I use the fancy-preview script but I don't get the desired result (error while compiling).
Perl 5 and Config::IniFiles module are well installed :
As well as the two AcroTeX packages :
fancytooltips v1.12 is also installed.
However, if I compile the file paper2.tex given by the author of the script on his website via the command perl fancy-preview paper2 I get the following errors during compilation :
A pdf file named texput.pdf is created during compilation and contains the equations and bibliographic references that the previews should contain:
Due to the error during compilation, however, no pdf file is generated. The error message is not very helpful (why look for a file named minimal.pdf?)...
Would you have a solution to this problem?
Here is a copy of the paper2.tex file used for compilation:
\documentclass[oneside,reqno]{elsarticle}
\frenchspacing
\usepackage{amsfonts,amsmath,amsthm}
\let\tilde\widetilde
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{Theorem}{Theorem}[section]
\renewcommand\theTheorem{\Alph{Theorem}}
\newtheorem{corollary}{Corollary}[section]
\newdefinition{remark}{Remark}[section]
\newdefinition{example}{Example}[section]
\def\sgn{\mathop{\hbox{\rm sgn}}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\dx{\,\hbox{\rm d}x\,}
\def\dS{\,\hbox{\rm dS}\,}
\def\dt{\,\hbox{\rm d}t\,}
\def\dr{\,\hbox{\rm d}r\,}
\def\eps{\varepsilon}
\def\lambdamin{\lambda_{\min}}
\def\lambdamax{\lambda_{\max}}
\let\epsilon\varepsilon
\usepackage{hyperref}
\begin{document}
\begin{frontmatter}
\title{A remark on connection between conjugacy of half-linear
differential equation and equation with mixed
nonlinearities\tnoteref{t1}}
\author{Robert Ma\v r\'\i k}
\ead{marik@mendelu.cz}
\address{Mendel University, Department of Mathematics\\
Zem\v ed\v elsk\'a 3, 613 00 Brno, Czech Republic
}
\tnotetext[t1]{Supported by the Grant 201/07/0145
of the Czech Grant Agency}
\begin{abstract}
In the paper new criteria for conjugacy of half-linear ordinary
differential equation are derived by using Riccati transformation.
These criteria are used to derive nonexistence and oscillation
results for equation with mixed nonlinearities, which is viewed as a perturbation of half-linear equation.
\end{abstract}
\begin{keyword}half-linear differential equation \sep second order equation \sep conjugacy \sep oscillation \sep mixed powers
\MSC 34C10
\end{keyword}
\end{frontmatter}
\section{Introduction}
In the paper we study equation
\begin{equation}
\label{eq:E}
(r(t)|x'|^{p-2}x')'+c(t)|x|^{p-2}x+\sum_{i=1}^m c_i(t)|x|^{p_i-2}x=e(t),
\end{equation}
where $p>1$ and $p_i>1$ are real numbers, $c(t)$, $c_i(t)$ and $e(t)$
are continuous functions and $r(t)$ is a positive continuous function.
Under solution of \eqref{eq:E} on the interval $I$ we understand
a smooth function $x(t)$ defined on $I$ such that
$r(t)|x'(t)|^{p-2}x'(t)$ is differentiable and $x(t)$ satisfies
\eqref{eq:E} everywhere on $I$. We suppose that $p_i\neq p$
for every $i$ and $p_i\neq p_j$ for every $i$, $j$ with property
$i\neq j$.
The paper is motivated by recent paper \cite{Zheng} and extends and
completes the results from this paper in several respects (see also
Remark \ref{final-remark}). In contrast to the paper \cite{Zheng} we
allow $p_i<p$ for some $i$ and do not assume anything about the fixed
sign of the functions $c_i$ in this case.
The oscillation of the half-linear equation has been studied using
generalized Riccati substitution in \cite[Theorem 2]{Li} and the
following theorem has been proved.
\begin{Theorem}[Li, Cheng, \cite{Li}]\label{th:Li}
Suppose that for any $T\geq t_0$ there exist $T\leq s_1<t_1\leq
s_2<t_2$ such that $e(t)<0$ for $t\in[s_1,t_1]$ and $e(t)\geq 0$ for
$t\in[s_2,t_2]$. Let $D(s_i,t_i)=\{u\in C^1[s_i,t_i]:u(t)\neq 0,
t\in(s_i,t_i)\text{ and } u(s_i)=0=u(t_i)\}$ for $i=1,\ 2$. If there
exist $H\in D(s_i,t_i)$ and a positive nondecreasing function
$\phi\in C^1([t_0,\infty),\R)$ such that
\begin{equation}
\label{eq:Li}
\int_{s_i}^{t_i}H^2(t)\phi(t)C(t)\dr>\frac{1}{p^p}\int_{s_i}^{t_i}\frac{r(t)\phi(t)}{|H(t)|^{p-2}}\left(2|H'(t)|+|H(t)|\frac{\phi'(t)}{\phi(t)}\right)^p\dt,
\end{equation}
for $i=1,2$, then
\begin{equation}
\label{eq:Ehl}
(r(t)|x'|^{p-2}x')'+C(t)|x|^{p-2}x=0
\end{equation}
is oscillatory.
