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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 174, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/174\hfil Oscillation criteria]
{Oscillation criteria for odd-order nonlinear
differential equations with advanced and delayed arguments}

\author[E. Thandapani, S. Padmavathi,  S.  Pinelas \hfil EJDE-2014/174\hfilneg]
{Ethiraju Thandapani, Sankarappan Padmavathy, Sandra Pinelas}  

\address{Ethiraju Thandapani \newline
Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai 600 005, India}
\email{ethandapani@yahoo.co.in}

\address{Sankarappan Padmavathy \newline
Ramanujan Institute for Advanced Study in Mathematics, 
University of Madras, Chennai 600 005, India}

\address{Sandra  Pinelas \newline
Academia Militar, Departamento de Ci\^encias Exactas e Naturais,
Av. Conde Castro Guimar\~aes, 2720-113 Amadora, Portugal}
\email{sandra.pinelas@gmail.com}

\thanks{Submitted March 14, 2014. Published August 14, 2014.}
\subjclass[2000]{34C15}
\keywords{Oscillation; odd order; neutral differential equation; mixed type}

\begin{abstract}
 This article presents oscillation criteria for n-th order nonlinear
 neutral mixed type differential equations of the form
 \begin{gather*}
 \big((x(t)+ax(t-\tau_1)-bx(t+\tau_2))^{\alpha}\big)^{(n)}
 =q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2), \\
 \big((x(t)-ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}\big)^{(n)}
 =q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2), \\
 \big((x(t)+ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}\big)^{(n)}
 =q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2)
 \end{gather*}
 where $n$ is an odd positive integer, $a$ and $b$ are nonnegative
 constants, $\tau_1,\tau_2,\sigma_1$ and $\sigma_2$ are positive
 real constants, $q(t),p(t)\in C([t_0,\infty),(0,\infty))$ and $\alpha,\beta$
 and $\gamma$ are ratios of odd positive integers with $\beta,\gamma\geq1$.
 Some examples are provided to illustrate the main results. 
\end{abstract}

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

\section{Introduction}

In this article, we study the oscillatory behavior
of all solutions of n-th order nonlinear neutral differential equations
of the forms
\begin{gather}
\left((x(t)+ax(t-\tau_1)-bx(t+\tau_2))^{\alpha}\right)^{(n)}
=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2), \label{e1.1}\\
\left((x(t)-ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}\right)^{(n)}
=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2), \label{e1.2}\\
\left((x(t)+ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}\right)^{(n)}
=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2) \label{e1.3}
\end{gather}
where $n$ is an odd positive integer, $a$ and $b$ are nonnegative
constants, $\tau_1,\tau_2,\sigma_1$ and $\sigma_2$ are positive
real constants, $q(t),p(t)\in C([t_0,\infty),(0,\infty))$ and $\alpha,\beta$
and $\gamma$ are ratios of odd positive integers with $\beta,\gamma\geq1$.

 As is customary, a solution is called oscillatory if it
has arbitrarily large zeros and non-oscillatory if it is eventually
positive or eventually negative. Equations \eqref{e1.1}, \eqref{e1.2} and \eqref{e1.3} are
called oscillatory if all its solutions are oscillatory.

 Differential equations with advanced and delayed  arguments
(also called mixed differential equations or equations with mixed
arguments) occur in many problems of economy, biology and physics
(see for example \cite{r3,r7,r11,r12,r19}), because differential equations
with mixed arguments are much more suitable than delay differential
equations for an adequate treatment of dynamic phenomena. The concept
of delay is related to a memory of system, the past events are importance
for the current behavior, and the concept of advance is related to
a potential future events which can be known at the current time which
could be useful for decision making. The study of various problems
for differential equations with mixed arguments can be seen in 
\cite{r4,r9,r18,r22,r23,r27}.

It is well known that the solutions of some of these equations cannot
be obtained in closed form. In the absence of closed form solutions
a rewarding alternative is to resort to the qualitative study of the
solutions of these types of differential equations. But it is not
quite clear how to formulate an initial value problem for such equations
and existence and uniqueness of solutions becomes a complicated issue.
To study the oscillation of solutions of differential equations, we
need to assume that there exists a solution of such equation on the
half line.

    The problem of asymptotic and oscillatory behavior
of solutions of n-th order delay and neutral type differential equations
has received great attention in recent years see for example \cite{r1}--\cite{r32},
and the references cited therein. However, there are few results regarding
the oscillatory properties of neutral differential equations with
mixed arguments.

   In \cite{r25} the author has obtained some oscillation theorems
for the odd order neutral differential equation
\begin{equation} \label{e1.4}
\big(x(t)+p_1x(t-\tau_1)+p_2x(t+\tau_2)\big)^{(n)}
=q_1x(t-\sigma_1)+q_2x(t+\sigma_2),t\geq t_0,
\end{equation}
where $n\geq1$ is odd.

   In \cite{r16} the authors established some oscillation criteria
for the following neutral equations
\begin{gather}
\left(x(t)+cx(t-h)-c^{*}x(t+h^{*})\right)^{(n)}=qx(t-g)+px(t+g^{*}), \label{e1.5}\\
\left(x(t)-cx(t-h)+c^{*}x(t+h^{*})\right)^{(n)}=qx(t-g)+px(t+g^{*}), \label{e1.6}\\
\left(x(t)+cx(t-h)-c^{*}x(t-h^{*})\right)^{(n)}=qx(t-g)+px(t+g^{*}), \label{e1.7}\\
\left(x(t)+cx(t+h)-c^{*}x(t+h^{*})\right)^{(n)}=qx(t-g)+px(t+g^{*}),\label{e1.8}
\end{gather}
where $t\geq t_0$ and $n$ is an odd positive integer, $c,c^{*},h,h^{*},p$
and $q$ are real numbers and $g$ and $g^{*}$ are positive constants.

