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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 169, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/169\hfil Oscillation of solutions]
{Oscillation of solutions to nonlinear forced fractional differential equations}

\author[Q. Feng, F. Meng\hfil EJDE-2013/169\hfilneg]
{Qinghua Feng, Fanwei Meng} 

\address{Qinghua Feng \newline
School of Science, Shandong University of Technology,
Zibo, Shandong, 255049, China}
\email{fqhua@sina.com}

\address{Fanwei Meng \newline
School of Mathematical Sciences, Qufu Normal University,
Qufu, 273165, China}
\email{fwmeng163@163.com}

\thanks{Submitted  April 1, 2013. Published July 26, 2013.}
\subjclass[2000]{34C10, 34K11}
\keywords{Oscillation; nonlinear
fractional differential equation; forced;\hfill\break\indent
 Riccati transformation}

\begin{abstract}
 In this article, we study the oscillation of solutions to a
 nonlinear forced fractional differential equation. The fractional
 derivative is defined in the sense of the modified Riemann-Liouville
 derivative. Based on a transformation of variables and properties
 of the modified Riemann-liouville derivative, the fractional
 differential equation is transformed into a second-order ordinary
 differential equation. Then by a generalized Riccati transformation,
 inequalities, and an integration average technique, we establish
 oscillation criteria for the fractional differential equation.
\end{abstract}

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


\section{Introduction}

 Recently, research on oscillation of various equations
including differential equations, difference equations and dynamic
equations on time scales has been a hot topic in the
literature. Much effort has been done to establish
oscillation criteria for these equations; see for example
the references in this article. We notice that
in these publications very little attention is paid
to oscillation of fractional differential equations.

 In this article, we are concerned with the  oscillation of solutons to
the nonlinear forced fractional differential equation
\begin{equation} \label{e1.1}
D_t^{\alpha}[r(t)\psi(x(t))D_t^{\alpha}x(t)]+q(t)f(x(t))=e(t) ,t
\geq t_0>0,\ 0<\alpha<1,
\end{equation}
 where
$D_t^{\alpha}(\cdot)$ denotes the modified  Riemann-Liouville
derivative \cite{j1} with respect to the variable $t$, the functions
$r\in C^{\alpha}([t_0,\infty),R_+)$, which is the set of functions with
continuous derivative of order $\alpha$, the functions
$q,e$ belong to $C([t_0,\infty),R)$, and  the functions $f,  \psi$
belong to $C(R,R)$, $0<\psi(x)\leq m$ for some positive constant $m$,
and $xf(x)>0$ for all $x\neq 0$.

The definition and some important properties for the Jumarie's
modified Riemann-Liouville derivative of order $\alpha$ are listed
next (see also in \cite{g3,z2,z3}):
\begin{gather}
D^{\alpha}_tf(t)
=\begin{cases}
 \frac{ 1}{ \Gamma(1-\alpha)}\frac{ d}{ dt}
\int_0^t(t-\xi)^{-\alpha}(f(\xi)-f(0))d\xi,& 0<\alpha<1,\\
 (f^{(n)}(t))^{(\alpha-n)},& 1\leq n\leq \alpha< n+1.
\end{cases} \nonumber
\\
D^{\alpha}_tt^{r}=\frac{ \Gamma(1+r)}{ \Gamma(1+r-\alpha)}t^{r-\alpha},
 \label{e1.2}
\\
D^{\alpha}_t(f(t)g(t))=g(t)D^{\alpha}_tf(t)+f(t)D^{\alpha}_tg(t),
\label{e1.3}
\\
D^{\alpha}_t f[g(t)]=f'_g[g(t)]D^{\alpha}_tg(t)=D^{\alpha}_gf[g(t)]
(g'(t))^{\alpha}. \label{e1.4}
\end{gather}

A solution of \eqref{e1.1} is called oscillatory if it has arbitrarily
large zeros, otherwise it is called non-oscillatory.
Equation \eqref{e1.1} is called oscillatory if all its solutions are
oscillatory.

For the sake of convenience, in this article, we denote:
\begin{gather*}
\xi_0=\frac{ t_0^{\alpha}}{\Gamma(1+\alpha)},\quad
\xi=\frac{ t^{\alpha}}{ \Gamma(1+\alpha)}, \quad
\widetilde{\rho}(\xi)=\rho(t),\quad
\widetilde{r}(\xi)=r(t),\\
 \widetilde{q}(\xi)=q(t),\quad
\xi_{a_i}=\frac{ a_i^{\alpha}}{\Gamma(1+\alpha)},\quad
\xi_{b_i}=\frac{b_i^{\alpha}}{ \Gamma(1+\alpha)},\quad
\mathbb{R}_{+}=(0,\infty).
\end{gather*}
Let $h_1,\ h_2,\ H\in C([\xi_0,\infty),R)$ satisfy
$$
H(\xi,\xi)=0, \quad H(\xi,s)>0, \quad
 \xi>s\geq\xi_0.
$$
Let $H$ have continuous partial derivatives
$\frac{ \partial H(\xi,s)}{\partial \xi}$ and
$\frac{ \partial H(\xi,s)}{\partial s}$
on $[\xi_0,\infty)$ such that
$$
\frac{ \partial H(\xi,s)}{\partial \xi}=-h_1(\xi,s)\sqrt{H(\xi,s)},\quad
\frac{ \partial H(\xi,s)}{ \partial s}=-h_2(\xi,s)\sqrt{H(\xi,s)}.
$$
For $\bf s,\ \xi\in [\xi_0,\infty)$, denote
\[
Q_1(s,\xi)=h_1(s,\xi)-\frac{ \widetilde{\rho}'(s)}{
\widetilde{\rho} (s)}\sqrt{H(s,\xi)}, \quad
Q_2(\xi,s)=h_2(\xi,s)-\frac{
\widetilde{\rho}'(s)}{ \widetilde{\rho} (s)}\sqrt{H(\xi,s)},
\]