\end{Theorem}
As pointed out in \cite{Zheng}, this result cannot be applied if
$p>2$. From this reason Han, Wang and Zheng presented in \cite{Zheng}
an extension of this theorem which removed the restriction $p\leq 2$
and also extends this theorem for equation with mixed
nonlinearities. However, the results from \cite{Zheng} do not include
Theorem~\ref{th:Li} as a special case.
In this paper we present another extension of Theorem \ref{th:Li}
which also removes the restriction $p\leq 2$ and in contrast to
\cite{Zheng} includes both results from \cite{Li,Zheng} as a special
case and deals with more general equation. Instead of to formulate
the results as oscillation criteria which are in fact consequence of
conjugacy criteria, we present our results in terms of nonexistence of
positive and negative solutions. The extension to oscillation
criterion is trivial and straightforward. This idea is also motivated
by the fact that as far as the author knows, we miss systematic
oscillation theory for the equation with mixed nonlinearities. The
reason of this lack is in the fact that the set of all solutions is
more comprehensive than in the half-linear case. In particular, the
solution may become infinite at some finite $t$ (see \cite{Sun} for
more detailed discussion).
Recall that $t_1$, $t_2\in I$ are said to be \textit{conjugate} point
relative to Eq. \eqref{eq:Ehl} if there exists a nontrivial solution
$x(t)$ of this equation which satisfies $x(t_1)=0=x(t_2)$. Since
oscillation theory attracts more attention than problems related to
conjugacy, the literature related to oscillation is much more
comprehensive. However many oscillation and nonoscillation criteria
are in fact conjugacy or disconjugacy criteria in a neighborhood of
infinity (see \cite{DF} for some recent progress in this field) or on
a sequence of intervals tending to infinity (see Theorem \ref{th:Li}).
We adopt the main idea of the paper \cite{Zheng} and we will consider
Eq. \eqref{eq:E} as a perturbation of half-linear differential
equation \eqref{eq:Ehl}. In contrast to \cite{Zheng} we do not use
the generalized Riccati transformation
\begin{equation*}
w(t)=\phi(t)r(t)\frac{|x'(t)|^{p-2}x'(t)}{|x(t)|^{p-2}x(t)},
\end{equation*}
but we consider the special case $\phi(x)=1$, i.e. we use the
transformation
\begin{equation}
\label{eq:RIC}
w(t)=r(t)\frac{|x'(t)|^{p-2}x'(t)}{|x(t)|^{p-2}x(t)},
\end{equation}
which converts Eq. \eqref{eq:Ehl} into
\begin{equation}
\label{eq:RICeq}
w'=-(p-1)r^{1-q}(t)|w|^q-C(t).
\end{equation}
There is no loss of generality in this approach, since the results
from \cite{Zheng} can be obtained from \eqref{eq:RICeq} by
transformation. As an advantage, some intermediate calculations like
proof of Theorem \ref{th1} are simpler and more transparent.
\section {Main results}
The following lemma is used to estimate terms involving powers $\alpha$
with term with power $\beta$. This estimate is necessary to collect
all terms into a term with power $p-1$. In contrast to \cite{Zheng} we
allow $\alpha<\beta$.
\begin{lemma}\label{lemma:ineq_cal}
The following inequalities hold for $a\geq 0$ and $x>0$.
\begin{enumerate}
\item If $\alpha<\beta$ and $b>0$, then $b-ax^\alpha\geq -x^\beta
\left(\frac{a(\beta-\alpha)}{\beta}\right)^{\frac\beta\alpha}
\frac{\alpha}{\beta-\alpha}b^{1-\frac\beta\alpha}$.
\label{pa}
\item If $\alpha>\beta$ and $b\geq0$, then $ax^\alpha+b\geq x^\beta
\left(\frac{a(\alpha-\beta)}{\beta}\right)^{\frac\beta\alpha}
\frac{\alpha}{\alpha-\beta}b^{1-\frac\beta\alpha}$.
