**4. The Case where ph**(**t**, **0**) = **0**

We'll see below that this restriction on *p* is the most common one and all fractional derivatives presented (i.e., (4)–(8), and (10)) satisfy this condition. The next result, when combined with Theorem 1, allows us to transform *p*-derivatives into ordinary derivatives.

**Proposition 3.** (See Theorem 2.4 in [1]) *Let p satisfy hypothesis (H). In addition, for t* ∈ *I, let*

$$\lim\_{\varepsilon \to 0} \frac{\varepsilon}{h(t, \varepsilon)} \neq 0. \tag{16}$$

*Then f is differentiable at t iff and only if f is p-differentiable at t. In addition,*

$$Df(t) = p\_h(t,0)f'(t). \tag{17}$$

**Remark 3.** *Violation of either* (16) *or the tacit assumption, ph*(*t*, 0) = 0*, can void* (17)*, see Remark 2.4 in [1] and the example therein. In other words, if* (16) *is not satisfied there may exist p-derivatives that are not necessarily representable as multiplication operators of the form* (17) *on the space of derivatives of differentiable functions. In fact, the function <sup>p</sup>*(*t*, *<sup>h</sup>*) = *<sup>t</sup>* <sup>+</sup> *<sup>h</sup>*<sup>3</sup> *on* (−1, 1) *is such an example with f*(*t*) = |*t*|*. It is easily seen that* (16) *does not hold and yet f is p-differentiable at t* = 0 *but not differentiable there.*

**Corollary 4.** (See Theorem 2.2 in [3]; Theorem 2.3 in [2]; and Theorem 1 in [10].) *Let the α-derivative be defined as in either* (4) *or* (5) *and let f be α-differentiable at t. Then,*

$$T\_0^\alpha f(t) = t^{1-\alpha} f'(t).$$

*If f is α-differentiable in the sense of* (6) *then,*

$$D^{GFD}f(t) = \frac{\Gamma(\beta)}{\Gamma(\beta - \alpha + 1)} \; t^{1-\alpha} \; f'(t). \tag{18}$$

*If f is α-differentiable in the sense of* (10) *then, for t* ∈ [0, *b*]*, b* < *π*/2*,*

$$Df(t) = \left(\cos t\right)^{1-\kappa} f'(t). \tag{19}$$

*Similar results hold for derivatives defined by either* (7) *or* (8) *(if F*(*t*, *α*) = 0.*)*

**Corollary 5.** *Let t* > 0 *(resp. t* ≥ 0*). Then f is α-differentiable at t in the sense of anyone of* (4)*,* (5)*, or* (6)*, (resp.* (10)*) if and only if f is differentiable at t.*

Stronger versions of a generalized mean value theorem follow in which we do not require the assumptions in Theorem 4 above but do require that *ph*(*t*, 0) = 0.

**Theorem 7.** A generalized mean value theorem. *Let p satisfy the conditions of Proposition 3 and let f be p-differentiable on* (*a*, *b*) *and continuous on* [*a*, *b*]*. Then there exists c* ∈ (*a*, *b*) *such that*

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

**Proof.** The proof is clear on account of the usual mean value theorem applied to *f* on (*a*, *b*) since *f* is necessarily differentiable there by Proposition 3. Since there exists *c* ∈ (*a*, *b*) such that *f*(*b*) − *f*(*a*)=(*b* − *a*) *f* (*c*) we get *Dp f*(*c*) = *ph*(*c*, 0) *f* (*c*) and the result follows.

**Theorem 8.** [Another generalized mean value theorem] *Let p satisfy the conditions of Proposition 3 and let f* , *g be p-differentiable on* (*a*, *b*)*, continuous on* [*a*, *b*]*, and Dp*(*g*(*t*)) = 0 *there. Then there exists c* ∈ (*a*, *b*) *such that*

$$\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{D\_p f(c)}{D\_p g(c)}.$$

**Proof.** Write

$$h(t) = f(t) - f(a) - \left[\frac{f(b) - f(a)}{g(b) - g(a)}\right](\mathcal{g}(t) - \mathcal{g}(a)).$$

then *h*(*a*) = *h*(*b*) = 0 and *h* satisfies the conditions of Theorem 7. So, there exists *c* ∈ (*a*, *b*) such that *Dph*(*c*) = 0. However,

$$D\_p h(c) = D\_p f(c) - \left[ \frac{f(b) - f(a)}{g(b) - g(a)} \right] D\_p g(c).$$

The result follows.

