Abstract
Based on the properties of nonlocal fractional calculus generated by conformable derivatives, we establish some sufficient conditions for oscillation of all solutions for fractional differential equations with damping term. Forced oscillation of conformable differential equations in the frame of Riemann, as well as of Caputo type, is established. Examples are provided to demonstrate the effectiveness of the main results.
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1 Introduction
Fractional differential equations gained considerable importance due to their various applications in viscoelasticity, electroanalytical chemistry, control theory, many physical problems, etc. The books [1,2,3,4,5,6] summarize and organize much of fractional calculus and many of theories and applications of fractional differential equations. Many authors have studied the existence and uniqueness of solutions for different types of fractional boundary value problems; see the papers [7,8,9,10,11,12,13,14,15,16,17,18] and the references cited therein.
The oscillation theory for fractional differential and difference equations has been studied by some authors (see [19,20,21,22,23,24,25,26,27,28,29]). In [23] the authors studied the oscillation theory for fractional differential equations by considering fractional initial value problem of the form
where \(\mathcal{D}^{q}_{a}\) denotes the Riemann–Liouville fractional derivative starting at a point a, of order q with \(0 < q\le 1\), \(J_{a}^{1-q}\) is the Riemann–Liouville fractional integral starting at a point a, of order \(1-q\), \(f_{1}\), \(f_{2}\) are continuous functions.
Recently, in [21] the authors studied the oscillation of a conformable initial value problem of the form
where \(m=\lceil \alpha \rceil =\min \{m\in \mathbb{Z}|m\geq \alpha \rbrace \), \({}_{a}{\mathfrak{D}}^{\alpha ,\rho }\) is the left conformable derivative of order \(\alpha \in {\mathbb{C}}\), \(\Re (\alpha )\ge 0\) in the Riemann–Liouville setting and \({}_{a}{\mathfrak{J}}^{\alpha , \rho }\) is the left conformable integral operator.
In [22] the authors studied forced oscillatory properties of solutions to the nonlinear fractional initial value problem with damping
where b is a real number, \(\alpha \in (0,1)\) is a given constant, and \(\mathcal{D}_{0^{+}}^{\alpha }\) is the Riemann–Liouville fractional derivative of order α.
In this paper, motivated by the above papers, we study forced oscillatory properties of solutions to the conformable initial value problem with damping in the Riemann–Liouville setting as follows:
where \(m=\lceil \alpha \rceil \), \(0<\rho \leq 1\), \(p\in \mathbb{C}( \mathbb{R}^{+},\mathbb{R})\), \(q\in \mathbb{C}(\mathbb{R}^{+}, \mathbb{R}^{+})\), \(g\in \mathbb{C}(\mathbb{R}^{+},\mathbb{R})\), \(f\in \mathbb{C}(\mathbb{R},\mathbb{R})\) are continuous functions, \({}_{a}\mathfrak{D}^{\alpha ,\rho }\) is the left conformable derivative of order \(\alpha \in \mathbb{C}\), \(\Re (\alpha )\geq 0\) in the Riemann–Liouville setting, and \({}_{a}\mathfrak{I}^{\alpha ,\rho }\) is the left conformable integral operator.
Moreover, we study the forced oscillation of conformable initial value problems in the Caputo setting of the form
where \(m=\lceil \alpha \rceil \), and \({}_{a}^{C}\mathfrak{D}^{\alpha , \rho }x\) is the left conformable derivative of order \(\alpha \in \mathbb{C}\), \(\Re (\alpha )\geq 0\) in the Caputo setting.
Definition 1.1
The solution x of problem (1.4) (respectively (1.5)) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory.
This paper is organized as follows. Section 2 introduces some notations and provides the definitions of conformable fractional integral and differential operators together with some basic properties and lemmas that are needed in the proofs of the main theorems. In Sect. 3, forced oscillation of conformable fractional differential equations in the frame of Riemann is presented, while in Sect. 4 forced oscillation of conformable fractional differential equations in the frame of Caputo is established. Examples are provided in Sect. 5 to demonstrate the effectiveness of the main theorems.
2 Preliminaries
The left conformable derivative starting from a of a function \(f:[a,\infty )\rightarrow \mathbb{R}\) of order \(0<\rho \leq 1\) is defined by
If \(({}_{a}D^{\rho }f )(t)\) exists on \((a,b)\), then \(({}_{a}D^{\rho }f )(a)=\lim_{t\rightarrow a^{+}} ({}_{a}D ^{\rho }f )(t)\). If f is differentiable, then
The corresponding left conformable integral is defined as
For the extension to the higher order \(\rho >1\), see [30].
Definition 2.1
([31])
The left conformable integral operator is defined by
where \(\alpha \in \mathbb{C}\), \(\Re (\alpha )\geq 0\).
