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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 151, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/151\hfil Forced oscillation]
{Forced oscillation for higher order functional differential equations}

\author[Y. G. Sun, T. S. Hassan \hfil EJDE-2012/151\hfilneg]
{Yuan Gong Sun, Taher S. Hassan} 

\address{Yuan Gong Sun \newline
School of Mathematical Sciences,
University of Jinan, Jinan 250022, China}
\email{sunyuangong@yahoo.cn}

\address{Taher S. Hassan \newline
Department of Mathematics, Faculty of Science\\
Mansoura University\\
Mansoura, 35516, Egypt}
\email{tshassan@mans.edu.eg}

\thanks{Submitted June 5, 2012. Published August 31, 2012.}
\subjclass[2000]{34C10, 34C15}
\keywords{Oscillation; higher order; differential equations}

\begin{abstract}
 We establish some oscillation criteria for the solutions to
 forced higher-order differential equations. We do not assume that
 the forcing term is the $n$-th derivative of an oscillatory function,
 and do not assume that the coefficients are of a definite sign.
 Our results are illustrated with examples.
\end{abstract}

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


\section{Introduction}

In the previous 50 years, there has been increasing interest in obtaining
sufficient conditions for the oscillation/nonoscillation of solutions of
different classes of differential equations; see for example
\cite{a,4,5,cc,ct,6,somaes1,soma,8,9} and
the references cited therein.
Regarding forced higher-order differential equations, one can use a
technique  introduced by Kartsatos \cite{8,9}, which assumes that the
forcing term $f(t)$ is the $n$-th derivative of an oscillatory
function $h(t)$ satisfying $\lim_{t\to \infty }h(t)=0$. Under
certain conditions, he found that the forced equation would remain
oscillatory if the unforced equation is oscillatory.
 Agarwal and Grace \cite{1} studied the superlinear differential equation
\begin{equation}
x^{(n)}(t)+p(t) x(t)|^{\alpha
-1}x(t)=f(t),\quad t\in [ t_0,\infty ),  \label{e1.0}
\end{equation}
where $p(t)<0$ and $\alpha >1$, using general means without
imposing the Kartsatos condition.
 Ou and Wong \cite{13}  investigated the  equation
\[
x^{(n)}(t)+p(t)g(x(t))=f(t),\quad t\in [t_0,\infty),
\]
assuming that $p(t)\geq 0$ ($<0$) on $[t_0,\infty )$, $xg(x)>0$ for
$x\neq 0$, and there exists a constant $c>0$ such that
$|g(x)|\geq c| x|^{\alpha }$, for $\alpha >1$,
or $| g(x)|\leq c| x|^{\alpha }$, for
$0<\alpha <1$. Sun and Wong \cite{sw} studied \eqref{e1.0} when $0<\alpha <1$,
and they do not assume that $p(t)$ is of definite sign.
Recently, \c{C}akmak and Tiryaki \cite{ct}
established some oscillation criteria for the forced higher-order differential
equation
\begin{equation} \label{e1.00}
x^{(n)}(t)+\sum_{i=1}^{n-1}a_ix^{(i)}(t)+q(t)f(x(g(t)))=e(t),
\end{equation}
where $a_i$ are real constants, $q(t)$, $f(t)$,
$e(t)$ and $g(t)$ are real continuous functions,
$xf(x)>0$ whenever $x\neq 0$ and $\lim_{t\to \infty}g(t)=\infty $.
We refer the reader for more oscillation results to \cite{ct,sm,yang}, and
to \cite{7,12,25} for oscillatory potentials.

The purpose of this paper is to extend the oscillation criteria to higher
order differential equations
\begin{equation}
\sum_{i=1}^{n}a_ix^{(i)}(t)+\Phi \Big(t,x(h(
t)),x(g(t)),x(l(t) )\Big)=f(t)  \label{e1.1}
\end{equation}
and
\begin{equation}
\sum_{i=1}^{n}a_ix^{(i)}(t)-\Phi (t,x(h(
t)),x(g(t)),x(l(t) ))=f(t),  \label{e1.2}
\end{equation}
where $a_i$ are real numbers with $a_{n}\equiv 1$, and $h$, $g$, $l$ and $
f $ are real continuous functions satisfying
\[
\lim_{t\to \infty }h(t)=\lim_{t\to \infty
}g(t)=\lim_{t\to \infty }l(t)=\infty \,.
\]
Here $\Phi:[t_0,\infty)\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous
function satisfying conditions \eqref{d1} and  \eqref{d2} below.
 It is easy to see that when $\Phi=p(t)| x(t)|^{\alpha-1}x(t)$,
$\Phi=p(t)g(x(t))$ and $\Phi=q(t)f(x(g(t)))$, Equation \eqref{e1.1}
reduces to  \eqref{e1.0} and \eqref{e1.00}, respectively.

A solution is said to be oscillatory if it has arbitrarily large zeros;
i.e., for any $T>0$ there exists a $t\geq T$ such that $x(t)=0$.

