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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 22, pp. 1--6.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/22\hfil Note on a non-oscillation theorem]
{Note on a non-oscillation theorem of Atkinson}
\author[S. Dub\'e \& A. B. Mingarelli\hfil EJDE-2004/22\hfilneg]
{Samuel G. Dub\'e \& Angelo B. Mingarelli} % in alphabetical order
\address{School of Mathematics and Statistics\\
Carleton University, Ottawa, Ontario, Canada, K1S\, 5B6}
\email[S. G. Dub\'e]{sdube@math.carleton.ca}
\email[A. B. Mingarelli]{amingare@math.carleton.ca}
\date{}
\thanks{Submitted January 7, 2004. Published February 16, 2004.}
\thanks{The second author was partially supported by NSERC Canada}
\subjclass[2000]{34A30, 34C10}
\keywords{Second order differential equations, nonlinear, non-oscillation}
\begin{abstract}
We present a general non-oscillation criterion for a second order
two-term scalar nonlinear differential equation in the spirit of a
classic result by Atkinson \cite{At}. The presentation is simpler
than most and can serve to unify many such criteria under one
common theme that eliminates the need for specfic techniques in
each of the classical cases (sublinear, linear, and superlinear).
As is to be expected in a result of this kind, the applications
are widespread and include, but are not limited to, linear,
sublinear, superlinear differential equations as well as some
transcendental cases and some possibly {\it mixed} cases.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
The study of oscillation and non-oscillation theory of second order
ordinary differential equations has a very long history.
The first theoretical results in this area were actually obtained by
Sturm himself in his abstract \cite{St} of his now classic 1836 memoir as
a consequence of his memorable comparison theorems. Thousands of papers
have been written since on all aspects of this theory ranging from
difference equations to partial differential equations and even integral
equations (cf., e.g., \cite{ABM} for further details). In this note we
revisit a classic theorem of F. V. Atkinson \cite{At}, probably the first
such theorem that exhibits a necessary and sufficient condition for the
existence of oscillatory solutions to (nonlinear) Emden-Fowler type equations.
We recall that a solution of a real second order differential equation
is said to be {\it oscillatory} on a half-axis provided it has an infinite
number of zeros on that semi-axis. By the standard existence and uniqueness
theorem we see that there must be a sign change at a zero and zeros cannot
accumulate on any finite interval. If the equation has at least one non-trivial
solution with a finite number of zeros it is termed non-oscillatory.
The question of interest here involves the determination of a general
criterion that will ensure the non-oscillation of a two-(or more)-term
nonlinear ordinary differential equation of the form (\ref{non}) where the
nonlinearity is of a general type. For the most part, work in this field
has centered mostly on establishing necessary and sufficient conditions
for the oscillation of all solutions of equations of the form
$y''(x) + f(x)g(y) = 0, x \in [0, \infty)$ in the superlinear
case after Atkinson's paper \cite{At}. This was followed by an important though
little referenced paper by G. Butler \cite{Bu} who dropped the non-negativity
assumption on $f(x)$ in the superlinear case. His necessary and sufficient
condition was extended to a framework that includes both differential and
difference equations in \cite{ABM}. Recent work in this general area has
centered around an extension of Atkinson's theorem \cite{At} to delay equations
and three-term equations with damping
(cf., \cite{Nasr}, \cite{Wong}, and \cite{Wong1}).
We present here a result on the existence of a positive solution of a
nonlinear two-term scalar differential equation of the form
\begin{equation}
\label{non}
y''(x) + F(x, y(x)) = 0, \quad x \in [0, \infty)
\end{equation}
where $F: \mathbb{R}^+ \times \mathbb{R} \longrightarrow \mathbb{R}$ is
always assumed
to be continuous on its domain along with some basic conditions that ensure
the existence and uniqueness of solutions to associated initial value problems.
There is no loss of generality in assuming that (\ref{non}) is defined on
$\mathbb{R}^+$ rather than on a half-axis such as $[a, \infty)$ where
$a >- \infty$, since a simple change of independent variable will transform
the equation on $[a, \infty)$ back into (\ref{non}).
Instead of specializing to various types of nonlinearities as is usually
done we will proceed directly to the fully nonlinear case, without the
additional standard asumption that the nonlinearity $F(x,y)$ in (\ref{non})
is variables separable. Using a fixed point theorem we find, as a special case,
a non-oscillation criterion that covers many of the different equations types
observed in the literature (linear, sublinear, superlinear) as well as some
rare transcendental cases and even mixed linear or semilinear cases.