Definition 2.2
([31])
The left conformable derivative of order \(\alpha \in \mathbb{C}\), \(\Re (\alpha )\geq 0\) in the Riemann–Liouville setting is defined by
where \(m=\lceil \Re (\alpha )\rceil \), \({}_{a}^{m}D^{\rho }=\underbrace{ {{}_{a}D^{\rho }} {{}_{a}D^{\rho }} \cdots {{}_{a}D^{\rho }}}_{{m \text{ times}}}\), and \({}_{a}D^{\rho }f\) is the left conformable differential operator presented in (2.1).
Definition 2.3
([31])
The left Caputo conformable derivative of order \(\alpha \in \mathbb{C}\), \(\Re (\alpha )\geq 0\) is defined by
where \(m=\lceil \Re (\alpha )\rceil \), \({}_{a}^{m}D^{\rho }=\underbrace{ {{}_{a}D^{\rho }} {{}_{a}D^{\rho }} \cdots {{}_{a}D^{\rho }}}_{{m \text{ times}}}\), and \({}_{a}D^{\rho }f\) is the left conformable differential operator presented in (2.1).
Lemma 2.1
([31])
Let \(\alpha \in \mathbb{C}\) and \({}_{a}\mathfrak{I}^{j- \alpha ,\rho }x(t)\) be the conformable integral (2.2) of order \(j-\alpha \), then
3 Forced oscillation of conformable differential equations in the frame of Riemann
In this section we study the oscillation theory for equation (1.4). We prove our result under the following assumption:
- \((H)\) :
-
\(p\in \mathbb{C}(\mathbb{R}^{+},\mathbb{R})\), \(q\in \mathbb{C}(\mathbb{R}^{+},\mathbb{R}^{+})\), \(g\in \mathbb{C}( \mathbb{R}^{+},\mathbb{R})\), \(f\in \mathbb{C}(\mathbb{R},\mathbb{R})\) and \(f(u)/u>0\) for all \(u\neq 0\).
We set
and
for \(a\leq t \leq T\), where \(M={}_{a}\mathfrak{D}^{\alpha ,\rho }x(t _{1})V(t_{1})\).
Theorem 3.1
Suppose that \((H)\) and for every sufficiently large T the following conditions hold:
and
where \(V(t)=\exp \int _{t_{1}}^{t}(s-a)^{\rho -1}p(s)\,ds\), \(t_{1}>a\), and M is an arbitrary constant. Then every solution of problem (1.4) is oscillatory.
Proof
Let x be a nonoscillatory solution of problem (1.4). Without loss of generality, suppose that \(T>a\) is large enough and \(t_{1}>T\) so that \(x(t)>0\) for \(t>t_{1}\). According to (1.4) and \((H)\), the following inequality is satisfied:
Taking the left conformable integral order ρ for the above inequality from \(t_{1}\) to t, we can obtain
From Lemma 2.1 and (3.5) we get
which leads to
So, we have
for every sufficiently large T. Multiplying both sides of the above inequality by \(\varGamma (\alpha )\), we can obtain
where Φ and Λ are defined in (3.1) and (3.2), respectively.
Multiplying (3.6) by \((\frac{t^{\rho }}{\rho } )^{1-\alpha }\), we get
Taking \(T_{1}>T\), we consider two cases as follows.
Case (1): Let \(0<\alpha \leq 1\). Then \(m=1\) and \((\frac{t^{\rho }}{\rho } )^{1-\alpha }\varPhi (t)=b_{1}t^{ \rho -\rho \alpha }(t-a)^{\rho \alpha -\rho }\). Since the function \(h_{1}(t)=t^{\rho -\rho \alpha }(t-a)^{\rho \alpha -\rho }\) is decreasing for \(\rho >0\) and \(\alpha <1\), we get for \(t\geq T_{1}\) (see [21])
The function \(h_{2}(t)=t^{\rho -\rho \alpha }[(t-a)^{\rho }-(w-a)^{ \rho }]^{\alpha -1}\) is decreasing for \(\rho >0\) and \(\alpha <1\). Thus, we get
Then, from equation (3.7) and \(t\geq T_{1}\), we get
hence
which is a contradiction to condition (3.3).