**Remark 4.** *Specializing to the case where*

$$p(t,h) = t + \frac{\Gamma(\beta) \, ht^{1-\alpha}}{\Gamma(\beta - \alpha + 1)}$$

*and g*(*t*) = *t <sup>α</sup>*/Γ(*α*) *with <sup>α</sup>* <sup>∈</sup> (0, 1) *we get [[10], Theorem 6]. The choice <sup>g</sup>*(*t*) = *<sup>t</sup> <sup>α</sup>*/*α gives [[2], Theorem 2.9].*

*Next, if ph*(*t*, 0) *exists everywhere on* (*a*, *b*) *and p*(*t*, 0) = 0*, then Theorem 6 gives us that k in* (15) *is given by k* = *ph*(*c*, 0)*. In this case, we note that the function f need not to be differentiable in the usual sense here (see Theorem 3) and ph*(*c*, 0) *may or may not be zero.*

**Example 1.** *Given I* = [*a*, *b*]*, f*(*t*) = |*t*| *and p*(*t*, *h*) = *t* + *th* + *t* <sup>3</sup>*h*3*. Then p satisfies the conditions of Theorem 5. Furthermore, ph*(*t*, 0) = 0 *for t* = 0*. A simple calculation shows that Dp f*(*t*) = |*t*| = *f*(*t*)*, for all t* ∈ (*a*, *b*)*. By Theorem 5, there exists c* ∈ (*a*, *b*) *that*

$$D\_p f(c) = \left[\frac{f(b) - f(a)}{b - a}\right] p\_h(c, 0),$$

*i.e.,*

$$|c| = \left[\frac{f(b) - f(a)}{b - a}\right] c.$$

*The existence of c can be calculated directly as follows. Let a* < *b* < 0*. Then,*

$$|c| = \left[\frac{-b+a}{b-a}\right]c\_{\prime}$$

*so that* |*c*| = −*c. So, we may choose any c such that a* < *c* < *b. Let* 0 < *a* < *b. In this case,*

$$|c| = \left[\frac{b-a}{b-a}\right]c.$$

*It suffices to choose c such that a* < *c* < *b again. Finally, let a* < 0 < *b. As*

$$|c| = \left[\frac{b+a}{b-a}\right]c\_r$$

*it suffices to choose c* = 0*.*

**Remark 5.** *In this example, ph*(0, 0) = 0 *and consequently Dp f*(0) = 0 *even though f* (0) *does not exist.*

*Of course, Rolle's theorem is obtained by setting f*(*a*) = *f*(*b*) = 0 *in Theorem 8. The latter then includes [[2], Theorem 2.8].*

**Definition 1.** *Let p satisfy (H), and let f* : [*a*, *b*] → R *be continuous. Then*

$$I\_{\mathcal{P}}(f)(t) = \int\_{a}^{t} \frac{f(\mathbf{x})}{p\_h(\mathbf{x}, 0)} \, d\mathbf{x}.$$

This definition includes the fractional integral considered in [2] (and Definition 3.1 therein). Observe that, since 1/*ph* ∈ *L*(*a*, *b*) and *f* is bounded, this integral always exists (absolutely). It follows that *Ip*(*f*) ∈ *AC*[*a*, *b*] and consequently *I <sup>p</sup>*(*f*) exists a.e. In this case, the continuity of *ph* guarantees that *Ip*(*f*) <sup>∈</sup> *<sup>C</sup>*1(*a*, *<sup>b</sup>*).

Next we state and prove a version of the generalized fundamental theorem of calculus for such *p*-derivatives. The first part is clear, i.e.,

**Theorem 9.** *Let p satisfy (H), and let f* : [*a*, *b*] → R *be continuous. Then Dp*(*Ip*(*f*)(*t*)) = *f*(*t*).

**Proof.** By Proposition 3, since *Ip*(*f*) is differentiable, we have,

$$D\_p(I\_p(f)(t)) \quad = \quad p\_h(t,0)I'\_p(f)(t) = f(t) \dots$$

**NOTE:** The preceding includes [2] (and Theorem 3.2 therein) as a special case.

**Theorem 10.** *Let p satisfy (H) and let F* : [*a*, *b*] → R *be continuous. If F is p-differentiable on* (*a*, *b*) *and DpF is continuous on* [*a*, *b*]*, then Ip*(*DpF*)(*b*) = *F*(*b*) − *F*(*a*).

