# Phase-Retrieval with Incomplete Autocorrelations Using Deep Convolutional
Autoencoders

Giovanni Pellegrini  Dipartimento di Fisica, Università degli studi di Pavia,
Via Bassi 6, Pavia 27100, Italy<EMAIL_ADDRESS>Jacopo
Bertolotti  Department of Physics and Astronomy, University of Exeter, Exeter,
Devon EX4 4QL, UK<EMAIL_ADDRESS>
###### Abstract

Phase-retrieval techniques aim to recover the original signal from just the
modulus of its Fourier transform, which is usually much easier to measure than
its phase, but the standard iterative techniques tend to fail if only part of
the modulus information is available. We show that a neural network can be
trained to perform phase retrieval using only incomplete information, and we
discuss advantages and limitations of this approach.

###### keywords:

American Chemical Society, LaTeX

##  1 Introduction

Scattering is one of the major limiting factors in imaging, as it scrambles
the light forming the image we desire into a shapeless blob, and even moderate
amounts of scattering in the optical path can easily deteriorate the
achievable quality to an unacceptable level 1, 2, 3, 4. A number of techniques
have been developed to deal with it, each with its own set of advantages and
disadvantages, and often working only in a very specific set of circumstances
5, 6, 7, 8, 9, 10, 11, 12, 13. One of them, based on the stellar speckle
interferometry technique developed for ground-based astronomy 14, exploit
speckle correlations to measure the autocorrelation of the desired image, and
then employs numerical techniques to invert the autocorrelation to yield the
image itself 15, 16, 17, a process known as “phase retrieval”, which is
usually achieved by employing one of the variations of the celebrated
Gerchberg–Saxton algorithm 18. A common limiting factor of this approach is
that it relies on the optical memory effect, i.e. the fact that the light from
different point sources will produce the same (but shifted) speckle pattern
when scattered. This is a very strong correlation, but it only works in a
limited range, which decays exponentially with the scattering medium thickness
19.

Among the approaches adopted to overcome the limitations imposed by the
optical memory effect, deep-learning methods have gained increasing popularity
in the recent years. In particular, especially if compared with traditional
retrieval algorithms, they have shown much promise in a variety of domains
ranging from non-line-of-sight imaging to ptychography 20, 21, 22, 23, 24.

In this paper, we consider a reconstruction problem where we wish to recover
the image of an hidden object $o$ from an estimate of the image
autocorrelation $o\star o$, obtained from the analysis of the spatial
correlations of the speckle image. Furthermore, we want to perform the image
retrieval in different conditions, where the autocorrelation information may
be either fully available ($o\star o$) or, more interestingly, partially
removed. In particular we will consider the case where only the
autocorrelation within a disk $D(r_{m})$ of radius $r_{m}$ (the ”mask radius”)
is available 20. This scenario may arise in different experimental contexts,
one of the most relevant being non-invasive imaging through strongly
scattering media 4, 25, 20. A straightforward approach to tackle this problem
would be to leverage the relationship $\mathcal{F}(o\star
o)=|\mathcal{F}(o)|^{2}$, where $\mathcal{F}$ stands for the Fourier transform
operation, and feed the obtained square amplitude to a standard phase
retrieval algorithm. This technique is known to work reasonably well in
presence of full information and in the absence of noise, but its performance
progressively degrades as the input autocorrelation data become progressively
non-ideal 20.

In the following we investigate to which extent deep learning, and in
particular Convolutional Neural Networks (CNN), can be employed to mitigate
the averse effect of information removal on phase retrieval attempts. As a
first step in this direction we verify how classic phase retrieval algorithms
can deal with partially available autocorrelations $D(r_{m})(o\star o)$.
Specifically, we adopt the Hybrid Input Output (HIO) algorithm as a reference
approach 26, 27, and we benchmark its performance on partial autocorrelations
$D(r_{m})(o\star o)$ by applying a circular mask to erode the periphery of the
full autocorrelation $o\star o$. Subsequently we devise a deep learning
workflow to tackle the same problem with tools traditionally provided by CNNs
and autoencorder architectures 20.

##  2 Methods

###  2.1 Traditional phase retrieval approach

Traditional approaches, such as the Hybrid Input Output (HIO) algorithm 26,
27, represent the state of the art in terms of classic image phase retrievals.
It is instead unclear whether they can efficiently retrieve original images
$o$ from partial autocorrelations $D(r_{m})(o\star o)$. We test this assertion
by implementing the HIO algorithm and, starting from a random guess, running
it for 400 iterations and 20 trial runs for each phase retrieval attempt, both
for full and partial autocorrelation inputs. As a sample input, we choose a
simple 128x128 pixels image representing two handwritten digits, and the
corresponding autocorrelation. The results of these retrieval attempts are
reported in Figure 1. In this instance, we obtain a successful phase retrieval
only when a complete autocorrelation is fed to the HIO algorithm (Figure 1b).
If instead we apply a circular erosion mask to the same input autocorrelation,
the reconstruction succeeds only if the mask application does not result in
any information removal (Figure 1c). On the other hand, as soon as the masking
leads to the smallest removal of information, the quality of the image
retrieval degrades dramatically (Figure 1d-f). Eventually, a large enough
information erosion degrades the phase retrieval process up to a point where
the original image is no longer recognizable in the retrieved solution. This
is because traditional approaches such as HIO can only target the available
portion of the autocorrelation, which in turn corresponds to a source image
completely different from the original one.