\label{pb}
\end{enumerate}
\end{lemma}
\begin{proof}
Divide both inequalities by $x^\beta$. Now the inequalities can be
proved directly by inspecting functions which appears on the left
hand sides.
\end{proof}
In the following theorem $[c_i(t)]_+=\max\{c_i(t),0\}$ denotes the
positive part of the function $c_i(t)$.
\begin{theorem}\label{th1}
Let $e(t)<0$ on $[a,b]$ and denote
\begin{multline}
\label{eq:C}
C(t)=c(t)+\sum_{i\in I_1}\frac{p_i-1}{p_i-p}\left[\frac{c_i(t)(p_i-p)}{p-1}\right]^{(p-1)/(p_i-1)} \bigl(\epsilon_i|e(t)|\bigr)^{\frac{p_i-p}{p_i-1}}\\
- \sum_{i\in
I_2}\frac{p_i-1}{p-p_i}\left [\frac{[-c_i(t)]_+(p-p_i)}{p-1}\right]^{(p-1)/(p_i-1)}\bigl(\epsilon_i|e(t)|\bigr)^{\frac{p_i-p}{p_i-1}},
\end{multline}
where
$I_1=\{i\in[1,m]\cap \N:p_i>p\}$, $I_2=\{i\in[1,m]\cap
\N:p_i<p\}$, $\epsilon_i>0$, $\sum_{i=1}^m\epsilon_i=1$.
If Eq. \eqref{eq:Ehl} has conjugate points on $[a,b]$, then Eq.
\eqref{eq:E} has no positive solution on $[a,b]$.
Moreover, if $I_2=\emptyset$, then the inequality $e(x)<0$ can be
relaxed to $e(x)\leq 0$.
\end{theorem}
\begin{proof}
Suppose that $x$ is a positive solution of \eqref{eq:E} on $[a,b]$ and let
the function $w$ be defined by the Riccati substitution
\eqref{eq:RIC}. Differentiating \eqref{eq:RIC} and using
\eqref{eq:E} we get
\begin{equation}
\begin{aligned}
w'(t)&=(1-p){|w(t)|^q}r^{1-q}(t) \\&\quad -c(t)-\sum_{i=1}^m
c_i (t) x^{p_i-p}(t)+\frac{e(t)}{x^{p-1}(t)} \\&=
(1-p){|w(t)|^q}r^{1-q}(t) \\&\quad
-c(t)-\frac{1}{x^{p-1}(t)}\sum_{i=1}^m \Bigl(c_i (t)
x^{p_i-1}(t)-{\epsilon_i e(t)}\Bigr).
\end{aligned}\label{eq:th1}
\end{equation}
If $p_i>p$, then using part (\ref{pb}) of Lemma \ref{lemma:ineq_cal}
we have
\begin{align*}
c_i(t)x^{p_i-1}(t)&-\epsilon_i e(t)= c_i(t) x^{p_i-1}(t)+\epsilon_i |e(t)|\\
&\geq x^{p-1}(t)
\left[\frac{c_i(t)(p_i-p)}{p-1}\right]^{(p-1)/(p_i-1)}\frac{p_i-1}{p_i-p}
\bigl(\epsilon_i|e(t)|\bigr)^{\frac{p_i-p}{p_i-1}}.
\end{align*}
If $p_i<p$, then using part (\ref{pa}) of Lemma \ref{lemma:ineq_cal}
we have
\begin{align*}
c_i(t)x^{p_i-1}(t)&-\epsilon_i e(t)= \epsilon_i |e(t)|-(-c_i(t))x^{p_i-1}(t)\\
&\geq \epsilon_i |e(t)|-[-c_i(t)]_+x^{p_i-1}(t)\\
&\geq - x^{p-1}(t)
\left[\frac{[-c_i(t)]_+(p-p_i)}{p-1}\right]^{(p-1)/(p_i-1)}\frac{p_i-1}{p-p_i}
\bigl(\epsilon_i|e(t)|\bigr)^{\frac{p_i-p}{p_i-1}}.