   In \cite{r30} the author has obtained some oscillation results
for third-order nonlinear neutral differential equation
\begin{equation} \label{e1.9}
\big((x(t)+b(t)x(t-\tau_1)+c(t)x(t+\tau_2))^{\alpha}\big)'''
=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2),
\end{equation}
for $t\geq t_0$, where $\alpha,\beta$ and $\gamma$ are ratios of odd positive integers,
$\tau_1,\tau_2,\sigma_1$ and $\sigma_2$ are positive constants.

   Clearly equations \eqref{e1.5} and \eqref{e1.6} with $\alpha=\beta=\gamma=1$
and $q(t)=q$, $p(t)=p$ are special cases of equations \eqref{e1.1} and \eqref{e1.2}.
Moreover equation \eqref{e1.9} with $n=3$ is special case of equation \eqref{e1.3}.
Motivated by the above observations in this paper we study the oscillatory
behavior of equations \eqref{e1.1},\eqref{e1.2} and \eqref{e1.3} for different values of $\beta\geq1$ and $\gamma\geq1$.

   In Section 2 we present some lemmas which are useful for
our main results. In Section 3, we present some sufficient conditions
for the oscillation of all solutions of equations \eqref{e1.1},\eqref{e1.2} and
\eqref{e1.3}. Examples are provided in Section 4 to illustrate the main results.



\section{Some preliminary lemmas}

   In this section we state the following lemmas which
are essential in the proofs of our oscillation theorems.

 \begin{lemma}[\cite{r20}] \label{lem2.1}
  Let $x(t)$ be a function such that it and
each of its derivative up to order $(n-1)$ inclusive are absolutely
continuous and of constant sign in an interval $(t_0,\infty)$.
If $x^{(n)}(t)$ is of constant sign and not identically zero on any
interval of the form $[t_1,\infty)$ for some $t_1\geq t_0$,
then there exists a $t_{x}\geq t_0$ and an integer $m$, $0\leq m\leq n$
with $n+m$ even for $x^{(n)}>0$, or $n+m$ odd for $x^{(n)}\leq0$,
and such that for every $t\geq t_{x}$,
\begin{gather*}
\text{$m>0$ implies $x^{k}(t)>0$ for $k=0,1,\dots ,m-1$}; and \\
\text{$m\leq n-1$ implies  $(-1)^{m+k}x^{(k)}(t)>0$ for $k=m,m+1,\dots ,n-1$.}
\end{gather*}
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.2.2]{r1}}] \label{lem2.2}
If  $x(t)$ is as in Lemma \ref{lem2.1} and $x^{(n-1)}(t)x^{(n)}(t)\leq0$ for all $t\geq t_{x}$,
then for every $\lambda$, $0<\lambda<1$, there exists a constant $M>0$
such that
\[
|x(\lambda t)|\geq Mt^{n-1}|x^{(n-1)}(t)|
\]
for all large $t$.
\end{lemma}

\begin{lemma}[\cite{r26}] \label{lem2.3}
 Let $x(t)$ be a function as in Lemma \ref{lem2.2}. 
If $\lim_{t\to\infty} x(t)\neq0$, then for every $\lambda\in(0,1)$,
\[
x(t)\geq\frac{\lambda}{(n-1)!}t^{n-1}x^{(n-1)}(t)
\]
for all large $t$.
\end{lemma}

\begin{lemma} \label{lem2.4}
Let $A\geq0$, $B\geq0$ and $\gamma\geq1$. Then
\[
A^{\gamma}+B^{\gamma}\geq\frac{1}{2^{\gamma-1}}(A+B)^{\gamma}.
\]
If $A\geq B$, then
$A^{\gamma}-B^{\gamma}\geq(A-B)^{\gamma}$.
\end{lemma}

A proof of the above lemma can be found in \cite{r29}.

 \begin{lemma}[\cite{r21}] \label{lem2.5}
Suppose
$q:[t_0,\infty)\to\mathbb{R}$ is a continuous and eventually
nonnegative function, and $\sigma$ is a positive real number. Then
the following hold.

(I) If \[
\limsup_{t\to\infty} \int_{t}^{t+\sigma}
\frac{(s-t)^{i}(t-s+\sigma)^{n-i-1}}{i!(n-i-1)!}q(s)ds>1,
\]
hold for some $i=0,1,\dots ,n-1$, then the inequality
\[
y^{(n)}(t)\geq q(t)y(t+\sigma)
\]
has no eventually positive solution $y(t)$ which satisfies $y^{(j)}(t)>0$
eventually, $j=0,1,\dots ,n$. 

(II) If
\[
\limsup_{t\to\infty} \int _{t-\sigma}^{t}
\frac{(t-s)^{i}(s-t+\sigma)^{n-i-1}}{i!(n-i-1)!}q(s)ds>1,
\]
hold for some of $i=0,1,\dots ,n-1$, then the inequality
\[
(-1)^{n}z^{(n)}(t)\geq q(t)z(t-\sigma)
\]
has no eventually positive solution $z(t)$ which satisfies $(-1)^{j}z^{(j)}(t)>0$
eventually, $j=0,1,\dots ,n$.
\end{lemma}

\begin{lemma}[\cite{r31}] \label{lem2.6}
 Assume that for large $t$,
\[
q(s)\neq0 \quad \text{for all }s\in[t,t^{*}],
\]
where $t^{*}$ satisfies $\sigma(t^{*})=t$. Then
\[
x'(t)+q(t)[x(\sigma(t))]^{\alpha}=0,\quad t\geq t_0,
\]
has an eventually positive solution if and only if the corresponding
inequality
\[
x'(t)+q(t)[x(\sigma(t))]^{\alpha}\leq0,\quad t\geq t_0,
\]
has an eventually positive solution. 
\end{lemma}

In \cite{r8,r13,r23,r32}, the authors investigated the oscillatory behavior of 
solutions to 
\begin{equation} \label{e2.1}
x'(t)+q(t)[x(\sigma(t))]^{\alpha}=0,\quad t\geq t_0,
\end{equation}
where $q\in C([t_0,\infty),\mathbb{R}^{+})$,
$\sigma\in C([t_0,\infty),\mathbb{R})$,
$\sigma(t)<t$, $\lim_{t\to\infty}\sigma(t)=\infty$
and $\alpha\in(0,\infty)$ is a ratio of odd positive integers.