Figure 1: Autocorrelation reconstruction with HIO algorithm. (a) Image ground
truth, (b) phase retrieval with full autocorrelation input, (c-f) phase
retrieval after information removal with a $r_{m}=56$ px, $r_{m}=46$ px,
$r_{m}=36$ px and $r_{m}=26$ px circular mask.

###  2.2 Deep learning: dataset

In the following, we devise an efficient deep learning workflow for the
solution of the phase retrieval problem. The first step is the construction of
an appropriate synthetic dataset. We choose the MNIST handwritten digits
database as a starting point to generate suitable samples to address the
partial autocorrelation reconstruction problem 28. Overall, each entry of the
synthetic dataset consists of an image, composed of one or more digits ($o$),
and the corresponding full autocorrelation ($o\star o$); a few representative
examples of the constructed dataset are reported in Figure 2. In practice,
half of the dataset are pairs of MNIST digits, randomly combined after a
rotation in the $[-\pi/4,\pi/4]$ interval, and matched with the corresponding
autocorrelation. The remaining half consists instead of single digit image-
autocorrelation pairs, where again each digit is randomly rotated in the same
angular interval. We generate a training dataset of 200000 samples and a
corresponding validation dataset of 10000 samples, and note that digits drawn
from the MNIST training set will exclusively populate our synthetic training
set, and likewise for the generation of the validation set. We finally
underline that such a dataset has no ambition to represent the generality of
all situations encountered in phase retrieval problems, but we believe that it
can provide useful indications regarding the potential of deep learning
approaches for the reconstruction of images starting from partially available
information.

Figure 2: Samples drawn from the synthetic dataset. Each sample of the dataset
is formed by and image-autocorrelation pair. We alternate two different color
scales to better distinguish the dataset entries. Each ground truth image
contains a single digit or a pair of digits drawn from the MNIST hand written
digit database.

###  2.3 Deep learning: architecture, loss and training procedure

We adopt the DeepLabV3+ model with a ResNet101 backbone for the phase
retrieval task 29. The DeeplabV3+ network is an encoder-decoder CNN
architecture, originally employed to tackle semantic segmentation tasks, which
can be adapted to several other applications (Figure 3). If compared to other
more traditional CNN models, DeepLabV3+ has the advantage to probe
convolutional features at multiple scales and to provide a denser and less
compressed feature extraction at the encoder level, while keeping the decoder
structure extremely simple 29.

Figure 3: DeepLabV3+ model schematics. Encoder: we employ a ResNet101
backbone, sample and stack the extracted features through atrous spatial
pyramid pooling (ASPP) and compress them with a 1x1 convolution. Decoder: we
concatenate the compressed ResNet101 features with the ASPP output and apply a
3x3 convolution followed by an upsampling to match the desired output size.
Loss: We compute the autocorrelation of the reconstructed image and compute an
$l_{1}$ loss with the input ground truth autocorrelation.

In our workflow, the model is simply fed either a full or a partial
autocorrelation, and outputs the corresponding ground truth image. We guide
the training computing an ordinary $l_{1}$ loss between the autocorrelation
ground truth and the reconstructed autocorrelation, to avoid the memorization
of the digit locations in the training data (Figure 3) 20.

We train our network using Stochastic Gradient Descent with momentum (SGD), a
variable learning rate and a batch size of 128. We implement the solution in
Pytorch and Pytorch-Lightning and train the network for about 5000 epochs
using 8 Nvidia V100 GPUs over the time of one week 30, 31.

The overall training procedure is divided in three main stages, as shown in
Figure 4. During the first training stage, lasting 700 epochs, we train the
network to reconstruct the ground truth image from the full autocorrelation
with a learning rate equal to $l_{r}=10^{-4}$, and scale it down to
$l_{r}=10^{-5}$ for the last 100 epochs to reduce the noise in the validation
loss.

We then transition to the second, 3200 epochs long, training phase. At this
stage, we apply a circular erosion mask to the input autocorrelation, starting
with a mask radius of $r_{m}=56$ px and terminating with a radius of
$r_{m}=26$ px, shrinking $r_{m}$ of one pixel every 100 epochs. During this
training stage it is necessary to reduce the learning rate from an initial
value of $l_{r}=10^{-6}$ to a final value of $l_{r}=10^{-8}$ in order to keep
the validation loss landscape sufficiently stable and reasonably free of
noise. It is clear that, as shown in Figure 4, each pixel erosion causes an
uptick in the training and validation loss, followed by a partial recovery
during the remaining training epochs. This means that the information removal
negatively impacts the reconstruction performance, but the adverse effect can
be partially recovered during the training process.