Case (2): Let \(\alpha >1\). Then \(m\geq 2\). Also \((\frac{t-a}{t} )^{\rho \alpha -\rho }<1\) for \(\alpha >1\) and \(\rho >0\). The function \(h_{3}(t)=(t-a)^{\rho -\rho {j}}\) is decreasing for \(j>1\) and \(\rho >0\). Thus, for \(t\geq T_{1}\), we have (see [21])
Also, since \((\frac{t^{\rho }}{\rho } )^{1-\alpha }<1\) and \((\frac{(t-a)^{\rho }-(w-a)^{\rho }}{t^{\rho }} )^{\alpha -1}<1\) for \(\alpha >1\) and \(\rho >0\), we get
From (3.7), (3.10), and (3.11), we conclude that
for \(t\geq T_{1}\). Hence
which is a contradiction to condition (3.3). Therefore, we get that \(x(t)\) is oscillatory. In case x is eventually negative, similar arguments lead to a contradiction with condition (3.4). The proof is completed. □
4 Forced oscillation of conformable differential equations in the frame of Caputo
In this section, we study the forced oscillation of conformable initial value problem (1.5).
We set
and
for \(a\leq t \leq T\), where \(M^{*}={{}_{a}^{C}}\mathfrak{D}^{\alpha , \rho }x(t_{1})V(t_{1})\).
Lemma 4.1
[31] Let \(f\in C_{\rho ,a}^{m}[a,b]\) and \(\alpha \in \mathbb{C}\), then
Lemma 4.2
[31] Let \(\alpha ,\beta \in \mathbb{C}\). If the conformable derivatives \({{}_{a}^{C}}\mathfrak{D}^{\alpha ,\rho }f(x)\) and \({{}_{a}^{C}}\mathfrak{D}^{\alpha +\beta ,\rho }f(x)\) exist, then
Lemma 4.3
[31] Let \(\alpha \in \mathbb{C}\), \(m=\lceil \Re (\alpha ) \rceil \). If \(\alpha \in \mathbb{N}\), then
Theorem 4.1
Suppose that \((H)\) and for every sufficiently large T the following conditions hold:
and
where \(V(t)=\exp \int _{t_{1}}^{t}(s-a)^{\rho -1}p(s)\,ds\), \(t_{1}>a\), and \(M^{*}\) is an arbitrary constant. Then every solution of problem (1.5) is oscillatory.
Proof
Let x be a nonoscillatory solution of problem (1.5). Without loss of generality, suppose that \(T>a\) is large enough and \(t_{1}>T\) so that \(x(t)>0\) for \(t>t_{1}\). According to (1.5) and \((H)\), the following inequality is satisfied:
Taking the left conformable integral of order ρ to the above inequality from \(t_{1}\) to t, we can obtain
From Lemma 4.1 and (4.8) we have
Then we get
So, we have
for every sufficiently large T. Multiplying both sides of the above inequality by a constant \(\varGamma (\alpha )\), we have
where Ψ and Ω are defined in (4.1) and (4.2), respectively.
Multiplying (4.9) by \((\frac{t^{\rho }}{\rho } )^{1-m}\), we get
Take \(T_{1}>T\). We consider two cases as follows.
Case (1): Let \(0<\alpha \leq 1\). Then \(m=1\) and \((\frac{t^{\rho }}{\rho } )^{1-m}\varPsi (t)=\varGamma (\alpha )b _{0}\).
The function \(h_{4}(t)= (\frac{(t-a)^{\rho }-(w-a)^{\rho }}{\rho } )^{ \alpha -1}\) is decreasing for \(\rho >0\), \(t>T_{1}>w\), and \(\alpha <1\). Thus, we get
Then, from equation (4.10) and \(t\geq T_{1}\), we get
hence
which contradicts condition (4.6).
Case (2): Let \(\alpha >1\). Then \(m\geq 2\). Also \((\frac{t-a}{t} )^{\rho m-\rho }<1\) for \(m\geq 2\) and \(\rho >0\). The function \(h_{5}(t)=(t-a)^{\rho (k-m+1)}\) is decreasing for \(k< m-1\) and \(\rho >0\). Thus, for \(t\geq T_{1}\), we have
Also, since \((\frac{t^{\rho }}{\rho } )^{1-m}<1\) and \((\frac{(t-a)^{\rho }-(w-a)^{\rho }}{t^{\rho }} )^{\alpha -1}<1\) for \(\alpha >1\) and \(\rho >0\), we get
From (4.10), (4.12), and (4.13), we conclude that
for \(t\geq T_{1}\). Hence
which contradicts condition (4.6). Therefore, we conclude that x is oscillatory. In case x is eventually negative, similar arguments lead to a contradiction with condition (4.7). The proof is completed. □
5 Examples
In this section, we present examples to illustrate our results.
