\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.\newline
{\em Electronic Journal of Differential Equations},
Conference 12, 2005, pp. 171--180.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{171}
\begin{document}
\title[\hfilneg EJDE/Conf/12 \hfil Oscillation criteria]
{Oscillation criteria for functional differential equations}
\author[I. P. Stavroulakis \hfil EJDE/Conf/12 \hfilneg]
{Ioannis P. Stavroulakis}
\address{Department of Mathematics, University of Ioannina \\
451 10 Ioannina, Greece}
\email{ipstav@cc.uoi.gr}
\date{}
\thanks{Published April 20, 2005.}
\subjclass[2000]{34K11, 34C10}
\keywords{Oscillation; delay differential equations}
\begin{abstract}
Consider the first-order linear delay differential equation
\[
x'(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0},
\]
and the second-order linear delay equation
\[
x''(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0},
\]
where $p$ and $\tau $ are continuous functions on $[t_{0},\infty )$,
$p(t)>0$, $\tau (t)$ is non-decreasing, $\tau (t)\leq t$ for $t\geq t_{0}$
and $\lim_{t\to \infty }\tau (t)=\infty $. Several oscillation
criteria are presented for the first-order equation when
\[
0<\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds\leq \frac{1}{e}
\quad \text{and}\quad \limsup_{t\to \infty }\int_{\tau
(t)}^{t}p(s)ds<1,
\]
and for the second-order equation when
\[
\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds
\leq \frac{1}{e}\quad \text{and}\quad \limsup_{t\to \infty }\int_{\tau
(t)}^{t}\tau (s)p(s)ds<1\,.
\]
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
The problem of establishing sufficient conditions for the oscillation of all
solutions to the first-order differential equation
\begin{equation}
x'(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0}, \label{e1}
\end{equation}
and to the second-order equation
\begin{equation}
x''(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0}, \label{e2}
\end{equation}
where $p\in C([t_{0,}\infty ),\mathbb{R}^{+})$ (here
$\mathbb{R}^{+}=[0,\infty ))$, $\tau \in C([t_{0},\infty )$, $\mathbb{R})$,
$\tau (t)$ is non-decreasing, $\tau (t)\leq t$ for $t\geq t_{0}$ and
$\lim_{t\to \infty }\tau (t)=\infty $, has been the subject of many
investigations. See, for example the references in this article and the
references cited therein.
By a solution of \eqref{e1} (resp. \eqref{e2}) we understand a continuously
differentiable function defined on $[\tau (T_{0}),\infty )$ for some
$T_{0}\geq t_{0}$ and such that \eqref{e1} (resp. \eqref{e2}) is satisfied
for $t\geq T_{0}$. Such a solution is called \textit{oscillatory} if it has
arbitrarily large zeros, and otherwise it is called \textit{nonoscillatory}.
In this paper our main purpose is to present the state of the art on the
oscillation of all solutions to \eqref{e1} especially in the case where
\[
0<\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds\leq
\frac{1}{e}\quad \text{ and}\quad \limsup_{t\to \infty }\int_{\tau
(t)}^{t}p(s)ds<1,
\]
and for \eqref{e2} when
\[
\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds
\leq \frac{1}{e}\quad \text{and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}\tau
(s)p(s)ds<1.
\]
\section{Oscillation criteria for the first-order equation}
In this section we study the delay equation \eqref{e1}. The first systematic
study for the oscillation of all solutions to \eqref{e1} was made by
Myshkis. In 1950 [42] he proved that every solution of \eqref{e1} oscillates
if
\begin{equation}
\limsup_{t\to \infty }[t-\tau (t)]<\infty \quad \mbox{and}\quad
\liminf_{t\to \infty }[t-\tau (t)]\liminf_{t\to \infty }p(t)>
\frac{1}{e}. \label{C1}
\end{equation}
In 1972, Ladas, Lakshmikantham and Papadakis [33] proved that the same
conclusion holds if
\begin{equation}
A:=\limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds>1. \label{C2}
\end{equation}
In 1979, Ladas [32] established integral conditions for the oscillation of
\eqref{e1} with constant delay. Tomaras [54-56] extended this result to
\eqref{e1} with variable delay. For related results see Ladde [36-38]. The
following most general result is due to Koplatadze and Canturija \cite{k5}.