The advantage lies in a unified framework for different settings, one that
provides a condition for the existence of a positive
(and so necessarily non-oscillatory) asymptotically constant solution to the
differential equation (\ref{non}). Due to its generality, it is to be noted
that when our results are specified to actual cases
(such as a supelinear equation) our results are generally stronger than
existing ones. But then we also require more than non-oscillation as a goal.
For related results dealing with fully nonlinear equations with damping
(e.g., $F(x, y, y')$ in (\ref{non}) see \cite{Yin}). In \cite{Zhao} the author
treats (\ref{non}) under an assumption of convexity in the second variable on
a majorant of $F$ along with some additional conditions
(e.g., $F(x,0)=0$, etc). As a result, Zhao concludes the existence of at least
one positive solution on $(0, \infty)$ that is asymptotically linear as
$x \to \infty$. For an equation of type (\ref{non}) for the Laplacian in
$\mathbb{R}^n$, see \cite{Con}, where Atkinson's condition (\ref{fva}), below,
is extended to this higher dimensional setting. Our conditions
(\ref{atk}), (\ref{teek}), (\ref{lip}) in the sequel appear to be weaker than
those presented in the literature and so the results may be of interest in
the study of the existence of positive solutions to semilinear elliptic
problems (cf., \cite{Con}, \cite{Yin}, \cite{Zhao}). These papers make use of
the Schauder-Tikhonov fixed point theorem and so generally there is no
uniqueness of the fixed point, in contrast with our technique which does
guarantee its uniqueness by virtue of the use of the contraction mapping
principle.
\section{The existence of a positive monotone solution}
Our techniques invoke the fixed-point theorem of Banach-Cacciopoli
(see \cite{Hale}) and are based on the simple premise that, basically,
in the variables separable case, the nonlinearity in the dependent variable
$y$ in (\ref{non}) maps a given compact interval back into
(and not necessarily onto) itself. This, along with the basic existence
and uniqueness theorems appealed to above, will ensure that a non-oscillatory
solution exists which, in fact, must be positive on the semi-axis
$(0, \infty)$. In the linear case, that is where $F(x,y) = f(x)y$,
this property results in the definition of a {\it disconjugate} equation by
Sturm theory (that is, an equation in which every non-trivial solution has at
most one zero).
\begin{theorem} \label{th}
Let $X = \{ u \in C[0, \infty) |: 0 \leq u(t) \leq M,\text{ for all }
t \geq 0\}$, where $M > 0$ is given but fixed. Assume that
$F: \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+$ and that for any
$u \in X$,
\begin{equation} \label{atk}
\int_{0}^{\infty} t\, F(t, u(t))\, dt \leq M
\end{equation}
and 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)\, dt < 1,
\end{equation}
such that for any $u, v \in \mathbb{R}^+$, we also have
\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) monotone solution
on $(0, \infty)$ such that $y(x) \to M$ as $x \to \infty$.
\end{theorem}
\begin{proof} %[Proof of Theorem~\ref{th}]
It is easy to see that the space $X$ defined in the statement of the theorem
is a closed subset of $C[0, \infty)$. It follows that $(X, \| \cdot \|_{X}) $
where $\| \cdot\|_{X}$ is defined as usual by the uniform norm,
$\|u\|_{X} = \sup_{t \in [0, \infty)} u(t)$ is a Banach space.
%\footnote{Corrigendum: Replace ``Banach space " by ``complete metric sapce''.}
Following Atkinson (cf., \cite{At}) we look for a uniformly bounded continuous
solution of the nonlinear integral equation
\begin{equation}\label{int}
y(x) = M - \int_{x}^{\infty} (t-x) F(t, y(t))\, dt
\end{equation}
for $x \in [0, \infty)$. Clearly, the existence of such a solution implies
that $y(x) \to M$ and $y'(x) \to 0$ as $x \to \infty$. Indeed, once such a
solution is found in $X$ we conclude that $y(x) \geq 0$ implies $y(x) > 0$
for all $x \in (0, \infty)$, on account of the tacit assumption of uniqueness
of solutions of initial value problems associated with (\ref{non}) and so
(\ref{int}).