   Let $\alpha=1$. Then  \eqref{e2.1} reduces to the linear
delay differential equation
\begin{equation} \label{e2.2}
x'(t)+q(t)x(\sigma(t))=0,\quad t\geq t_0,
\end{equation}
and it is shown that every solution of equation \eqref{e2.2} oscillates if
\begin{equation} \label{e2.3}
\liminf_{t\to\infty} \int _{\sigma{(t)}}^{t}q(s)ds>\frac{1}{e}.
\end{equation}


\section{Oscillation results}

   In this section we shall obtain some sufficient conditions
for the oscillation of all solutions of \eqref{e1.1}, \eqref{e1.2}
and \eqref{e1.3}. 
First we study the oscillation of all solutions of equation
\eqref{e1.1}. 

\begin{theorem} \label{thm3.1}
Assume that
\[
\int_{t_0}^{+\infty}\left(q(t)+p(t)\right)dt=+\infty
\]
hold, and $\tau_2>\sigma_2,(1+a^{\beta})>0,a,b\leq1$, and $1\leq\beta\leq\gamma$,
and $q(t)$ and $p(t)$ are positive and non-increasing functions for
$t\geq t_0$. If the differential inequalities either
\begin{equation} \label{e3.1}
y^{(n)}(t)+\frac{q(t)}{b^{\beta}}y^{\beta/\alpha}(t-\sigma_1-\tau_2)\leq0,
\end{equation}
or
\begin{equation} \label{e3.2}
y^{(n)}(t)+\frac{q(t)+p(t)}{b^{\gamma}}y^{\beta/\alpha}(t-(\tau_2-\sigma_2))\leq0
\end{equation}
and
\begin{equation} \label{e3.3}
y^{(n)}(t)-\frac{p(t)}{2^{\gamma-1}(1+a^{\beta})^{\gamma/\alpha}}
y^{\gamma/\alpha}(t+\sigma_2)\geq0,
\end{equation}
have no eventually positive solution and no eventually positive increasing
solution respectively then every solution of equation \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Let $x(t)$ be a non-oscillatory solution of \eqref{e1.1}. 
Without loss of generality we may assume that $x(t)$ is eventually
positive; i.e., there exists a $t_1\geq t_0$ such that $x(t)>0$
for $t\geq t_1$. Set
\[
z(t)=(x(t)+ax(t-\tau_1)-bx(t+\tau_2))^{\alpha}.
\]
Then
\begin{equation} \label{e3.4}
z^{(n)}(t)=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2)>0\quad
\text{for all } t\geq t_1\geq t_0.
\end{equation}
Thus $z^{(i)}(t)$, $i=0,1,\dots ,n$, are of one sign on 
$[t_2,\infty);t_2\geq t_1$.
There are two possibilities: (a) $z(t)<0$ for $t\geq t_2$, 
(b) $z(t)>0$ for $t\geq t_2$.

 Case 1: Assume $z(t)<0$ for $t\geq t_2$. In this case, we let
\[
0<v(t)=-z(t)=(bx(t+\tau_2)-ax(t-\tau_1)-x(t))^{\alpha}
\leq b^{\alpha}x^{\alpha}(t+\tau_2).
\]
Then in view of the last inequality, we obtain
\begin{equation} \label{e3.5}
x(t)\geq\frac{1}{b}v^{1/\alpha}(t-\tau_2)\quad \text{for }t\geq t^{*}\geq t_2.
\end{equation}
Thus by \eqref{e3.4} and \eqref{e3.5},
\begin{equation} \label{e3.6}
v^{(n)}(t)+\frac{q(t)}{b^{\beta}}v^{\beta/\alpha}(t-\sigma_1-\tau_2)
+\frac{p(t)}{b^{\gamma}}v^{\gamma/\alpha}(t+\sigma_2-\tau_2)\leq0,\quad
t\geq t^{*}.
\end{equation}
By Lemma \ref{lem2.1}, it is easy to check that there exists a $T_0\geq t^{*}$
such that $v^{(n-1)}(t)>0$ for $t\geq T_0$. Now, if $v'(t)>0$
for $t\geq T_0$ then there exist a constant $k>0$ and a $T\geq T_0$
such that
\[
v(t-\sigma_1-\tau_2)\geq k,\quad v(t+\sigma_2-\tau_2)\geq k
\quad \text{for } t\geq T.
\]
Thus
\[
v^{(n)}(t)\leq-k^{\beta/\alpha}\frac{p(t)+q(t)}{b^{\gamma}},\quad
\text{for } t\geq T,
\]
and hence
\[
0<v^{(n-1)}(t)\leq v^{(n-1)}(T)-\frac{k^{\beta/\alpha}}{b^{\gamma}}
\int _{T}^{t}(p(s)+q(s))ds\to-\infty\quad \text{as } t\to\infty,
\]
a contradiction. Thus, $v'(t)<0$ for $t\geq T$ and the function
satisfies $(-1)^{i}v^{(i)}(t)>0$ eventually for $i=0,1,\dots ,n$ and
$t\geq T$. From \eqref{e3.6}, we have either
\[
v^{(n)}(t)+\frac{q(t)}{b^{\beta}}v^{\beta/\alpha}(t-\sigma_1-\tau_2)\leq0,\quad
t\geq T
\]
or
\[
v^{(n)}(t)+\frac{q(t)+p(t)}{b^{\gamma}}v^{\beta/\alpha}
(t-(\tau_2-\sigma_2))\leq0,\quad t\geq T,
\]
has a positive solution, which is a contradiction.