**Proof.** Let *a* = *x*<sup>0</sup> < *x*<sup>1</sup> < *x*<sup>2</sup> < .... < *xn* = *b* be a partition of [*a*, *b*]. Applying Corollary 7 to each [*xi*−1, *xi*] we get, for some *ti*,

$$D\_p F(t\_i) = \frac{F(\varkappa\_i) - F(\varkappa\_{i-1})}{\varkappa\_i - \varkappa\_{i-1}} p\_h(t\_i, 0)$$

or

$$F(\mathbf{x}\_i) - F(\mathbf{x}\_{i-1}) = (\mathbf{x}\_i - \mathbf{x}\_{i-1}) \frac{D\_p F(t\_i)}{p\_h(t\_i, 0)}.$$

Thus,

$$F(b) - F(a) \quad = \sum\_{i=1}^{n} F(\mathbf{x}\_i) - F(\mathbf{x}\_{i-1}) = \sum\_{i=1}^{n} \frac{D\_p F(t\_i)}{p\_h(t\_i, 0)} \Delta \mathbf{x}\_i.$$

Now since *<sup>f</sup>* is continuous on every subinterval [*xi*−1, *xi*] of [*a*, *<sup>b</sup>*], we can pass to the limit as Δ*xi* → 0. This gives,

$$F(b) - F(a) = \int\_{a}^{b} \frac{D\_p F(t)}{p\_h(t, 0)} dt.$$

This shows that *Ip*(*DpF*)(*b*) = *F*(*b*) − *F*(*a*) and we are done.

Combining Theorems 9 and 10 we get the generalized fundamental theorem of calculus. In addition, using the above relation, we can get a generalized integration by parts formula, i.e.,

**Corollary 6.** *If f* , *g are both p-differentiable on* (*a*, *b*) *and continuous on* [*a*, *b*]*, then,*

$$I\_p(fD\_p(\emptyset)) = [f\emptyset] - I\_p(D\_p(f)\emptyset).$$

This is clear on account of the product rule in Proposition 1(b) and Theorem 10, above.

$$\text{5. The Case where } \mathbf{p\_h}(\mathbf{t}, \mathbf{0}) = \mathbf{0}$$

In this section we consider the exceptional case

$$p\_h(t,0) = 0.\tag{20}$$

The effect of (20) is that it tends to smooth out discontinuities in the ordinary derivative of functions. A glance at (3) would lead one to guess that whenever (20) holds we have *Dp f*(*t*) = 0 but that is not the case, in general.

**Example 2.** *Consider the special case p*(*t*, *h*) = *t* + *h*<sup>2</sup> *which satisfies* (20)*. Then the function <sup>f</sup>*(*x*) = <sup>√</sup>*x, <sup>x</sup>* <sup>&</sup>gt; <sup>0</sup>*, although not differentiable at <sup>x</sup>* <sup>=</sup> <sup>0</sup>*, is clearly right-p-differentiable at <sup>x</sup>* <sup>=</sup> <sup>0</sup> *with D*<sup>+</sup> *<sup>p</sup> f*(0) = 1*.*

**Theorem 11.** *Let* (20) *hold, p*(*t*, 0) = *t, and assume that* (13)*,* (14) *are satisfied for each t, as well. If f is continuous on* [*a*, *b*] *and Dp f*(*t*) *exists, then Dp f*(*t*) = 0*.*

**Proof.** Note that *f* is continuous on (*a*, *b*) on account of the hypothesis and Theorem 3. Using the proofs of Theorems 5 and 6 we observe that the function *h* defined there is continuous on [*a*, *b*], as *f* is continuous there, and therefore its maximum value is attained at *x* = *c*. Thus *k* = *ph*(*c*, 0) = 0 by (15).