If
\begin{equation}
\alpha :=\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds>\frac{1}{e},
\label{C3}
\end{equation}
then all solutions of \eqref{e1} oscillate; If
\begin{equation}
\limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds<\frac{1}{e},
\label{N1}
\end{equation}
then \eqref{e1} has a nonoscillatory solution.
In 1982 Ladas, Sficas and Stavroulakis \cite{l4} and in 1984 Fukagai and
Kusano [13] established oscillation criteria (of the type of conditions
\eqref{C2} and \eqref{C3}) for \eqref{e1} with \textit{oscillating }
coefficient $p(t)$.
It is obvious that there is a gap between the conditions \eqref{C2} and
\eqref{C3} when the limit $\lim_{t\to \infty} \int_{\tau (t)}^{t}p(s)ds$
does not exist. How to fill this gap is an interesting problem which has
been recently investigated by several authors.
In 1988, Erbe and Zhang \cite{e6} developed new oscillation criteria by
employing the upper bound of the ratio $x(\tau (t))/x(t)$ \ for possible
nonoscillatory solutions $x(t)$ of \eqref{e1}. Their result says that all
the solutions of \eqref{e1} are oscillatory, if $0<\alpha \leq \frac{1}{e}$
and
\begin{equation}
A>1-\frac{\alpha ^{2}}{4}. \label{C4}
\end{equation}
Since then, several authors tried to obtain better results by improving the
upper bound for $x(\tau (t))/x(t)$. In 1991, Jian [20] derived the condition
\begin{equation}
A>1-\frac{\alpha ^{2}}{2(1-\alpha )}, \label{C5}
\end{equation}
while in 1992, Yu and Wang [63] and Yu, Wang, Zhang and Qian [64] obtained
the condition
\begin{equation}
A>1-\frac{1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}. \label{C6}
\end{equation}
In 1990, Elbert and Stavroulakis \cite{e2} and in 1991 Kwong \cite{k10},
using different techniques, improved \eqref{C4}, in the case where $0<\alpha
\leq \frac{1}{e}$, to the conditions
\begin{equation}
A>1-(1-\frac{1}{\sqrt{\lambda _{1}}})^{2} \label{C7}
\end{equation}
and
\begin{equation}
A>\frac{\ln \lambda _{1}+1}{\lambda _{1}}, \label{C8}
\end{equation}
respectively, where $\lambda _{1}$ is the smaller real root of the equation
$\lambda =e^{\alpha \lambda }$.
In 1994, Koplatadze and Kvinikadze \cite{k6} improved \eqref{C6}, while in
1998, Philos and Sficas \cite{p1} and in 1999, Zhou and Yu [65] and Jaro\v{s}
and Stavroulakis \cite{j1} derived the conditions
\begin{gather}
A>1-\frac{\alpha ^{2}}{2(1-\alpha )}-\frac{\alpha ^{2}}{2}\lambda _{1},
\label{C9} \\
A>1-\frac{1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}-(1-\frac{1}{\sqrt{
\lambda _{1}}})^{2}, \label{C10} \\
A>\frac{\ln \lambda _{1}+1}{\lambda _{1}}-\frac{1-\alpha -\sqrt{1-2\alpha
-\alpha ^{2}}}{2}, \label{C11}
\end{gather}
respectively.