To the best of our knowledge \eqref{e1.1} and \eqref{e1.2} have not been
considered earlier. We hope to kindle the reader's interest in further
research on the oscillation of these equations that arise, for
example, in population growth with competitive species.
Also, we want to present interesting examples that illustrate the importance of our
results.

\section{Main results}

In the following, we consider  a nonnegative kernel  $H(t,s)$ defined on the set
$\mathbb{D}:=\{(t,s):t\geq s\geq t_0\} $. We shall assume that $H(t,s)$
is sufficiently smooth in the variable $s$, so that the following conditions
are satisfied:
\begin{itemize}
\item[(H1)] $H(t,t)=0$, $H(t,s)\geq 0$  for $t\geq s\geq t_0$,

\item[(H2)] The partial derivatives satisfy:
\[
H_i(t,s)=(-1)^i\frac{\partial ^iH}{\partial
s^i},\quad i=0,1,\dots ,n\text{ for}t>s\geq t_0,
\]

\item[H3)] $H_i(t,t)=0$, $i=0,1,\dots ,n-1$,

\item[(H4)] $H^{-1}(t,t_0)H_i(t,t_0)=O(1)$
 as $t\to \infty $  for $i=1,2,\dots ,n-1$.

\end{itemize}

Throughout the paper, for $t\geq s\geq t_0$,  we let
\begin{gather*}
d_{+}:=\max \{0,d\},\quad d_{-}:=\max \{ 0,-d\} ,\\
h(t,s):=\sum_{i=1}^{n}a_i\frac{H_i(t,s)}{H^{1/\gamma }(t,s)}.
\end{gather*}

\begin{theorem}\label{t1}
Assume that there exists a nonnegative function $G_{1}(t,s)$ defined
on  $[t_0,\infty )\times [t_0,\infty )$ such that
\begin{equation}
\sum_{i=1}^{n}a_iH_i(t,s)x+H(t,s)\Phi(s,y,z,w)
\begin{cases}
\geq G_{1}(t,s), & \text{if } x,y,z,w>0 \\
\leq -G_{1}(t,s), & \text{if } x,y,z,w<0
\end{cases} \label{d1}
\end{equation}
for $t>s\geq t_0$,
\begin{equation}
\limsup_{t\to \infty }\frac{1}{H(t,t_0)}
\int_{t_0}^{t}(H(t,s)f(s)+G_{1}(t,s))ds=\infty
\label{uu1}
\end{equation}
and
\begin{equation}
\liminf_{t\to \infty }\frac{1}{H(t,t_0)}
\int_{t_0}^{t}(H(t,s)f(s)-G_{1}(t,s))ds=-\infty .
\label{uu2}
\end{equation}
Then every solution of \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Let $x$ be a non-oscillatory solution of \eqref{e1.1} on $[t_0,\infty)$.
First assume that $x(t)>0$ on some interval  $[T,\infty )$, $t\geq t_0$.
Multiplying both sides of \eqref{e1.1}  by $H(t,s)$, with $t$
replaced by $s$, for $t\geq s\geq 0$ integrating with respect
to $s$ from $T$ to $t$, we have
\begin{align*}
&\int_{T}^{t}H(t,s)f(s)ds \\
&=\int_{T}^{t}H(t,s)\sum_{i=1}^{n}a_ix^{(i)}(s)ds
 +\int_{T}^{t}H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s)))ds \\
&=\sum_{i=1}^{n}a_i\int_{T}^{t}H(t,s)x^{(i)}(s)ds
+\int_{T}^{t}H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s)))ds.
\end{align*}
Integrating by parts and using (H1), (H2) and (H3).
For $i=1,2,3,\dots ,n$, we obtain
\begin{align*}
&\int_{T}^{t}H(t,s)x^{(i)}(s)ds\\
&=-H(t,T)x^{(i-1)}(T)-
\sum_{j=1}^{i-1}H_{j}(t,T)x^{(i-j-1)}(T)+\int_{T}^{t}H_i(t,s)x(s)ds,
\end{align*}
where $\sum_{j=1}^{0}=0$. This implies
\begin{align*}
&\int_{T}^{t}H(t,s)f(s)ds\\
&=-\sum_{i=1}^{n}a_i\Big[H(t,T)x^{(i-1)}(T)+
\sum_{j=1}^{i-1}H_{j}(t,T)x^{(i-j-1)}(T)\Big]
\\
&\quad +\int_{T}^{t}\Big[\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi
(s,x(h(s)),x(g(s)),x(l(s)))\Big] ds.
\end{align*}
In view of (H4), there exists a constant $M$ such that
\[
-\sum_{i=1}^{n}a_i\Big[H(t,T)x^{(i-1)
}(T)+\sum_{j=1}^{i-1}H_{j}(t,T)x^{(i-j-1)}(T)\Big] \geq MH(t,T),
\]
which implies
\begin{align*}
&\int_{T}^{t}H(t,s)f(s)ds\\
&\geq MH(t,T) +\int_{T}^{t}\Big[\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi
(s,x(h(s)),x(g(s)),x(l(s)))\Big] ds.
\end{align*}
Then \eqref{d1} implies
\[
\frac{1}{H(t,T)}\int_{T}^{t}(H(t,s)f(s)-G_{1}(t,s))ds\geq M,
\]
which leads to a contradiction to \eqref{uu2}.