We define a map on $X$ as usual by
\begin{equation} \label{map}
(Tu)(x) = M - \int_{x}^{\infty} (t-x) \, F(t, u(t))\, dt
\end{equation}
where $u \in X$. Note that the right-side of (\ref{map}) clearly converges
for each $x \geq 0$, on account of (\ref{atk}). Indeed, for given $u \in X$ and
$x \geq 0$,
\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 M,
\end{equation}
since $F(t, u(t)) \geq 0$ for such $u$ (which implies that $(Tu)(x) \leq M$)
and the indefinite integral is a non-increasing function of $x$ on
$[0, \infty)$. Thus, $(Tu)(x) \geq 0$ for any $x \geq 0$. This shows that
$TX \subseteq X$. Finally, we show that $T$ is a contraction on $X$
(and so in particular, $T$ is continuous there). 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 \|_{X} \, \int_{0}^{\infty} t\, k(t)\, dt,
\end{align*}
where we have used (\ref{lip}) and the fact that
$\int_{x}^{\infty}(t-x)k(t)\, dt$ is a non-increasing function of $x$ for
$x \in [0, \infty)$, since $k(t) \geq 0$. From this we readily see that
$$
\| Tu - Tv \|_{X} \leq \alpha \| u - v \|_{X},
$$
where $\alpha < 1$ is given by the left-side of (\ref{teek}). Hence $T$ is a
contraction on $X$ and so $T$ has a fixed point $u=y$ in $X$ which must
satisfy (\ref{int}). The monotonicity is clear since all quantities in the
integral in (\ref{map}) are non-negative. This proves the theorem.
\end{proof}
For a pointwise criterion on $F(t, u)$ we can formulate the following
result.
\begin{corollary} \label{coro}
Let $M > 0$ be given. Assume that
$F: \mathbb{R}^+ \times {\bf R^+} \to {\bf R^+}$ and that there is a
continuous function $f :\mathbb{R}^+\to \mathbb{R}^+$ such that
\begin{equation} \label{effo}
\int_{0}^{\infty} t f(t)\, dt \leq 1,
\end{equation}
and
\begin{equation}\label{atko}
F(t, u) \leq f(t) g(u),\quad t \geq 0, u \in \mathbb{R}^+,
\end{equation}
for some function $g$ where $g : [0, M] \to [0, M] $ is continuous on
$[0,M]$. 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)\, dt < 1,
\end{equation*}
such that for any $u, v \in \mathbb{R}^+$, we also have
\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) monotone solution
on $(0, \infty)$ such that $y(x) \to M$ as $x \to \infty$.
\end{corollary}
\begin{proof} % Proof of Corollary~\ref{coro}:}
Proceed as in the theorem up to (\ref{bound}). At this point, the estimate
(\ref{atko}) is used in lieu of (\ref{atk}) to show that
(\ref{atk}) may be replaced by (\ref{effo}) the remaining argument being similar.
\end{proof}
\section{Applications}
Although conditions (\ref{atk}), (\ref{lip}) and (\ref{teek}) are stronger than
existing corresponding conditions for specific choices of the nonlinearity
$F(x,y)$, Theorem~\ref{th} basically eliminates the distinction between the
classical {\it sublinear} and {\it superlinear} cases determined by a growth
condition on $F(x, \cdot)$, (cf., \cite{Wong1} for detailed definitions).
We will show that Theorem~\ref{th} applies in various cases using the simple
conditions enunciated there, bearing in mind that these hypotheses can be
weakened considerably in specific cases.
\begin{example}[A sublinear case] \label{one} \rm
Let $F(x,y) = f(x)g(y)$ where $g(y) = {|y+1|^{\nu}}$, and $0 < \nu < 1$.
This case is characterized by the convergence of the reciprocal of $g(y)$
away from zero (cf., \cite{Wong}). Let $f \in C[0, \infty), f(x) \geq 0$ on
$[0, \infty)$. Then the function $k(t)$ in Theorem~\ref{th} may be chosen as
$k(t) = f(t)\nu$ since $g$ is Lipschitzian with Lipschitz constant at most
$\nu$ on $[0, M] \equiv [0,1]$. An appropriate choice of $f(x)$ leads to a
verification of each of (\ref{atk}) and (\ref{teek}) provided we choose $M=1$ and
\begin{equation} \label{subnu}
\int_{0}^{\infty} t f(t)\, dt \leq \frac{1}{2^\nu}.
\end{equation}
In this case,
$$
y'' +f(x) | y +1|^{\nu} =0, \quad x \in [0, \infty),
$$
has a unique positive solution $y(x) \to 1$ as $ x \to \infty$. Of course,
oscillations in the sublinear case have been completely characterized by
Belohorec in \cite{Be} and others in recent times (cf., \cite{Wong} for further
details). Note that, in this case, the (necessary and sufficient) Belohorec
condition
$$
\int_{0}^{\infty} t^{\nu} f(t)\, dt < \infty
$$
is automatically satisfied on account of (\ref{subnu}).
