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Forced oscillation of certain fractional differential equations
Advances in Difference Equations volume 2013, Article number: 125 (2013)
Abstract
The paper deals with the forced oscillation of the fractional differential equation
with the initial conditions ({D}_{a}^{qk}x)(a)={b}_{k} (k=1,2,\dots ,m1) and {lim}_{t\to {a}^{+}}({I}_{a}^{mq}x)(t)={b}_{m}, where {D}_{a}^{q}x is the RiemannLiouville fractional derivative of order q of x, m1<q\le m, m\ge 1 is an integer, {I}_{a}^{mq}x is the RiemannLiouville fractional integral of order mq 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 RiemannLiouville 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.
1 Introduction
The aim of the paper is to establish several oscillation theorems for forced fractional differential equation with initial conditions of the form
where {D}_{a}^{q}x is the RiemannLiouville fractional derivative of order q of x, m1<q\le m, m\ge 1 is an integer, {I}_{a}^{mq}x is the RiemannLiouville fractional integral of order mq 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 RiemannLiouville 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.
However, to the best of our knowledge, very little is known regarding the oscillation of fractional differential equations up to now. Recently, Chen [19] established some oscillation criteria for the fractional differential equation
where q\in (0,1) is a constant, \eta >0 is a quotient of odd positive integers, {D}_{}^{q}x is the Liouville rightsided fractional derivative of order q of x defined by
here Γ is the gamma function defined by
For details of the Liouville fractional integrals and fractional derivatives, one can refer to [[10], Sections 2.2 and 2.3].
Grace et al. [20] discussed the oscillation of a forced fractional differential equation with initial conditions of the form (1.1) under the conditions
and
where {p}_{1},{p}_{2}\in C([a,\mathrm{\infty}),(0,\mathrm{\infty})) and \beta ,\gamma >0 are constants. Grace 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 RiemannLiouville fractional operator is replaced by the Caputo fractional operator.
Obviously, Grace 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 RiemannLiouville 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 wellknown 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]
The RiemannLiouville fractional integral of order q>0 of a function x:[a,\mathrm{\infty})\to \mathbb{R} is defined by
provided the righthand 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]
The RiemannLiouville fractional derivative of order q>0 of a function x:[a,\mathrm{\infty})\to \mathbb{R} is defined by
provided the righthand side is pointwise defined on [a,\mathrm{\infty}), where m1<q\le m and m\ge 1 is an integer. Furthermore, we set {D}_{a}^{0}x:=x.
Definition 2.3 [7]
The Caputo fractional derivative of order q>0 of a function x:[a,\mathrm{\infty})\to \mathbb{R} is defined by
provided the righthand side is pointwise defined on [a,\mathrm{\infty}), where m1<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}^{u1}.

(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}^{u1}.
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
where H is defined as in Theorem 3.1. From (1.4), (1.2), (1.3) and (3.3), we obtain
where
and
Multiplying (3.4) by {t}^{1q}, we have, for t\ge {T}_{1},
Take {T}_{2}>{T}_{1}. Next, we consider the cases 0<q\le 1 and q>1, respectively.
Case (i). Let 0<q\le 1. Then we get m=1, \mathrm{\Phi}(t)={b}_{1}{(ta)}^{q1},
and
It follows from (3.7)(3.9) that {t}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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).
Case (ii). Let q>1. Then we have m\ge 2,
and
From (3.7), (3.10) and (3.11), we conclude {t}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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 2.22.4 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 RiemannLiouville fractional differential equation
where 0<q<1.
In (3.12), a=0, m=1, {f}_{1}(t,x)={x}^{5}ln(\mathrm{e}+t), v(t)=\frac{2{t}^{2q}}{\mathrm{\Gamma}(3q)}+({t}^{10}{t}^{2/3})ln(\mathrm{e}+t), {f}_{2}(t,x)={x}^{1/3}ln(\mathrm{e}+t) and {b}_{1}=0. Taking {p}_{1}(t)={p}_{2}(t)=ln(\mathrm{e}+t), \beta =5 and \gamma =1/3, we find that the conditions (1.2) and (1.3) are satisfied. But the condition (3.1) is not satisfied since for every sufficiently large T\ge 1 and all t\ge T, we have v(t)>0 and
where H is defined as in Theorem 3.1. Taking x(t)={t}^{2}, by Definition 2.1 we get
Integrating by parts twice, we obtain
Therefore, by Definition 2.2 we conclude
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}^{1q}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
Firstly, we suppose 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
where 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}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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}^{1q}\ge {c}_{3}({T}_{2}){c}_{4}({T}_{1})+{t}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[v(s)+H(s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s for t\ge {T}_{2}. Since {lim}_{t\to \mathrm{\infty}}{t}^{1q}=0, we obtain {lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}{t}^{1q}{\int}_{{T}_{1}}^{t}{(ts)}^{q1}[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 RiemannLiouville 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 RiemannLiouville derivatives have certain disadvantages when trying to model realworld phenomena with fractional differential equations. When comparing the RiemannLiouville 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 integerorder differential equations, i.e., they contain the limit values of integerorder derivatives of unknown functions at the lower terminal t=a.
In this section, we study the oscillation of (1.1) when the RiemannLiouville fractional operator is replaced by the Caputo fractional operator, i.e., the oscillation of the initial value problem
where {}^{C}D_{a}^{q}x is the Caputo fractional derivative of order q of x defined by (2.3), m1<q\le m, m\ge 1 is an integer, {b}_{k} (k=0,1,\dots ,m1) 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 3.23.4 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.
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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).
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The first author discovered the topic and the main ideas for the proof of the paper and made the actual writing. All authors discussed the paper together. The second and the third authors discovered some helpful ideas for the proof of this paper and checked the proof of the paper. All authors read and approved the final manuscript.
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Chen, DX., Qu, PX. & Lan, YH. Forced oscillation of certain fractional differential equations. Adv Differ Equ 2013, 125 (2013). https://doi.org/10.1186/168718472013125
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DOI: https://doi.org/10.1186/168718472013125