Next, we assume that $x$ satisfies \eqref{e1.1} and is eventually negative.
The same argument as above leads to a contradiction, provided
\[
\limsup_{t\to \infty }\frac{1}{H(t,T)}\int_{T}^{t}(
H(t,s)f(s)+G_{1}(t,s))ds=\infty ,
\]
which indeed holds due to  condition \eqref{uu1}.
\end{proof}

\begin{theorem}\label{t2}
Assume there exists a nonnegative function $G_{2}(t,s)$  defined
on $[t_0,\infty )\times [ t_0,\infty )$ such that
\begin{equation}
\sum_{i=1}^{n}a_iH_i(t,s)x-H(t,s)\Phi (s,y,z,w)
\begin{cases}
\leq G_{2}(t,s), & \text{if } x,y,z,w>0 \\
\geq-G_{2}(t,s), & \text{if } x,y,z,w<0
\end{cases}
 \label{d2}
\end{equation}
for $t>s\geq t_0$,
\begin{equation}
\limsup_{t\to \infty }\frac{1}{H(t,t_0)}
\int_{t_0}^{t}(H(t,s)f(s)-G_{2}(t,s))ds=\infty
\label{pp}
\end{equation}
and
\begin{equation}
\liminf_{t\to \infty }\frac{1}{H(t,t_0)}
\int_{t_0}^{t}(H(t,s)f(s)+G_{2}(t,s))ds=-\infty .
\label{ppp}
\end{equation}
Then every solution of  \eqref{e1.2} is oscillatory.
\end{theorem}

\begin{proof}
Let $x$ be a non-oscillatory solution of \eqref{e1.2} on $[t_0,\infty)$.
First assume that $x(t)>0$ on some interval $[T,\infty)$, $T\geq t_0$.
 As in the proof of Theorem \ref{t1}, we obtain
\begin{align*}
&\int_{T}^{t}H(t,s)f(s)ds\\
&=-\sum_{i=1}^{n}a_i\Big[H(t,T)x^{(i-1)}(T)+
\sum_{j=1}^{i-1}H_{j}(t,T)x^{(i-j-1)}(T)\Big]
\\
&\quad +\int_{T}^{t}\Big[\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (
s,x(h(s)),x(g(s)),x(l(s)))\Big] ds.
\end{align*}
Again, by (H4), there exists a constant $C$ such that
\[
-\sum_{i=1}^{n}a_i\Big[H(t,T)x^{(i-1)}(T)+
\sum_{j=1}^{i-1}H_{j}(t,T)x^{(i-j-1)}(T)\Big] \leq CH(t,T),
\]
which implies
\begin{align*}
&\int_{T}^{t}H(t,s)f(s)ds\\
&\leq CH(t,T) +\int_{T}^{t}\Big[\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (
s,x(h(s)),x(g(s)),x(l(s)))\Big] ds.
\end{align*}
From this  inequality and \eqref{d2} we obtain
\[
\frac{1}{H(t,T)}\int_{T}^{t}[H(t,s)f(s)-G_{2}(t,s)] ds\leq C,
\]
which leads to a contradiction to \eqref{pp}.

The proof for $x(t)<0$ is similar to the first part.
\end{proof}

As particular cases, we present  oscillation criteria for the
differential equations
\begin{equation}
\sum_{i=1}^{n}a_ix^{(i)}(t)+r(t)\psi _{\gamma }(x(
h(t)))+p(t)\psi _{\alpha }(x(g(
t)))+q(t)\psi _{\beta }(x(l(t)
))=f(t)  \label{111}
\end{equation}
and
\begin{equation}
\sum_{i=1}^{n}a_ix^{(i)}(t)-r(t)\psi _{\gamma }(x(
h(t)))+p(t)\psi _{\alpha }(x(g(
t)))+q(t)\psi _{\beta }(x(l(t)
))=f(t)  \label{122}
\end{equation}
which satisfy conditions \eqref{d1} and  \eqref{d2} respectively,
where $\psi _{\gamma }(u):=| u|^{\gamma -1}u$, $\gamma >0 $, and where
$r$ and $q$ are real continuous functions and $p$ is a
positive function.
 Our interest is to establish oscillation criteria for
 \eqref{111} and \eqref{122} without assuming that $r,q,f$
are of definite sign and without assuming that $f(t)$ is  the $n$-th derivative
of an oscillatory function.