Case 2: Assume $z(t)>0$ for $t\geq t_2$. By the Lemma \ref{lem2.1}, there
exists a $t_3\geq t_2$ such that $z'(t)>0$ for $t\geq t_3$.
Next, we let
\begin{equation}
y(t)=z(t)+a^{\beta}z(t-\tau_1)-\frac{b^{\gamma}}{2^{\gamma-1}}z(t+\tau_2),\quad
t\geq t_3.
\end{equation}
Then
\begin{align*}
y^{(n)}(t)
&=z^{(n)}(t)+a^{\beta}z^{(n)}(t-\tau_1)-\frac{b^{\gamma}}{2^{\gamma-1}}z^{(n)}
 (t+\tau_2)\\
&=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2)
 +a^{\beta}\Big(q(t-\tau_1)x^{\beta}(t-\sigma_1-\tau_1)\\
&\quad +p(t-\tau_1)x^{\gamma}(t+\sigma_2-\tau_1)\Big)
 -\frac{b^{\gamma}}{2^{\gamma-1}}
 \Big(q(t+\tau_2)x^{\beta}(t-\sigma_1+\tau_2)\\
&\quad +p(t+\tau_2)x^{\gamma}(t+\sigma_2+\tau_2)\Big).
\end{align*}
Using the monotonicity of $q(t)$ and $p(t)$, $a,b\leq1,1\leq\beta\leq\gamma$
and Lemma \ref{lem2.4} in the above inequality, we obtain
\begin{align*}
y^{(n)}(t)
&\geq\frac{q(t)}{2^{\beta-1}}\left(x(t-\sigma_1)
 +ax(t-\sigma_1-\tau_1)-bx(t-\sigma_1+\tau_2)\right)^{\beta}\\
 &\quad +\frac{p(t)}{2^{\gamma-1}}\left(x(t+\sigma_2)+ax(t+\sigma_2-\tau_1)
 -bx(t+\sigma_2+\tau_2)\right)^{\gamma}.
\end{align*}
Now using $z(t)>0$ for $t\geq t_2$ in the above inequality, we
obtain
\begin{equation} \label{e3.8}
y^{(n)}(t)\geq\frac{q(t)}{2^{\beta-1}}z^{\beta/\alpha}(t-\sigma_1)
+\frac{p(t)}{2^{\gamma-1}}z^{\gamma/\alpha}(t+\sigma_2)>0,\quad t\geq t_3.
\end{equation}
If $y(t)<0$ eventually, we can get same conclusion as in Case 1.
Thus we observe that $y(t)>0$ eventually. Now, if $z'(t)>0$ eventually
for $t\geq t_2$ then there exist a positive constant $c$ and a
$T\geq t_2$ such that, $z(t-\sigma_1)\geq c$,
$z(t+\sigma_2)\geq c$.
Thus using last inequality in \eqref{e3.8}, we obtain
\[
y^{(n)}(t)\geq\frac{q(t)}{2^{\beta-1}}c^{\beta/\alpha}
+\frac{p(t)}{2^{\gamma-1}}c^{\gamma/\alpha}>0.
\]
Then $y^{(n-1)}(t)\to\infty$ and $y^{(i)}(t)\to\infty$
for $i=0,1,\dots ,n-2$ as $t\to\infty$. Therefore, one can conclude
that
\begin{equation} \label{e3.9}
y^{(i)}(t)>0\quad \text{eventually for } i=0,1,\dots ,n.
\end{equation}
Now, using the monotonicity of $z(t)$, we obtain
\[
y(t)=z(t)+a^{\beta}z(t-\tau_1)
-\frac{b^{\gamma}}{2^{\gamma-1}}z(t+\tau_2)\leq(1+a^{\beta})z(t).
\]
then from the above inequality and \eqref{e3.8}, we have
\begin{equation}  \label{e3.10}
y^{(n)}(t)\geq\frac{p(t)}{2^{\gamma-1}(1+a^{\beta})^{\gamma/\alpha}}
y^{\gamma/\alpha}(t+\sigma_2),\quad t\geq t_3.
\end{equation}
This inequality  admits a solution that satisfies \eqref{e3.9}, thus $y(t)$
is a positive increasing solution of the inequality \eqref{e3.3}, which is
a contradiction. The proof is now complete. 
\end{proof}

\begin{corollary} \label{coro3.1}
Assume that
\[
\int_{t_0}^{+\infty}\left(q(t)+p(t)\right)dt=+\infty
\]
hold, and $\tau_2>\sigma_2,(1+a^{\alpha})>0,a,b\leq1$ and 
$\alpha=\beta=\gamma\geq1$
and $q(t)$ and $p(t)$ are non-increasing functions for $t\geq t_0$.
If either
\begin{equation} \label{e3.11}
\liminf_{t\to\infty} \int _{t-(\sigma_1+\tau_2)}^{t}
(s-\sigma_1-\tau_2)^{n-1}q(s)ds>\frac{b^{\alpha}(n-1)!}{\lambda e},
\quad \lambda\in(0,1),
\end{equation}
or
\begin{equation} \label{e3.12}
\underset{t\to\infty}{\liminf}\int _{t-\tau_2
+\sigma_2}^{t}(s-\tau_2+\sigma_2)^{n-1}(p(s)+q(s))ds
>\frac{b^{\alpha}(n-1)!}{\lambda e},\quad \lambda\in(0,1),
\end{equation}
and
\begin{equation} \label{e3.13}
\underset{t\to\infty}{\limsup}\int _{t}^{t+\sigma_2}\frac{(s-t)^{i}
(t-s+\sigma_2)^{n-i-1}}{i!(n-i-1)!}p(s)ds>2^{\alpha-1}(1+a^{\alpha}),\quad
i=0,1,\dots ,n-1,
\end{equation}
then every solution of \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof}
Let $y(t)$ be a positive solution of  \eqref{e3.2}, for $t\geq t_1\geq t_0$.
Then we have $y^{(n)}(t)\leq0$ for all $t\geq t_1$. More over,
$(-1)^{i}y^{(i)}(t)>0$ for $i=1,2,\dots ,n$ for all $t\geq t_1$.
Then from Lemma \ref{lem2.3} we obtain
\[
y(t)\geq\frac{\lambda}{(n-1)!}t^{n-1}y^{(n-1)}(t),\quad \lambda\in(0,1).
\]
From \eqref{e3.2}, we have
\[
y^{(n)}(t)+\frac{p(t)+q(t)}{b^{\alpha}}y(t-\tau_2+\sigma_2)\leq0,\quad t\geq t_2.
\]
Combining the last two inequalities, we obtain
\[
y^{(n)}(t)+(p(t)+q(t))\frac{\lambda}{b^{\alpha}(n-1)!}
(t-\tau_2+\sigma_2)^{n-1}y^{(n-1)}(t-\tau_2+\sigma_2)\leq0,\quad t\geq t_2.
\]
Let $w(t)=y^{(n-1)}(t)$. Then we see that $w(t)$ is a positive solution
of
\begin{equation} \label{e3.14}
w'(t)+(p(t)+q(t))\frac{\lambda}{b^{\alpha}(n-1)!}
(t-\tau_2+\sigma_2)^{n-1}w(t-\tau_2+\sigma_2)\leq0,\quad t\geq t_2
\end{equation}