We organize  this article as follows.
In Section 2, we  establish some new oscillation criteria for \eqref{e1.1}
under the condition that $f(x)$ is increasing.
In Section 3, we  establish  oscillation criteria for \eqref{e1.1} without
the condition $f(x)$ being increasing.
In the proof for the main results in Sections 2 and 3,
we use a generalized Riccati transformation method.
This Riccati transformation and the function $H$ defined above
are widely used for proving oscillation of ordinary differential equations
of integer order; see for example  \cite{h3,l1,s5,t1,z1}.
  Yet this approach has scarcely been used to prove oscillation of
fractional differential equations.
 In Section 4, we present some examples that apply the results
established. Finally, some conclusions
 are presented at the end of this article.

\section{Oscillation criteria when $f(x)$ is increasing}

\begin{theorem} \label{thm2.1}
 Assume $f'(x)$ exists and $f'(x) \geq \mu $ for
some $\mu> 0$ and for all $x\neq 0$. Also assume that for any
$T \geq t_0$, there exist $a_1, b_1, a_2, b_2$ such that
$T \leq a_1 <b_1\leq a_2< b_2$ satisfying
\begin{equation} \label{e2.1}
e(t)  \begin{cases}
\leq0, & t\in [a_1,b_1], \\
\geq0, & t\in [a_2,b_2].
\end{cases}
\end{equation}
 If there exist  $y_i \in
(\xi_{a_i},\xi_{b_i})$ and
$\rho \in C^{\alpha}([t_0,\infty),\mathbb{R}_{+})$ such that
\begin{equation} \label{e2.2}
\begin{aligned}
&\frac{ 1}{ H(y_i,\xi_{a_i})}\int _{\xi_{a_i}} ^{y_i}
\widetilde{\rho}(s)[H(s,\xi_{a_i})\widetilde{q}(s)
-\frac{ \widetilde{r}(s)}{ 4k_1}Q_1^2(s,\xi_{a_i})]ds\\
&\quad +\frac{ 1}{ H(\xi_{b_i},y_i)}\int_{y_i}
^{\xi_{b_i}}\widetilde{\rho}(s)[H(\xi_{b_i},s)\widetilde{q}(s)
-\frac{ \widetilde{r}(s)}{ 4k_1}Q_2^2(\xi_{b_i},s)]ds>0
\end{aligned}
\end{equation}
for $i=1,2$,
 where $k_1=\mu / m$, then every solution
of  \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that $x(t)$ be a non-oscillatory solution of
 \eqref{e1.1}, say $x(t)\neq 0$ on $[T_0,\infty)$ for some sufficient
large $T_0\geq t_0$.
 Define the following  Riccati transformation function:
\begin{equation} \label{e2.3}
\omega(t)=\rho(t)\frac{r(t)\psi (x(t))D_t^{\alpha}x(t)}{f(x(t))}, \quad
t \geq T_0.
\end{equation}
Then for $t \geq T_0$, from \eqref{e1.2}-\eqref{e1.4} we deduce that
\begin{equation} \label{e2.4}
\begin{aligned}
D_t^{\alpha}\omega(t)
&=-\rho(t)q(t)+\frac{ D_t^{\alpha}\rho(t)}{ \rho(t)}\omega (t)
-\frac{ f'(x(t)) }{\rho(t)r(t)\psi(x(t))}\omega^2(t)+\frac{
e(t)\rho(t)}{ f(x(t))}\\
&\leq -\rho(t)q(t)+\frac{ D_t^{\alpha}\rho(t)}{
\rho(t)}\omega (t)-k_1\frac{ \omega
^2(t)}{ \rho(t)r(t)}+\frac{e(t)\rho(t)}{ f(x(t))}.
\end{aligned}
\end{equation}
By assumption, if $ x(t)>0$, then we can choose
 $a_1,b_1\geq T_0$ with $a_1<b_1$ such that $e(t)\leq 0$ on the interval
$[a_1,b_1]$.  If $x(t)<0$, then we can choose
$a_2,b_2\geq T_0$ with $a_2<b_2$ such that $e(t)\geq 0$ on the interval
$[a_2,b_2]$. So $\frac{ e(t)\rho(t)}{f(x(t))}\leq 0$, $t\in [a_i,b_i],\ i=1,2$,
and from \eqref{e2.4} one can deduce that
\begin{equation} \label{e2.5}
D_t^{\alpha}\omega(t)\leq -\rho(t)q(t)+\frac{ D_t^{\alpha}\rho(t)}{
\rho(t)}\omega (t)-k_1\frac{ \omega
^2(t)}{ \rho(t)r(t)},\quad t\in [a_i,b_i],\; i=1,2.
\end{equation}
Let
 $w(t)=\widetilde{w}(\xi)$. Then  $D_t^{\alpha}w(t)=\widetilde{w}'(\xi)$ and
$ D_t^{\alpha}\rho(t)=\widetilde{\rho}'(\xi)$. So
  \eqref{e2.5} is transformed into
\begin{equation} \label{e2.6}
\widetilde{\omega}'(\xi) \leq -\widetilde{\rho}(\xi)\widetilde{q}(\xi)+\frac{
\widetilde{\rho}'(\xi)}{\widetilde{\rho}(\xi)}\widetilde{\omega}
(\xi)-k_1\frac{\widetilde{\omega}^2(\xi)}{
\widetilde{\rho}(\xi)\widetilde{r}(\xi)},\quad \xi\in
[\xi_{a_i},\xi_{b_i}],\; i=1,2.
 \end{equation}