Consider \eqref{e1} and assume that $\tau (t)$ is continuously
differentiable and that there exists $\theta >0$ such that $p(\tau(t))\tau
^{\prime}(t)\geq \theta p(t)$ eventually for all $t$. Under this additional
condition, in 2000, Kon, Sficas and Stavroulakis \cite{k2} and in 2003,
Sficas and Stavroulakis \cite{s1} established the conditions
\begin{equation}
A>\frac{\ln \lambda _{1}+1}{\lambda _{1}}-\frac{1-\alpha -\sqrt{(1-\alpha
)^{2}-4\Theta }}{2} \label{e2.1}
\end{equation}
and
\begin{equation}
A>\frac{\ln \lambda _{1}}{\lambda _{1}}-\frac{1+\sqrt{1+2\theta -2\theta
\lambda _{1}M}}{\theta \lambda _{1}}\text{ } \label{e2.2}
\end{equation}
respectively, where
\[
\Theta =\frac{e^{\lambda _{1}\theta \alpha }-\lambda _{1}\theta \alpha -1}{
(\lambda _{1}\theta )^{2}}
\]
and
\[
M=\frac{1-\alpha -\sqrt{(1-\alpha )^{2}-4\Theta }}{2}.
\]
\begin{remark}[\cite{k2,s1}] \label{rmk2.1} \rm
Observe that when $\theta =1$, then
$\Theta =\frac{\lambda _{1}-\lambda _{1}\alpha -1}{\lambda _{1}{}^{2}}$, and
\eqref{e2.1} reduces to
\begin{equation}
A>2\alpha +\frac{2}{\lambda _{1}}-1,\label{C12}
\end{equation}
while in this case it follows that $M=1-\alpha -\frac{1}{\lambda _{1}}$
and \eqref{e2.2} reduces to
\begin{equation}
A>\frac{\ln \lambda _{1}-1+\sqrt{5-2\lambda _{1}
+2\alpha \lambda _{1}}}{\lambda _{1}}. \label{C13}
\end{equation}
In the case where $\alpha =\frac{1}{e}$, then $\lambda _{1}=e$, and
\eqref{C13} leads to
\[
A>\frac{\sqrt{7-2e}}{e}\approx 0.459987065.
\]
It is to be noted that as $\alpha \to 0$, then all the previous
conditions \eqref{C4}-\eqref{C12} reduce to the condition \eqref{C2}, i.e.
$A>1$.
However, the condition \eqref{C13} leads to
\[
A>\sqrt{3}-1\approx 0.732,
\]
which is an essential improvement. Moreover \eqref{C13} improves all the
above conditions when $0<\alpha \leq 1/e$ as well. Note that the
value of the lower bound on $A$ can not be less than
\[
\frac{1}{e}\approx 0.367879441.
\]
Thus the aim is to establish a condition which leads to a value {\it as
close as possible to} $1/e$. For illustrative purpose, we give
the values of the lower bound on $A$ under these conditions when
$\alpha=1/e$.
\begin{center}
\begin{tabular}{|cc|} \hline
Condition & Lower bound\\ \hline
\eqref{C4} & 0.966166179 \\
\eqref{C5} & 0.892951367 \\
\eqref{C6} & 0.863457014 \\
\eqref{C7} & 0.845181878 \\
\eqref{C8} & 0.735758882 \\
\eqref{C9} & 0.709011646 \\
\eqref{C10}& 0.708638892 \\
\eqref{C11}& 0.599215896 \\
\eqref{C12}& 0.471517764 \\
\eqref{C13}& 0.459987065 \\ \hline
\end{tabular}
\end{center}
We see that the condition \eqref{C13} essentially
improves all the known results in the literature.
\end{remark}
\begin{example} [\cite{s1}] \label{exa2.1} \rm
Consider the delay differential equation
\[
x'(t)+px\big(t-q\sin ^{2}\sqrt{t}-\frac{1}{pe}\big)=0,
\]
where $p>0$, $q>0$ and $pq=0.46-\frac{1}{e}$. Then
\[
\alpha =\liminf_{t\to \infty }\int_{\tau
(t)}^{t}pds=\liminf_{t\to \infty }p(q\sin ^{2}\sqrt{t}+\frac{1}{pe})=
\frac{1}{e}
\]
and
\[
A=\limsup_{t\to \infty }\int_{\tau
(t)}^{t}pds=\limsup_{t\to \infty }p(q\sin ^{2}\sqrt{t}+\frac{1}{pe}
)=pq+\frac{1}{e}=0.46\,.
