*Article* **On the Basic Theory of Some Generalized and Fractional Derivatives**

**Leila Gholizadeh Zivlaei and Angelo B. Mingarelli \***

School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada **\*** Correspondence: angelo@math.carleton.ca

**Abstract:** We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle's theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which provides simple proofs of the corresponding version for the so-called conformable fractional derivatives considered by many. Finally, we show that for each *α* > 1 there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line.

**Keywords:** fractional differential equation; p-derivative

**MSC:** Primary 34B24, 34C10; Secondary 47B50

### **1. Introduction**

In a previous paper [1] generalized derivatives of the form

$$D\_p f(t) = \lim\_{h \to 0} \frac{f(p(t, h)) - f(t)}{h} \tag{1}$$

were introduced whenever the limit exists and is finite. Here *p* is a real valued function of the two real variables (*t*, *h*) where *t* is defined on some interval *I* ⊆ **R**. Recently, two special cases of fractional derivatives of order *α* (with 0 < *α* < 1) were defined by means of the limit definition,

$$D^\alpha f(t) = \lim\_{h \to 0} \frac{f(p(t, h, a)) - f(t)}{h} \tag{2}$$

where *p*(*t*, *h*, *α*) = *t* + *ht*1−*<sup>α</sup>* see Katugampola [2], and where *p*(*t*, *h*, *α*) = *t* exp(*ht*−*α*), see Khalil [3]. Under mild conditions on *p* it can be shown (see [1]) that such *p*-derivatives satisfy all the basic rules of differentiation (Product rule, etc.) and that indeed, the derivatives defined in [2,3] are special cases of these where, however, *ph*(*t*, 0) = 0 (*ph*(*t*, *h*) is the partial derivative of *p* with respect to its second variable). We note that these generalized derivatives, that are not of themselves fractional derivatives, may include some recently considered fractional derivatives. For example, criteria for considering a derivative as fractional can be found in [4] where it was noticed that, according to that criteria, the conformable fractional derivative defined in [3] would not, per se, be a fractional derivative. However, the notion of generalized derivatives defined herein should not be made to adhere to the strict assumptions outlined in [4] as these *p*-derivatives are not, strictly speaking, fractional and thus we take as basic the properties accepted in [5] and other books in analysis as our starting point.

We will show below that the fractional derivatives considered in [2,3] among others specified later are essentially integer valued ordinary derivatives except possibly in at most one point. This is another reason, in addition to [4], to suggest that they should not be considered "fractional".

**Citation:** Gholizadeh Zivlaei, L.; Mingarelli, A.B. On the Basic Theory of Some Generalized and Fractional Derivatives. *Fractal Fract.* **2022**, *6*, 672. https://doi.org/10.3390/ fractalfract6110672

Academic Editor: Carlo Cattani

Received: 8 October 2022 Accepted: 9 November 2022 Published: 14 November 2022

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We now know (see [1]) that whenever *f* is a differentiable function (in the ordinary sense, i.e., when *p*(*t*, *h*) = *t* + *h*) and *ph*(*t*, 0) = 0, then the generalized derivative (1) exists at *t* and

$$D\_p f(\mathbf{t}) = p\_h(\mathbf{t}, \mathbf{0}) \, f'(\mathbf{t}). \tag{3}$$

where the prime is the ordinary derivative of *f* at *t*. This shows that the existence of a derivative implies the existence of a *p*-derivative so long as *ph*(*t*, 0) = 0. Indeed, (3) shows that generalized or even the so-called fractional derivatives defined by either (1) and (2), or by [2,3], are simply multiplication operators on the space of derivatives of differentiable functions, or just *weighted* first order differential operators where the weight, represented by *ph*(*t*, 0), may itself be sign-indefinite on *I*.

Possible physical applications of the foregoing theory can be found in [1], where it is shown that in non-relativistic celestial mechanics elliptical orbits prevail independently of the fractional derivative chosen. The resulting theory can then be included within the framework of weighted operator theory of ordinary differential operators, not without its own difficulties especially when the indicated partial derivative changes sign on intervals or, more generally, on sets of positive measure.