 Let $c_i$ be an arbitrary point in $(\xi_{a_i},\xi_{b_i})$.
Substituting $\xi$ with $s$, multiplying both sides of \eqref{e2.6} by $H(\xi,s)$
and integrating it  over $[c_i,\xi)$ for $\xi\in [c_i,\xi_{b_i})$, $i=1,2$,
 we obtain
\begin{equation} \label{e2.7}
\begin{aligned}
& \int _{c_i} ^{\xi}H(\xi,s)\widetilde{\rho}(s)\widetilde{q}(s)ds  \\
&\leq -\int _{c_i} ^{\xi}H(\xi,s)\widetilde{\omega}'(s)ds+\int _{c_i}
^{\xi}H(\xi,s)\big[\frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}
\widetilde{\omega}(s)-k_1\frac{\widetilde{\omega}^2(s)}{
\widetilde{\rho}(s)\widetilde{r}(s)} \big]ds\\
&=H(\xi,c_i)\widetilde{\omega}(c_i)
 -\int_{c_i}^{\xi}\widetilde{\omega}(s)h_2(\xi,s)\sqrt{H(\xi,s)}ds\\
&\quad +\int _{c_i}^{\xi}H(\xi,s)
 \big[\frac{ \widetilde{\rho}'(s)}{ \widetilde{\rho}
(s)}\widetilde{\omega}(s)-k_1\frac{
\widetilde{\omega}^2(s)}{\widetilde{\rho}(s)\widetilde{r}(s)}\big]ds\\
&=H(\xi,c_i)\widetilde{\omega}(c_i)-\int_{c_i}^{\xi}
\Big[\Big(\frac{ H(\xi,s)k_1}{\widetilde{\rho}(s)\widetilde{r}(s)}\Big)^{1/2}
 \widetilde{\omega}(s)-\frac{1}{ 2}
 \Big(\frac{\widetilde{\rho}(s)\widetilde{r}(s)}{k_1}\Big)^{1/2}
 Q_2(\xi,s)\big]^2ds\\
&\quad +\int_{c_i}^{\xi}\frac{ \widetilde{\rho}(s)
  \widetilde{r}(s)}{4k_1}Q_2^2(\xi,s)ds \\
&\leq H(\xi,c_i)\widetilde{\omega}(c_i)
 +\int_{c_i}^{\xi}\frac{\widetilde{\rho}(s)\widetilde{r}(s)}{
4k_1}Q_2^2(\xi,s)ds.
\end{aligned}
\end{equation}
 Letting $\xi\to \xi_{b_i}^{-}$  and dividing it
by $H(\xi_{b_i},c_i)$, we obtain
\begin{equation} \label{e2.8}
\begin{aligned}
&\frac{ 1}{ H(\xi_{b_i},c_i)}\int _{c_i} ^{\xi_{b_i}}H(\xi_{b_i},s)
\widetilde{\rho}(s)\widetilde{q}(s)ds \\
&\leq \widetilde{\omega}(c_i)
+\frac{ 1}{ H(\xi_{b_i},c_i)}\int _{c_i}^{\xi_{b_i}}
\frac{\widetilde{\rho}(s)\widetilde{r}(s)}{4k_1}Q_2^2(\xi_{b_i},s)ds.
\end{aligned}
\end{equation}