\end{align*}
Summing up the last two estimates over all $i\in I_1$ and $i\in I_2$,
respectively, dividing by $|x|^{p-1}$ and using definition of $C(t)$
we get
\begin{equation*}
C(t)-c(t)\leq \frac{1}{x^{p-1}(t)}\sum_{i=1}^m \Bigl(c_i (t) x^{p_i-1}(t)-{\epsilon_i e(t)}\Bigr)
\end{equation*}
and from \eqref{eq:th1} it follows that
\begin{equation*}
w'(t)\leq(1-p){|w(t)|^q}r^{1-q}(t)-C(t)
\end{equation*}
holds on $[a,b]$. Using simple comparison argument or using \cite[Theorem
2.2.1]{DR} it can be shown, that the generalized Riccati equation
\begin{equation*}
v'(t)=(1-p){|v(t)|^q}r^{1-q}(t) -C(t)
\end{equation*}
has solution on $[a,b]$. Hence by half-linear Roundabout theorem (see \cite{DR}), Eq. \eqref{eq:Ehl} has no conjugate points on $[a,b]$. Theorem is proved.
\end{proof}
\begin{remark}
Note that we have no sign restriction on the functions $c_i(t)$ if
$p_i<p$ and the negative parts of the functions $c_i(t)$ play a role
in the function $C(t)$.
\end{remark}
\begin{corollary}\label{col0}
Theorem \ref{th1} remains valid, if we replace the condition
$e(t)<0$ ($e(t)\leq 0$) by $e(t)>0$ ($e(t)\geq 0$) and the words
``positive solution'' by ``negative solution''.
\end{corollary}
\begin{proof}
Follows from the fact that if $x(t)$ is a solution of \eqref{eq:E},
then $(-x(t))$ is a solution of equation in the same form but with
the right-hand side $(-e(t))$.
\end{proof}
\begin{theorem}\label{th2}
Suppose that there exist a real number $\alpha$, $\alpha\geq p$ and
smooth functions $h$, $\phi$, such that $h(a)=0=h(b)$, $h(t)>0$ on
$(a,b)$, $\phi$ is positive on $[a,b]$ and
\begin{equation}\label{eq:condition}
\int_a^{b}h^{\alpha}(t)\phi(t)C(t)\dt>\frac{1}{p^p}\int_a^b \left|\alpha h'(t)+h(t)\frac{\phi'(t)}{\phi(t)}\right|^p r(t)\phi(t) h^{\alpha-p}(t)\dt.
\end{equation}
Then Eq. \eqref{eq:Ehl} has conjugate points on $[a,b]$.
\end{theorem}
\begin{proof}
Suppose that Eq. \eqref{eq:Ehl} has no conjugate
points on $[a,b]$. Then there exists a positive solution $x(t)$ of
this equation on $[a,b]$ and the Riccati type transformation
\eqref{eq:RIC} defines a function $w(t)$ which solves the Riccati
type equation \eqref{eq:RICeq}
on $[a,b]$. The function $W(t)=\phi(t)w(t)$ satisfies
\begin{equation*}
W'(t)=\frac{\phi'(t)}{\phi(t)}W(t)+(1-p){|W(t)|^q}r^{1-q}(t)\phi^{1-q}(t)
-\phi(t)C(t)
\end{equation*}
on $[a,b]$.
Rearranging terms, multiplying by $h^{\alpha}(t)$ and integrating over
the interval $[a,b]$ we get
\begin{align*}
\int_a^b h^\alpha(t)\phi(t)C(t)\dt&= -\int_a^b h^{\alpha}(t)W'(t)\dt+\int_a^b h^{\alpha}(t)\frac{\phi'(t)}{\phi(t)}W(t)\dt\\&\qquad-(p-1)\int_a^bh^{\alpha}(t)r^{1-q}(t)\phi^{1-q}(t){|W(t)|^q}\dt.
\end{align*}
Integrating by parts and using the conditions $h(a)=0=h(b)$ we get
\begin{equation*}
-\int_a^b h^\alpha(t) W'(t)\dt= \alpha\int_a^bh^{\alpha-1}(t)h'(t) W(t)\dt.
\end{equation*}
Hence
\begin{equation}\label{eq:pom}
\begin{aligned}
\int_a^b h^\alpha(t)\phi(t)C(t)\dt&\leq \int_a^b\left|\alpha
h^{\alpha-1}(t)h'(t)+h^\alpha(t)\frac{\phi'(t)}{\phi(t)}
\right||W(t)|\dt\\& - (p-1)\int_a^b h^\alpha(t)
r^{1-q}(t)\phi^{1-q}(t) |W(t)|^q \dt.
