Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives
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Abstract
Keywords
Fractional conformable integrals Fractional conformable derivatives Fractional differential equations Oscillation theoryMSC
34A08 34C10 26A331 Introduction
The objective of this paper is to study the oscillation of conformable fractional differential equations of the form (1). This will generalize the results obtained in [13, 14] when we take \(a=0\).
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, the main theorems are presented. Section 4 is devoted to the results obtained for the conformable fractional operators in the Caputo setting where we also remark the oscillation of Katugampola-type fractional operators. Examples are provided in Sect. 5 to demonstrate the effectiveness of the main theorems.
2 Notations and preliminary assertions
Definition 2.1
([1])
Definition 2.2
([1])
Definition 2.3
([1])
Lemma 2.1
(Young’s inequality [26])
- (i)
Let\(X,Y\geq0\), \(u>1\), and\(\frac{1}{u}+\frac{1}{v}=1\), then\(XY\leq\frac{1}{u}X^{u}+\frac{1}{v}Y^{v}\).
- (ii)
Let\(X\geq0\), \(Y>0\), \(0< u<1\), and\(\frac{1}{u}+\frac{1}{v}=1\), then\(XY\geq\frac{1}{u}X^{u}+\frac{1}{v}Y^{v}\), where equalities hold if and only if\(Y=X^{u-1}\).
3 Oscillation of conformable fractional differential equations in the frame of Riemann
In this section we study the oscillation theory for equation (1).
Lemma 3.1
([1])
A solution of (1) is said to be oscillatory if it has arbitrarily large zeros on \((0,\infty)\); otherwise, it is called nonoscillatory. An equation is said to be oscillatory if all of its solutions are oscillatory.
Theorem 3.2
Proof
Case (1): Let \(0<\alpha\leq1\). Then \(m=1\) and \((\frac{t^{\rho}}{\rho} )^{1-\alpha}\Phi(t)=b_{1}t^{\rho-\rho\alpha}(t-a)^{\rho\alpha -\rho}\).
Theorem 3.3
Proof
Theorem 3.4
Proof
4 Oscillation of conformable fractional differential equations in the frame of Caputo
Lemma 4.1
Theorem 4.2
Proof
Case (1): Let \(0<\alpha\leq1\). Then \(m=1\) and \((\frac{t^{\rho}}{\rho} )^{1-m}\chi(t)=\Gamma(\alpha)b_{0}\).
We state the following two theorems without proof.
Theorem 4.3
Theorem 4.4
5 Examples
In this section, we construct numerical examples to illustrate the effectiveness of our theoretical results.
Example 5.1
Example 5.2
Example 5.3
Remark 5.1
The oscillation of fractional differential equations in the frame of Katugampola-type fractional derivatives studied in [23, 24, 25] can be investigated in a similar way as we have done in this article for CFDs and their Caputo settings. The reader can verify sufficient conditions and the proofs by observing the kernel which is free from the starting point a.
6 Conclusion
In this article, the oscillation theory for conformable fractional differential equations was studied. Sufficient conditions for the oscillation of solutions of Riemann conformable fractional differential equations of the form (1) were given in three theorems in Sect. 3. As \(\rho\rightarrow1\) in these theorems, we get the results obtained in [13] and [14] when \(a=0\). The main approach is based on applying Young’s inequality which will help us in obtaining sharper conditions. The oscillation for the Caputo conformable fractional differential equations has been investigated as well. Numerical examples have been presented to demonstrate the effectiveness of the obtained results. We shall discuss the case when \(\rho\rightarrow0\) in the future work. Namely, we shall discuss the oscillation of Hadamard-type fractional differential equations with kernels both depending or not depending on the starting point.
Notes
Acknowledgements
The author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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