But according to the Lemma \ref{lem2.6} and the condition \eqref{e2.3}, condition
\eqref{e3.12} guarantees that inequality \eqref{e3.14} has no positive solution,
which is a contradiction. Hence \eqref{e3.2} has no eventually positive solution.
Moreover condition \eqref{e3.11} is sufficient for the inequality \eqref{e3.1} has
no eventually positive solution,which is a contradiction. Moreover
in view of Lemma \ref{lem2.5} (I) and the condition \eqref{e3.13}, inequality \eqref{e3.10}
has no eventually positive solution which satisfies \eqref{e3.9}, which is
a contradiction. Hence \eqref{e3.3} has no eventually positive increasing
solution. 
\end{proof}  

 Next we consider \eqref{e1.2}, and present sufficient conditions 
 for the oscillation of all solutions.
 
 \begin{theorem} \label{thm3.2}
 Assume that
\[
\int_{t_0}^{+\infty}\left(q(t)+p(t)\right)dt=+\infty
\]
hold, and $\sigma_{i}>\tau_{i}$ for $i=1,2,(1+b^{\gamma})>0,a,b\leq1$,
and $1\leq\gamma\leq\beta$, and $q(t)$ and $p(t)$ are positive
and nondecreasing functions for $t\geq t_0$. If the differential
inequalities
\begin{equation} \label{e3.15}
y^{(n)}(t)+\frac{q(t)}{a^{\beta}}y^{\beta/\alpha}(t-\sigma_1+\tau_1)\leq0,
\end{equation}
and
\begin{equation} \label{e3.16}
y^{(n)}(t)-\frac{p(t)}{2^{\gamma-1}(1+b^{\gamma})^{\gamma/\alpha}}y^{\gamma/\alpha}(t+\sigma_2-\tau_2)\geq0,
\end{equation}
have no eventually positive solution and no eventually positive increasing
solution respectively. Then every solution of equation \eqref{e1.2} is oscillatory.
\end{theorem}

\begin{proof}
Let $x(t)$ be a non-oscillatory solution of \eqref{e1.2}. Without loss of 
generality we may assume that $x(t)$ is eventually
positive; i.e., there exists a $t_1\geq t_0$ such that $x(t)>0$
for $t\geq t_1$. Set
\[
z_1(t)=(x(t)-ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}
\]
and proceeding as in the proof of Theorem \ref{thm3.1}, we that the function
$z_1^{(i)}(t)$, $i=0,1,\dots ,n$, are of one sign on $[t_2,\infty)$, $t_2\geq t_1$.
There are two possibilities:
(1) $z_1(t)<0$ for $t\geq t_2$, (2) $z_1(t)>0$ for $t\geq t_2$.

Case 1: Assume $z_1(t)<0$ for $t\geq t_2$. In this case, we
let
\[
0<v_1(t)=-z_1(t)=(ax(t-\tau_1)-bx(t+\tau_2)-x(t))^{\alpha}
\leq a^{\alpha}x^{\alpha}(t-\tau_1).
\]
Then in view of the last inequality, we obtain
\begin{equation} \label{e3.17}
x(t)\geq\frac{1}{a}v_1^{1/\alpha}(t+\tau_1)\quad \text{for } t\geq t^{*}\geq t_2.
\end{equation}
Thus by \eqref{e1.2} and \eqref{e3.17},
\begin{equation} \label{e3.18}
v_1^{(n)}(t)+\frac{q(t)}{a^{\beta}}v_1^{\beta/\alpha}(t-\sigma_1+\tau_1)
+\frac{p(t)}{a^{\gamma}}v_1^{\gamma/\alpha}(t+\sigma_2+\tau_1)\leq0,\quad
t\geq t^{*}.
\end{equation}
By Lemma \ref{lem2.1}, it is easy to check that there exists a $T_0\geq t^{*}$
such that $v_1^{(n-1)}(t)>0$ for $t\geq T_0$. Now, if $v_1'(t)>0$
for $t\geq T_0$ then there exist a constant $k_1>0$ and a $T\geq T_0$
such that
\[
v_1(t-\sigma_1+\tau_1)\geq k_1,\quad v_1(t+\sigma_2+\tau_1)\geq k_1\quad
\text{for }t\geq T.
\]
Thus
\[
v_1^{(n)}(t)\leq-k_1^{\gamma/\alpha}\frac{p(t)+q(t)}{a^{\beta}},\quad
\text{for }t\geq T,
\]
and hence
\[
v_1^{(n-1)}(t)\leq v_1^{(n-1)}(T)-\frac{k_1^{\gamma/\alpha}}{a^{\beta}}
\int _{T}^{t}(p(s)+q(s))ds\to-\infty\quad \text{as } t\to\infty,
\]
a contradiction. Thus, $v_1'(t)<0$ for $t\geq T$ and the function
satisfies $(-1)^{i}v_1^{(i)}(t)>0$ eventually for $i=0,1,\dots ,n$
and $t\geq T$. From \eqref{e3.18}, we have
\[
v_1^{(n)}(t)+\frac{q(t)}{a^{\beta}}v_1^{\beta/\alpha}(t-\sigma_1+\tau_1)\leq0,
\quad t\geq T,
\]
has a positive solution, which is a contradiction.