On the other hand, substituting $\xi$ by $s$, multiplying both
sides of \eqref{e2.6} by $H(s,\xi)$ and integrating it over $(\xi,c_i)$
for $\xi\in[\xi_{a_i},c_i)$, we obtain
\begin{equation} \label{e2.9}
\begin{aligned}
& \int _{\xi} ^{c_i}H(s,\xi)\widetilde{\rho}(s)\widetilde{q}(s)ds\\
&\leq -\int _{\xi} ^{c_i}H(s,\xi)\widetilde{\omega}'(s)ds+\int _{\xi}
 ^{c_i}H(s,\xi)\Big[\frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}
 \widetilde{\omega}(s)-k_1\frac{\widetilde{\omega}^2(s)}{
 \widetilde{\rho}(s)\widetilde{r}(s)}\Big]ds \\
&=-H(\xi,c_i)\omega (c_i)-\int_{\xi}^{c_i}\widetilde{\omega}(s)h_1(s,\xi)
 \sqrt{H(s,\xi)}ds\\
&\quad +\int _{\xi}^{c_i}H(s,\xi)
 \Big[\frac{ \widetilde{\rho}'(s)}{ \widetilde{\rho} (s)}
 \widetilde{\omega}(s)-k_1\frac{\widetilde{\omega}^2(s)}{
\widetilde{\rho}(s)\widetilde{r}(s)}\Big]ds \\
&\leq -H(c_i,\xi)\widetilde{\omega}(c_i)+\int_{\xi}^{c_i}\frac{
\widetilde{\rho}(s)\widetilde{r}(s)}{4k_1}Q_1^2(s,\xi)ds.
\end{aligned}
 \end{equation}
Letting $\xi\to \xi_{a_i}^{+}$  and dividing by $H(c_i,\xi_{a_i})$, we obtain
\[   %\label{e2.10}
\frac{ 1}{ H(c_i,\xi_{a_i})}\int _{\xi_{a_i}} ^{c_i}
 H(s,\xi_{a_i})\widetilde{\rho}(s)\widetilde{q}(s)ds\\
\leq -\widetilde{\omega}(c_i)+\frac{ 1}{
H(c_i,\xi_{a_i})}\int _{\xi_{a_i}}^{c_i}\frac{
\widetilde{\rho}(s)\widetilde{r}(s)}{
4k_1}Q_1^2(s,\xi_{a_i})ds.
\]
A combination of \eqref{e2.8} and the above inequality yields
\begin{align*}
&\frac{ 1}{ H(c_i,\xi_{a_i})}\int _{\xi_{a_i}} ^{c_i}H(s,\xi_{a_i})
 \widetilde{\rho}(s)\widetilde{q}(s)ds
+\frac{ 1}{ H(\xi_{b_i},c_i)}\int_{c_i}
^{\xi_{b_i}}H(\xi_{b_i},s)\widetilde{\rho}(s)\widetilde{q}(s)ds\\
&\leq \frac{ 1}{ 4H(c_i,\xi_{a_i})}\int_{\xi_{a_i}} ^{c_i}
\frac{ \rho(s)\widetilde{r}(s)}{ k_1}Q_1^2(s,\xi_{a_i})ds
+\frac{ 1}{ 4H(\xi_{b_i},c_i)}\int_{c_i} ^{\xi_{b_i}}
\frac{ \rho(s)\widetilde{r}(s)}{k_1}Q_2^2(\xi_{b_i},s)ds.
\end{align*}
which contradicts to \eqref{e2.2} since
$c_i$ is arbitrary in  $(\xi_{a_i},\xi_{b_i})$.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm2.2}
Under the conditions of Theorem \ref{thm2.1}, suppose \eqref{e2.2} does
not hold, and $\widetilde{q}(\xi)>0$ for any $\xi \geq \xi_0$. If
for some $u\in C[\xi_{a_i},\xi_{b_i}]$ satisfying
$u'\in L^2[\xi_{a_i},\xi_{b_i}]$, $u(\xi_{a_i})=u(\xi_{b_i})=0$,
$i=1,2$, and $u$ is not identically zero, there exists
$\rho \in C^{1}([\xi_0,\infty),R_+)$ such that
\begin{equation} \label{e2.11}
\int _{\xi_{a_i}} ^{\xi_{b_i}}
\Big\{u^2(s)\widetilde{\rho}(s)\widetilde{q}(s)
-\Big[\frac{ \widetilde{\rho}(s)\widetilde{r}(s)}{ k_1}\Big(u'(s)
+\frac{ 1}{ 2}u(s)\frac{
\widetilde{\rho}'(s)}{
\widetilde{\rho}(s)}\Big)\Big]^2 \Big\}ds>0
\end{equation}
for $i=1,2$, where $\widetilde{\rho}, \widetilde{r},
\widetilde{q}, \xi_{a_i}, \xi_{b_i}, k_1$ are defined as in
Theorem \ref{thm2.1}, then  \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that $x(t)$ be a non-oscillatory solution of
 \eqref{e1.1}, say $x(t)\neq 0$ on $[T_0,\infty)$ for some sufficient
large $T_0\geq t_0$.  Let $\widetilde{\omega}$ be defined as in 
Theorem \ref{thm2.1}.
Then  we obtain \eqref{e2.6}. Substituting $\xi$ by $s$, multiplying both
sides of \eqref{e2.6} by $u^2(s)$, integrating it with respect to $s$
from $\xi_{a_i}$ to $\xi_{b_i}$ and using
$u(\xi_{a_i})=u(\xi_{b_i})=0$, we obtain
\begin{align*}
&\int _{\xi_{a_i}} ^{\xi_{b_i}}u^2(s)\widetilde{\rho}(s)\widetilde{q}(s)ds\\
&\leq -\int _{\xi_{a_i}} ^{\xi_{b_i}}u^2(s)\widetilde{\omega}'(s)ds
 +\int _{\xi_{a_i}}^{\xi_{b_i}}u^2(s)
 \Big[\frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}
 \widetilde{\omega}(s)-k_1\frac{\widetilde{\omega}^2(s)}{
\widetilde{\rho}(s)\widetilde{r}(s)}\Big]ds\\
&=2 \int_{\xi_{a_i}}^{\xi_{b_i}}\widetilde{\omega}(s)u(s)u'(s)ds
 +\int _{\xi_{a_i}}^{\xi_{b_i}}u^2(s)\Big[\frac{ \widetilde{\rho}'(s)}{ \rho
(s)}\widetilde{\omega}(s)-k_1\frac{
\widetilde{\omega}^2(s)}{
\widetilde{\rho}(s)\widetilde{r}(s)}\Big]ds
\\
&=-\int _{\xi_{a_i}}^{\xi_{b_i}}\Big[\sqrt{\frac{k_1}{
\widetilde{\rho}(s)\widetilde{r}(s)}}u(s)\widetilde{\omega}(s)-\sqrt{\frac{
\widetilde{\rho}(s)\widetilde{r}(s)}{
k_1}}\Big(u'(s)+\frac{ 1}{2}u(s)\frac{ \widetilde{\rho}'(s)}{
\widetilde{\rho}(s)}\Big)\Big]^2ds
\\
&\quad +\int_{\xi_{a_i}}^{\xi_{b_i}}
\Big[\frac{\widetilde{\rho}(s)\widetilde{r}(s)}{ k_1}\Big(u'(s)
+\frac{ 1}{ 2}u(s)\frac{
\widetilde{\rho}'(s)}{
\widetilde{\rho}(s)}\Big)\Big]^2ds.
\end{align*}
$$
\int _{\xi_{a_i}} ^{\xi_{b_i}}
\Big\{u^2(s)\widetilde{\rho}(s)\widetilde{q}(s)
-\Big[\frac{ \widetilde{\rho}(s)\widetilde{r}(s)}{ k_1}\Big(u'(s)
+\frac{ 1}{ 2}u(s)\frac{\widetilde{\rho}'(s)}{
\widetilde{\rho}(s)}\Big)\Big]^2\Big\}ds\leq 0
$$
 which contradicts to  \eqref{e2.11}. So every solution of \eqref{e1.1} is
oscillatory. The proof is complete.
\end{proof}

