\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 $$\label{e1.1} D_t^{\alpha}[r(t)\psi(x(t))D_t^{\alpha}x(t)]+q(t)f(x(t))=e(t) ,t \geq t_0>0,\ 0<\alpha<1,$$ 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 0 \end{aligned} fori=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: $$\label{e2.3} \omega(t)=\rho(t)\frac{r(t)\psi (x(t))D_t^{\alpha}x(t)}{f(x(t))}, \quad t \geq T_0.$$ Then for$t \geq T_0, from \eqref{e1.2}-\eqref{e1.4} we deduce that \label{e2.4} \begin{aligned} D_t^{\alpha}\omega(t) &=-\rho(t)q(t)+\frac{ D_t^{\alpha}\rho(t)}{ \rho(t)}\omega (t) -\frac{ f'(x(t)) }{\rho(t)r(t)\psi(x(t))}\omega^2(t)+\frac{ e(t)\rho(t)}{ f(x(t))}\\ &\leq -\rho(t)q(t)+\frac{ D_t^{\alpha}\rho(t)}{ \rho(t)}\omega (t)-k_1\frac{ \omega ^2(t)}{ \rho(t)r(t)}+\frac{e(t)\rho(t)}{ f(x(t))}. \end{aligned} By assumption, if x(t)>0$, then we can choose$a_1,b_1\geq T_0$with$a_10$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 $$\label{e2.11} \int _{\xi_{a_i}} ^{\xi_{b_i}} \Big\{u^2(s)\widetilde{\rho}(s)\widetilde{q}(s) -\Big[\frac{ \widetilde{\rho}(s)\widetilde{r}(s)}{ k_1}\Big(u'(s) +\frac{ 1}{ 2}u(s)\frac{ \widetilde{\rho}'(s)}{ \widetilde{\rho}(s)}\Big)\Big]^2 \Big\}ds>0$$ 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, \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} then \eqref{e1.1} is oscillatory. \end{corollary} The proof of the above corollary is done by settingu(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\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} 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 $$\label{e3.2} \omega(t)=\rho(t)\frac{r(t)\psi (x(t))D_t^{\alpha}x(t)}{x(t)}, \quad t \geq T_0.$$ Then for $t \geq T_0$, from \eqref{e1.2}-\eqref{e1.4} we deduce that \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} By assumption, if $x(t)>0$, then we can choose $a_1,b_1\geq T_0$ with $a_10$ 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 $$\label{e3.9} \int _{\xi_{a_i}} ^{\xi_{b_i}}\Big\{u^2(s)k_2\widetilde{\rho}(s)\widetilde{q}(s) -\Big[m\widetilde{\rho}(s)\widetilde{r}(s)\Big(u'(s) +\frac{ 1}{ 2}u(s)\frac{ \widetilde{\rho}'(s)}{ \widetilde{\rho}(s)}\Big)\Big]^2 \Big\}ds>0$$ 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 $$\label{e4.1} D_t^{\alpha}\Big(\sin^2(\frac{t^{\alpha}}{\Gamma(1+\alpha)})e^{-x^2(t)} D_t^{\alpha}x(t)\Big)+(x(t)+x^{3}(t)) =\sin(\frac{ t^{\alpha}}{\Gamma(1+\alpha)}),$$ $t\geq 2$, $0<\alpha<1$. 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: $$\label{e4.2} D_t^{\alpha}\Big(\sin ^2(\frac{t^{\alpha}}{\Gamma(1+\alpha)})\frac{ 1}{ 1+x^2(t)}D_t^{\alpha}x(t)\Big)+ \frac{ x(t)(2+x^2(t))}{1+x^2(t)} =\sin(\frac{ t^{\alpha}}{ \Gamma(1+\alpha)}),$$ $t\geq 2$, $0<\alpha<1$. 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. \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal, M. 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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}