Case 2: Assume $z_1(t)>0$ for $t\geq t_2$. By the Lemma \ref{lem2.1},
there exists a $t_3\geq t_2$ such that $z_1'(t)>0$ for $t\geq t_3$.
Next, we let
\begin{equation}
y_1(t)=z_1(t)-\frac{a^{\beta}}{2^{\beta-1}}z_1(t-\tau_1)
+b^{\gamma}z_1(t+\tau_2),\quad t\geq t_3.
\end{equation}
Then
\begin{align*}
y_1^{(n)}(t)
&=z_1^{(n)}(t)-\frac{a^{\beta}}{2^{\beta-1}}z_1^{(n)}(t-\tau_1)
 +b^{\gamma}z_1^{(n)}(t+\tau_2)\\
 &=q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2)
 -\frac{a^{\beta}}{2^{\beta-1}}
  \Big(q(t-\tau_1)x^{\beta}(t-\sigma_1-\tau_1)\\
&\quad +p(t-\tau_1)x^{\gamma}  (t+\sigma_2-\tau_1)\Big)
 +b^{\gamma}\Big(q(t+\tau_2)x^{\beta}(t-\sigma_1+\tau_2)\\
&\quad +p(t+\tau_2)x^{\gamma}(t+\sigma_2+\tau_2)\Big).
\end{align*}
Using the monotonicity of $q(t)$ and $p(t)$, $a,b\leq1,1\leq\gamma\leq\beta$
and Lemma \ref{lem2.4} in the above inequality, we obtain
\begin{align*}
y_1^{(n)}(t) & \geq\frac{q(t)}{2^{\beta-1}}\left(x(t-\sigma_1)
-ax(t-\sigma_1-\tau_1)+bx(t-\sigma_1+\tau_2)\right)^{\beta}\\
&\quad  +\frac{p(t)}{2^{\gamma-1}}\left(x(t+\sigma_2)
 -ax(t+\sigma_2-\tau_1)+bx(t+\sigma_2+\tau_2)\right)^{\gamma}.
\end{align*}
Now using $z_1(t)>0$ for $t\geq t_2$ in the above inequality,
we obtain
\begin{equation} \label{e3.20}
y_1^{(n)}(t)\geq\frac{q(t)}{2^{\beta-1}}z_1^{\beta/\alpha}(t-\sigma_1)
+\frac{p(t)}{2^{\gamma-1}}z_1^{\gamma/\alpha}(t+\sigma_2)>0,\quad t\geq t_3.
\end{equation}
If $y(t)<0$ eventually, we can get same conclusion as in Case 1.
Thus we observe that $y(t)>0$ eventually. Now, if $z'(t)>0$ eventually
for $t\geq t_2$ then there exist a positive constant $c_1$ and
a $T\geq t_2$ such that, $z(t-\sigma_1)\geq c_1$, $z(t+\sigma_2)\geq c_1$.
Thus using last inequality in \eqref{e3.20}, we obtain
\[
y^{(n)}(t)\geq\frac{q(t)}{2^{\beta-1}}c_1^{\beta/\alpha}
+\frac{p(t)}{2^{\gamma-1}}c_1^{\gamma/\alpha}>0.
\]
Then $y^{(n-1)}(t)\to\infty$ and $y^{(i)}(t)\to\infty$
for $i=0,1,\dots ,n-2$ as $t\to\infty$. Therefore one can conclude
that
\begin{equation} \label{e3.21}
y_1^{(i)}(t)>0\quad \text{eventually for }i=0,1,\dots ,n,\; t\geq t_3.
\end{equation}
Now,
\[
y_1(t)=z_1(t)-\frac{a^{\beta}}{2^{\beta-1}}z_1(t-\tau_1)
+b^{\gamma}z_1(t+\tau_2)\leq(1+b^{\gamma})z_1(t+\tau_2).
\]
then from the above inequality and \eqref{e3.20}, we have
\begin{equation} \label{e3.22}
y_1^{(n)}(t)\geq\frac{p(t)}{2^{\gamma-1}(1+b^{\gamma})^{\gamma/\alpha}}
y_1^{\gamma/\alpha}(t+\sigma_2-\tau_2),\quad t\geq t_3.
\end{equation}
Inequality \eqref{e3.22} admits a solution that satisfies \eqref{e3.21}, 
thus $y_1(t)$ is a positive increasing solution of the inequality \eqref{e3.16}, 
which is a contradiction. The proof is now complete.
\end{proof} 

\begin{corollary} \label{coro3.2}
Let $\sigma_{i}>\tau_{i}$ for $i=1,2,(1+b^{\alpha})>0,a,b\leq1$
and $\alpha=\beta=\gamma\geq1$. If
\begin{equation} \label{e3.23}
\liminf_{t\to\infty} \int _{t-(\sigma_1-\tau_1)}^{t}(s-\sigma_1
+\tau_1)^{n-1}q(s)ds>\frac{a^{\alpha}(n-1)!}{\lambda_1e},\quad \lambda_1\in(0,1)
\end{equation}
and
\begin{equation} \label{e3.24}
\underset{t\to\infty}{\limsup}\int _{t}^{t+\sigma_2
-\tau_2}\frac{(s-t)^{i}(t-s+\sigma_2-\tau_2)^{n-i-1}}{i!(n-i-1)!}
p(s)ds>2^{\alpha-1}(1+b^{\alpha}),
\end{equation}
for $i=0,1,\dots ,n-1$,
then every solution of  \eqref{e1.2} is oscillatory.
\end{corollary}