\begin{corollary} \label{coro2.3}
Under the conditions of Theorem \ref{thm2.1},  suppose that \eqref{e2.2} does
not hold, and $\widetilde{q}(\xi)>0$ for any $\xi \geq \xi_0$.
If for each $r\geq \xi_0$,
\begin{equation} \label{e2.12}
\begin{aligned}
&\limsup_{\xi\to \infty} \int_{r} ^{\xi}
\{\widetilde{\rho}(s)(\xi-s)^2(s-r)^2\widetilde{q}(s)\\
&- \Big[ \frac{\widetilde{\rho}(s)\widetilde{r}(s)}{k_1}
 \Big((\xi+r-2s)+\frac{ 1}{2}(\xi-s)(s-r)
 \frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}\Big)\Big]^2 \}ds>0,
\end{aligned}
\end{equation}
then  \eqref{e1.1} is oscillatory.
\end{corollary}

The proof of the above corollary  is done by setting
$u(s)=(\xi_{b_i}-s)(s-\xi_{a_i})$ in the proof of Theorem \ref{thm2.2}.


\begin{remark} \label{rmk2.1} \rm
 The results established above provide sufficient conditions for oscillation
of \eqref{e1.1} with $f(x)$ increasing. These results are similar
to those for ordinary differential equations of integer order.
The Riccati transformation methods are similar, However, they are
essentially different.
The most significant difference lies in the fact that the
functions $\widetilde{\rho}, \widetilde{r}, \widetilde{q}$
are compound functions, the variable $\xi$  has a special form
$\xi=\frac{t^{\alpha}}{ \Gamma(1+\alpha)}$.
The main difficulty to overcome in using the Riccati transformation
for \eqref{e1.1} can be summarized in two aspects.
One is the computation of the $\alpha$-order derivative for the Riccati
transformation function $\omega(t)$,
in which two important properties \eqref{e1.3} and \eqref{e1.4}
for the modified Riemann-Liouville derivative are used
The other is how to transform \eqref{e2.5} into \eqref{e2.6},
in which the property \eqref{e1.4} for the modified Riemann-Liouville
derivative and a suitable variable transformation from the original
variable $t$ to a new variable $\xi$ denoted by $\xi=\frac{
t^{\alpha}}{ \Gamma(1+\alpha)}$ are used. In summary,
the oscillation criteria presented above are established under
the combination of the Riccati transformation method and the
properties of the modified Riemann-Liouville derivative.
\end{remark}

\section{Oscillation criteria with $f(x)$ not necessarily increasing}
\begin{theorem} \label{thm3.1}
Suppose  $ f(x)/ x\geq k_2> 0$
for all $x\neq 0$, and for any $T \geq \xi_0$, there exist
$T \leq a_1<b_1\leq a_2<b_2$ such that \eqref{e2.1} holds.
If there exist
$y_i \in (\xi_{a_i},\xi_{b_i})$ and
$\rho \in C^{1}([\xi_0,\infty),R_+)$ such that
\begin{equation} \label{e3.1}
\begin{aligned}
&\frac{ 1}{ H(y_i,\xi_{a_i})}\int _{\xi_{a_i}} ^{y_i}H(s,\xi_{a_i})
 k_2\widetilde{\rho}(s)\widetilde{q}(s)ds
+\frac{ 1}{ H(\xi_{b_i},y_i)}\int_{y_i} ^{\xi_{b_i}}H(\xi_{b_i},s)
k_2\widetilde{\rho}(s)\widetilde{q}(s)ds\\
&>\frac{ 1}{ 4H(y_i,\xi_{a_i})}\int _{\xi_{a_i}}
^{y_i}m\rho (s)\widetilde{r}(s)Q_1^2(s,\xi_{a_i})ds\\
&\quad +\frac{ 1}{ 4H(\xi_{b_i},y_i)}\int_{y_i} ^{\xi_{b_i}}m\rho
(s)\widetilde{r}(s)Q_2^2(\xi_{b_i},s)ds.
\end{aligned}
\end{equation}
for $i=1,2$, where
$Q_1, Q_2, \widetilde{q}, \widetilde{r}, \widetilde{\rho},
\xi_{a_i}, \xi_{b_i}$ are defined as in Theorem \ref{thm2.1}.
Then every solution of \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that $x(t)$ be a non-oscillatory solution of
\eqref{e1.1}, say $x(t)\neq 0$ on $[T_0,\infty)$ for some sufficient
large $T_0\geq t_0$. Define the  Riccati
transformation function
\begin{equation} \label{e3.2}
\omega(t)=\rho(t)\frac{r(t)\psi (x(t))D_t^{\alpha}x(t)}{x(t)}, \quad
 t \geq T_0.
 \end{equation}
Then for $t \geq T_0$, from \eqref{e1.2}-\eqref{e1.4} we deduce that
\begin{equation}
\begin{aligned} \label{e3.3}
&D_t^{\alpha}\omega(t)\\
&=-\frac{f(x(t))\rho(t)q(t) }{ x(t)}+\frac{
D_t^{\alpha}\rho(t)}{ \rho(t)}\omega (t)
-\frac{ 1}{\rho(t)r(t)\psi(x(t))}\omega^2(t)
+\frac{e(t)\rho(t)}{ x(t)}\\
&\leq -k_2\rho(t)q(t)+\frac{
D_t^{\alpha}\rho(t)}{ \rho(t)}\omega
(t)-\frac{ \omega ^2(t)}{m\rho(t)r(t)}+\frac{ e(t)\rho(t)}{ x(t)}.
\end{aligned}
\end{equation}

