without loss of generality,
$x_{1}<x_{2}$. Then
\begin{equation}
\begin{aligned}
|Ay(x_{2})-Ay(x_{1})|
&=\Big|\int_{x_{2}}^{\infty}(t-x_{2})F(t,y(t))d\sigma(t)-
\int_{x_{2}}^{\infty}(t-x_{1})F(t,y(t))d\sigma(t) \\
&\quad +  \int_{x_{2}}^{\infty}(t-x_{1})F(t,y(t))d\sigma(t)-
\int_{x_{1}}^{\infty}(t-x_{1})F(t,y(t))d\sigma(t)\Big| \\
&\leq \Big|\int_{x_{2}}^{\infty}(x_{1}-x_{2})F(t,y(t))d\sigma(t)
 +\int_{x_{1}}^{x_{2}}(t-x_{1})F(t,y(t))d\sigma(t)\Big|\\
&\leq \int_{x_{1}}^{x_{2}}(t-x_{1})F(t,y(t))d\sigma(t)+
\int_{x_{2}}^{\infty}(x_{2}-x_{1})F(t,y(t))d\sigma(t)\\
&\leq  (x_{2}-x_{1})\int_{x_{1}}^{\infty}F(t,y(t))d\sigma(t).
\end{aligned}\label{mnc0}
\end{equation}
Next, we note that for any $x_0 \geq 0$, $y \in X$,
\begin{equation} \label{mnc}
\begin{aligned}
\int_{x_{0}}^{\infty}F(t,y(t))\,d\sigma(t)
&\leq \int_{0}^{\infty}F(t,y(t))\, d\sigma(t) \\
&\leq \int_{0}^{1}F(t,y(t))\,d\sigma(t)+\int_{1}^{\infty}t\,F(t,y(t))\,d\sigma(t)\\
&\leq \|F\|_{[0,1]\times [0,M]}(\sigma(1)-\sigma(0))+M,
\end{aligned}
\end{equation}
by hypothesis (a) and since $\sigma$ is nondecreasing.
Thus, use of \eqref{mnc} and \eqref{mnc0} give us that
$$
w(Ay,\varepsilon)\leq\varepsilon[\|F\|_{[0,1]\times
[0,M]}(\sigma(1)-\sigma(0))+M];
$$
 consequently,
$$
w(AX,\varepsilon)\leq \varepsilon[\|F\|_{[0,1]\times
[0,M]}(\sigma(1)-\sigma(0))+M],
$$
 so that
\begin{equation} \label{w0a}
w_{0}(AX)=0.
\end{equation}
 Finally, let $y,z\in X$, $x\geq 0$. Then the previous continuity
argument also yields the estimate
\begin{equation} \label{ayxaz}
|(Ay)(x)-(Az)(x)|\leq \|y-z\|_{\infty}\int_{x}^{\infty}(t-x)\,k(t)\,dt
\end{equation}
On the other hand, for $y, z\in X$,  $\|y-z\|_{\infty}\leq 2M$ and so
\[
|(Ay)(x)-(Az)(x)|\leq  2M\int_{x}^{\infty}(t-x)k(t)\,d\sigma(t)
\leq  2M\int_{x}^{\infty}t\,k(t)\, d\sigma(t),
\]
since $x \in I$. It follows that the diameter of the set $AX$ can be
estimated by
$$
\mathop{\rm diam} AX(x)\leq 2M\int_{x}^{\infty}sk(s)d\sigma(s),
$$
and taking the limit as $x\to \infty$, we get
\begin{equation}\label{axx}
\lim_{x\to\infty}\mathop{\rm diam} (AX)(x)=0.
\end{equation}
Therefore, \eqref{w0a} and \eqref{axx} give us that $AX$ is compact.
 An application of Schauder's fixed point theorem now gives the
desired conclusion.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{coro}]
 Observe that for $y\in X$, conditions \eqref{effo} and \eqref{atko}
together imply condition (a) of  Theorem~\ref{th6}.
\end{proof}