The  proof of the above corollary is similar to that of Corollary \ref{coro3.1} 
and hence it is omitted.
 Next we consider equation \eqref{e1.3} and present sufficient conditions 
for the oscillation of all solutions.

\begin{theorem} \label{thm3.3}
Let $\sigma_2>\tau_2,a\leq1,b\geq1$ and $1\leq\beta\leq\gamma$, and
\begin{gather*}
Q(t)=\min{\{q(t-\tau_1),q(t),q(t+\tau_2)\}}, \\
P(t)=\min{\{p(t-\tau_1),p(t),p(t+\tau_2)\}},
\end{gather*}
be positive functions for $t\geq t_0$. If the differential inequality
\begin{equation} \label{e3.25}
y^{(n)}(t)-\frac{P(t)}{4^{\gamma-1}(1+a^{\beta}
+b^{\gamma})^{\gamma/\alpha}}y^{\gamma/\alpha}(t+\sigma_2-\tau_2)\geq0,
\end{equation}
has no eventually positive solution. Then every solution of 
\eqref{e1.3} is oscillatory.
\end{theorem}

\begin{proof}
Let $x(t)$ be an eventually positive solution of equation \eqref{e1.3}, 
then there exists a $t_1\geq t_0$ such that $x(t)>0$ for $t\geq t_1$. Set
\[
z_2(t)=(x(t)+ax(t-\tau_1)+bx(t+\tau_2))^{\alpha},\quad t\geq t_1.
\]
and proceeding as in the proof of Theorem \ref{thm3.1}, we see that the function
$z_2^{(i)}(t),i=0,1,\dots ,n$ are of one sign on $[t_2,\infty)$,
for some $t_2\geq t_1$. Now we define
\begin{equation} \label{e3.26}
y_2(t)=z_2(t)+a^{\beta}z_2(t-\tau_1)+b^{\gamma}z_2(t+\tau_2),\quad t\geq t_2.
\end{equation}
Then $y_2(t)>0$ for $t\geq t_2$ and then
\begin{align*}
y_2^{(n)}(t)
&= z_2^{(n)}(t)+a^{\beta}z_2^{(n)}(t-\tau_1)+b^{\gamma}z_2^{(n)}(t+\tau_2)\\
&=  q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2)
 +  a^{\beta}\Big(q(t-\tau_1)x^{\beta}(t-\sigma_1-\tau_1)\\
&\quad +p(t-\tau_1)  x^{\gamma}(t+\sigma_2-\tau_1)\Big)
  +  b^{\gamma}\Big(q(t+\tau_2)x^{\beta}(t-\sigma_1+\tau_2)\\
&\quad +p(t+\tau_2)x^{\gamma}(t+\sigma_2+\tau_2)\Big).
\end{align*}
Since $a\leq1,b\geq1,1\leq\beta\leq\gamma$ and using Lemma \ref{lem2.4} in
the above inequality, we obtain
\begin{align*}
y_2^{(n)}(t) 
& \geq  \frac{Q(t)}{4^{\beta-1}}\left(x(t-\sigma_1)
 +ax(t-\sigma_1-\tau_1)+bx(t-\sigma_1+\tau_2)\right)^{\beta}\\
&\quad  +  \frac{P(t)}{4^{\gamma-1}}\left(x(t+\sigma_2)
 +ax(t+\sigma_2-\tau_1)+bx(t+\sigma_2+\tau_2)\right)^{\gamma}.
\end{align*}
Now using $z_2(t)>0$ for $t\geq t_2$ in the above inequality,
we obtain
\begin{equation} \label{e3.27}
y_2^{(n)}(t)\geq\frac{Q(t)}{4^{\beta-1}}z_2^{\beta/\alpha}
(t-\sigma_1)+\frac{P(t)}{4^{\gamma-1}}z_2^{\gamma/\alpha}(t+\sigma_2)>0,\quad
t\geq t_2.
\end{equation}
Since $z_2(t)>0$ and $z'_2(t)>0$ are eventually positive increasing
functions. From  \eqref{e3.26} we see that $y_2(t)>0$ and 
$y'_2(t)>0$ and also from inequality \eqref{e3.27}, $y_2^{(n)}(t)>0$ 
for $t\geq t_2$. As a result of this
\begin{equation} \label{e3.28}
y_2^{(i)}(t)>0,\quad \text{for $t\geq t_2$  and $i=0,1,\dots ,n$.}
\end{equation}
Using the monotonicity of $z_2(t)$, we obtain
\[
y_2(t)=z_2(t)+a^{\beta}z_2(t-\tau_1)+b^{\gamma}z_2(t+\tau_2)
\leq(1+a^{\beta}+b^{\gamma})z_2(t+\tau_2).
\]
Then from the above inequality and \eqref{e3.27}, we have
\begin{align*}
y_2^{(n)}(t) 
& \geq  \frac{P(t)}{4^{\gamma-1}}z_2^{\gamma/\alpha}(t+\sigma_2)\nonumber \\
& \geq  \frac{P(t)}{4^{\gamma-1}(1+a^{\beta}+b^{\gamma})^{\gamma/\alpha}}
 y_2^{\gamma/\alpha}(t-\tau_2+\sigma_2),\quad t\geq t_2.
\end{align*}
This inequality  admits a solution that satisfies \eqref{e3.28}, thus $y_2(t)$
is a positive increasing solution of the inequality \eqref{e3.25}, which
is a contradiction. The proof is now complete.
\end{proof}

\begin{corollary} \label{coro3.3}
Let $\sigma_2>\tau_2,a\leq1,b\geq1$ and $\alpha=\beta=\gamma\geq1$.
If
\begin{equation}
\limsup_{t\to\infty} \int _{t}^{t+\sigma_2-\tau_2}\frac{(s-t)^{i}
(t-s+\sigma_2-\tau_2)^{n-i-1}}{i!(n-i-1)!}P(s)ds
>4^{\alpha-1}(1+a^{\alpha}+b^{\alpha}),
\end{equation}
where $i=0,1,\dots ,n-1$, then every solution of equation \eqref{e1.3} 
is oscillatory.
\end{corollary}

The proof of the above corollary is similar to that of Corollary  \ref{coro3.1} 
and hence it is omitted.