Finally, in the third and last stage, we anneal the network for an additional
1500 training epochs with a learning rate of $l_{r}=10^{-8}$.

We extract several network checkpoints during the training process to monitor
the network phase retrieval performance at different stages of information
erosion. We archive the first checkpoint at the end of the full
autocorrelation training, and save three more for an autocorrelation mask size
of 56, 46 and 36 pixels. Finally, we archive the checkpoints at the end of the
training, corresponding to a mask size of 26 pixel. All of our training and
inference code is made available in a public repository 32.

##  3 Results and Discussion

It is now interesting to explore the phase retrieval performance of the deep
learning approach at different stages of information erosion, and compare it
with that of a traditional algorithm.

Figure 5 examines the phase reconstruction performance of the deep learning
and the HIO approaches for a random sample drawn from our validation dataset.
When dealing with a full autocorrelation input, Figure 5b shows that both
techniques display a similar performance, even though we notice that, also in
this trivial case, the CNN output is less noisy and closer to the ground truth
image. In this instance, we underline that the image reconstructed by neural
network, while correct, is flipped by a point symmetry if compared to the
original one, since the loss functions targets the difference between
autocorrelations and is invariant under center point transformations.

As we progress with the information erosion, and shrink the mask radius below
the $r_{m}=56$ px limit, the difference in the quality of the phase retrieval
between the deep learning and the traditional approach becomes apparent. If,
as an example, we explore the phase retrieval attempt for $r_{m}=46$ px
(Figure 5d), it is indeed clear that the image reconstruction with the deep
learning approach has undergone little to no degradation, while the
traditional solution relying on the HIO algorithm has completely broken down.

The same is true for the $r_{m}=36$ px mask application (Figure 5e), with the
deep learning reconstruction virtually identical to the ground truth, and the
HIO algorithm unable to produce a meaningful output.

Figure 4: Training curves. (a) Training loss (black line) and validation loss
(light gray line) added for comparison. (b) Validation loss (black line) and
training loss (light gray line) added for comparison. (c) Learning rate (black
line) and mask radius (blue line) schedules. Figure 5: Performance comparison
between HIO and deep learning algorithms. (a) Image ground truth, (b) phase
retrieval with full autocorrelation input, (c-f) phase retrieval after
information removal with a $r_{m}=56$ px, $r_{m}=46$ px, $r_{m}=36$ px and
$r_{m}=26$ px circular mask.

The situation is instead qualitatively different for the largest information
erosion obtained applying the $r_{m}=26$ px mask. Figure 5f highlights a
situation where, while the traditional phase retrieval is still unable to
provide a proper reconstruction, the deep learning approach also struggles to
perform accurately. In this instance the CNN correctly locates the position of
the digits inside the image, but cannot accurately reconstruct the details of
each one, resulting in an overall blurred reconstruction. This lack of
accuracy is also reflected in the computed autocorrelation, which is correctly
reconstructed in broad strokes, but lacks the fine details of the target
ground truth. The fact that the degradation in the output coincides with the
next to complete loss of the side lobes in the autocorrelation, suggests that,
contrary to more traditional algorithms, a deep learning approach can still
produce reliable results as long as some information from the autocorrelation
side lobes (which encode the spatial relation between different objects in the
scene) is retained.

As a final investigation, we wish to study the behavior of the deep learning
phase retrieval approach for a variety of different inputs, including single
digit images and out of distribution samples, such as images containing never
observed letters from the alphabet.

Figure 6: Examples of deep learning reconstruction for different degrees of
masking: (a,c) digit reconstruction degradation for the $r_{m}=26$ px mask,
(b) single digit flip for the $r_{m}=26$ px mask, (d) collapse of the image
reconstruction for the $r_{m}=26$ px mask, (e) successful reconstruction of a
single digit image, (f) successful reconstruction of an out of distribution
sample containing letters of the alphabet.

Figure 6 displays a selection of these phase retrieval attempts. The first
four examples reported in Figure 6a-d are drawn from the validation set, and
roughly follow the trends already observed in Figure 5. The last two cases of
Figure 6e-f are qualitatively different single digit and out of distribution
inputs. As before, the deep learning approach provides a satisfactory
performance up to the $r_{m}=46$ px and $r_{m}=36$ px erosion stage. For more
extremes crops of the autocorrelation, some information is irremediably lost,
and the reconstruction suffers from that. This might show up as a deviation
from the shape of the ground truth (e.g. Figure 6a, c, or f), lack of
information about the relative distance between the different parts of the
image and thus a collapse of two digits into a single shape (e.g. Figure 6d),
or sometimes more subtle mistakes like the flip of one digit, but not the
whole figure, in Figure 6b. This analysis is consistent with recent results
reported in the literature, where a CNN can successfully reconstruct an image
ground truth composed of two digits from the center lobe of the
autocorrelation, but only when given constraints about the number and shape
the of included digits 33. We finally remark that the network successfully
reconstructs single digits autocorrelations without allucinating any spurious
side lobes (Figure 6e) and that even out of distribution samples are handled
correctly (Figure 6f).