\begin{proof}[Proof of Lemma~\ref{lem1}]
 Since $F \geq 0$ and $\sigma$ is nondecreasing we see that
$y'(x) \leq y'(0)$ for every $x \geq 0$ (by \eqref{nlvs2}).
In addition, for $x_2> x_1 >0$,
$$
y'(x_2)-y'(x_1) = - \int_{x_1}^{x_2}F(t,y(t))\,d\sigma(t) \leq 0,
$$
and so $y'(x)$ is nonincreasing. It follows that $y'(x)\to L$ where
the limit $L \leq y'(0)$. The possibility that $-\infty \leq L < 0$
is excluded by the assumption that $y(x) > 0$ for all large $x$.
Hence $L$ is finite and non-negative. Suppose that $L \neq 0$.
Then, for $\varepsilon > 0$ we can choose $X$ so large that
$L-\varepsilon < y'(x) < L + \varepsilon$, for every $x \geq X$.
Integrating this last expression over $[X, x)$ we get the
inequality $y(X) + (x-X)(L-\varepsilon) < y(x) < y(X) + (x-X)(L+\varepsilon)$.
Dividing the latter by $x$ and letting $x \to \infty$ we get $y(x) \sim Lx$
as $x \to \infty$. On the other hand, if $L=0$, then the same argument
gives us $y(x)/x \to 0$ as $x \to \infty$.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{th7}]
 Assume that \eqref{nlvs2} has a bounded eventually positive solution $y(x)$,
with $y(x) > 0$, for all $x \geq x_0$. An application of Lemma~\ref{lem1}
gives that $y'(x) \to L$ where $L$ is finite (otherwise $y(x)$ cannot
remain bounded at infinity). In addition, passing to the limit as
$x \to \infty$ in \eqref{nlvs2}, and rearranging terms, we obtain
\begin{equation} \label{ypx}
y'(x) = L + \int_{x}^{\infty}F(t,y(t))\,d\sigma(t)
\end{equation}
If $L \neq 0$ then Lemma~\ref{lem1} implies that $y(x)\sim Lx$ as
$x \to \infty$ which contradicts the boundedness of $y(x)$. Hence $L=0$.
For $x_2>x_1>0$, we integrate \eqref{ypx} over $[x_1,x_2)$ to find that
$y(x_2)-y(x_1)>0$ (by our assumptions on $F$ and $\sigma$), that is,
$y(x)$ is nondecreasing. Since $y(x)$ is bounded by assumption,
we get that $y(x) \to c$ for some finite $c > 0$. Integrating \eqref{ypx}
over $[x, \infty)$ and rearranging terms we obtain the existence
and finiteness of all integrals involved and, in fact, for all
$x\geq x_0$ there holds,
\begin{equation} \label{fub}
y(x) = c - \int_{x}^{\infty}(t-x)\,F(t,y(t))\,d\sigma(t),
\end{equation}
after an application of Fubini's Theorem. Consolidating our results we
have that $0 < y(x_0) < y(x) \leq c$, for all $x \geq x_0$. Observe that
the integral \eqref{fub} is finite for $x=x_0$. This, along with the
hypothesis that $F(x,\cdot)$ is nondecreasing gives us
\begin{equation} \label{fub1}
\int_{x_0}^{\infty}(t-x_0)\,F(t,y(x_0))\,d\sigma(t) < \infty,
\end{equation}
and this equivalent to the convergence of \eqref{zn1} with $M=y(x_0)$.
Note that we can also replace $M$ by any number smaller than $c$.

For the sufficiency we assume that \eqref{zn1} holds for some $M>0$.
Fix $A > 0$, $A < M$ and choose $x=a$ so large that
$$
\int_{a}^{\infty}(t-a)\,F(t,M)\,dt \leq A/2.
$$
We set up the iterative scheme
\begin{equation} \label{iter}
y_{n+1}(x) = A - \int_{x}^{\infty}(t-x)\,F(t, y_{n}(t))\, d\sigma(t),
\quad x\geq a,
\end{equation}
with $y_0(x)=A$, for each $x\geq a$. Since $F(t,y_0(t))=F(t,A)\leq F(t,M)$
we obtain
\begin{align*}
y_{1}(x) &\geq  A - \int_{x}^{\infty}(t-x)\,F(t, M)\, d\sigma(t)\\
& \geq  A - \int_{a}^{\infty}(t-a)\,F(t, M)\, d\sigma(t)\\
&\geq  A - A/2 = A/2.
\end{align*}
Thus, $A/2 \leq y_1(x) \leq A$ for every $x \geq a$. A similar argument
shows that if $A/2 \leq y_n(x) \leq A$ for every $x \geq a$, then the
same is true of $y_{n+1}(x)$. An induction argument gives us that
\begin{equation} \label{a2yn}
A/2 \leq y_n(x) \leq A, \quad x\geq a, n\geq 1.
\end{equation}
Next, we show that each $y_n(x)$ is nondecreasing and the family
$\{y_n(x)\}_{n=1}^{\infty}$ is equicontinuous on every interval  $[a,b]$.
Let $x_2 > x_1 > a$. Since
\begin{equation} \label{yn1x}
y_{n+1}(x_2)-y_{n+1}(x_1) = \int_{x_1}^{x_2}(t-x_1)\,F(t,y_n(t))\,d\sigma(t)
+ \int_{x_2}^{\infty}(x_2-x_1)\,F(t,y_n(t))\,d\sigma(t) ,
\end{equation}
and $F\geq 0$, $\sigma$ is nondecreasing, it follows that the right side
of \eqref{yn1x} is non-negative; thus for each $n$, the $y_{n}(x)$ are
increasing over $[a, \infty)$. Next, estimating the integrals in
\eqref{yn1x} using \eqref{a2yn} and the basic estimates on $A$;
for $[x_1, x_2] \in [a,b]$, we have
\begin{align*}
|y_{n+1}(x_2)-y_{n+1}(x_1) |
&\leq |x_2-x_1|\Big\{\int_{x_1}^{x_2}F(t,M)\,d\sigma(t)
+ \int_{x_2}^{\infty}F(t,M)\,d\sigma(t)\Big\}\\
&\leq |x_2-x_1|\int_{a}^{\infty}F(t,M)\, d\sigma(t).
\end{align*}
This last integral, being finite on account of \eqref{zn1}, shows that
the family is equicontinuous on $[a,b]$ for every $b > a$.
Thus, passing to a subsequence if necessary, we can say that the
limit $y(x) = \lim_{n\to \infty}y_n(x)$ exists and is a continuous
function on every interval $[a,b]$.