One may think of these generalized derivatives as momentum operators where the *ph*-term in (3) can serve both as a classical positive mass or hypothetically as a negative mass. We note that such negative mass particles are a subject of current interest; see, for example, [6] and the references therein. Negative masses can be interpreted as the negative mass density, which belongs to a kind of exotic hypothetical matter distribution like dark matter or dark energy. This could happen conceptually, for example, due to a region of space where the stress component of the Einstein energy-momentum tensor is larger in magnitude than the mass density or due to the Casimir effect. More importantly, from the physical point of view, a negative mass violates energy conditions in general relativity and has its own consequence [7–9]. However, positive energy is not a required condition for the mathematical consistency of the theory, and this concept can be built up only mathematically.

By way of examples, there has been a flurry of activity of late in the area of locally defined fractional differential operators and corresponding equations. Among these we cite [2,3,10–18]. We cannot begin to cite all the references related to these as, for example, Google Scholar refers to more than 2000 references to paper [3] alone! Clearly, our bibliography cannot be exhaustive. In the sequel, unless otherwise specified, *α*, *β* will denote real parameters with 0 < *α* < 1, *β* = 0, and *β* ∈ **Z**−.

Khalil, et al. [3] define a function *f* to be *α*−**differentiable** at *t* > 0 if the limit,

$$T\_0^\mathbf{x} f(t) = \lim\_{h \to 0} \frac{f(t + ht^{1-n}) - f(t)}{h}. \tag{4}$$

exists and is finite. Katugampola [2] presented another (locally defined) derivative by requiring that, for *t* > 0,

$$T\_0^\chi f(t) = \lim\_{h \to 0} \frac{f(te^{ht^{-n}}) - f(t)}{h}. \tag{5}$$

exist and be finite. Recently, other authors, e.g., [10], considered minor variations in the definition (4) by assuming that, for *t* > 0,

$$D^{GFD}f(t) = \lim\_{h \to 0} \frac{f(t + \frac{\Gamma(\beta) \, ht^{1-a}}{\Gamma(\beta - a + 1)}) - f(t)}{h}.\tag{6}$$

exist and be finite. Note that in each of the three definitions, the case *α* = 1 (and *β* = 1 in (6)) leads to the usual definition of a derivative (see Section 2 in the case of (5)).

Observe that, in each case, (4)–(6), the derivatives are defined for *t* > 0. As many authors have noticed one merely needs to replace *t* <sup>1</sup>−*<sup>α</sup>* by (*<sup>t</sup>* <sup>−</sup> *<sup>a</sup>*)1−*<sup>α</sup>* in each of these definitions to allow for a derivative to be defined on an a given interval (*a*, *b*), whether finite or infinite. In the case of (5) the point *t* = 0 must be excluded as it is not in the domain of definition of the exponential.

More recently, Gúzman et al. [19], considered a fractional derivative called the *Nderivative* defined for *t* > 0 by

$$N\_F^a f(t) = \lim\_{h \to 0} \frac{f(t + h e^{t^{-a}}) - f(t)}{h}. \tag{7}$$

The function *f* is said to be *N*-differentiable if (7) exists and is finite. In a later paper [20], the authors stated slight extensions of the previous definition to another *N<sup>α</sup> <sup>F</sup>* (*t*) derivative defined by the existence and finiteness of the limit,

$$N\_F^{\alpha}f(t) = \lim\_{h \to 0} \frac{f(t + hF(t, \alpha)) - f(t)}{h}. \tag{8}$$

Here *F* seems to have been left generally unspecified, yet the authors obtain the relation (see [20], Equation (2))

$$N\_{\rm F}^{a}f(t) = F(t, a) \, f'(t), \tag{9}$$

also a consequence of Theorem 3 below and the results in [1] but whose validity is sensitive to the behaviour of the function *F*. We will show below that the fractional derivatives defined by anyone of (4)–(8) are, in fact, integer ordered differential operators multiplied by a weight function.