##  4 Conclusions

The presented results indicate that, whereas traditional phase retrieval
algorithms struggle to perform efficiently if presented partial information,
CNNs can overcome the detrimental effects of the information removal, and
largely exceed the performance achievable with conventional procedures. The
deep learning approach easily performs phase retrievals in regimes where
classical methods fail to deliver any meaningful result. In practice, CNNs can
reconstruct multiple digits images from partial autocorrelations in which the
information about the number and relative position of the digits has been
almost completely removed. At the same time it is important to always remember
that the training dataset biases the output, so such a system can only ever be
reliable on cases close enough to the one it was trained on, and can easily
hallucinate superficially plausible reconstructions very different from the
ground truth 34.

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# On the basic theory of some generalized and fractional derivatives

Leila Gholizadeh<EMAIL_ADDRESS>and Angelo B.
<EMAIL_ADDRESS>School of Mathematics and Statistics,
Carleton University, Ottawa, Canada

###### Abstract

We continue the development of the basic theory of generalized derivatives as
introduced in [10] and give some of their applications. In particular, we
formulate versions of a weak maximum principle, 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 includes, as corollaries, the corresponding version
for conformable fractional derivatives considered by [7], [8] among many
others. Finally, we show that for each $\alpha>1$ 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:

[2010] Primary 34B24, 34C10; Secondary 47B50

##  1 Introduction

In a previous paper [10] generalized (fractional) derivatives of the form

$D_{p}f(t)=\lim_{h\to 0}\frac{f(p(t,h))-f(t)}{h}$ (1.1)

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 $I\subseteq\mathbf{R}$. Recently, two special cases of fractional
derivatives of order $\alpha$ (with $0<\alpha<1$) were defined by means of the
limit definition,

$D^{\alpha}f(t)=\lim_{h\to 0}\frac{f(p(t,h,\alpha))-f(t)}{h}$ (1.2)

where $p(t,h,\alpha)=t+ht^{1-\alpha}$ see Katugampola [7], and where
$p(t,h,\alpha)=t\exp(ht^{-\alpha})$, see Khalil [8]. Under mild conditions on
$p$ it can be shown (see [10]) that such $p$-derivatives satisfy all the basic
rules of differentiation (Product rule, etc.) and that indeed, the derivatives
defined in [7] and [8] are special cases of these where, however,
$p_{h}(t,0)\neq 0$ ($p_{h}$ is the partial derivative of $p$ with respect to
its second variable).

We know (see [10]) that whenever $f$ is a differentiable function (in the
ordinary sense, i.e., when $p(t,h)=t+h$) and $p_{h}(t,0)\neq 0$, then its
generalized derivative (1.1) exists at $t$ and

$D_{p}f(t)=p_{h}(t,0)\,f^{\prime}(t).$ (1.3)

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
$p_{h}(t,0)\neq 0$. Indeed, (1.3) shows that generalized or even fractional
derivatives defined by either (1.1) and (1.2) are simply multiplication
operators on the space of derivatives of differentiable functions, or just
weighted first order differential operators where the weight, represented by
$p_{h}(t,0)$, may itself be sign-indefinite on $I$.

Physically, one may think of these generalized derivatives as momentum
operators with the $p$-term serving as a mass of arbitrary sign. Applications
of the foregoing theory can be found in classical mechanics, see [10], where
it is shown, in part, that in 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.

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 chronologically, [8, 7, 1]. We cannot begin to cite all
the references related to these as, for example, Google Scholar refers to more
than two-thousand references to paper [8] alone! Still, the articles referred
to here will suffice for our purpose. In the sequel, unless otherwise
specified, $\alpha,\beta$ will denote real parameters with $0<\alpha<1$,
$\beta\neq 0$, and $\beta\not\in\mathbf{Z}^{-}$.