\begin{corollary}\label{c1}
Let $h(t)\equiv g(t)\equiv l(t)\equiv t$ on $[t_0,\infty )$.
Assume that\ $0<\gamma <1$ and $\alpha >\beta >\gamma $ hold and
$\sum_{i=1}^{n}a_iH_i(t,s)>0$, for $t\geq s\geq t_0$. If \eqref{uu1} and \eqref{uu2}
 are satisfied,
where
\begin{gather*}
G_{1}(t,s):=(\gamma -1)\gamma ^{\gamma /(1-\gamma )
}| Q_{1}(s)|^{1/(1-\gamma )}|h(t,s)|^{\gamma /(\gamma -1)},
\\
Q_{1}(s):=r_{-}(s)+\sigma _{1}p^{(\gamma -\beta
)/(\alpha -\beta )}(s)q_{-}^{(\alpha -\gamma )/(\alpha -\beta)}(s),
\\
\sigma _{1}:=(\alpha -\beta )(\alpha -\gamma )^{(\gamma -\alpha )/(\alpha
-\beta )}(\beta -\gamma )^{(\beta -\gamma )/(\alpha -\beta )},
\end{gather*}
then every solution of  \eqref{111} is oscillatory.
\end{corollary}

\begin{proof}
Let $x$ be non-oscillatory solution of \eqref{111} on $[t_0,\infty)$.
First assume that $x(t)>0$ on some interval $[T,\infty )$.
 We claim that \eqref{d1} is satisfied with $x=x(s)$, $y=x(h(s))$, $z=x(g(s))$ and
$w=x(l(s))$. As in the proof of
Theorem \ref{t1}, we obtain
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&=\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Big[r(s)x^{\gamma }(s)+(
p(s)x^{\alpha }(s)+q(s)x^{\beta }(s))\Big]  \\
&\geq \sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)r_{-}(s)x^{\gamma}(s)
+H(t,s)x^{\gamma }(s)(p(s)x^{\alpha -\gamma }(s)\\
&\quad -q_{-}(s)x^{\beta -\gamma }(s))
\end{align*}
For a given $s$, set $F(x):=px^{\alpha -\gamma }-q_{-}x^{\beta
-\gamma },\ $for $x>0$ and $\alpha >\beta >\gamma >0$. Thus $F$ obtains its
minimum at
\[
x=(\alpha -\gamma )^{1/(\beta -\alpha )}(\beta -\gamma )^{1/(\alpha -\beta
)}p^{(\gamma -\beta )/((\alpha -\beta )
(\alpha -\gamma ))}(q_{-})^{1/(\alpha
-\beta )},
\]
and
\[
F_{\rm min}=-\sigma _{1}p^{(\gamma -\beta )/(\alpha -\beta )}
(q_{-})^{(\alpha -\gamma )/(\alpha -\beta )}.
\]
Then
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&\geq \sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)Q_{1}(s)
x^{\gamma }(s).
\end{align*}
Define $X\geq 0$ and $Y>0$ by
\[
X^{\gamma }:=HQ_{1}x^{\gamma },\quad Y^{\gamma -1}:=\gamma
^{-1}Q_{1}^{-1/\gamma }h.
\]
Then, using the inequality (see \cite{w})
\begin{equation}
\gamma XY^{\gamma -1}-X^{\gamma }\geq (\gamma -1))Y^{\gamma },\qquad
0<\gamma <1,  \label{0h}
\end{equation}
we obtain
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&\geq \sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)Q_{1}(s)
x^{\gamma }(s) \\
&\geq (\gamma -1)\gamma ^{\gamma /(1-\gamma )
}Q_{1}^{1/(1-\gamma )}(s)h^{\gamma /(\gamma -1)
}(t,s)=G_{1}(t,s).
\end{align*}
Then from Theorem \ref{t1}, we obtain the desired result.
The proof for $x(t)<0$ similar to the first part of this proof.
\end{proof}

\begin{corollary}\label{c2}
Let $h(t)\equiv t$ and $l(t)\equiv g(t)$ on $[t_0,\infty )$.
 Assume that $0<\gamma <1$ and $\alpha >\beta >0$ hold and
$\sum_{i=1}^{n}a_iH_i(t,s)>0$, for $t\geq s\geq t_0$.
If \eqref{uu1} and \eqref{uu2} are satisfied, where
\begin{gather*}
\begin{aligned}
G_{1}(t,s)&:=(\gamma -1)\gamma ^{\gamma /(1-\gamma )
}r_{-}^{1/(1-\gamma )}(s)h^{\gamma /(\gamma -1)}(t,s)\\
&\quad +\delta H(t,s)p^{\beta /(\beta -\alpha )}(
s)q_{-}^{\alpha /(\alpha -\beta )}(s),
\end{aligned}\\
\delta :=(\beta -\alpha )\alpha ^{\alpha /(\beta -\alpha )}
\beta ^{\beta/(\alpha -\beta )},
\end{gather*}
then every solution of  \eqref{111} is oscillatory.
\end{corollary}