\end{aligned}
\end{equation}
Since the Young inequality implies
$
A |W|-(p-1)B|W|^q\leq \frac 1{p^p}A^p B^{1-p},
$
we get the following estimate on $[a,b]$
\begin{multline*}
|W(t)|\left|\alpha h^{\alpha-1}(t)h'(t)+h^\alpha(t)\frac{\phi'(t)}{\phi(t)}\right|
- (p-1)h^\alpha(t)r^{1-q}(t)\phi^{1-q}(t)|W(t)|^q\\
\leq
\frac {1}{p^p}\left|\alpha h'(t)+h(t)\frac{\phi'(t)}{\phi(t)}\right|^p r(t)\phi(t) h^{\alpha-p}(t).
\end{multline*}
Integrating over $(a,b)$ and using this estimate in \eqref{eq:pom} we get an inequality which contradicts \eqref{eq:condition}.
\end{proof}
Summarizing Theorem \ref{th1} and \ref{th2} we get the following
oscillation result. Recall that, adopting terminology of \cite{Zheng},
Eq. \eqref{eq:E} is said to be oscillatory if all its nontrivial
solutions (i.e. the solutions extensible up to infinity which are not
identically equal to zero in a neighborhood of infinity) have
arbitrarily large zeros.
\begin{corollary}\label{cor}
Assume that for every $T\geq t_0$ there exist $T\leq s_1<t_1\leq
s_2<t_2$ such that $e(t)< 0$ for $t\in[s_1,t_1]$ and $e(t)> 0$ for
$t\in[s_2,t_2]$. Let $D(s_i,t_i)=\{u\in C^1[s_i,t_i]: u(t)>0 \text{
for }t\in(s_i,t_i)\text{ and } u(t_i)=0=u(s_i)\}$ for $i=1,2$. Let
$C$ be defined by \eqref{eq:C}. If there exists $H\in D(s_i,t_i)$
and a positive function $\phi$ such that \eqref{eq:condition} holds
with $a=s_i$, $b=t_i$ for $i=1,2$, then Eq. \eqref{eq:E} is
oscillatory.
Moreover, if $p_i>p$ for every $i$, then the inequalities $e(t)<0$
and $e(t)>0$ can be relaxed to $e(t)\leq 0$ and $e(t)\geq 0$,
respectively.
\end{corollary}
\begin{proof}
Using Theorems \ref{th1} and \ref{th2} on the interval $[s_1,t_1]$
we can see that there is no positive solution on $[s_1,t_1]$ and
thus there is no positive solution on $[s_1,t_2]$. Taking into
account Corollary \ref{col0} we can prove in a similar way that
there exists no negative solution on $[s_2,t_2]$ and thus there is
no negative solution on $[s_1,t_2]$.
\end{proof}
\begin{remark}\label{final-remark}
The main result of the paper \cite{Zheng} is a special case of
Corollary \ref{cor}. Really, if we require $p_i>p$ (and thus also
$c_i(t)\geq 0$ on $(s_i,t_i)$) for every $i$ and if we put
$\alpha=p$ and put $\epsilon_i=\frac 1m$, then inequality
\eqref{eq:condition} in Corollary \ref{cor} implies (12) of
\cite[Theorem 2.2]{Zheng}. Since $s_1$ can be arbitrarily large, Eq.
\eqref{eq:E} is oscillatory. Remark also that we use more accurate
estimates when handling absolute values and thus
\eqref{eq:condition} is less restrictive than (12) of \cite{Zheng}
and thus yields more general criterion. Remark also that Theorem
\ref{th:Li} is another special case of Corollary \ref{cor} (for
$\alpha =2$ and $c_i\equiv 0$).
\end{remark}
\textbf{Acknowledgment} The author would like to thank to the
anonymous referees for valuable comments and remarks to the manuscript
of this paper.
\begin{thebibliography}{99}
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criteria: perturbation in term involving derivative, preprint.
\bibitem{DR} O. Do\v sl\'y, P. \v Reh\' ak, {Half-linear Differential
Equations}, North-Holland Mathematics Studies 202., Amsterdam:
Elsevier Science (2005), 517 p.
\bibitem{Li} W. T. Li, S. S. Cheng, An oscillation criterion for
nonhomogeneous half-linear differential equations, Appl. Math. Lett.
15 (2002), 259--263.
\bibitem{Sun} Y. G. Sun, J. S.W. Wong, Oscillation criteria for second
order forced ordinary differential equations with mixed
nonlinearities, J. Math. Anal. Appl. \textbf{334} (2007), 549--560.
\bibitem{Zheng} Z. Zheng, X. Wang, H. Han, Oscillation criteria for
forced second order differential equations with mixed
nonlinearities, Appl. Math. Lett. \textbf{22} (2009), 1096--1101.
\end{thebibliography}
\end{document}
For all intents and purposes, I use an up-to-date MikTeX distribution on windows 11.