Example 5.1
Consider the conformable initial value problem
Here \(\alpha =1/2\), \(\rho =1\), \(a=0\), \(p(t)=-1\), \(q(t)=(t+5)^{2}\), \(f(x)=(2x+5)e^{\sin 2x}\), \(g(t)=e^{2t}\cos {t}\), and \(V(s)=e^{t_{1}-s}\). It is easy to verify that assumption \((H)\) is satisfied if \(x(t)>0\). Then
Set \(t_{1}=\pi /2\). Hence, we can obtain
Set \(t-w=s^{2}\), then the above integral can be written as the following form:
Let \(t \rightarrow +\infty \), as the result of \(|e^{-2s^{2}}\cos {s^{2}}|\leq e^{-2s^{2}}\), \(|e^{-2s^{2}}\sin {s^{2}}|\leq e^{-2s^{2}}\) and \(\lim_{t \rightarrow +\infty }\int _{0}^{\sqrt{t}}e^{-2s^{2}}\,ds=\frac{\sqrt{2 \pi }}{4}\). So, we know that
are convergent.
Thus, we can set \(\lim_{t \rightarrow +\infty }\int _{0}^{\sqrt{t}}e ^{-2s^{2}}\cos {s^{2}}\,ds=A\), \(\lim_{t \rightarrow +\infty }\int _{0} ^{\sqrt{t}}e^{-2s^{2}}\sin {s^{2}}\,ds=B\). Select the sequence \(\lbrace t_{k}\rbrace = \lbrace \frac{7\pi }{2} - \frac{\pi }{4} + 2k\pi -\arctan {\frac{-B}{A}} \rbrace \), \(\lim_{k \rightarrow \infty }t_{k}=\infty \), then we calculate the following term:
Firstly, we consider the following limit:
Secondly, we know that \(\lim_{k\rightarrow \infty }t_{k}^{\frac{1}{2}}e ^{t_{k}}=+\infty \) and \(\lim_{k\rightarrow \infty } (2M-e^{ \pi } ){e^{{-\frac{\pi }{2}}}}\int _{0}^{\sqrt{t_{k}}}e^{-s ^{2}}\,ds= (2M-e^{\pi } ){e^{{-\frac{\pi }{2}}}}\frac{\sqrt{ \pi }}{2}\). Hence, for (5.2), we have
Then we obtain
Similarly, selecting the sequence \(\lbrace t_{l}\rbrace = \lbrace \frac{5\pi }{2} - \frac{\pi }{4} + 2l\pi -\arctan {\frac{-B}{A}} \rbrace \), we can obtain
Hence, by Theorem 3.1 all solutions of (5.1) are oscillatory.
Example 5.2
Consider the Caputo conformable initial value problem
Here \(\alpha =1/2\), \(\rho =1\), \(a=0\), \(m=1\), \(p(t)=-1\), \(q(t)=e^{t ^{2}}\), \(f(x)=\ln (x+e)\), \(g(t)=e^{2t}\sin {t}\), and \(V(s)=e^{t_{1}-s}\). Thus assumption \((H)\) is satisfied. Then we have
By setting \(t_{1}=\pi /4\) and \(t-w=s^{2}\), we obtain
and
respectively. Using the method in Example 5.1, we choose a sequence
where the constants A and B are defined in Example 5.1. Then we calculate
and
Then we obtain
Similarly, by selecting the sequence \(\lbrace t_{l}\rbrace = \lbrace \frac{\pi }{2} + \frac{\pi }{4} + 2l \pi -\arctan {\frac{-B}{A}} \rbrace \), we can obtain
Hence, by Theorem 4.1 all the solutions of (5.3) are oscillatory.
Example 5.3
By direct computation, we can find that the function \(x(t)=-t^{2}\) is a nonoscillatory solution of problem
Next we will show that condition (3.3) does not hold by setting \(\alpha =1/2\), \(\rho =1\), \(a=0\), \(p(t)=0\), \(q(t)=\sqrt{t}\), \(f(x)=((4/\sqrt{\pi })+(e^{\sqrt{x}}/x^{1/4}))\), \(g(t)=e^{t}\), and \(V(s)=1\). It is obvious that \((H)\) is satisfied. Therefore, we get
By setting \(t_{1}=1\), we obtain
which yields
6 Conclusion
In this paper force oscillatory properties of solutions of conformable differential equations with damping term are established. The cases of conformable differential equations in the frame of Riemann and Caputo type are considered. A sufficient condition for oscillation of all solutions is given. The obtained results are illustrated by numerical examples. Moreover, a counterexample is presented to show the existence of a nonoscillatory solution in case the conditions do not hold.
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The authors express their deep gratitude to the referees for their valuable suggestions and comments for improvement of the paper.
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A. Aphithana is supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program (Grant No. PHD/0134/2558).
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Aphithana, A., Ntouyas, S.K. & Tariboon, J. Forced oscillation of fractional differential equations via conformable derivatives with damping term. Bound Value Probl 2019, 47 (2019). https://doi.org/10.1186/s13661-019-1162-8
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DOI: https://doi.org/10.1186/s13661-019-1162-8