Of course, the previous example had an ordinary derivative with an infinite discontinuity at *x* = 0 but still simple discontinuities in the ordinary derivative can lead to the existence of their *p*-derivative for certain *p*.

**Remark 6.** *Incidentally, Example 2 also shows that* (13) *cannot be waived in the statement of Theorem 11.*

**Example 3.** *As before we let <sup>p</sup>*(*t*, *<sup>h</sup>*) = *<sup>t</sup>* <sup>+</sup> *<sup>h</sup>*2*. Then the function <sup>f</sup> , defined by <sup>f</sup>*(*x*) = <sup>|</sup>*x*|*, although not differentiable at x* = 0 *it is clearly p-differentiable at x* = 0 *with Dp f*(0) = 0*.*

Below we study the consequences of this extraordinary assumption (20) and its impact on the study of such *p*-derivatives.

### *5.1. Consequences of ph*(*t*, 0) = 0

We have seen that the notion of *p*-differentiability can be used to turn non-differentiable functions into *p*-differentiable ones, for some exceptional *p*'s and these can have a pderivative equal to zero, as well. We first look at some simple special cases of *p* satisfying (20).

As is usual we define a polygonal function as a function whose planar graph is composed of line segments only, i.e., it is piecewise linear.

**Theorem 12.** *Let p*(*t*, *h*) = *t* + *h*2*. Then every polygonal function f on* **R** *is p-differentiable everywhere and Dp f*(*x*) = 0 *for all x* ∈ **R***.*

**Proof.** Since the graph of every polygonal function consists of an at most countable and discrete set of simple discontinuities in its ordinary derivative, it is easy to show that its *p*-derivative at the cusp points must be zero (just like the absolute value function above). The curve being linear elsewhere it is easy to see that at all such points its *p*-derivative exists and must be equal to zero (see Example 3.1 in [1]).

**Remark 7.** *In contrast with the case where ph*(*t*, 0) = 0 *where an integral can be defined via Definition 1, in this case such an inverse cannot be constructed, in general, as the preceding example shows.*

There must be limitations to this theory of generalized or *p*-derivatives. Thus, we investigate the non-existence of *p*-derivatives under condition (20) for a class of power functions defining the derivative. Our main result is Theorem 1 below which states that for power-like *p*-functions there are functions that are nowhere *p*-differentiable on the real line. In the event that *ph*(*t*, 0) = 0, Proposition 3 makes it easy to construct functions that are nowhere *p*-differentiable on the whole real axis simply by choosing, in particular, any function with *ph*(*t*, 0) = 1. For a fascinating historical survey of classical nowhere differentiable functions, the reader is encouraged to look at [22]

At this point one may think that *p*-differentiability is normal and that most functions have a zero *p*-derivative if *ph*(*t*, 0) = 0. This motivates the next question: Does there exist a function *p* satisfying (20) such that it is continuous and nowhere *p*-differentiable on **R**? The answer is yes and is in the next theorem.

#### *5.2. Weierstrass' Continuous, Nowhere Differentiable Function*

In this subsection we show that the series (21), first considered by Weierstrass, and one that led to a continuous nowhere differentiable function, [5], can also serve as the basis for a continuous nowhere *p*-differentiable function for a large class of functions *p* satisfying (20), namely power functions. Below we show that for each *α* > 1 there is a function *p* satisfying (20) and a function *f* that is nowhere *p*-differentiable on the whole line.

**Theorem 13.** *Let p*(*t*, *h*) = *t* + *h<sup>α</sup> where α* > 1*. Then Weierstrass' continuous and nowhere differentiable function*

$$f(\mathbf{x}) = \sum\_{n=0}^{\infty} b^n \cos(a^n \pi \mathbf{x}) \tag{21}$$

*where* 0 < *b* < 1*, a is a positive integer, and*

$$
\sqrt[4]{a}b > 1 + \frac{3}{2}\pi,
$$

*is also nowhere p-differentiable on* **R***.*

**Proof.** Observe that the cases where *α* ≤ 1 are excluded by (20), so we let *α* > 1. We will show, as usual, that there exists a sequence of *<sup>h</sup>* <sup>→</sup> 0 along which <sup>|</sup>(*f*(*<sup>x</sup>* <sup>+</sup> *<sup>h</sup>α*) <sup>−</sup> *<sup>f</sup>*(*x*))/*h*| → ∞. Now, for fixed *x* ∈ **R**,