\begin{proof}
Let $x$ be a nonoscillatory solution of \eqref{111}  on $[t_0,\infty)$.
First assume that $x(t)>0$ on some interval $[T,\infty )$.
We claim that \eqref{d1} holds for $x=x(s)$, $y=x(h(s))$, $z=x(g(s))$ and $w=x(l(s))$.
As in the proof of Theorem \ref{t1}, we obtain
\begin{equation} \label{k3}
\begin{split}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&= \sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Big[r(s)x^{\gamma }(s)+(
p(s)x^{\alpha }(s)+q(s)x^{\beta }(s))\Big]   \\
&\geq \sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)r_{-}(s)x^{\gamma
}(s)+H(t,s)(p(s)x^{\alpha }(g(s))\\
&\quad -q_{-}(s)x^{\beta}(g(s)))
\end{split}
\end{equation}
As in the proof of Corollary \ref{c1}, we obtain
\begin{equation}
px^{\alpha }-q_{-}x^{\beta }\geq \delta p^{\beta /(\beta -\alpha
)}q_{-}^{\alpha /(\alpha -\beta )},  \label{k2}
\end{equation}
where $\delta =(\beta -\alpha )\alpha ^{\alpha /(\beta -\alpha )}\beta
^{\beta /(\alpha -\beta )}$
and
\begin{equation}
\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)r_{-}(s)x^{\gamma
}(s)\geq (\gamma -1)\gamma ^{\gamma /(1-\gamma )
}r_{-}^{1/(1-\gamma )}(s)h^{\gamma /(\gamma
-1)}(t,s).  \label{k1}
\end{equation}
Using \eqref{k2} and \eqref{k1} in \eqref{k3}, we have
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)+H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&\geq (\gamma -1)\gamma ^{\gamma /(1-\gamma )
}r_{-}^{1/(1-\gamma )}(s)h^{\gamma /(\gamma
-1)}(t,s)+\delta H(t,s)p^{\beta /(\beta -\alpha
)}q_{-}^{\alpha /(\alpha -\beta )}\\
&=G_{1}(t,s).
\end{align*}
Then from Theorem \ref{t1}, we obtain that every solution of equation
\eqref{111} is oscillatory.
\end{proof}

\begin{example} \rm
Consider the equation
\begin{equation}
\begin{split}
&\sum_{i=1}^{n}a_ix^{(i)}(t)+\phi _{1}(t)\psi
_{\gamma }(x(t))+t^{\frac{\mu }{3}}\psi _{3}(
x(g(t)))+t^{\frac{8\mu }{27}}\cos ^{1/9}(t)\psi _{\frac{8}{3}}(x(g(t)
))\\
&=t^{\mu }\cos (s),
\end{split} \label{f1}
\end{equation}
where $0<\gamma <1$, $\phi _{1}(t)\geq 0$, for $t\geq t_0$,
$a_i\geq 0$, $i=1,2,\dots ,n-1$ and $a_{n}\equiv 1$.
By taking $H(t,s)=(t-s)^{n}$. It is easy to see that (H1)-(H4) are satisfied
 and $\sum_{i=1}^{n}a_iH_i(t,s)>0$. Applying Corollary \ref{c2},  every
solution of  \eqref{f1} is oscillatory if $\mu >n$.
\end{example}

\begin{corollary}\label{c3}
Let $q(t)$ is a nonnegative function and $h(t)\equiv t$ on $[t_0,\infty )$.
 Assume that $0<\gamma <1$ and $\sum_{i=1}^{n}a_iH_i(t,s)>0$, for
$t\geq s\geq t_0$ hold. If \eqref{uu1} and \eqref{uu2} are satisfied, where
\[
G_{1}(t,s):=(\gamma -1)\gamma ^{\gamma /(1-\gamma )
}r_{-}^{1/(1-\gamma )}(s)h^{\gamma /(\gamma-1)}(t,s)
\]
then every solution of  \eqref{111} is oscillatory.
\end{corollary}

\begin{corollary} \label{c4}
Let $r(t)$ and $q(t)$ be nonnegative functions on $[t_0,\infty )$.
Assume that $\sum_{i=1}^{n}a_iH_i(t,s)\geq 0$, for $t\geq s\geq t_0$ holds.
If \eqref{uu1} and
\eqref{uu2} are satisfied, then every solution of
\eqref{111} is oscillatory.
\end{corollary}

\begin{example} \rm
Consider the equation \eqref{111} with $f(t)=t^{\mu }\sin (t)$, where
 $n\geq 2$, $\mu $, $a_i\geq 0$, $i=1,2,\dots ,n-1$
and $a_{n}\equiv 1$. Also assume $r(t)$ and $q(t)$ be nonnegative functions on
$[t_0,\infty ).$ By taking $H(t,s)=(t-s)^{\lambda },\ \lambda >n-1$.
It is easy to see that $H(t,s)$ satisfies all of (H1)--(H4) and
$\sum_{i=1}^{n}a_iH_i(t,s)\geq 0$. Then, by Corollary \ref{c4}, every
solution of equation \eqref{111} is oscillatory if $\mu >\lambda$.
\end{example}