By assumption, if $ x(t)>0$, then we can choose
$a_1,b_1\geq T_0$ with $a_1<b_1$ such that $e(t)\leq 0$ on the interval
$[a_1,b_1]$.  If $x(t)<0$, then we can choose $a_2,b_2\geq T_0$ with
$a_2<b_2$ such that $e(t)\geq 0$ on the interval
$[a_2,b_2]$. So $\frac{ e(t)\rho(t)}{x(t)}\leq 0$, $t\in [a_i,b_i]$,
$i=1,2$, and from \eqref{e3.3} one can deduce that
\begin{equation} \label{e3.4}
D_t^{\alpha}\omega(t)\leq -k_2\rho(t)q(t)+\frac{
D_t^{\alpha}\rho(t)}{ \rho(t)}\omega
(t)-\frac{ \omega ^2(t)}{m\rho(t)r(t)},\quad t\in [a_i,b_i],\; i=1,2.
 \end{equation}
Let
 $w(t)=\widetilde{w}(\xi)$. Then  we have $D_t^{\alpha}w(t)=\widetilde{w}'(\xi)$ and $
 D_t^{\alpha}\rho(t)=\widetilde{\rho}'(\xi)$. So
  \eqref{e3.4} is transformed into
\begin{equation} \label{e3.5}
\widetilde{\omega}'(\xi) \leq \frac{
\widetilde{\rho}'(\xi)}{\widetilde{\rho}(\xi)}\widetilde{\omega}(\xi)
-k_2\widetilde{\rho}(\xi)\widetilde{q}(\xi)
-\frac{1}{m\widetilde{\rho}(\xi)\widetilde{r}(\xi)}\widetilde{\omega}^2(\xi),\quad
\xi\in [\xi_{a_i},\xi_{b_i}],\; i=1,2.
\end{equation}

Let $c_i$ be selected from $(\xi_{a_i},\xi_{b_i})$ arbitrarily.
Substituting $\xi$ with $s$, multiplying both sides of \eqref{e3.5} by
$H(\xi,s)$ and integrating it
over $[c_i,\xi)$ for $\xi\in [c_i,\xi_{b_i})$, after similar computation
to \eqref{e2.7}, we obtain
\begin{equation} \label{e3.6}
\begin{aligned}
& \int _{c_i} ^{\xi}H(\xi,s)k_2\widetilde{\rho}(s)\widetilde{q}(s)ds \\
&\leq -\int _{c_i} ^{\xi}H(\xi,s)\widetilde{\omega}'(s)ds+\int _{c_i}
^{\xi}H(\xi,s)\Big[\frac{
\widetilde{\rho}'(s)}{ \widetilde{\rho}
(s)}\widetilde{\omega}(s)-\frac{\widetilde{\omega}^2(s)}{
m\widetilde{\rho}(s)\widetilde{r}(s)}\Big]ds\\
&\leq H(\xi,c_i)\widetilde{\omega}(c_i)
+\int_{c_i}^{\xi}\frac{m\widetilde{\rho}(s)\widetilde{r}(s)}{4}Q_2^2(\xi,s)ds.
\end{aligned}
\end{equation}
 Letting $\xi\to \xi_{b_i}^{-}$ in $\eqref{e3.6}$ and dividing it
by $H(\xi_{b_i},c_i)$, we obtain
\begin{equation} \label{e3.7}
\begin{aligned}
&\frac{ 1}{ H(\xi_{b_i},c_i)}\int _{c_i} ^{\xi_{b_i}}H(\xi_{b_i},s)
 k_2\widetilde{\rho}(s)\widetilde{q}(s)ds \\
&\leq \widetilde{\omega}(c_i)+\frac{ 1}{
H(\xi_{b_i},c_i)}\int _{c_i}^{\xi_{b_i}}\frac{
m\widetilde{\rho}(s)\widetilde{r}(s)}{4}Q_2^2(\xi_{b_i},s)ds.
\end{aligned}
\end{equation}

On the other hand, substituting $\xi$ with $s$, multiplying both
sides of \eqref{e3.5} by $H(s,\xi)$, and integrating it  over $(\xi,c_i)$
for $\xi\in[\xi_{a_i},c_i)$, we deduce that
$$
\int _{\xi} ^{c_i}H(s,\xi)k_2\widetilde{\rho}(s)\widetilde{q}(s)ds
\leq -H(c_i,\xi)\widetilde{\omega}(c_i)
+\int_{\xi}^{c_i}\frac{m\widetilde{\rho}(s)\widetilde{r}(s)}{
4}Q_1^2(s,\xi)ds
$$
Letting $\xi\to \xi_{a_i}^{+}$ and dividing
by $H(c_i,\xi_{a_i})$, we obtain
\begin{equation} \label{e3.8}
\begin{aligned}
&\frac{ 1}{ H(c_i,\xi_{a_i})}\int _{\xi_{a_i}} ^{c_i}
H(s,\xi_{a_i})k_2\widetilde{\rho}(s)\widetilde{q}(s)ds\\
&\leq -\widetilde{\omega}(c_i)+\frac{ 1}{
H(c_i,\xi_{a_i})}\int _{\xi_{a_i}}^{c_i}\frac{
m\widetilde{\rho}(s)\widetilde{r}(s)}{
4}Q_1^2(s,\xi_{a_i})ds.
\end{aligned}
\end{equation}
A combination of \eqref{e3.7} and \eqref{e3.8} yields the inequality
\begin{align*}
&\frac{ 1}{ H(c_i,\xi_{a_i})}\int _{\xi_{a_i}} ^{c_i}
H(s,\xi_{a_i})k_2\widetilde{\rho}(s)\widetilde{q}(s)ds
+\frac{ 1}{ H(\xi_{b_i},c_i)}\int_{c_i}
^{\xi_{b_i}}H(\xi_{b_i},s)k_2\widetilde{\rho}(s)\widetilde{q}(s)ds\\
&\leq \frac{ 1}{ 4H(c_i,\xi_{a_i})}\int_{\xi_{a_i}} ^{c_i} m\rho
(s)\widetilde{r}(s)Q_1^2(s,\xi_{a_i})ds +\frac{
1}{ 4H(\xi_{b_i},c_i)}\int _{c_i} ^{\xi_{b_i}}
m\widetilde{\rho}(s)\widetilde{r}(s)Q_2^2(\xi_{b_i},s)ds,
\end{align*}
which contradicts to $\eqref{e3.1}$ since $c_i$ is selected from
$(\xi_{a_i},\xi_{b_i})$ arbitrarily. Therefore, every
solution of \eqref{e1.1} is oscillatory, and the proof is complete.
\end{proof}