Khalil, et al. [8] define a function $f$ to be $\bm{\alpha}$${\bf-
differentiable}$ at $t>0$ if the limit,

$T_{0}^{\alpha}f(t)=\lim_{h\to 0}\frac{f(t+ht^{1-\alpha})-f(t)}{h}.$ (1.4)

exists and is finite. Katugampola [7] presented another (locally defined)
derivative by requiring that, for $t>0$,

$T_{0}^{\alpha}f(t)=\lim_{h\to 0}\frac{f(te^{ht^{-\alpha}})-f(t)}{h}.$ (1.5)

exist and be finite. Recently, other authors, e.g., [1], considered minor
variations in the definition (1.4) by assuming that, for $t>0$,

$D^{GFD}f(t)=\lim_{h\to
0}\frac{f(t+\frac{\Gamma(\beta)\,ht^{1-\alpha}}{\Gamma(\beta-\alpha+1)})-f(t)}{h}.$
(1.6)

exist and be finite. Note that in each of the three definitions, the case
$\alpha=1$ (and $\beta=1$ in (1.6)) leads to the usual definition of a
derivative (see Section 2 in the case of (1.5)).

Observe that, in each case, (1.4), (1.5), (1.6), the derivatives are defined
for $t>0$. As many authors have noticed one merely needs to replace
$t^{1-\alpha}$ by $(t-a)^{1-\alpha}$ 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 (1.5) the point $t=0$ must be excluded as it is not
in the domain of definition of the exponential.

More recently, Gúzman et al., [5], considered a fractional derivative called
the N-derivative defined for $t>0$ by

$N_{F}^{\alpha}f(t)=\lim_{h\to 0}\frac{f(t+he^{t^{-\alpha}})-f(t)}{h}.$ (1.7)

The function $f$ is said to be $N$-differentiable if (1.7) exists and is
finite. In a later paper [13], the authors stated slight extensions of the
previous definition to another $N_{F}^{\alpha}(t)$ derivative defined by the
existence and finiteness of the limit,

$N_{F}^{\alpha}f(t)=\lim_{h\to 0}\frac{f(t+hF(t,\alpha))-f(t)}{h}.$ (1.8)

Here $F$ seems to have been left generally unspecified, yet the authors obtain
the relation (see [13], Equation (2))

$N_{F}^{\alpha}f(t)=F(t,\alpha)\,f^{\prime}(t),$ (1.9)

also a consequence of Theorem 4.1 below and the results in [10] but whose
validity is sensitive to the behaviour of the function $F$.

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

$D^{\alpha}f(t)=\lim_{h\to 0}\,\frac{f(t+\sin(h)\,(\cos
t)^{1-\alpha})-f(t)}{h},$ (1.10)

where $0<\alpha<1$ and the limit exists and is finite for $t\in[0,b]$,
$b<\pi/2$, can also be handled by our methods and will lead to results that
are a consequence of the definition in [10]. Observe, in passing, that the
case where $\alpha=1$ in (1.10), i.e., $D^{\alpha}f(t)=f^{\prime}(t)$ 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., [6]. The previous, though very
popular, derivatives defined in terms of singular integral operators shall not
be considered here.

In this paper we continue the study of the generalized derivatives introduced
in [10] and show that a large number of definitions such as (1.4), (1.5),
(1.6), (1.7), (1.8) and (1.10) can be included in the more general framework
of [10]. Thus, the results obtained in papers using either of the fabove
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.

##  2 Preliminary observations

The authors in each of [8, 7, 1] point out that the notion of
differentiability represented by either one of the definitions (1.4), (1.5),
(1.6) considered, is more general than the usual one by showing that there are
examples whereby $\alpha-$differentiability does not imply differentiability
in the usual sense, although the converse holds. However, note that for $t>0$
the change of variable $s=t+ht^{1-\alpha}$ 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
(1.6).

Insofar as (1.5) is concerned, observe that solving for $h$ after the change
of variable $s=te^{ht^{-\alpha}}$ is performed, and $t>0$, leads one to

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

It follows that both definitions, (1.4) and (1.5), coincide for $t>0$ (see
also [10]). Definition (1.6) is simply a re-scaling of (1.4) by a constant as
can be verified by replacing $h$ in (1.4) by $\eta=h\,c$ where
$c=\Gamma(\beta)/\Gamma(\beta-\alpha+1).$ Thus, strictly speaking, although it
appears to be more general than (1.4), it isn’t really so.

Next, observe that, for $t>0$, the $N$-derivative defined by (1.7),

$\lim_{h\to
0}\frac{f(t+he^{t^{-\alpha}})-f(t)}{h}=\frac{1}{e^{-t^{\alpha}}}\left(\lim_{s\to
t}\frac{f(s)-f(t)}{s-t}\right)=e^{t^{\alpha}}\,f^{\prime}(t)$

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 [[5],
Theorem 2.3 (f)], $N$-differentiability is equivalent to ordinary
differentiability (see the Remark in [5], p.91).

Finally, for definition (1.10), note that, for $t\in[0,b]$, $b<\pi/2$,

$\lim_{h\to 0}\,\frac{f(t+\sin(h)\,\cos(t)^{1-\alpha})-f(t)}{h}=\lim_{s\to
t}\,\frac{f(s)-f(t)}{s-t}\lim_{s\to
t}\,\frac{s-t}{\arcsin((s-t)\sec^{1-\alpha}(t)}=\cos^{1-\alpha}(t)\,f^{\prime}(t),$

if either limit on the left or right exists.