\begin{corollary} \label{c5}
Let $q(t)$ be a nonnegative function and $h(t)\equiv g(t)\equiv l(t)\equiv t$ on
$[t_0,\infty )$. Assume that $\gamma >1$, $\alpha >\gamma >\beta >0$ and
$Q_{2}(t)<0$, for $t\geq t_0$. If \eqref{pp} and
\eqref{ppp} are satisfied, where
\begin{gather*}
G_{2}(t,s):=(\gamma -1)\gamma ^{\gamma /(1-\gamma )
}| Q_{2}(s)|^{1/(1-\gamma )}|h(t,s)|^{\gamma /(\gamma -1)},
\\
Q_{2}(s):=-r(s)-\sigma _{2}p^{(\gamma -\beta )
/(\alpha -\beta )}(s)q^{(\alpha -\gamma )/(\alpha -\beta)}(s),
\\
\sigma _{2}:=(\alpha -\beta )(\alpha -\gamma )^{(\gamma -\alpha )/(\alpha
-\beta )}(\gamma -\beta )^{(\beta -\gamma )/(\alpha -\beta )},
\end{gather*}
then every solution of  \eqref{122} is oscillatory.
\end{corollary}

\begin{proof}
Let $x$ a non-oscillatory solution of \eqref{122}  on $[t_0,\infty)$.
First assume that $x(t)>0$ on some interval $[T,\infty )$.
We claim that \eqref{d2} is satisfied with $x=x(s)$, $y=x(h(s))$, $z=x(g(s))$
and $w=x(l(s))$. As in the proof of
Theorem \ref{t2}, we obtain
\begin{equation} \label{j1}
\begin{split}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s)))
 \\
&=\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\big[r(s)x^{\gamma }(s)+(
p(s)x^{\alpha }(s)+q(s)x^{\beta }(s))\big]    \\
&\leq \big| \sum_{i=1}^{n}a_iH_i(t,s)\big| x(s)-H(t,s)r(s)x^{\gamma }(s)   \\
&\quad -H(t,s)x^{\gamma }(s)(p(s)x^{\alpha -\gamma }(s)+q(s)x^{\beta
-\gamma }(s)).
\end{split}
\end{equation}
For a given $s$, set $K(x):=px^{\alpha -\gamma }+qx^{\beta-\gamma }$,
for $x>0$ and $\alpha >\gamma >\beta >0$. Thus $K$ obtains its minimum at
\[
x=(\alpha -\gamma )^{1/(\beta -\alpha )}(\gamma -\beta
)^{1/(\alpha -\beta )}p^{(\gamma -\beta )/((\alpha
-\beta )(\alpha -\gamma ))}q^{1/(\alpha -\beta )}
\]
and
\[
K_{\rm min}=\sigma _{2}p^{(\gamma -\beta )/(\alpha -\beta
)}q^{(\alpha -\gamma )/(\alpha -\beta )},
\]
where $\sigma _{2}=(\alpha -\beta )(\alpha -\gamma )^{(\gamma -\alpha
)/(\alpha -\beta )}(\gamma -\beta )^{(\beta -\gamma )/(\alpha -\beta )}$.
Then, from this and \eqref{j1}, we obtain
\begin{equation} \label{e1}
\begin{split}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&\leq | \sum_{i=1}^{n}a_iH_i(t,s)|x(s)-H(t,s)| Q_{2}(s)|x^{\gamma }(s).
\end{split}
\end{equation}
Define $X\geq 0$ and $Y\geq 0$ by
\[
X^{\gamma }:=H| Q_{2}|x^{\gamma },\quad
Y^{\gamma-1}:=\gamma ^{-1}| Q_{2}|^{-1/\gamma }|h|.
\]
Then, using the inequality (see \cite{w})
\begin{equation}
\gamma XY^{\gamma -1}-X^{\gamma }\leq (\gamma -1)Y^{\gamma },\qquad \gamma
>1,  \label{0h0}
\end{equation}
we obtain
\begin{align*}
&| \sum_{i=1}^{n}a_iH_i(t,s)|x(s)-H(t,s)|
Q_{2}(s)|x^{\gamma }(s) \\
&\leq (\gamma -1)\gamma ^{\gamma /(1-\gamma )}| Q_{2}(
s)|^{1/(1-\gamma )}| h(t,s)
|^{\gamma /(\gamma -1)}=G_{2}(t,s).
\end{align*}
Then from Theorem \ref{t2}, we obtain the desired result.
The proof for $x<0$ is similar to the case above.
\end{proof}