$$\begin{aligned} \frac{f(x+h^a) - f(x)}{h} &= \sum\_{n=0}^{\infty} b^n \frac{\cos(a^n \pi(x+h^a)) - \cos(a^n \pi x)}{h} \\ &= \sum\_{n=0}^{m-1} b^n \frac{\cos(a^n \pi(x+h^a)) - \cos(a^n \pi x)}{h} + \sum\_{n=m}^{\infty} b^n \frac{\cos(a^n \pi(x+h^a)) - \cos(a^n \pi x)}{h} \\ &:= \quad S\_m + R\_m. \end{aligned}$$

Estimating *Sm* by the mean value theorem shows that for some 0 < *θ* < 1,

$$|\cos(a^n \pi(\mathbf{x} + h^a)) - \cos(a^n \pi \mathbf{x})| = |h^a a^n \pi \sin(a^n \pi(\mathbf{x} + \theta h^a))| \le a^n \pi |h|^a,\tag{23}$$

so that

$$|S\_m| \le \pi |h|^{a-1} \sum\_{n=0}^{m-1} (ab)^n < \frac{\pi |h|^{a-1} (ab)^m}{ab - 1}. \tag{24}$$

Recall that *x* is fixed at the outset. Now, for any positive integer *m*, we can write *amx* in the form *<sup>a</sup>mx* <sup>=</sup> *<sup>α</sup><sup>m</sup>* <sup>+</sup> *tm* where *<sup>α</sup><sup>m</sup>* is an integer and <sup>|</sup>*tm*| ≤ 1/2. Define a sequence *hm* <sup>&</sup>gt; 0 by

$$h\_m = \sqrt[n]{\frac{1 - t\_m}{a^m}}.$$

Then 0 < *h<sup>α</sup> <sup>m</sup>* <sup>≤</sup> 3/(2*am*). From this choice of a sequence and (24) we get the estimate,

$$|S\_m| < \pi \left(\frac{1 - t\_m}{a^m}\right)^{\frac{a - 1}{a}} \frac{(ab)^m}{ab - 1} \le \pi \left(\frac{3}{2a^m}\right)^{\frac{a - 1}{a}} \frac{(ab)^m}{ab - 1} = \pi \left(\frac{3}{2}\right)^{\frac{a - 1}{a}} \frac{a^{\frac{m}{a}} b^m}{ab - 1}.\tag{25}$$

The next step is to show that the remainder term, *Rm*, remains bounded away from 0. To this end note that *a<sup>n</sup> π* (*x* + *h<sup>α</sup> <sup>m</sup>*) = *an*−*mamπ* (*x* + *h<sup>α</sup> <sup>m</sup>*) = *an*−*mπ*(*α<sup>m</sup>* + 1). It follows that since *a* is odd, then for *n* ≥ *m*, we have

$$\cos\left(a^{\mathfrak{n}}\,\pi\left(\mathfrak{x}+h\_{\mathfrak{m}}^{\mathfrak{a}}\right)\right)=(-1)^{a\_{\mathfrak{m}}+1}.\tag{26}$$

A similar calculation shows that

$$\cos(a^{\mathfrak{n}}\,\pi\,\mathbf{x}) = \cos(a^{\mathfrak{n}-\mathfrak{m}}\,\pi\,(a\_{\mathfrak{m}}+t\_{\mathfrak{m}})) = (-1)^{a\_{\mathfrak{m}}}\cos(a^{\mathfrak{n}-\mathfrak{m}}\,\pi t\_{\mathfrak{m}}).\tag{27}$$

Combining (26) and (27) we see that

$$\begin{array}{rcl} R\_m &=& \sum\_{n=m}^{\infty} b^n \frac{(-1)^{a\_m+1} - (-1)^{a\_m} \cos(a^{n-m} \pi t t\_m)}{l\_{\text{th}}} \\ &=& \frac{(-1)^{a\_m+1}}{l\_{\text{th}}} \sum\_{n=m}^{\infty} b^n (1 + \cos(a^{n-m} \pi t\_m)) \\ \text{i.e., } |R\_m| &=& \frac{1}{|l\_{\text{th}}|} \sum\_{n=m}^{\infty} b^n (1 + \cos(a^{n-m} \pi t\_m)). \end{array}$$