Definition (1.10) gives something new that hasn’t been studied before but
falls within the framework of the theory developed in [10]. These observations
lead to the following theorem (see also Theorem 2.4 in [10]).

Finally, for the generalized derivative defined by (1.8), it can be shown that
this and the previous results are consequences of Proposition 4.1 below.

######  Theorem 2.1.

For $t>0$ (resp. $t\geq 0$) a function is differentiable in the sense of
either (1.4), (1.5), or (1.6), (1.8), (1.9) (resp. (1.10)) if and only if it
is differentiable in the usual sense.

######  Remark 2.1.

It follows that the conformable fractional derivatives defined above (except
for (1.10)) are actually differentiable in the ordinary sense whenever $t>0$,
the only possible point of non-differentiability being at $t=0$.

##  3 Generalized derivatives

Let $f:I\rightarrow\mathbb{R}$ , $I\subseteq\mathbb{R}$ and
$p:U_{\delta}\rightarrow\mathbb{R}$ where $U_{\delta}=\\{(t,h):t\in
I,|h|<\delta\\}$ for some $\delta>0$ is some generally unspecified
neighborhood of $(t,0)$. Unless $\delta$ 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, $L(I)\equiv
L_{1}(I)$ is the usual space of Lebesgue integrable functions on $I$.

For a given $\alpha$, the generalized derivative or $\mathbf{p}$-derivative in
[10] is defined by

$D_{p}f(t)=\lim_{h\to 0}\frac{f(p(t,h))-f(t)}{h}.$ (3.1)

whenever the limit exists and is finite. Occasionally, we’ll introduce the
parameter $\alpha$ mentioned above into the definition so that the limit

$D_{p}^{\alpha}f(t)=\lim_{h\to 0}\frac{f(p(t,h,\alpha))-f(t)}{h}.$ (3.2)

will then be called the $\mathbf{p}$-derivative of order ${\bm{\alpha}}$.
Since $\alpha$ is a parameter (3.2) is actually a special case of (3.1).

The main hypotheses on the function $p$ are labeled $H1^{\pm}$ and $H2$ in
[10] and can be summarized together as follows:

Hypothesis (H). Given an interval $I\subset\mathbf{R}$, in addition to
requiring that for $t\in I$, the function $p(t,h)$, $p_{h}(t,h)$ are
continuous in a neighborhood of $h=0$, we ask that for $t\in I$ and for all
sufficiently small $\varepsilon>0$, $p(t,h)=t\pm\varepsilon$ has a solution
$h=h(t,\varepsilon)$ such that $h\rightarrow 0$ as $\varepsilon\rightarrow 0$.

Occasionally, we will assume, in addition, that $\frac{1}{p_{h}(\cdot,0)}\in
L(I).$

NOTATION: The notations $D,D^{\alpha},D_{p},D_{p}^{\alpha}$ will be used
interchangeably, occasionally for emphasis.

It is easy to see that the notion of $p$-differentiability as defined in (3.1)
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 [10].

######  Theorem 3.1.

Each of the derivatives defined by (1.4), (1.5), (1.6), (1.7), (1.8), and
(1.10) 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 [10].

######  Theorem 3.2.

(See Theorem 2.1 in [10].) Let $p$ satisfy (H). If $f$ is $p$-differentiable
at $a$ then $f$ is continuous at $a$.

######  Corollary 3.1.

Let $f$ be $\alpha$-differentiable where the $\alpha$-derivative is defined in
either (1.4), (1.5), (1.6), (1.7), (1.8) or (1.10). Then $f$ is continuous
there.

######  Remark 3.1.

We note that Corollary 3.1 includes Theorem 2.1 in [8]; Theorem 2.2 in [7],
and Theorem 2.2 in [5]).

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
$D_{p}:=D$ or, if there is a parameter dependence, $D_{p}^{\alpha}:=D$, for
simplicity.

######  Proposition 3.1.

(See Theorem 2.2 in [10] for proofs.)

  * (a)

(The Sum Rule) If $f,g$ are both $p$-differentiable at $t\in I$ then so is
their sum, $f+g$, and

$D(f+g)(t)=Df(t)+Dg(t).$

  * (b)

(The Product Rule) Assume that $p$ satisfies (H) and that for $t\in I$,
$p(t,h)$ is continuous at $h=0$. If $f,g$ are both $p$-differentiable at $t\in
I$ then so is their product, $f\cdot g$, and

$D(f\cdot g)(t)=f(t)\cdot Dg(t)+g(t)\cdot Df(t).$

  * (c)

(The Quotient Rule) Assume that $p$ satisfies (H). If $f,g$ are both
$p$-differentiable at $t\in I$ and $g(t)\neq 0$ then so is their quotient,
$f/g$, and

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

As a result we obtain,

######  Corollary 3.2.

(See Theorem 2.2 in [8]; Theorem 2.3 in [7]; Theorem 4 in [1], and Theorem 2.3
in [5].) For each of the definitions (1.4), (1.5), (1.6), (1.7), (1.8) and
(1.10) there holds an analog of the sum/product/and quotient rule for
differentiation of corresponding $p$-derivatives.