\]
Thus, according to Remark 2.1, all solutions of this equation oscillate.
Observe that none of the conditions \eqref{C4}-\eqref{C12} apply to this
equation.
\end{example}
Following this historical (and chronological) review we also mention that in
the case where
\[
\int_{\tau (t)}^{t}p(s)ds\geq \frac{1}{e}\quad \text{and}\quad
\lim_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds=\frac{1}{e}
\]
this problem has been studied in 1995, by Elbert and Stavroulakis \cite{e3},
by Kozakiewicz \cite{k8}, Li \cite{l9,l10}, and in 1996, by Domshlak and
Stavroulakis \cite{d5}.
\section{Oscillation criteria for the second-order equation}
In this section we study the second-order delay equation \eqref{e2}:
\[
x^{\prime\prime}(t)+p(t)x(\tau (t))=0,\;t\geq t_{0},
\]
For the case of ordinary differential equations, i.e. when $\tau (t)\equiv
t, $ the history of the problem began as early as in 1836 by the work of
Sturm \cite{s2} and was continued in 1893 by Kneser \cite{k1}. Essential
contribution to the subject was made by Hille, Wintner, Hartman, Leighton,
Nehari, and others (see the monograph by C. Swanson \cite{s3} and the
references cited therein). In particular, in 1948 Hille \cite{h2} obtained
the following well-known oscillation criteria. Let
\begin{equation}
\limsup_{t\to \infty }t\int_{t}^{+\infty }p(s)ds>1 \label{e3.1}
\end{equation}
or
\begin{equation}
\liminf_{t\to \infty }t\int_{t}^{+\infty }p(s)ds>\frac{1}{4}, \label{e3.2}
\end{equation}
the conditions being assumed to be satisfied if the integral diverges. Then
\eqref{e2} with $\tau (t)\equiv t$ is oscillatory.
For the delay differential equation \eqref{e2} earlier oscillation results
can be found in the monographs by Myshkis \cite{m2} and Norkin \cite{n1}. In
1968 Waltman \cite{w1} and in 1970 Bradley \cite{b1} proved that \eqref{e2}
is oscillatory if
\[
\int^{+\infty }p(t)dt=+\infty .
\]
Proceeding in the direction of generalization of Hille's criteria, in 1971
Wong \cite{w4} showed that if $\tau (t)\geq \alpha t$ for $t\geq 0$ with
$0<\alpha \leq 1$, then the condition
\begin{equation}
\liminf_{t\to \infty }t\int_{t}^{+\infty }p(s)ds>\frac{1}{4\alpha }
\label{e3.3}
\end{equation}
is sufficient for the oscillation of \eqref{e2}. In 1973, Erbe \cite{e4}
generalized this condition to
\begin{equation}
\liminf_{t\to \infty }t\int_{t}^{+\infty }\frac{\tau (s)}{s}p(s)ds>
\frac{1}{4} \label{e3.4}
\end{equation}
without any additional restriction on $\tau $. In 1987, Yan \cite{y1}
obtained some general criteria improving the previous ones.
An oscillation criterion of different type is given in 1986 by Koplatadze
\cite{k3} and in 1988 by Wei \cite{w3}, where it is proved that \eqref{e2}
is oscillatory if
\begin{equation}
\limsup_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds>1
\label{C2'}
\end{equation}
or
\begin{equation}
\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds>\frac{1}{e}.
\label{C3'}
\end{equation}
The conditions \eqref{C2'} and \eqref{C3'} are analogous to the oscillation
conditions \eqref{C2} and \eqref{C3} respectively, for the first order delay
equation \eqref{e1}. The essential difference between \eqref{e3.3},
\eqref{e3.4} and \eqref{C2'}, \eqref{C3'} is that the first two can
guarantee oscillation for ordinary differential equations as well, while the
last two work only for delay equations. Unlike first-order differential
equations, where the oscillatory character is due to the delay only,
equation \eqref{e2} can be oscillatory without any delay at all, i.e., in
the case $\tau (t)\equiv t$. Figuratively speaking, two factors contribute
to the oscillatory character of \eqref{e2}: the presence of the delay and
the second order nature of the equation. The conditions \eqref{e3.3},
\eqref{e3.4} and \eqref{C2'}, \eqref{C3'} illustrate the role of these
factors taken separately.