\begin{theorem} \label{thm3.2}
Under the conditions of Theorem \ref{thm3.1}, furthermore, suppose \eqref{e3.1} does
not hold, and $\widetilde{q}(\xi)>0$ for any $\xi \geq \xi_0$. If
for some $u\in C[\xi_{a_i},\xi_{b_i}]$ satisfying
$u'\in L^2[\xi_{a_i},\xi_{b_i}]$, $u(\xi_{a_i})=u(\xi_{b_i})=0$,
$i=1,2$, and $u$ is not identically zero, there exists  $\rho \in
C^{1}([\xi_0,\infty),R_+)$ such that
\begin{equation} \label{e3.9}
\int _{\xi_{a_i}} ^{\xi_{b_i}}\Big\{u^2(s)k_2\widetilde{\rho}(s)\widetilde{q}(s)
-\Big[m\widetilde{\rho}(s)\widetilde{r}(s)\Big(u'(s)
+\frac{ 1}{ 2}u(s)\frac{
\widetilde{\rho}'(s)}{
\widetilde{\rho}(s)}\Big)\Big]^2 \Big\}ds>0
\end{equation}
for $i=1,2$, then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that $x(t)$ be a non-oscillatory solution of
\eqref{e1.1}, say $x(t)\neq 0$ on $[T_0,\infty)$ for some sufficient
large $T_0\geq t_0$.  Let $\omega$ be defined as in Theorem \ref{thm3.1}.
Then we obtain \eqref{e3.5}. Substituting $\xi$ by $s$, multiplying both
sides of  \eqref{e3.5} by $u^2(s)$, integrating it with respect to $s$
from $\xi_{a_i}$ to $\xi_{b_i}$ and using
$u(\xi_{a_i})=u(\xi_{b_i})=0$,  we deduce that
\begin{align*}
&\int _{\xi_{a_i}} ^{\xi_{b_i}}u^2(s)k_2\widetilde{\rho}(s)\widetilde{q}(s)ds \\
&\leq -\int _{\xi_{a_i}} ^{\xi_{b_i}}u^2(s)\widetilde{\omega}'(s)ds
 +\int _{\xi_{a_i}}^{\xi_{b_i}}u^2(s)
 \Big[\frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}\widetilde{\omega}(s)
-\frac{\widetilde{\omega}^2(s)}{m\widetilde{\rho}(s)\widetilde{r}(s)}
\Big]ds\\
&=-\int_{\xi_{a_i}}^{\xi_{b_i}}\Big[\sqrt{\frac{1}{m\widetilde{\rho}(s)
 \widetilde{r}(s)}}u(s)\widetilde{\omega}(s)
 -\sqrt{m\widetilde{\rho}(s)\widetilde{r}(s)}\Big(u'(s)
 +\frac{1}{ 2}u(s)\frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}\Big)\Big]^2ds\\
&\quad +\int_{\xi_{a_i}}^{\xi_{b_i}}
 \Big[m\widetilde{\rho}(s)\widetilde{r}(s)\Big(u'(s)
 +\frac{ 1}{ 2}u(s)\frac{\widetilde{\rho}'(s)}{
\widetilde{\rho}(s)}\Big)\Big]^2ds .
\end{align*}
Then
$$
\int _{\xi_{a_i}} ^{\xi_{b_i}}\Big\{u^2(s)k_2\widetilde{\rho}(s)\widetilde{q}(s)
-\Big[m\widetilde{\rho}(s)\widetilde{r}(s)\Big(u'(s)
+\frac{ 1}{ 2}u(s)\frac{
\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}\Big)\Big]^2 \Big\}ds\leq 0,
$$
which contradicts  \eqref{e3.9}. The proof is complete.
\end{proof}

The following corollary has a proof similar to the one of 
Corollary \ref{coro2.3}.

\begin{corollary} \label{coro3.3}
Under the conditions of Theorem \ref{thm3.2}, if for each $r\geq \xi_0$,
\begin{align*}
&\limsup _{\xi\to \infty} \int _{r} ^{\xi}
\Big\{k_2\widetilde{\rho}(s)(\xi-s)^2(s-r)^2\widetilde{q}(s)\\
&-\Big[m\widetilde{\rho}(s)\widetilde{r}(s)\Big((\xi+r-2s)+\frac{
1}{ 2}(\xi-s)(s-r)\frac{\widetilde{\rho}'(s)}{\widetilde{\rho}(s)}\Big)\Big]^2
\Big\}ds>0,
\end{align*}
then  \eqref{e1.1} is oscillatory.
\end{corollary}