Finally, we show that the limit $y(x)$ is a solution of \eqref{nlvs2} on
$[a,\infty)$. Let $\varepsilon > 0$. Rearranging terms in \eqref{iter}
 we can write, for $x \in [a,b]$,
\begin{eqnarray} \label{ii1}
| y_{n+1}(x) - A + \int_{x}^{b}(t-x)\,F(t, y_{n}(t))\, d\sigma(t)|
&\leq &  \int_{b}^{\infty}(t-x)\,F(t, y_{n}(t))\, d\sigma(t) \nonumber\\
&\leq & \int_{b}^{\infty}(t-a)\,F(t, y_{n}(t))\, d\sigma(t) \nonumber\\
&\leq & \int_{b}^{\infty}(t-a)\,F(t, M)\, d\sigma(t) \nonumber \\
& < & \varepsilon,
\end{eqnarray}
provided $b$ is sufficiently large (this is possible on account of
\eqref{zn1}). We can now pass to the limit as $n \to \infty$ in
\eqref{ii1} to find
\begin{equation*}
| y(x) - A + \int_{x}^{b}(t-x)\,F(t, y(t))\, d\sigma(t)| \leq \varepsilon,
\end{equation*}
holds for every sufficiently large $b$, which is equivalent to saying that
\begin{equation} \label{ii0}
y(x) = A - \int_{x}^{\infty}(t-x)\,F(t, y(t))\, d\sigma(t),
\end{equation}
Finally, an application of Fubini's theorem shows that the latter
is equivalent to
\begin{equation} \label{ii2}
y(x) = A - \int_{x}^{\infty}\int_{t}^{\infty}F(s, y(s))\, d\sigma(s)\,dt.
\end{equation}
Hence, $y$ is locally absolutely continuous and its (right-) derivative
is given at every point $x \geq a$ by differentiating \eqref{ii2}, that is,
\begin{equation} \label{ii3}
y'(x) =  \int_{x}^{\infty}F(t, y(t))\, d\sigma(t).
\end{equation}
Writing $y'(0)=\int_{0}^{\infty}F(t, y(t))\, d\sigma(t)$,
(which necessarily exists and is finite because of \eqref{zn1}) we can
 rewrite \eqref{ii3} in the form
\begin{align*}
y'(x) &= y'(0) + \int_{x}^{\infty}F(t, y(t))\, d\sigma(t) - \int_{0}^{\infty}F(t, y(t))\, d\sigma(t)\\
&= y'(0) - \int_{0}^{x}F(t, y(t))\, d\sigma(t),
\end{align*}
as desired (see \eqref{nlvs2}).
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor001}]
 First, we note that the function $F(x,y) := yG(x,y^2)$ satisfies all
the conditions of the theorem. In addition, $y$ is a solution
of \eqref{nlvs2p} if and only if $-y$ is. Furthermore, \eqref{zn18}
is equivalent to \eqref{zn1} for appropriate choices of $c,M$
(indeed, \eqref{zn1} implies \eqref{zn18} with $c=M^2$, and \eqref{zn18}
implies \eqref{zn1} with $M=\sqrt{c}$).
Since, for a given solution $y$ of \eqref{nlvs2p} its counterpart $-y$
is also a solution, we can assume without loss of generality that this
bounded nonoscillatory solution is eventually positive and so proceed,
with no other important changes, as in the proof of the necessity in
the theorem to arrive at \eqref{zn18}.
The sufficiency proceeds along similar lines.
 \end{proof}

\begin{proof}[Proof of Theorem~\ref{th8}]
The sufficiency follows the proof of the sufficiency of Theorem~\ref{th7},
which is applicable since \eqref{sts} holds for $\varepsilon =0$,
as required by the theorem. Since the solution in Theorem~\ref{th7}
is asymptotically a positive constant, it is eventually positive.

