# Forced oscillation of certain fractional differential equations

- Da-Xue Chen
^{1}Email author, - Pei-Xin Qu
^{2}and - Yong-Hong Lan
^{3}

**2013**:125

https://doi.org/10.1186/1687-1847-2013-125

© Chen et al.; licensee Springer 2013

**Received: **2 December 2012

**Accepted: **19 April 2013

**Published: **2 May 2013

## Abstract

The paper deals with the forced oscillation of the fractional differential equation

with the initial conditions $({D}_{a}^{q-k}x)(a)={b}_{k}$ ($k=1,2,\dots ,m-1$) and ${lim}_{t\to {a}^{+}}({I}_{a}^{m-q}x)(t)={b}_{m}$, where ${D}_{a}^{q}x$ is the Riemann-Liouville fractional derivative of order *q* of *x*, $m-1<q\le m$, $m\ge 1$ is an integer, ${I}_{a}^{m-q}x$ is the Riemann-Liouville fractional integral of order $m-q$ of *x*, and ${b}_{k}$ ($k=1,2,\dots ,m$) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young’s inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results.

**MSC:**34A08, 34C10.

## Keywords

## 1 Introduction

where ${D}_{a}^{q}x$ is the Riemann-Liouville fractional derivative of order *q* of *x*, $m-1<q\le m$, $m\ge 1$ is an integer, ${I}_{a}^{m-q}x$ is the Riemann-Liouville fractional integral of order $m-q$ of *x*, ${b}_{k}$ ($k=1,2,\dots ,m$) are/is constants/constant, ${f}_{i}:[a,\mathrm{\infty})\times \mathbb{R}\to \mathbb{R}$ ($i=1,2$) are continuous functions, and $v:[a,\mathrm{\infty})\to \mathbb{R}$ is a continuous function.

By a solution of (1.1), we mean a nontrivial function $x\in C([a,\mathrm{\infty}),\mathbb{R})$ which has the property ${D}_{a}^{q}x\in C([a,\mathrm{\infty}),\mathbb{R})$ and satisfies (1.1) for $t\ge a$. Our attention is restricted to those solutions of (1.1) which exist on $[a,\mathrm{\infty})$ and satisfy $sup\{|x(t)|:t>{t}_{\ast}\}>0$ for any ${t}_{\ast}\ge a$. A solution *x* of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, *etc.*; see, for example, [1–6]. There has been a significant development in ordinary and partial differential equations involving both Riemann-Liouville and Caputo fractional derivatives in recent years. The books on the subject of fractional integrals and fractional derivatives by Diethelm [7], Miller and Ross [8], Podlubny [9] and Kilbas *et al.* [10] summarize and organize much of fractional calculus and many of theories and applications of fractional differential equations. Many papers have studied some aspects of fractional differential equations such as the existence and uniqueness of solutions to Cauchy type problems, the methods for explicit and numerical solutions, and the stability of solutions, and we refer to [11–18] and the references quoted therein.

*q*of

*x*defined by

For details of the Liouville fractional integrals and fractional derivatives, one can refer to [[10], Sections 2.2 and 2.3].

*et al.*[20] discussed the oscillation of a forced fractional differential equation with initial conditions of the form (1.1) under the conditions

*et al.*gave several oscillation results for (1.1) by reducing the equation to the equivalent Volterra fractional integral equation (see [7, Lemma 5.2])

when $\beta >1=\gamma $, $\beta =1>\gamma >0$ and $\beta >1>\gamma >0$, respectively. The results are also stated when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator.

*et al.*[20] did not consider the cases $\beta >\gamma >1$ and $1>\beta >\gamma >0$ for (1.1). In this paper, we establish several oscillation criteria for (1.1) under the conditions (1.2), (1.3) and $\beta >\gamma >0$ by using Young’s inequality. Furthermore, we obtain some oscillation theorems for (1.1) without the condition (1.3) but with the condition (1.2) and the following conditions:

where ${p}_{1},{p}_{2}\in C([a,\mathrm{\infty}),(0,\mathrm{\infty}))$ and $\beta ,\gamma >0$ are constants. We also get some new oscillatory properties of (1.1) when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. Our results improve and extend some of those in [20].

## 2 Preliminaries and a lemma

In this section, we recall several definitions of fractional integrals and fractional derivatives and the well-known Young’s inequality, which will be used throughout this paper. More details on fractional calculus can be found in [7–10].

**Definition 2.1** [7]

provided the right-hand side is pointwise defined on $[a,\mathrm{\infty})$, where Γ is the gamma function. Furthermore, we set ${I}_{a}^{0}x:=x$.

**Definition 2.2** [7]

provided the right-hand side is pointwise defined on $[a,\mathrm{\infty})$, where $m-1<q\le m$ and $m\ge 1$ is an integer. Furthermore, we set ${D}_{a}^{0}x:=x$.