In what follows it will be assumed that the condition
\begin{equation}
\int^{+\infty }\tau (s)p(s)ds=+\infty \label{e3.5}
\end{equation}
is fulfilled. As it follows from \cite[Lemma 4.1]{k4}, this condition is
necessary for \eqref{e2} to be oscillatory. The study being devoted to the
problem of oscillation of \eqref{e2}, the condition \eqref{e3.5} does not
affect the generality.
In this section oscillation results are obtained for \eqref{e2} by reducing
it to a first order equation. Since for the latter the oscillation is due
solely to the delay, the criteria hold for delay equations only and do not
work in the ordinary case.
\begin{theorem}[\cite{k7}] \label{thm3.1}
Let \eqref{e3.5} be fulfilled and the
differential inequality
\[
x'(t)+\Big( \tau (t)+\int_{T}^{\tau (t)}\xi \tau (\xi )p(\xi )d\xi
\Big) p(t)x(\tau (t))\leq 0
\]
have no eventually positive solution. Then \eqref{e2} is oscillatory.
\end{theorem}
Note that Theorem \ref{thm3.1} reduces the question of oscillation of
\eqref{e2} to that of the absence of eventually positive solutions of the
differential inequality
\begin{equation}
x^{\prime}(t)+\Big( \tau (t)+\int_{T}^{\tau (t)}\xi \tau (\xi )p(\xi )d\xi
\Big) p(t)x(\tau (t))\leq 0. \label{e3.6}
\end{equation}
So oscillation results for first order delay differential equations can be
applied since the oscillation of the equation
\begin{equation}
u^{\prime}(t)+g(t)u(\delta (t))=0 \label{e3.7}
\end{equation}
is equivalent to the absence of eventually positive solutions of the
inequality
\begin{equation}
u^{\prime}(t)+g(t)u(\delta (t))\leq 0. \label{e3.8}
\end{equation}
This fact is a simple consequence of the following comparison theorem
deriving the oscillation of \eqref{e3.7} from the oscillation of the
equation
\begin{equation}
v^{\prime}(t)+h(t)v(\sigma (t))=0. \label{e3.9}
\end{equation}
We assume that $g,h:\mathbb{R}^{+}\to \mathbb{R}^{+}$ are locally
integrable, $\delta ,\sigma :\mathbb{R}^{+}\to \mathbb{R}$ are
continuous, $\delta (t)\leq t,\sigma (t)\leq t$ for $t\in \mathbb{R}^{+}$,
and $\delta (t)\to +\infty ,$ $\sigma (t)\to +\infty $ as
$t\to +\infty $.
\begin{theorem} \label{thm3.2}
Let
$g(t)\geq h(t)$ and $\delta (t)\leq \sigma (t)$ for $t\in R^{+}$
and let \eqref{e3.9} be oscillatory. Then \eqref{e3.7} is also oscillatory.
\end{theorem}
\begin{corollary} \label{coro 3.1}
Let \eqref{e3.7} be oscillatory.
Then the inequality \eqref{e3.8} has no eventually positive solution.
\end{corollary}
Turning to applications of Theorem \ref{thm3.1}, we will use it together
with the criteria \eqref{C2} and \eqref{C3} to get the following result.
\begin{theorem}[\cite{k7}] \label{thm3.3}
Let
\begin{equation}
K:=\limsup_{t\to \infty }\int_{\tau (t)}^{t}\Big( \tau
(s)+\int_{0}^{\tau (s)}\xi \tau (\xi )p(\xi )d\xi \Big) p(s)ds>1, \label{C2''}
\end{equation}
or
\begin{equation}
k:=\liminf_{t\to \infty }\int_{\tau (t)}^{t}\Big( \tau
(s)+\int_{0}^{\tau (s)}\xi \tau (\xi )p(\xi )d\xi \Big) p(s)ds>\frac{1}{e}.