Since the previous series is a series of non-negative terms we can drop all terms except the first. In this case note that cos(*πtm*) ≥ 0 since |*tm*| ≤ 1/2. So,

$$|R\_{\mathfrak{m}}| > \frac{b^{m}}{|h\_{\mathfrak{m}}|} > \sqrt[n]{\frac{2}{3}} a^{m/a} \ b^{m}. \tag{28}$$

Finally, using (28) and (25) we get

$$\left|\frac{f(\mathbf{x} + h\_m^a) - f(\mathbf{x})}{h\_m}\right| \ge |R\_m| - |S\_m| > \sqrt[a]{\frac{2}{3}}a^{\frac{m}{a}}b^m - \pi\left(\frac{3}{2}\right)^{\frac{a-1}{a}}\frac{a^{\frac{m}{a}}b^m}{ab-1} = \left(\sqrt[a]{\frac{2}{3}} - \frac{\pi}{ab-1}\left(\frac{3}{2}\right)^{\frac{a-1}{a}}\right)a^{\frac{m}{a}}b^m \tag{29}$$

Since *<sup>b</sup>* <sup>&</sup>lt; 1 we must have *<sup>a</sup>* <sup>≥</sup> 3 so that *ab* <sup>&</sup>gt; <sup>√</sup>*<sup>α</sup> ab*. The stronger hypothesis (22) forces both <sup>√</sup>*<sup>α</sup> ab* <sup>&</sup>gt; 1 and the term in the parentheses in (29) to be positive. Since *hm* <sup>→</sup> 0 as *<sup>m</sup>* <sup>→</sup> <sup>∞</sup>, the left hand side of (29) tends to infinity, so that the resulting *p*-derivative cannot exist at *x*. Since *x* is arbitrary, the conclusion follows.

#### **6. An Existence and Uniqueness Theorem**

In the final section we give conditions under which an initial value problem for a generalized Riccati equation with *p*-derivatives has a solution that exists and is unique.

**Theorem 14.** *Let p satisfy (H) and q* : [0, *T*] → R, *T* < ∞ *be continuous. Assume that for some b* > 0, *we have*

$$\|\|\frac{1}{p\_h}\|\_{L^1[0,T]} < \min\left\{\frac{b}{\|q\|\_{\infty} + b^2}, \frac{1}{2b}\right\}.\tag{30}$$

*Then the initial value problem for the (generalized) Riccati differential equation*

$$D\_{\mathcal{P}}u(t) + u^2(t) = q(t), \quad u(0) = u\_{0\prime} \tag{31}$$

*has a unique continuous solution u*(*t*) *on* [0, *T*]*.*

**Proof.** Let *B* = {*u* ∈ *C*[0, *T*], *u*<sup>∞</sup> ≤ *b*}. Then *B* is a complete metric space. Define an operator *<sup>F</sup>* on *<sup>B</sup>* by *<sup>F</sup>*(*u*) = *Ip*(*q*(*t*) <sup>−</sup> *<sup>u</sup>*2(*t*)) + *<sup>u</sup>*0. Then for every *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>B</sup>*.

$$\begin{aligned} |Fu - Fv| &= \left| I\_p(q(t) - u^2(t)) - I\_p(q(t) - v^2(t)) \right| \\ &= \left| \int\_0^t \frac{q(s) - u^2(s)}{p\_h(s, 0)} - \frac{q(s) - v^2(s)}{p\_h(s, 0)} \, ds \right| \\ &= \left| \int\_0^t \frac{(v(s) - u(s))(v(s) + u(s))}{p\_h(s, 0)} \, ds \right| \\ &\le \|2b\| \|u - v\|\_{\infty} \int\_0^t \left| \frac{1}{p\_h(s, 0)} \right| \, ds \end{aligned}$$

It follows that *Fu* <sup>−</sup> *Fv*<sup>∞</sup> <sup>&</sup>lt; *<sup>k</sup><sup>u</sup>* <sup>−</sup> *<sup>v</sup>* is a contraction on *<sup>B</sup>*, with *<sup>k</sup>* <sup>=</sup> <sup>2</sup>*b* <sup>1</sup> *ph L*1[0,*T*] < 1, by hypothesis.