By way of another example, we show that the definition of the generalized derivative,

$$D^\mathfrak{a}f(t) = \lim\_{h \to 0} \frac{f(t + \sin(h)(\cos t)^{1-\mathfrak{a}}) - f(t)}{h},\tag{10}$$

where 0 < *α* < 1 and the limit exists and is finite for *t* ∈ [0, *b*], *b* < *π*/2, can also be handled by our methods and will lead to results that are a consequence of the definition in [1]. Observe, in passing, that the case where *α* = 1 in (10), i.e., *D<sup>α</sup> f*(*t*) = *f* (*t*) whenever *f* is differentiable in the ordinary sense.

These definitions of a generalized derivative function (sometimes called a fractional derivative) on the real line will be referred to occasionally as *locally defined* derivatives as, in each case, knowledge of *f* is required merely in a neighborhood of the point *t* under consideration. In contrast, in the case of the more traditional Riemann-Liouville or Caputo fractional derivatives, knowledge of the function *f* is required on a much larger interval including the point *t*, see e.g., [21]. The previous, though very popular, derivatives defined in terms of singular integral operators will not be considered here.

In this paper we continue the study of the generalized derivatives introduced in [1] and show that a large number of definitions such as (4)–(8) and (10) can be included in the more general framework of [1]. Thus, the results obtained in papers using either of the above definitions are actually a corollary of our results. We also find versions of Rolle's theorem, the mean value theorem, and the fundamental theorem of calculus, a result on the nowhere fractional differentiability of a class of fractional derivatives and corresponding functions, as well as an existence and uniqueness theorem for a generalized Riccati differential equation, all of which generalize many results in the literature.

In Section 2 we show that a large class of known *fractional* derivatives are actually integer order derivatives at all points except at most one. In Section 3 we summarize briefly the notion of *p*-derivatives as introduced in [1] and show that these derivatives all satisfy the usual properties (such as linearity, the product rule, the quotient rule, and the chain rule). We then formulate general versions of the mean value theorem (see Theorem 5) and study the cases where *ph*(*t*, 0) = 0 and *ph*(*t*, 0) = 0, separately as each leads to widely differing results. Thus, in Section 4 we formulate a fundamental theorem of calculus and an integration by parts formula in the former case where *ph*(*t*, 0) = 0 while in Section 5 we consider the same results in the anomalous case where *ph*(*t*, 0) = 0 and show that, generally speaking, such fundamental results are not possible. In the same section we consider the problem of determining classes of functions that are *p*-differentiable everywhere and nowhere *p*-differentiable everywhere. In Section 6 we apply the preceding theory to the formulation of an existence and uniqueness theorem for a *p*-differential equation of Riccati-type on a finite interval whose solutions can be approximated arbitrarily closely and uniformly by the method of successive approximations. We conclude by asking a question to which the answer evades us, namely, whether there is a function *p*, satisfying (20), and a function *f* such that *f* is *p*-differentiable and such that *Dp f*(*t*) = 0 for all t in some interval.

#### **2. Preliminary Observations**

The authors in each of [2,3,10] point out that the notion of differentiability represented by either one of the definitions (4)–(6), is more general than the usual one by showing that there are examples whereby *α*−differentiability does not imply differentiability in the usual sense, although the converse holds. However, note that for *t* > 0 the change of variable *s* = *t* + *ht*1−*<sup>α</sup>* shows that

$$\lim\_{h \to 0} \frac{f(t + ht^{1-\alpha}) - f(t)}{h} = \frac{1}{t^{\alpha - 1}} \lim\_{s \to t} \frac{f(s) - f(t)}{s - t}.$$

Consequently, the left hand limit exists if and only if the right hand limit exists. Thus, for *t* > 0, *p*-differentiability is equivalent to ordinary differentiability and the only possible exception is at *t* = 0 (which is excluded anyway, by definition). A similar argument applies in the case of (6).

Insofar as (5) is concerned, observe that solving for *h* after the change of variable *s* = *teht*−*<sup>α</sup>* is performed, and *t* > 0, leads one to

$$\lim\_{h \to 0} \frac{f(te^{ht^{-a}}) - f(t)}{h} = \frac{1}{t^a} \left( \lim\_{s \to t} \frac{f(s) - f(t)}{s - t} \right) \left( \lim\_{s \to t} \frac{s - t}{\ln s - \ln t} \right) = \frac{1}{t^{a - 1}} \lim\_{s \to t} \frac{f(s) - f(t)}{s - t}.$$

It follows that both definitions, (4) and (5), coincide for *t* > 0 (see also [1]). Definition (6) is simply a re-scaling of (4) by a constant as can be verified by replacing *h* in (4) by *hc*where *c* = Γ(*β*)/Γ(*β* − *α* + 1). Thus, strictly speaking, although it appears to be more general than (4), it isn't really so.