######  Proposition 3.2.

(See [10], Theorem 2.4) Assume (H). Let $f$ be continuous and non-constant on
$I$, and let $f$ be $p$-differentiable at $t\in I$. Let $g$ be defined on the
range of $f$ and let $g$ be differentiable at $f(t)$. Then the composition
$g\circ f$ is $p$-differentiable at $t$ and

$D(g\circ f)(t)=g^{\prime}(f(t))\,Df(t).$

######  Corollary 3.3.

(See [8], p.66 (iv), although the Chain Rule is not stated correctly there,
and Theorem 2.3 in [7].) For each of the definitions (1.4), (1.5), (1.6),
(1.7), (1.8) and (1.10) there holds an analog of the Chain rule for
differentiation of corresponding $p$-derivatives in the form

$D(g\circ f)(t)=g^{\prime}(f(t))\,Df(t),$

where $D:=D^{\alpha}$ (resp. $D:=D_{p}^{\alpha}$) is the corresponding
$p$-derivative in question defined in (3.1) (resp. (3.2)).

We now proceed to formulating fundamental theorems of a calculus for
generalized derivatives. We start with a weak maximum principle.

######  Theorem 3.3.

Let $f:[a,b]\to\mathbb{R}$ be such that $D_{p}f(t)$ exists for every
$t\in(a,b)$. If $f$ has a local maximum (resp. minimum) at $c\in(a,b)$ and for
each $t\in(a,b)$, and for all sufficiently small $|h|$ we have,

$p(t,h)<t,\quad h<0\quad(resp.\ p(t,h)>t,\quad h>0),$ (3.3) $p(t,h)>t,\quad
h>0\quad(resp.\ p(t,h)<t,\quad h<0),$ (3.4)

then $D_{p}f(c)=0$.

###### Proof.

We give the proof in the case where $f(c)$ is a local maximum. By hypothesis
(3.4), there exists $\delta_{0}$ such that

$p(t,h)>t,\quad 0<h<\delta_{0}.$

Since

$p(c,h)>c,\quad 0<h<\delta_{0},$

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

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

Therefore

$\lim_{h\to 0^{+}}\frac{f(p(c,h))-f(c)}{h}\leq 0.$

Hence, $D_{p}f(c)\leq 0$. Next, by hypothesis (3.3), there exists $\delta_{1}$
such that

$p(c,h)<c,\quad-\delta_{1}<h<0.$

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

$\lim_{h\to 0^{-}}\frac{f(p(c,h))-f(c)}{h}\geq 0.$

So, $D_{p}f(c)\geq 0$. However, since $D_{p}f(c)$ exists, it follows that
$D_{p}f(c)=0$. 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 3.4.

[A generalized Mean Value Theorem] Let $p$ satisfy the conditions of Theorem
3.3 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 $t\in(a,b)$. Then there is a constant $c\in(a,b)$ such that

$D_{p}f(c)=\frac{f(b)-f(a)}{b-a}k,$ (3.5)

where the constant $k$ is the $p$-derivative of the function whose values are
$t-a$ evaluated at $t=c$, i.e., $k=D_{p}(t-a)\bigg{|}_{t=c}.$

###### 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 $D_{p}h(t)=0$ for all $t\in(a,b)$, $c$ can be chosen to be
any point in $(a,b)$. If $D_{p}h(t)\neq 0$ 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, $t=c\in(a,b)$. By
Theorem 3.3, $D_{p}h(c)=0$. The result follows. ∎

######  Theorem 3.5.

If, in addition to the hypotheses in Theorem 3.4, we have $p(t,0)=t$, for
every $t$, and $p_{h}(t,0)$ exists, then

$D_{p}f(c)=\frac{f(b)-f(a)}{b-a}\,p_{h}(c,0).$

###### Proof.

Since

$D_{p}(t-a)\bigg{|}_{t=c}=\lim_{h\to 0}\frac{p(t,h)-a-(t-a)}{h}=\lim_{h\to
0}\frac{p(t,h)-p(t,0)}{h}=p_{h}(t,0),$

the result follows. ∎

In the next two sections we can formulate more precise results by considering
the two cases, $p_{h}(t,0)\neq 0$ and $p_{h}(t,0)=0.$

##  4 The case where $p_{h}(t,0)\neq 0$.

We’ll see below that this restriction on $p$ is the most common one and all
fractional derivatives presented (i.e., (1.4), (1.5), (1.6), (1.7), (1.8), and
(1.10)) satisfy this condition.

The next result, when combined with Theorem 2.1, allows us to transform
$p$-derivatives into ordinary derivatives.

