Comment on Vatsala et al. Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications. Foundations 2022, 2, 1129–1142

1. Introduction

In the paper by Vatsala et al. [1], Analysis of sequential Caputo fractional differential equations versus non-sequential Caputo fractional differential equations with applications, Foundations 2022, 2(4), 1129–1142, it is claimed that some results from other mathematicians are wrong. As “proof”, two “counterexamples” are given. Here, we state that the given “counterexamples” are not correct.

2. Results under Attack

Let $[a,b](-\infty <a<b<\infty )$ be a finite interval and let $AC[a,b]$ be the space of functions $f:[a,b]\to \mathbb{R}$ which are absolutely continuous on $[a,b]$. For $n\in \mathbb{N}=1,2,3,\dots$ we denote $A{C}^n[a,b]$ as the space of all real valued functions $f\left(t\right)$ which have continuous derivatives up to order $n-1$ on $[a,b]$, such that $f^{(n-1)}\left(t\right)\in AC[a,b]$. So in particular, $A{C}^1[a,b]=AC[a,b]$. For $m\in {\mathbb{N}}_0=0,1,2,\dots$ we denote $C^m[a,b]$ as the space of all real valued functions $f\left(t\right)$ which are m-times continuously differentiable on $[a,b]$. In particular, $C^0[a,b]=C[a,b]$. Let us also denote $L_1^{loc}(\mathbb{R},\mathbb{R})$ as the linear space of all locally Lebesgue integrable functions $f:\mathbb{R}\to \mathbb{R}$. Let also $\Gamma (.)$ be the Gamma-function, $\alpha >0$ and

$$n=\left[\alpha \right]+1\mathrm{for}\alpha \notin {\mathbb{N}}_0;n=\alpha \mathrm{for}\alpha \in {\mathbb{N}}_0.$$

(1)

The first result under attack in [1] is point $\left(a\right)$, equality (4) of the following theorem in the famous monograph by Kilbas et al. [2]. For convenience and also to avoid possible misunderstandings, we present the whole theorem below.

Theorem 1. 

(Theorem 2.2 in Kilbas et al. [2], p. 93) Let $\alpha \ge 0$ and n be given by (1). Also, let $f\left(t\right)\in C^n[a,b]$. Then, the Caputo fractional derivatives ${(}_C{D}_{a+}^{\alpha }f)\left(t\right)$ and ${(}_C{D}_{b-}^{\alpha }f)\left(t\right)$ are continuous on $[a,b]$: ${(}_C{D}_{a+}^{\alpha }f)\left(t\right)\in C[a,b]$ and ${(}_C{D}_{b-}^{\alpha }f)\left(t\right)\in C[a,b]$.

(a) If $\alpha \notin {\mathbb{N}}_0$, then

$$C^{D_{a+}^{\alpha }}f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}\underset{a}{\overset{t}{\int }}{(t-s)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)ds,$$

(2)

$$C^{D_{b-}^{\alpha }}f\left(t\right)={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha )}}\underset{t}{\overset{b}{\int }}{(s-t)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)ds.$$

(3)

Moreover,

$${(}_C{D}_{a+}^{\alpha }f)\left(a\right)={(}_C{D}_{b-}^{\alpha }f)\left(b\right)=0.$$

(4)

For $0<\alpha <1$ in particular, they have, respectively, the forms

$$C^{D_{a+}^{\alpha }}f\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}\underset{a}{\overset{t}{\int }}{(t-s)}^{-\alpha }{f}^{\prime }\left(s\right)ds,$$

(5)

$$C^{D_{b-}^{\alpha }}f\left(t\right)=-{\displaystyle \frac{1}{\Gamma (1-\alpha )}}\underset{t}{\overset{b}{\int }}{(s-t)}^{-\alpha }{f}^{\prime }\left(s\right)ds.$$

(6)

(b) If $\alpha \in {\mathbb{N}}_0$, then ${(}_C{D}_{a+}^{\alpha }f)\left(t\right)$ and ${(}_C{D}_{b-}^{\alpha }f)\left(t\right)$ are presented by

$${(}_C{D}_{a+}^{\alpha }f)\left(t\right)=f^{\left(n\right)}\left(t\right)\mathrm{and}{(}_C{D}_{b-}^{\alpha }f)\left(t\right)={(-1)}^n{f}^{\left(n\right)}\left(t\right)$$

The second result, attacked in [1], is the article [3], Kiskinov et al, Remarks on the coincidence of the left-side and right-side fractional derivatives on an interval and some consequences, AIP Conf. Proc. 2021, 2333, 080003, where it is proved that a function $f\in C^2([a,b],\mathbb{R})$, whose left-side and right-side Caputo fractional derivatives of order $\alpha \in (0,1)$ coincide on an given finite interval $[a,b]$ (i.e., $C^{D_{a+}^{\alpha }}f\left(t\right)=C^{D_{b-}^{\alpha }}f\left(t\right)$ for any $t\in [a,b]$), can be only a constant on $[a,b]$.