\begin{corollary}
Let $h(t)\equiv g(t)\equiv l(t)\equiv t$ on $[t_0,\infty )$.
Assume that $\gamma >1$, $\alpha >\beta >\gamma >0$
and $Q_{3}(t)<0$, for $t\geq t_0$. If \eqref{pp} and
\eqref{ppp} are satisfied, where
\begin{gather*}
G_{2}(t,s):=(\gamma -1)\gamma ^{\gamma /(1-\gamma )
}| Q_{3}(s)|^{1/(1-\gamma )}|
h(t,s)|^{\gamma /(\gamma -1)},
\\
Q_{3}(s):=-r(s)+\sigma _{1}p^{(\gamma -\beta )/(\alpha -\beta
)}(s)q_{-}^{(\alpha -\gamma )/(\alpha -\beta )}(s),
\\
\sigma _{1}:=(\alpha -\beta )(\alpha -\gamma )^{(\gamma -\alpha )/(\alpha
-\beta )}(\beta -\gamma )^{(\beta -\gamma )/(\alpha -\beta )},
\end{gather*}
then every solution of  \eqref{122} is oscillatory.
\end{corollary}

\begin{proof}
Let $x$ be a non-oscillatory solution of  \eqref{122}  on some interval
$[t_0,\infty)$.  We claim that \eqref{d2} holds for $x=x(s)$, $y=x(h(s))$,
$z=x(g(s))$ and $w=x(l(s))$. As in the proof of Theorem \ref{t2}, we obtain
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&=\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\big[r(s)x^{\gamma }(s)+(
p(s)x^{\alpha }(s)+q(s)x^{\beta }(s))\big]  \\
&\leq | \sum_{i=1}^{n}a_iH_i(t,s)|
x(s)-H(t,s)r(s)x^{\gamma }(s) \\
&\quad -H(t,s)x^{\gamma }(s)(p(s)x^{\alpha -\gamma }(s)-q_{-}(s)x^{\beta
-\gamma }(s)).
\end{align*}
As in the proof of Corollary \ref{c5}, we have
\[
px^{\alpha -\gamma }-p_{-}x^{\beta -\gamma }\geq -\sigma _{1}p^{(\gamma
-\beta )/(\alpha -\beta )}q_{-}^{(\alpha -\gamma )/(\alpha -\beta )},
\]
which implies
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s)) \\
&\leq | \sum_{i=1}^{n}a_iH_i(t,s)|
x(s)ds-H(t,s)| Q_{3}(s)|x^{\gamma }(s).
\end{align*}
The rest of the proof is the same as the proof of Corollary \ref{c5} with
$Q_{2}$ replaced by $Q_{3}$.
\end{proof}

\begin{corollary}\label{c6}
 Let $r(t)$ be a positive function, and $h(t)\equiv t$ and $l(t)\equiv g(t)$ on
$[t_0,\infty )$. Assume that $\gamma >1$ and $\alpha >\beta >0$.
If \eqref{pp} and \eqref{ppp} are satisfied, where
\begin{gather*}
\begin{aligned}
G_{2}(t,s) &:=(\gamma -1)\gamma ^{\gamma /(1-\gamma )}| r(s)|^{1/(1-\gamma )}
| h(t,s)|^{\gamma /(\gamma -1)} \\
&\quad -\delta H(t,s)p^{\beta /(\beta -\alpha )}(s)q_{-}^{\alpha/(\alpha -\beta )}(s),
\end{aligned}\\
\delta :=(\beta -\alpha )\alpha ^{\alpha /(\beta -\alpha )}\beta
^{\beta/(\alpha -\beta )},
\end{gather*}
then every solution of \eqref{122} is oscillatory.
\end{corollary}

\begin{proof}
Let $x$  a nonoscillatory solution of  \eqref{122} on some interval
$[t_0,\infty)$.
 As in the proof of Theorem \ref{t2}, we obtain
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&=\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\big[r(s)x^{\gamma }(s)+(
p(s)x^{\alpha }(s)+q(s)x^{\beta }(s))\big]  \\
&\leq | \sum_{i=1}^{n}a_iH_i(t,s)|
x(s)-H(t,s)| r(s)|x^{\gamma }(s) \\
&\quad -H(t,s)(p(s)x^{\alpha }(g(s))-q_{-}(s)x^{\beta}(g(s))).
\end{align*}
As in the proof of Corollary \ref{c2}, we have
\[
px^{\alpha }-q_{-}x^{\beta }\geq \delta p^{\beta /(\beta -\alpha
)}q_{-}^{\alpha /(\alpha -\beta )},
\]
and as in the proof of Corollary \ref{c5}, we obtain
\[
| \sum_{i=1}^{n}a_iH_i(t,s)|x(s)-H(t,s)|
r(s)|x^{\gamma }(s)\leq (\gamma -1)\gamma ^{\gamma
/(1-\gamma )}| r(s)|^{1/(1-\gamma
)}| h(t,s)|^{\gamma /\gamma -1}.
\]
Then
\begin{align*}
&\sum_{i=1}^{n}a_iH_i(t,s)x(s)-H(t,s)\Phi (s,x(h(s)),x(g(s)),x(l(s))) \\
&\leq (\gamma -1)\gamma ^{\gamma /(1-\gamma )}| r(s)
|^{1/(1-\gamma )}| h(t,s)|
^{\gamma /\gamma -1}-H(t,s)\delta p^{\beta /(\beta -\alpha )}q_{-}^{\alpha
/(\alpha -\beta )}=G_{2}(t,s),
\end{align*}
which implies that every solution of  \eqref{122} is
oscillatory.
\end{proof}