\label{C3''}
\end{equation}
Then \eqref{e2} is oscillatory.
\end{theorem}
To apply Theorem \ref{thm3.1} it suffices to note that: (i) \eqref{e3.5} is
fulfilled since otherwise $k=K=0$; (ii) since $\tau (t)\to +\infty $ as
$t\to +\infty $, the relations \eqref{C2''}, \eqref{C3''} imply the same
relations with $0$ changed by any $T\geq 0$.
\begin{remark}[\cite{k7}] \label{rmk3.1} \rm
Theorem \ref{thm3.3} improves the criteria \eqref{C2'}, \eqref{C3'} by
Koplatadze \cite{k3} and Wei \cite{w3} mentioned above.
This is directly seen from \eqref{C2''}, \eqref{C3''}
and can be easily checked if we take
$\tau(t)\equiv t-\tau _{0}$ and $p(t)\equiv p_{0}/(t-\tau _{0})$ for
$t\geq 2\tau_{0}$, where the constants $\tau _{0}>0$ and $p_{0}>0$ satisfy
\[
\tau _{0}p_{0}<1/e.
\]
In this case neither of \eqref{C2'}, \eqref{C3'} is applicable
for \eqref{e2} while both \eqref{C2''}, \eqref{C3''}
give the positive conclusion about its oscillation. Note also that this is
exactly the case where the oscillation is due to the delay since the
corresponding equation without delay is non-oscillatory.
\end{remark}
\begin{remark}[\cite{k7}] \label{rmk3.2} \rm
Criteria \eqref{C2''}, \eqref{C3''} look like \eqref{C2},
\eqref{C3} but there is an
essential difference between them as pointed out in the introduction. The
condition \eqref{C3} is close to be the necessary one, since according to
\cite{k5} if $A\leq 1/e$, then \eqref{e3.7} is nonoscillatory. On the other hand,
for an oscillatory equation \eqref{e2} without delay we have $k=K=0$. Nevertheless,
the constant $1/e$ in Theorem \ref{thm3.3} is also the best possible in the sense
that for any $\varepsilon \in (0,1/e]$ it can not be replaced by
$1/e-\varepsilon $ without affecting the validity of the theorem. This is
illustrated as follows.
\end{remark}
\begin{example}[\cite{k7}] \label{exa3.1} \rm
Let $\varepsilon \in (0,1/e]$, $1-e\varepsilon <\beta <1$,
$\tau (t)\equiv \alpha t$ and $p(t)\equiv \beta (1-\beta )\alpha ^{-\beta }t^{-2}$,
where $\alpha =\exp(\frac{1}{\beta -1})$.
Then \eqref{C3''} is fulfilled with $1/e$ replaced by $1/e-\varepsilon $.
Nevertheless \eqref{e2} has a nonoscillatory solution,
namely $u(t)\equiv t^{\beta }$. Indeed, denoting
$c=\beta (1-\beta )\alpha^{-\beta }$, we see that the expression under the
limit sign in \eqref{C3''} is constant and equals
$$
\alpha c\vert \ln \alpha \vert (1+\alpha c)
=(\beta /e)(1+(\beta (1-\beta ))/e)>\beta
/e>1/e-\varepsilon .
$$
\end{example}
Note that there is a gap between conditions \eqref{C2''} and \eqref{C3''}
when $0\leq k\leq 1/e$, $kk+\frac{1}{\lambda (k)}-\frac{1-k-\sqrt{1-2k-k^{2}}}{2} \label{C11'}
\end{equation}
where $\lambda (k)$ is the smaller root of the equation
$\lambda =\exp (k\lambda )$. Then \eqref{e2} is oscillatory.
\end{theorem}
Note that condition \eqref{C11'} is analogous to condition \eqref{C11}.
\subsection*{Acknowledgment}
The author would like to thank the referee for some useful remarks.
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