More precisely, in [1], the following theorems are attacked:

• Let $\alpha \in (0,1)$.

Theorem 2. 

(Theorem 4 in [3], p. 4). Let the following conditions hold:

1. $f\in C^2([a,b],\mathbb{R})$.

2. f is nondecreasing on $[a,b]$.

3. For each $x\in [a,b]$, the relation $C^{D_{a+}^{\alpha }}f\left(x\right){=}_C{D}_{b-}^{\alpha }f\left(x\right)$ holds.

Then, the function f is constant on $[a,b]$.

Theorem 3. 

(Theorem 5 in [3], p. 5). Let the following conditions hold:

1. $f\in C^2([a,b],\mathbb{R})$.

2. For each $x\in [a,b]$ the relation $C^{D_{a+}^{\alpha }}f\left(x\right){=}_C{D}_{b-}^{\alpha }f\left(x\right)$ holds.

Then, the function f is constant on $[a,b]$.

3. What Is Claimed in [1] and Why It Is Wrong

Point 1

In Remark 5 in [1], page 1138, the function $f\left(x\right)={(x-a)}^q$ is presented as a “counterexample”. The authors calculate $C_{D_{a+}^q}{(x-a)}^q=\Gamma (q+1)$ on $(a,b)$ and prolonging by continuity to $[a,b]$ because “they need that”, obtain $C_{D_{a+}^q}{(x-a)}^q{|}_{x=a}=\Gamma (q+1)\ne 0$. That is why the authors of [1] claim that (4) (and also Theorems 2 and 3, because their proofs use (4)) are not true.

But it is obviously that $f\left(x\right)$ does not have first derivative in $x=a$ and, hence, $f\left(x\right)\notin C^1[a,b]$ (i.e., continuously differentiable on $[a,b]$) as required in Theorem 1, and also $f\left(x\right)\notin C^2([a,b],\mathbb{R})$ (i.e., two-times continuously differentiable on $[a,b]$) as required in Theorems 2 and 3.

That is an undisputable reason why this function can not be considered as a “counterexample” and, hence, why the claim from the authors of [1] remains without proof.

Point 2

In Remark 6 in [1], page 1139, the function $f\left(x\right)={(x-a)}^{nq}{(b-x)}^{nq}$ is presented as an “example for symmetric function, where the left derivative at any point $x=x_1$ will be exactly equal to the right derivative at $x=x_1$” (as claimed in [1]).

Since $f\left(x\right)\in C[a,b]$ and $f\left(x\right)\in C^1(a,b)$ and its first derivative $f^{\prime }$ is Lebesgue integrable on $[a,b]$, then $f\left(x\right)\in AC[a,b]$ and then according Theorem 2.1 in Kilbas et al [2], p.92, we can use formulas (2) and (3). A simply calculation with Wolfram Mathematica using (2) and (3) shows for $n=1,q=0.25,a=0,b=1$ and $x_1=0.2$ that

$$C_{D_{a+}^q}\left({(x-a)}^{nq}{(b-x)}^{nq}\right){|}_{x=x_1}={\displaystyle \frac{1.03457}{\Gamma \left(0.75\right)}}$$

and

$$C_{D_{b-}^q}\left({(x-a)}^{nq}{(b-x)}^{nq}\right){|}_{x=x_1}={\displaystyle \frac{-0.607571}{\Gamma \left(0.75\right)}},$$

i.e. $C_{D_{a+}^q}f\left(x\right){{|}_{x=x_1}\ne C^{D_{b-}^q}f\left(x\right)|}_{x=x_1}$.

Hence, the behaviour of the presented “example” does not match the claimed expectation of the authors of [1].

4. Conclusions

Since the “counterexample” and “example” presented from the authors of [1] are not correct, the results shown in Section 2 and proved in [2,3] remain true.