\begin{example} \rm
Consider the equation
\begin{equation}
\begin{split}
&x'(t)+\phi _{2}(t)\psi _{\gamma }(x(t))\\
&=e^{\frac{5t}{3}}\psi _{\frac{7}{3}}(x(g(
t)))+e^{\frac{10}{21}t}\sin ^{\frac{5}{7}}(
t)\psi _{\frac{2}{3}}(x(g(t)))
+e^{t}\sin (t),
\end{split}  \label{z1}
\end{equation}
where $\gamma >1$ and $\phi _{2}(t)<0$, for $t\geq t_0$. By
taking $H(t,s)=1$ and $t>s\geq t_0$,  it is easy to see that
$H(t,s)$ satisfies all of (H1)--(H4).
Applying Corollary \ref{c6}, every solution of \eqref{z1} is
oscillatory.
\end{example}

\begin{corollary} \label{c7}
Let $r(t)$ be a nonnegative function and $h(t)\equiv t$ on $[t_0,\infty )$.
 Assume that $\gamma >1$ holds. If \eqref{pp} and \eqref{ppp} are satisfied, where
\[
G_{2}(t,s):=(\gamma -1)\gamma ^{\gamma /(1-\gamma )}r^{1/(1-\gamma )}(
s)| h(t,s)|^{\gamma /\gamma -1},
\]
then every solution of  \eqref{122} is oscillatory.
\end{corollary}

\begin{corollary} \label{c8}
Let $r(t)$ and $q(t)$ be nonnegative functions , and assume that \\
$\sum_{i=1}^{n}a_iH_i(t,s)\leq 0$, for $t\geq s\geq t_0$ holds.
If \eqref{pp} and \eqref{ppp} are satisfied,
then every solution of  \eqref{122} is oscillatory.
\end{corollary}

\begin{example} \rm
Consider  equation \eqref{122} with
 $f(t)=e^{t}\sin (t)$, where $n\geq 2$, $a_i\leq 0$, $i=1,2,\dots ,n-1$ and
$a_{n}\equiv 1$. Also, we assume that $r(t)$ and $q(t)$ be
nonnegative functions. By taking $H(t,s)=(t-s)^{\lambda }$ and $\lambda >n-1$,
it is easy to see that $H(t,s)$ satisfies (H1)--(H4) and
$\sum_{i=1}^{n}a_iH_i(t,s)\leq 0$. Then, by Corollary \ref{c8}, every solution
of  \eqref{122} is oscillatory.
\end{example}

The above results are extendable to neutral equation in this from
\begin{align*}
&\sum_{i=1}^{n}a_iy^{(i)}(t)+r(t)\Phi _{\gamma }(
x(h(t)))+\sum_{i=1}^{n}\Big[p_i(t)\Phi
_{\alpha _i}(x(g_i(t)))
+q_i(t)\Phi _{\beta _i}(x(l_i(t)))\Big] \\
&\quad +\overline{r}(t)\Phi _{\overline{\gamma }}(x(\overline{h}
(t)))+\sum_{j=1}^{m}\Big[\overline{p}
_{j}(t)\Phi _{\overline{\alpha }_{j}}(x(\overline{g}_{j}(
t)))+\overline{q}_{j}(t)\Phi _{\overline{\beta }
_{j}}(x(\overline{l}_{j}(t)))\Big]
\\
&= a(t)x(t)+b(t)x(t-\tau)+f(t)
\end{align*}
where $y(t):=x(t)+\delta (t)x(\tau (t))$
with $\Phi _{\eta }(u):=| u|^{\eta -1}u$, $\eta >0$, and where $a_i$ are
real numbers with $a_{n}\equiv 1$ and $r$, $\overline{r}$ $p_i$,
$\overline{p}_{j}$, $q_{j}$, $\overline{q}_{j}$, $h$, $\overline{h}$, $g_i$,
$\overline{g}_{j}$, $l_i$, $\overline{l}_{j}$, $a$, $b$ and $f$ are real continuous
functions such that $a, b, p_i, \overline{p}_{j}$ are positive, and
and
$\lim_{t\to \infty }h(t)=\lim_{t\to \infty }\overline{h}(t)
=\lim_{t\to \infty }g_i(t)=\lim_{t\to \infty }\overline{g}_{j}(t)
=\lim_{t\to \infty }l_i(t)=\lim_{t\to \infty }\overline{l}_{j}(t)=\infty $.
The details are left to the reader to check them.

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\end{document}