**Definition 2.3** [7]

provided the right-hand side is pointwise defined on $[a,\mathrm{\infty})$, where $m-1<q\le m$, $m\ge 1$ is an integer and ${x}^{(m)}$ denotes the usual derivative of integer order *m* of *x*. Furthermore, we set ${}^{C}D_{a}^{0}x:=x$.

**Lemma 2.1**(Young’s inequality)

- (i)
*Let*$X,Y\ge 0$, $u>1$*and*$\frac{1}{u}+\frac{1}{v}=1$,*then*$XY\le \frac{1}{u}{X}^{u}+\frac{1}{v}{Y}^{v}$,*where the equality holds if and only if*$Y={X}^{u-1}$. - (ii)
*Let*$X\ge 0$, $Y>0$, $0<u<1$*and*$\frac{1}{u}+\frac{1}{v}=1$,*then*$XY\ge \frac{1}{u}{X}^{u}+\frac{1}{v}{Y}^{v}$,*where the equality holds if and only if*$Y={X}^{u-1}$.

## 3 Main results

**Theorem 3.1**

*Suppose that*(1.2)

*and*(1.3)

*hold with*$\beta >\gamma $.

*If*

*and*

*for every sufficiently large* *T*, *where* $H(s):=(\beta /\gamma -1){[\gamma {p}_{2}(s)/\beta ]}^{\beta /(\beta -\gamma )}{p}_{1}^{\gamma /(\gamma -\beta )}(s)$, *then every solution of* (1.1) *is oscillatory*.

*Proof*Let

*x*be a nonoscillatory solution of (1.1). Firstly, we suppose that

*x*is an eventually positive solution of (1.1). Then there exists ${T}_{1}>a$ such that $x(t)>0$ for $t\ge {T}_{1}$. Let $s\ge {T}_{1}$ and take $X={|x|}^{\gamma}(s)$, $Y=\gamma {p}_{2}(s)/(\beta {p}_{1}(s))$, $u=\beta /\gamma $ and $v=\beta /(\beta -\gamma )$, then from Part (i) of Lemma 2.1 we conclude

*H*is defined as in Theorem 3.1. From (1.4), (1.2), (1.3) and (3.3), we obtain

Take ${T}_{2}>{T}_{1}$. Next, we consider the cases $0<q\le 1$ and $q>1$, respectively.

It follows from (3.7)-(3.9) that ${t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s>-[{c}_{1}({T}_{2})+{c}_{2}({T}_{1},{T}_{2})]$ for $t\ge {T}_{2}$. Therefore, we find ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty}}{t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge -[{c}_{1}({T}_{2})+{c}_{2}({T}_{1},{T}_{2})]>-\mathrm{\infty}$, which contradicts (3.1).

From (3.7), (3.10) and (3.11), we conclude ${t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s>-[{c}_{3}({T}_{2})+{c}_{4}({T}_{1})]$ for $t\ge {T}_{2}$. Hence, we obtain ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty}}{t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge -[{c}_{3}({T}_{2})+{c}_{4}({T}_{1})]>-\mathrm{\infty}$, which contradicts (3.1).

Finally, we assume that *x* is an eventually negative solution of (1.1). Then a similar argument leads to a contradiction with (3.2). The proof is complete. □

**Remark 3.1** In [20], the plus sign ‘+’ in (2.9) in Theorem 2.2, (2.13) in Theorem 2.3, (2.17) in Theorem 2.4, (3.6) in Theorem 3.2, (3.8) in Theorem 3.3 and (3.10) in Theorem 3.4 should be the minus sign ‘−’.

**Remark 3.2** Theorems 2.2 and 2.3 in [20] are the special cases of our Theorem 3.1 with $\beta >1=\gamma $ and $\beta =1>\gamma >0$, respectively. Our Theorem 3.1 improves and extends the results of Theorems 2.2-2.4 in [20] since these theorems did not include the cases $\beta >\gamma >1$ and $1>\beta >\gamma >0$ for (1.1).

The following example shows that the condition (3.1) cannot be dropped.

**Example 3.1**Consider the Riemann-Liouville fractional differential equation

where $0<q<1$.

*H*is defined as in Theorem 3.1. Taking $x(t)={t}^{2}$, by Definition 2.1 we get

which implies that $x(t)={t}^{2}$ satisfies the first equality in (3.12). From (3.13) we get ${lim}_{t\to {0}^{+}}({I}_{0}^{1-q}x)(t)=0$, which yields that $x(t)={t}^{2}$ satisfies the second equality in (3.12). Hence, $x(t)={t}^{2}$ is a nonoscillatory solution of (3.12).

Next, we consider the case when (1.5) holds, which was not considered in [20].

**Theorem 3.2**

*Let*$q\ge 1$

*and suppose that*(1.2)

*and*(1.5)

*hold with*$\beta <\gamma $.

*If*

*and*

*for every sufficiently large* *T*, *where* *H* *is defined as in Theorem * 3.1, *then every bounded solution of* (1.1) *is oscillatory*.