Next, we show that *F* : *B* → *B*. Clearly, for *u* ∈ *B*, *Fu* is continuous on [0, *T*]. Next, observe that

$$\|Fu\|\_{\infty} \le \int\_0^T \frac{\|q(s) - u^2(s)\|\_{\infty}}{|p\_h(s, 0)|} ds \le (\|q\|\_{\infty} + b^2) \|\frac{1}{p\_h}\|\_{L^1[0, T]} \le b\_{\nu}$$

by hypothesis. Hence *F* maps *B* into itself. Applying the contraction principle we get that *F* has a unique fixed point *u* ∈ *C*[0, *T*] such that *Fu* = *u*. Theorem 9 gives us the final result.

**Remark 8.** *Observe that there are no sign restrictions on ph*(*t*, 0)*. Note that Dp may in fact depend on a parameter α, subject only to the L*1*-condition on* 1/*ph at the outset. For example, if we choose p*(*t*, *h*) = *t* + *ht*1−*<sup>α</sup> as in [3], the hypothesis* (30) *above becomes,*

$$\frac{T^{\alpha}}{\alpha} \le \min \left\{ \frac{b}{||q||\_{\infty} + b^2}, \frac{1}{2b} \right\}\_{\prime}$$

*so we can see that the assumption that α* ∈ (0, 1) *is not necessary, just that α* > 0*. Of course, T will generally decrease as α grows. Finally, this solution can always be found using the method of successive approximations as implied by the contraction principle.*

*Similarly, if p*(*t*, *h*) = *t* + <sup>Γ</sup>(*β*) <sup>Γ</sup>(*β*−*α*+1) *ht*1−*<sup>α</sup> with <sup>β</sup>* <sup>&</sup>gt; <sup>−</sup>1*, <sup>β</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *and* <sup>0</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> <sup>1</sup> *as in [10], the generalized Rolle's theorem, mean value theorem and Riccati differential equation studied here include the corresponding theorems in [10]. In addition, this existence theorem clarifies the purely numerical results obtained in [10] when solving a special Riccati equation of the form* (31) *using the fractional derivative* (6)*, which, as we have shown, is contained in our theory.*

#### **7. Open Question**

1. Is there 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 (or, more generally, some set of positive measure)?

#### **8. Conclusions**

In this paper we have extended the theory of *p*-derivatives in [1] to include results such as the mean-value theorem, Rolle's theorem and integration by parts. In so doing we showed that the so-called *conformable fractional derivative* of a given function, as considered by [2,3], is actually an ordinary (integer-valued) derivative of the first order except in at most one point. We expanded on the cases where the partial derivative *ph*(*t*, 0) either vanishes or doesn't and in so doing showed that in the former case there exists, for each *α* > 1, a fractional derivative and a function whose fractional derivative exists nowhere on the real line. In the case where *ph*(*t*, 0) = 0 many of the previous results have no analogues and an inverse of the *p*-derivative generally does not exist. We also presented an existence and uniqueness theorem for a Riccati-type equation involving a *p*-derivative whose solution may always be found using successive approximations. The results presented here extend many of the results found in the literature as referred to in the text.

**Author Contributions:** Conceptualization, L.G.Z. and A.B.M.; methodology, L.G.Z. and A.B.M.; software, L.G.Z. and A.B.M.; validation, L.G.Z. and A.B.M.; formal analysis, L.G.Z. and A.B.M.; investigation, L.G.Z. and A.B.M.; resources, L.G.Z. and A.B.M.; data curation, Not applicable; writing?original draft preparation, L.G.Z. and A.B.M.; writing?review and editing, L.G.Z. and A.B.M.; visualization, Not applicable; supervision, L.G.Z. and A.B.M.; project administration, L.G.Z. and A.B.M.; funding acquisition, L.G.Z. and A.B.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by a grant from the Office of the Dean of Science, Carleton University.

**Acknowledgments:** We thank the many referees for comments that have led to an improved presentation of the results herein.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**