\end{example}
\begin{example}[A linear case] \label{two} \rm
Consider
$$
y'' + f(x)y =0, \quad x\in [0, \infty),
$$
where $f$ is chosen so that
\begin{equation}\label{tft}
\int_{0}^{\infty}t\, f(t)\, dt < 1.
\end{equation}
The choice $M=1$ and $k(t) = f(t)$ in Theorem~\ref{th} gives the basic result
that our linear equation is non-oscillatory (in fact, disconjugate)
since it has an asymptotically constant solution $y(x) \to 1$ as
$ x \to \infty$. In the linear case, this conclusion is classical.
For example, using \cite[p. 255, Exercise 3]{CL} we see that (\ref{tft})
implies that the linear equation is disconjugate. Furthermore,
\cite[p. 35, Exercise 6.3]{Hale} indicates that our equation must have
asymptotically constant solutions under the milder assumption that the
integral appearing in (\ref{tft}) is merely finite.
\end{example}
\begin{example}[A superlinear case] \label{three} \rm
Once again, we let $F(x,y) = f(x)g(y)$ where, say, $g(y) = y^{2n-1}$, and
$n > 1$ is an integer. This case, generally motivated by the convergence
of the reciprocal of the integral of $g(y)$ at infinity was inspired by
Atkinson's paper \cite{At}, thus leading to hundreds of papers in the subject.
In this case, note that $g$ is a {\it self-map} (i.e., $g([0,1]) = [0,1]$) and
$g$ is Lipschitzian with Lipschitz constant equal to $2n-1$.
It then follows from our theorem that if $f$ is chosen so that
$$
\int_{0}^{\infty}t\, f(t)\, dt < \frac{1}{2n-1},
$$
then $y''+f(x)y^{2n-1} = 0$ will have a positive solution on $(0, \infty)$
(and so is non-oscillatory there) since it will be asymptotically constant:
$y(x) \to 1$ as $ x \to \infty$. Of course, our condition is stronger than
Atkinson's original necessary and sufficient condition for non-oscillation,
namely
\begin{equation}\label{fva}
\int_{0}^{\infty}t f(t)\, dt < \infty,
\end{equation}
which implies the same result (of non-oscillation but not necessarily of
positivity on $(0, \infty)$).
\end{example}
\begin{example}[A transcendental case] \label{four} \rm
The following example is rarely covered in the literature since the reciprocal
of the function $g(y)= \sin (\frac{\pi}{2}y)$ does not converge at all at
infinity (to either a finite or extended real number) due to oscillations
in the nonlinearity, nor does it converge at $0$ being infinite there.
Thus, it is neither sublinear nor superlinear. Fixing this $g$ we choose
$f \in C[0, \infty)$, $f(x) \geq 0$ so that
$$
\int_{0}^{\infty}t f(t)\, dt < \frac{2}{\pi}.
$$
Note that $g$ is, once again, a self-map on $[0,1]$. In this case, for our
choice of $f$, we see that use of the theorem shows that the equation
$$
y''+f(x)\,\sin (\frac{\pi}{2}y) = 0, \quad x \in [0, \infty),
$$
admits a positive asymptotically constant solution $y(x) \to 1$ as
$x \to \infty$.
\end{example}
\begin{example}[A mixed case] \label{five} \rm
Our final example covers a mixed situation, one where $F(x,y)$ is a sum of
a linear and nonlinear term in the second variable. It appears that the
literature involving such problems is scarce as well. We choose
$$
F(x,y) = \frac{e^{-x}y}{4} + \frac{e^{-2x}}{1+y}, \quad (x,y) \in \mathbb{R}^+
\times \mathbb{R}^+.
$$
The choice $M=1$ in the theorem gives
$\int_{0}^{\infty}t\, F(t, u(t))\, dt \leq \frac{1}{2}$ for $u \in X$.
In fact, (\ref{lip}) is satisfied provided we put
$$
k(t) = \frac{e^{-t}}{4} + \frac{e^{-2t}}{2}.
$$
It follows that (\ref{teek}) is verified as well and so the nonlinear
equation
$$
y''+ \frac{e^{-x}}{4}y+ \frac{e^{-2x}}{1+y} = 0, \quad x \in [0, \infty),
$$
has a positive solution that tends asymptotically to one as $x \to \infty$.
\end{example}
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\section*{April 27, 2004: Corrigendum}
In the proof of Theorem 2.1, replace ``Banach space'' by
``complete metric space''.
Thanks,
Angelo Mingarelli.
\end{document}