Next, observe that, for *t* > 0, the *N*-derivative defined by (7),

$$\lim\_{h \to 0} \frac{f(t + he^{t^{-a}}) - f(t)}{h} = \frac{1}{e^{-t^a}} \left( \lim\_{s \to t} \frac{f(s) - f(t)}{s - t} \right) = e^{t^a} f'(t)$$

if either limit exists. Thus, for *t* > 0, *f* is *N*-differentiable if and only if *f* is differentiable in the ordinary sense. This means that in [19] (and Theorem 2.3 (f) therein), *N*-differentiability is equivalent to ordinary differentiability (see the Remark in [19], p.91).

For definition (10), note that, for *t* ∈ [0, *b*], *b* < *π*/2,

$$\lim\_{h \to 0} \frac{f(t + \sin(h)\cos(t)^{1-\mathfrak{a}}) - f(t)}{h} = \lim\_{s \to t} \frac{f(s) - f(t)}{s - t} \lim\_{s \to t} \frac{s - t}{\arcsin((s - t)\sec^{1-\mathfrak{a}}(t))} = \cos^{1-\mathfrak{a}}(t) \, f'(t), s \in [0, \infty)$$

if either limit on the left or right exists.

Definition (10) gives something new that hasn't been studied before but falls within the framework of the theory developed in [1]. These observations lead to the following theorem (see also Theorem 2.4 in [1]).

Finally, for the generalized derivative defined by (8), it can be shown that this and the previous results are consequences of Proposition 3 below.

**Theorem 1.** *For t* > 0 *(resp. t* ≥ 0*) a function is differentiable in the sense of either* (4)*,* (5)*, or* (6)*,* (8)*,* (9) *(resp.* (10)*) if and only if it is differentiable in the usual sense.*

**Remark 1.** *It follows that the conformable fractional derivatives defined above (except for* (10)*) are actually differentiable in the ordinary sense whenever t* > 0*, the only possible point of nondifferentiability being at t* = 0*.*

#### **3. Generalized Derivatives**

Let *f* : *I* → R , *I* ⊆ R and *p* : *U<sup>δ</sup>* → R where *U<sup>δ</sup>* = {(*t*, *h*) : *t* ∈ *I*, |*h*| < *δ*} for some *δ* > 0 is some generally unspecified neighborhood of (*t*, 0). Unless *δ* is needed in a calculation we shall simply assume that this condition is always met. Of course, we always assume that the range of *p* is contained in *I*. In the sequel, *L*(*I*) ≡ *L*1(*I*) is the usual space of Lebesgue integrable functions on *I*.

For a given *α*, the **generalized derivative** or **p**-**derivative** in [1] is defined by

$$D\_p f(t) = \lim\_{h \to 0} \frac{f(p(t, h)) - f(t)}{h}. \tag{11}$$

whenever the limit exists and is finite. Occasionally, we'll introduce the parameter *α* mentioned above into the definition so that the limit

$$D\_p^\alpha f(t) = \lim\_{h \to 0} \frac{f(p(t, h, \alpha)) - f(t)}{h}. \tag{12}$$

will then be called the **p-derivative of order** *α*. Since *α* is a parameter (12) is actually a special case of (11).

The main hypotheses on the function *p* are labeled *H*1± and *H*2 in [1] and can be summarized together as follows:

**Hypothesis 1.** *Given an interval I* ⊂ **R***, in addition to requiring that for t* ∈ *I, the function p*(*t*, *h*)*, ph*(*t*, *h*) *are continuous in a neighborhood of h* = 0*, we ask that for t* ∈ *I and for all sufficiently small ε* > 0*, p*(*t*, *h*) = *t* ± *ε has a solution h* = *h*(*t*,*ε*) *such that h* → 0 *as ε* → 0*.*

Occasionally, we will assume, in addition, that <sup>1</sup> *ph*(·,0) <sup>∈</sup> *<sup>L</sup>*(*I*).