\section{Applications}

\begin{example} \label{examp1} \rm
 Consider the  nonlinear fractional
differential equation with forced term
\begin{equation} \label{e4.1}
D_t^{\alpha}\Big(\sin^2(\frac{t^{\alpha}}{\Gamma(1+\alpha)})e^{-x^2(t)}
D_t^{\alpha}x(t)\Big)+(x(t)+x^{3}(t))
=\sin(\frac{ t^{\alpha}}{\Gamma(1+\alpha)}),
\end{equation}
$t\geq 2$, $0<\alpha<1$.
This corresponds to \eqref{e1.1} with
$t_0=2$, $r(t)=\sin ^2(\frac{t^{\alpha}}{\Gamma(1+\alpha)})$,
$\psi(x)=e^{-x^2}$, $q(t)\equiv 1$, $f(x)=x+x^3$,
$e(t)=\sin (\frac{t^{\alpha}}{ \Gamma(1+\alpha)})$.
Therefore,  $\psi(x)\leq 1$, $f'(x)=1+3x^2\geq 1$, which implies $\mu=m=1$.
Since $\xi=\frac{ t^{\alpha}}{
\Gamma(1+\alpha)}$,  it follows that
$\widetilde{r}(\xi)=r(t)=\sin^2(\frac{ t^{\alpha}}{
\Gamma(1+\alpha)})=\sin ^2\xi$.

 In \eqref{e2.11}, we have $k_1=\mu/m=1$. Furthermore, letting
$u(s)=\sin s$, $\xi_{a_i}=(2k+i)\pi,\ \xi_{b_i}=(2k+i)\pi +\pi$ such that
 $\xi_{a_i}$, $\xi_{b_i}$ is sufficiently large, we obtain
$u(\xi_{a_i})=u(\xi_{b_i})=0$, and
 considering $\widetilde{q}(s)\equiv 1$, $\widetilde{\rho}(s)\equiv 1$,
it holds that
 $$
\int_{(2k+i)\pi}^{(2k+i)\pi+\pi}\left(\sin^2s-\sin^2s\cos^2 s
\right)ds
= \int _{(2k+i)\pi}^{(2k+i)\pi +\pi}\sin^{4}sds >0.
$$
On the other hand, by the connection between $a_i, b_i$ and
$\xi_{a_i}, \xi_{b_i}$ we have
\[
a_i=[\Gamma(1+\alpha)(2k+i)\pi]^{\frac{1}{\alpha}},\quad
b_i=[\Gamma(1+\alpha)(2k+i)\pi +\pi]^{\frac{1}{\alpha}}.
\]
So for $e(t)=\sin (\frac{ t^{\alpha}}{\Gamma(1+\alpha)})$,  one can
see \eqref{e2.1} holds with $k$ selected
enough large. Therefore, by Theorem \ref{thm2.2} Equation
\eqref{e4.1} is oscillatory.
\end{example}


\begin{example} \label{examp2} \rm
 Consider the  nonlinear fractional
differential equation with forced term:
\begin{equation} \label{e4.2}
 D_t^{\alpha}\Big(\sin
^2(\frac{t^{\alpha}}{\Gamma(1+\alpha)})\frac{
1}{ 1+x^2(t)}D_t^{\alpha}x(t)\Big)+
\frac{ x(t)(2+x^2(t))}{1+x^2(t)}
=\sin(\frac{ t^{\alpha}}{
\Gamma(1+\alpha)}),
\end{equation}
$t\geq 2$, $0<\alpha<1$.
This corresponds to  \eqref{e1.1} with
$t_0=2$, $r(t)=\sin^2(\frac{t^{\alpha}}{\Gamma(1+\alpha)})$,
$\psi(x)=\frac{ 1}{ 1+x^2}$, $q(t)\equiv 1$,
$f(x)= \frac{ 2x+x^{3}}{ 1+x^2}$,
$e(t)=\sin (\frac{ t^{\alpha}}{\Gamma(1+\alpha)})$.

Therefore,  $\widetilde{r}(\xi)=r(t)
=\sin ^2(\frac{t^{\alpha}}{ \Gamma(1+\alpha)})=\sin ^2\xi$,
$\psi(x)\leq 1$, which implies $m=1$. Furthermore, we notice that it
is complicated in obtaining the lower bound of $f'(x)$. So Theorems
\ref{thm2.1} and \ref{thm2.2} are not applicable, 
while one can easily see $f(x)/x\geq 1$,
which implies $k_2=1$. Then by Theorem \ref{thm3.2}, and analysis similar
to the last paragraph in Example \ref{examp1},
 Equation \eqref{e4.2} is oscillatory.
\end{example}



\subsection*{Conclusions}
We have established some new oscillation criteria for a nonlinear
forced fractional differential equation. As one can see, the
variable transformation used in $\xi$  is very important, transforms
a fractional  differential equation into an  ordinary differential
equation of integer order, whose oscillation criteria can be established
using a generalized Riccati transformation,
inequalities, and an integration average technique.
Finally, we note that this approach can also be applied to the
oscillation for other fractional differential equations involving
the modified Riemann-liouville derivative.

\subsection*{Acknowledgements}
The authors would like to thank the anonymous reviewers 
for their valuable suggestions on improving the content of this
article.


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\section{Addendum posted on November 17, 2016}

In response to a message from a reader, the authors want to point out
that the first equality in \eqref{e1.4} is the chain rule for
Jumarie's modified Riemann-Liouville derivative obtained in \cite{j1}.
However. this rule is icorrect, a shown in the article

Cheng-shi Liu;
Counterexamples on Jumarie’s two basic fractional calculus
formulae, Commun Nonlinear Sci Numer Simulat 22 (2015) 92–94.

Therefore the main result of this article is icorrect.
End of addendum.


\end{document}