*Proof*Let

*x*be a bounded nonoscillatory solution of (1.1). Then there exist constants ${M}_{1}$ and ${M}_{2}$ such that

*x*is a bounded eventually positive solution of (1.1). Then there exists ${T}_{1}>a$ such that $x(t)>0$ for $t\ge {T}_{1}$. Similar to the proof of (3.3), by Part (ii) of Lemma 2.1 we find

*H*is defined as in Theorem 3.1. Define Φ and Ψ as in (3.5) and (3.6), respectively. Similar to the proof of (3.7), from (1.4), (1.2), (1.5) and (3.17), we get, for $t\ge {T}_{1}$,

Take ${T}_{2}>{T}_{1}$. Next, we consider the cases $q=1$ and $q>1$, respectively.

Case (i). Let $q=1$. Then (3.8) and (3.9) are still true. From (3.16), (3.8), (3.9) and (3.18), we conclude ${M}_{2}\mathrm{\Gamma}(q)\ge -{c}_{1}({T}_{2})-{c}_{2}({T}_{1},{T}_{2})+{t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$ for $t\ge {T}_{2}$. Thus, we see ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}{t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le {c}_{1}({T}_{2})+{c}_{2}({T}_{1},{T}_{2})+{M}_{2}\mathrm{\Gamma}(q)<\mathrm{\infty}$, which contradicts (3.14).

Case (ii). Let $q>1$. Then (3.10) and (3.11) are still valid. From (3.16), (3.10), (3.11) and (3.18), we conclude ${M}_{2}\mathrm{\Gamma}(q){t}^{1-q}\ge -{c}_{3}({T}_{2})-{c}_{4}({T}_{1})+{t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$ for $t\ge {T}_{2}$. Since ${lim}_{t\to \mathrm{\infty}}{t}^{1-q}=0$, we obtain ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}{t}^{1-q}{\int}_{{T}_{1}}^{t}{(t-s)}^{q-1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le {c}_{3}({T}_{2})+{c}_{4}({T}_{1})<\mathrm{\infty}$, which contradicts (3.14).

Finally, we suppose that *x* is a bounded eventually negative solution of (1.1). Then a similar argument leads to a contradiction with (3.15). The proof is complete. □

## 4 Results with the Caputo fractional derivative

The Riemann-Liouville fractional derivatives played an important role in the development of the theory of fractional derivatives and integrals and for their applications in pure mathematics. But it turns out that the Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations. When comparing the Riemann-Liouville definition and the Caputo definition of fractional derivatives, we will see this second one seems to be better suited to such tasks. The main advantages of the Caputo fractional derivatives is that the initial conditions for fractional differential equations with Caputo fractional derivatives take on the same form as for integer-order differential equations, *i.e.*, they contain the limit values of integer-order derivatives of unknown functions at the lower terminal $t=a$.

*i.e.*, the oscillation of the initial value problem

*q*of

*x*defined by (2.3), $m-1<q\le m$, $m\ge 1$ is an integer, ${b}_{k}$ ($k=0,1,\dots ,m-1$) are/is constants/constant, ${f}_{i}:[a,\mathrm{\infty})\times \mathbb{R}\to \mathbb{R}$ ($i=1,2$) are continuous functions, and $v:[a,\mathrm{\infty})\to \mathbb{R}$ is a continuous function. The corresponding Volterra fractional integral equation (see [7, Lemma 6.2]) becomes

Similar to the proof of Theorems 3.1 and 3.2, we can prove the following theorems.

**Theorem 4.1**

*Suppose that*(1.2)

*and*(1.3)

*hold with*$\beta >\gamma $.

*If*

*and*

*for every sufficiently large* *T*, *where* *H* *is defined as in Theorem * 3.1, *then every solution of* (4.1) *is oscillatory*.

**Theorem 4.2**

*Let*$q\ge 1$

*and suppose that*(1.2)

*and*(1.5)

*hold with*$\beta <\gamma $.

*If*

*and*

*for every sufficiently large* *T*, *where* *H* *is defined as in Theorem * 3.1, *then every bounded solution of* (4.1) *is oscillatory*.

**Remark 4.1** Theorems 3.2 and 3.3 in [20] are the special cases of our Theorem 4.1 with $\beta >1=\gamma $ and $\beta =1>\gamma >0$, respectively. Our Theorem 4.1 improves and extends the results of Theorems 3.2-3.4 in [20] since these theorems did not include the cases $\beta >\gamma >1$ and $1>\beta >\gamma >0$ for (4.1). The case considered in our Theorem 4.2 was not discussed in [20] and hence our Theorem 4.2 is a new result.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are very grateful to the anonymous referees for their valuable suggestions and comments, which helped the authors to improve the previous manuscript of the article. This work was supported by the National Natural Science Foundation of P.R. China (Grants No. 11271311 and No. 61104072) and the Natural Science Foundation of Hunan of P.R. China (Grant No. 11JJ3010).

## Authors’ Affiliations

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