For the necessity we apply the proof of Lemma~\ref{lem1} to find that
if $y(x)$ is eventually positive, say for $x \geq a$, then
$c:=\lim_{x\to \infty}y'(x) \geq 0$. Thus, with right-derivatives,
$$
y'(x) = c + \int_{x}^{\infty}F(t,y(t))\,d\sigma(t) \geq \int_{x}^{\infty}
F(t,y(t))\,d\sigma(t),
$$
and since $y$ is non-decreasing for $x\geq a$ (see the proof of
Theorem~\ref{th7}), this gives
\[
{y(x)}^{-\varepsilon}\,y'(x)
\geq \int_{x}^{\infty}{y(t)}^{-\varepsilon}{F(t,y(t))}\,d\sigma(t).
\]
Now since $y$ is positive and locally absolutely continuous for
$x\geq a$ so is the function $y(x)^{1-\varepsilon}$, since $y(x)$
is bounded away from zero on finite intervals. Hence, for $b > a$,
writing $M:=y(a)$,
\begin{align*}
\int_{a}^{b}{y(x)}^{-\varepsilon}\,y'(x)\,dx
&\geq \int_{a}^{b}\int_{x}^{\infty}{y(t)}^{-\varepsilon}\,{F(t,y(t))}
 \,d\sigma(t)\,dx\\
&\geq \int_{a}^{b}\int_{x}^{\infty}{M}^{-\varepsilon}\,F(t, M)\,d\sigma(t)\,dx.
\end{align*}
Since this is valid for any $b > a$ we can let $b\to \infty$ and simplify
the right to get
\[
\int_{a}^{\infty}{y(x)}^{-\varepsilon}y'(x)\,dx
\geq {M}^{-\varepsilon}\int_{a}^{\infty}(t-a)\,F(t,M)\,d\sigma(t).
\]
The left side is finite since $\varepsilon > 1$ and so the right must
be finite, that is, so is \eqref{zn1} for this choice of $M$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem2}]
 By assumption, there exists a number $c>0$ such that for every $x\geq a$,
say, we have $y(x)\geq cx$ and $y'(x) > 0$ (recall that $y'(x)$ tends
to a limit as $x \to \infty$, cf., Lemma~\ref{lem1}).
Applying \eqref{nlvs2} over $[a,x]$ and rearranging terms we get
\[
y'(a) = y'(x) + \int_{a}^{x}F(t, y(t))\,d\sigma(t), \geq
\int_{a}^{x}F(t, ct)\,d\sigma(t),
\]
for every $x \geq a$. Since $x$ is arbitrary, we can pass to the limit
as $x \to \infty$ and thus obtain \eqref{tmt} with $M=c>0$.
\end{proof}

\begin{proof}[Proof Theorem~\ref{th9}]
 This follows directly from an application of both Lemma~\ref{lem2}
 and an application of Theorem~\ref{th5} in the special case where
$\sigma$ is nondecreasing.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th15}]
We need only sketch the details as they are similar to those included
above in the proof of Theorem~\ref{th1}. For $y\in X$ where
$X=\{ y\in Y: \|y(x)/f(x)\| \leq 2\}$ consider the map on $X$ defined
by \eqref{fnon}. Minor changes in the proof of said Theorem show that,
indeed, $T$ is a self-map on $X$ (since $\sigma$ is non-decreasing).
In addition, $T$ is a contraction on account of \eqref{t000}.
Hence the theorem follows.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th11}]
Define a right-continuous step-function by defining its jumps to be at
the integers $n$, of magnitude $\sigma(n)-\sigma(n-0) = - b_n$,
for $n\geq 0$  and so $\sigma(t) = {\rm constant}$ in the interval
$(n, n+1)$, for $n\in \mathbb{N}$. The integral condition (3) in
Theorem~\ref{th5} is equivalent to the above condition on the sum
above and the solution $y(x)$ of the Volterra-Stieltjes integro-differential
equation \eqref{nlvs2} is such that $y(n):=y_n$ satisfies \eqref{2bdef}
for each $n$. The conclusion is a consequence of said theorem.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{th12}]
We omit the proof as it is similar to that introduced in
Theorem~\ref{th11}, with the necessary modifications.
\end{proof}


\subsection*{Acknowledgment} The first author wants to
express his gratitude to the Department of Mathematics of the
University of Las Palmas in Gran Canaria for its hospitality
during a visit there in December 2004, and in 2006.



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