**NOTATION:** The notations *D*, *Dα*, *Dp*, *D<sup>α</sup> <sup>p</sup>* will be used interchangeably, occasionally for emphasis.

It is easy to see that the notion of *p*-differentiability as defined in (11) is very general in that it includes all the above definitions. To this end, it suffices to show that hypothesis (H) is satisfied and this is a simple matter, see also [1].

**Theorem 2.** *Each of the derivatives defined by* (4)*–*(8)*, and* (10) *above are p-derivatives for an appropriate function p satisfying (H).*

The following basic property is expected of a generalized derivative and indeed holds for the class considered here and in [1].

**Theorem 3.** (See Theorem 2.1 in [1]). *Let p satisfy (H). If f is p-differentiable at a then f is continuous at a.*

**Corollary 1.** *Let f be α-differentiable where the α-derivative is defined in either* (4)*–*(8) *or* (10)*. Then f is continuous there.*

**Remark 2.** *We note that Corollary 1 includes Theorem 2.1 in [3]; Theorem 2.2 in [2], and Theorem 2.2 in [19]).*

The usual rules for differentiation are also valid in this more general scenario. In the sequel for a given function *p* satisfying (H) we will write *Dp* := *D* or, if there is a parameter dependence, *D<sup>α</sup> <sup>p</sup>* := *D*, for simplicity.

#### **Proposition 1.** (See Theorem 2.2 in [1] for proofs.)

*(a)* (Linearity) *If f* , *g are both p-differentiable at t* ∈ *I and α is a constant, then then so is their sum, f* + *αg, and*

$$D(f + \mathfrak{a}\mathfrak{g})(t) = Df(t) + \mathfrak{a}D\mathfrak{g}(t).$$

*(b)* (The product rule) *Assume that p satisfies (H) and that for t* ∈ *I, p*(*t*, *h*) *is continuous at h* = 0*. If f* , *g are both p-differentiable at t* ∈ *I then so is their product, f* · *g, and*

$$D(f \cdot \mathbb{g})(t) = f(t) \cdot D\mathbb{g}(t) + \mathbb{g}(t) \cdot Df(t).$$

*(c)* (The quotient rule) *Assume that p satisfies (H). If f* , *g are both p-differentiable at t* ∈ *I and g*(*t*) = 0 *then so is their quotient, f* /*g, and*

$$D\left(\frac{f}{\mathcal{S}}\right)(t) = \frac{\mathcal{g}(t) \cdot Df(t) - f(t) \cdot D\mathcal{g}(t)}{\mathcal{g}(t)^2}.$$

As a result we obtain,

**Corollary 2.** (See Theorem 2.2 in [3]; Theorem 2.3 in [2]; Theorem 4 in [10], and Theorem 2.3 in [19].) *For each of the definitions* (4)*–*(8) *and* (10) *there holds an analog of the sum/product/and quotient rule for differentiation of corresponding p-derivatives.*

**Proposition 2.** (See [1], Theorem 2.4) *Assume (H). Let f be continuous and non-constant on I, and let f be p-differentiable at t* ∈ *I. Let g be defined on the range of f and let g be differentiable at f*(*t*)*. Then the composition g* ◦ *f is p-differentiable at t and*

$$D(\mathcal{g}\circ f)(t) = \mathcal{g}'(f(t))\,Df(t).$$

**Corollary 3.** (See [3], p.66 (iv), although the chain rule is not stated correctly there, and Theorem 2.3 in [2].) *For each of the definitions* (4)*–*(8) *and* (10) *there holds an analog of the chain rule for differentiation of corresponding p-derivatives in the form*

$$D(\mathcal{g}\circ f)(t) = \mathcal{g}'(f(t))\,Df(t),$$

*where D* := *D<sup>α</sup> (resp. D* := *D<sup>α</sup> p) is the corresponding p-derivative in question defined in* (11) *(resp.* (12)*).*

We now proceed to formulating fundamental theorems of a calculus for generalized derivatives. We begin by stating a necessary condition for extrema in terms of generalized derivatives.

