Abstract
In this paper, we present the \( \psi \)-Hilfer fractional derivatives of variable order (FDVO) of 3 types I, II and III, versions A and B, as well as their combinations. In addition, we propose approximations and relations between both derivatives, i.e., \( \psi \)-Hilfer FDVO and \( \psi \)-Caputo FDVO. With regard to the \( \psi \)-Hilfer FDVO type II, we discuss the stability of the FVO nonlinear systems 7 solutions by means of one-parameter Mittag-Leffler functions of variable order. Examples 8 involving the FDVO Lü and Chen systems are also presented.
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Acknowledgements
JVCS acknowledges the financial support of a PNPD-CAPES (process number no. 88882.305834/2018-01) scholarship of the Postgraduate Program in Applied Mathematics of IMECC-Unicamp. We are grateful to the anonymous referees for the suggestions that improved the manuscript.
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Appendix
Appendix
From Theorems 11 and 12, we now discuss the operator limitation \({\mathbf {I}}_{a+}^{\alpha \left( x,t\right) ;\psi }\left( \cdot \right) \) and a version of the integration by parts.
Theorem 11
Let \(\dfrac{1}{n}<\alpha (x,t)<1\), for all \((x,t)\in \left[ a,b\right] ^{2} \) with a number \(n\in {\mathbb {N}}\) greater or equal than two and an increasing function \(\psi \in C^{1}([a,b],{\mathbb {R}})\), for \(t\in [a,b]\). The \(\psi \)-Riemann–Liouville FIVO \(\alpha (x,t)\),
is a linear and bounded operator.
Proof
First, note that the operator \({\mathbf {I}}_{a^{+}}^{\alpha (x,t);\psi }\left( \cdot \right) \) is linear. Let \(\dfrac{1}{n}<\alpha (x,t)<1\) and \(f\in L_{1}(\left[ a,b\right] ,{\mathbb {R}})\). We define the following function:
\(\forall (x,t,s)\in \varOmega :=\left[ a,b\right] \times \left[ a,b\right] \times \left[ a,b\right] \).
By hypothesis, \(\dfrac{1}{n}< \alpha (x,t) < 1\), and in this sense, for \(s+1\le t\) we have \((\psi (t)-\psi (s))^{\alpha (x,t)-1}<1\). On the other hand, for \(s< t < s+1\), we have \((\psi (t)-\psi (s))^{\alpha (x,t)-1}<(\psi (t)-\psi (s))^{\frac{1}{n}-1}\).
Therefore, we can write
Remembering the inequality
and using the Fubini’s theorem, we have that h is integrable on \(\varOmega \) and
Therefore, result
completes the proof.
Theorem 12
Let \(\dfrac{1}{n}<\alpha (x,t)<1\), for all \((x,t)\in \left[ a,b\right] ^{2} \) with a number \(n\in {\mathbb {N}}\) greater or equal than two, \(f,g\in C([a,b],{\mathbb {R}})\) and \(\psi \in C^{1}([a,b],{\mathbb {R}})\) than two an the increasing functions such that \(\psi ^{\prime }(\cdot )\ne 0\), for all \(t\in [a,b]\). Then
Proof
First, note that the operator \({\mathbf {I}}_{a+}^{\alpha (x,t);\psi }(\cdot )\) is linear. We define the following function
\(\forall (x,t,s)\in \varOmega :=\left[ a,b\right] \times \left[ a,b\right] \times \left[ a,b\right] \).
As \(f,g\in C(\left[ a,b\right] ,{\mathbb {R}})\) and using the Bolzano’s theorem, f and g have maximum and minimum. Therefore, there are constants \(c_{1},c_{2}>0\), such that \(\left| g\left( t\right) \right| \le c_{1}\) and \(\left| f\left( t\right) \right| \le c_{2}\), with \(t\in \left[ a,b\right] \).
By hypothesis, \(\dfrac{1}{n}<\alpha (x,t)<1\), and, in this sense, for \(1\le t-s\), we have \((\psi (t)-\psi (s))^{\alpha (x,t)-1}<1\). On the other hand, for \( 1>t-s\), we have \(\left( \psi (t)-\psi (s)\right) ^{\alpha (x,t)-1}< \left( \psi (t)-\psi (s) \right) ^{\frac{1}{n}-1}\).
We can write
On the other hand, knowing the inequality
we can write
and using the Fubini’s theorem, we can obtain
which is the desired result.
Theorem 13
(Tavares et al. 2016, 2018; Pooseh et al. 2013; Odzijewicz et al. 2013; Tavares et al. 2017) Assuming the same conditions as Theorem 12 and choosing \(\psi \left( t\right) =t\), we have
Proof
The proof follows directly from Theorem (12).
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Sousa, J.V.d.C., Machado, J.A.T. & de Oliveira, E.C. The \(\psi \)-Hilfer fractional calculus of variable order and its applications. Comp. Appl. Math. 39, 296 (2020). https://doi.org/10.1007/s40314-020-01347-9
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DOI: https://doi.org/10.1007/s40314-020-01347-9