\section{Examples}

   In this section we present some examples to illustrate
the main results. 

\begin{example} \label{examp4.1} \rm
Consider the differential equation
\begin{equation} \label{e4.1}
\Big((x(t)+\frac{1}{4}x(t-\pi)-\frac{1}{4}x(t+2\pi))^3\Big)^{(v)}
=\frac{1}{4}x^3(t-3\pi/2)+\frac{1}{8}x^3(t+3\pi/2),
\end{equation}
for $t\geq0$.
Here $a=1/4$, $b=1/4$, $\alpha=\beta=\gamma=3$, $\tau_1=\pi$,
$\tau_2=2\pi$, $\sigma_1=3\pi/2$, $\sigma_2=3\pi/2$, 
$q(t)=1/4$, $p(t)=1/8$.
Then one can see that all conditions of Corollary  \ref{coro3.1} are satisfied.
Therefore all the solutions of equation \eqref{e4.1} are oscillatory. In
fact $x(t)=\sin^{1/3}t$ is one such oscillatory solution of equation
\eqref{e4.1}.
\end{example} 

\begin{example} \label{examp4.2} \rm
Consider the differential equation
\begin{equation} \label{e4.2}
\Big((x(t)-\frac{e^{\pi/3}}{9}x(t-\pi)+\frac{1}{e^{\pi/3}}x(t+\pi))^3\Big)^{(v)}
=\frac{4e^{5\pi/2}}{729}x^3(t-5\pi/2)+\frac{4}{729e^{3\pi}}x^3(t+3\pi),
\end{equation}
where $t\geq0$. Here $a=e^{\pi/3}/9$, $b=1/e^{\pi/3}$, 
$\alpha=\beta=\gamma=3$, $\tau_1=\pi$, $\tau_2=\pi$, $\sigma_1=5\pi/2$,
$\sigma_2=3\pi$, $q(t)=4e^{5\pi/2}/729$, $p(t)=4/(729e^{3\pi})$.
Then one can see that all conditions of Corollary  \ref{coro3.2} are satisfied.
Therefore, all the solutions of equation \eqref{e4.2} are oscillatory. In
fact $x(t)=e^{t/3}\sin^{1/3}t$ is one such oscillatory solution of
equation \eqref{e4.2}.
\end{example} 

\begin{example} \label{examp4.3} \rm
Consider the differential equation
\begin{equation} \label{e4.3}
\left(x(t)+x(t-\pi)+x(t+\pi)\right)^{(v)}
=\frac{5}{t-\pi}x(t-\pi)+\frac{t}{t+3\pi/2}x(t+3\pi/2),
\end{equation}
for $t\geq0$. 
Here $a=b=1$, $\alpha=\beta=\gamma=1$, $\tau_1=\tau_2=\pi$,
$\sigma_1=\pi$, $\sigma_2=3\pi/2$, $q(t)=\frac{5}{t-\pi},p(t)=\frac{t}{t+3\pi/2}$.
Then one can see that all conditions of Corollary  \ref{coro3.3} are satisfied.
Therefore, all the solutions of equation \eqref{e4.3} are oscillatory. In
fact $x(t)=t\sin t$ is one such oscillatory solution of equation
\eqref{e4.3}.
\end{example}

\begin{example} \label{examp4.4} \rm
Consider the differential equation
\begin{equation} \label{e4.4}
\Big(x(t)+\frac{1}{2}x(t-\pi/2)-\frac{1}{2}x(t+2\pi)\Big)^{(vii)}
=\frac{1}{2}x(t-7\pi/2)+\frac{1}{2}x(t+4\pi),
\end{equation}
for $t\geq0$.
Here $a=1/2$, $b=1/2$, $\alpha=\beta=\gamma=1$, $\tau_1=\pi/2$,
$\tau_2=2\pi$, $\sigma_1=5\pi/2$, $\sigma_2=4\pi$,
 $q(t)=1/2$, $p(t)=1/2$.
Then one can see that all conditions of Corollary  \ref{coro3.1} are satisfied.
Therefore, all the solutions of equation \eqref{e4.4} are oscillatory. In
fact $x(t)=\sin t+\cos t$ is one such oscillatory solution of equation
\eqref{e4.4}.
\end{example} 

\begin{example} \label{examp4.5} \rm
Consider the differential equation
\begin{equation} \label{e4.5}
\Big((x(t)-\frac{e}{3}x(t-1)+\frac{1}{e^{2}}x(t+2))^3\Big)^{(v)}
=1000e^{9}x^3(t-3)+\frac{125}{e^{9}}x^3(t+3),
\end{equation}
for $t\geq0$.  Here $a=e/3$, $b=1/e^2$, $\alpha=\beta=\gamma=3$,
$\tau_1=1$, $\tau_2=2$, $\sigma_1=\sigma_2=3$, $q(t)=1000e^{9}$,
$p(t)=\frac{e^9}{125}$.
Then one can see that all conditions of Corollary  \ref{coro3.2} are satisfied
except the condition \eqref{e3.24}. Therefore, not all  solutions of 
\eqref{e4.5} are oscillatory. 
In fact $x(t)=e^{t}$ is one such
non-oscillatory solution, since it satisfies equation \eqref{e4.5}.
\end{example}

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\end{document}