**Theorem 4.** *Let f* : [*a*, *b*] → R *be such that Dp f*(*t*) *exists for every t* ∈ (*a*, *b*)*. If f has a local maximum (resp. minimum) at c* ∈ (*a*, *b*) *and for each t* ∈ (*a*, *b*)*, and for all sufficiently small* |*h*| *we have,*

$$p(t,h) < t, \quad h < 0 \quad (resp. \ p(t,h) > t, \quad h > 0), \tag{13}$$

$$p(t, h) > t, \quad h > 0 \quad (resp. \ p(t, h) < t, \quad h < 0), \tag{14}$$

*then Dp f*(*c*) = 0*.*

**Proof.** We give the proof in the case where *f*(*c*) is a local maximum. By hypothesis (14), there exists *δ*<sup>0</sup> such that

$$p(t, h) > t, \quad 0 < h < \delta\_0.$$

Since

$$p(c, h) > c, \quad 0 < h < \delta\_{0\prime}$$

and *f*(*c*) is a maximum, we can conclude that

$$f(p(c,h)) \le f(c).$$

Therefore

$$\lim\_{h \to 0^+} \frac{f(p(c, h)) - f(c)}{h} \le 0.$$

Hence, *Dp f*(*c*) ≤ 0. Next, by hypothesis (13), there exists *δ*<sup>1</sup> such that

$$p(c, h) < c, \quad -\delta\_1 < h < 0.$$

As before, since *f*(*c*) is a maximum,

$$\lim\_{h \to 0^{-}} \frac{f(p(c, h)) - f(c)}{h} \ge 0.$$

So, *Dp f*(*c*) ≥ 0. However, since *Dp f*(*c*) exists, it follows that *Dp f*(*c*) = 0. The proof in the case where *f*(*c*) is a minimum is similar, and therefore omitted.

Next, we present a weak form of a general mean value theorem.

**Theorem 5.** A generalized mean value theorem. *Let p satisfy the conditions of Theorem 4 and let f be continuous on* [*a*, *b*]*. As an additional condition on p, assume that the p-derivative of the function whose values are t* − *a exists for all t* ∈ (*a*, *b*)*. Then there is a constant c* ∈ (*a*, *b*) *such that*

$$D\_p f(c) = \frac{f(b) - f(a)}{b - a} k\_\prime \tag{15}$$

*where the constant k is the p-derivative of the function whose values are t* − *a evaluated at t* = *c, i.e., k* = *Dp*(*t* − *a*) *t*=*c* .

**Proof.** Define

$$h(t) = f(t) - f(a) - \frac{f(b) - f(a)}{b - a}(t - a).$$

Then, by hypothesis, *h* is *p*-differentiable on (*a*, *b*), continuous on [*a*, *b*] and

$$D\_p h(t) = D\_p f(t) - \frac{f(b) - f(a)}{b - a} D\_p(t - a).$$

In the event that *Dph*(*t*) = 0 for all *t* ∈ (*a*, *b*), *c* can be chosen to be any point in (*a*, *b*). If *Dph*(*t*) = 0 for all such *t* we may assume, without loss of generality, that *h*(*t*) > 0 somewhere. Since *h*(*a*) = *h*(*b*) = 0, and *h* is continuous, it must achieve its maximum at, say, *t* = *c* ∈ (*a*, *b*). By Theorem 4, *Dph*(*c*) = 0. The result follows.

**Theorem 6.** *If, in addition to the hypotheses in Theorem 5, we have p*(*t*, 0) = *t, for every t, and ph*(*t*, 0) *exists, then*

$$D\_p f(c) = \frac{f(b) - f(a)}{b - a} \, p\_h(c, 0).$$

**Proof.** Since

$$D\_p(t - a)\Big|\_{t = a} = \lim\_{h \to 0} \frac{p(t, h) - a - (t - a)}{h} = \lim\_{h \to 0} \frac{p(t, h) - p(t, 0)}{h} = p\_h(t, 0),$$

the result follows.

In the next two sections we can formulate more precise results by considering the two cases, *ph*(*t*, 0) = 0 and *ph*(*t*, 0) = 0.