######  Proposition 4.1.

(See Theorem 2.4 in [10]) Let $p$ satisfy hypothesis (H). In addition, for
$t\in I$, let

$\lim_{\varepsilon\to 0}\frac{\varepsilon}{h(t,\varepsilon)}\neq 0.$ (4.1)

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

$Df(t)=p_{h}(t,0)\,f^{\prime}(t).$ (4.2)

######  Remark 4.1.

Violation of either (4.1) or the tacit assumption, $p_{h}(t,0)\neq 0$, can
void (4.2), see Remark 2.4 in [10] and the example therein.

######  Corollary 4.1.

(See Theorem 2.2 in [8]; Theorem 2.3 in [7]; and Theorem 1 in [1].) Let the
$\alpha$-derivative be defined as in either (1.4) or (1.5) and let $f$ be
$\alpha$-differentiable at $t$. Then,

$T_{0}^{\alpha}f(t)=t^{1-\alpha}\,f^{\prime}(t).$

If $f$ is $\alpha$-differentiable in the sense of (1.6) then,

$D^{GFD}f(t)=\frac{\Gamma(\beta)}{\Gamma(\beta-\alpha+1)}\,t^{1-\alpha}\,f^{\prime}(t).$
(4.3)

If $f$ is $\alpha$-differentiable in the sense of (1.10) then, for
$t\in[0,b]$, $b<\pi/2$,

$Df(t)=(\cos t)^{1-\alpha}\,f^{\prime}(t).$ (4.4)

Similar results hold for derivatives defined by either (1.7) or (1.8) (if
$F(t,\alpha)\neq 0.$)

######  Corollary 4.2.

Let $t>0$ (resp. $t\geq 0$). Then $f$ is $\alpha$-differentiable at $t$ in the
sense of anyone of (1.4), (1.5), or (1.6), (resp. (1.10)) 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 3.3 above but do require that
$p_{h}(t,0)\neq 0$.

######  Theorem 4.1.

[A generalized Mean Value Theorem] Let $p$ satisfy the conditions of
Proposition 4.1 and let $f$ be $p$-differentiable on $(a,b)$ and continuous on
$[a,b]$. Then there exists $c\in(a,b)$ 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 4.1.
Since there exists $c\in(a,b)$ such that $f(b)-f(a)=(b-a)\,f^{\prime}(c)$ we
get $D_{p}f(c)=p_{h}(c,0)\,f^{\prime}(c)$ and the result follows. ∎

######  Theorem 4.2.

[Another generalized Mean Value Theorem] Let $p$ satisfy the conditions of
Proposition 4.1 and let $f,g$ be $p$-differentiable on $(a,b)$, continuous on
$[a,b]$, and $D_{p}(g(t))\neq 0$ there. Then there exists $c\in(a,b)$ such
that

$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{D_{p}f(c)}{D_{p}g(c)}.$

###### Proof.

Write

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

then $h(a)=h(b)=0$ and $h$ satisfies the conditions of Theorem 4.1. So, there
exists $c\in(a,b)$ such that $D_{p}h(c)=0$. 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 4.2.

Specializing to the case where

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

and $g(t)=t^{\alpha}/\Gamma(\alpha)$ with $\alpha\in(0,1)$ we get [[1],
Theorem 6]. The choice $g(t)=t^{\alpha}/\alpha$ gives [[7], Theorem 2.9].

Next, if $p_{h}(t,0)$ exists everywhere on $(a,b)$ and $p(t,0)=0$, then
Theorem 3.5 gives us that $k$ in (3.5) is given by $k=p_{h}(c,0)$. In this
case, we note that the function $f$ need not to be differentiable in the usual
sense here (see Theorem 4.1) and $p_{h}(c,0)$ may or may not be zero.

######  Example 4.1.

Given $I=[a,b]$, $f(t)=|t|$ and $p(t,h)=t+th+t^{3}h^{3}$. Then $p$ satisfies
the conditions of Theorem 3.4. Furthermore, $p_{h}(t,0)\neq 0$ for $t\neq 0$.
A simple calculation shows that $D_{p}f(t)=|t|=f(t)$, for all $t\in(a,b)$. By
Theorem 3.4, there exists $c\in(a,b)$ 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,

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

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

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

it suffices to choose $c=0$.

######  Remark 4.3.

In this example, $p_{h}(0,0)=0$ and consequently $D_{p}f(0)=0$ even though
$f^{\prime}(0)$ does not exist.

Of course, Rolle’s theorem is obtained by setting $f(a)=f(b)=0$ in Theorem
4.2. The latter then includes [[7], Theorem 2.8].

######  Definition 4.1.

Let $p$ satisfy (H), and let $f:[a,b]\to\mathbb{R}$ be continuous. Then

$I_{p}(f)(t)=\int_{a}^{t}\frac{f(x)}{p_{h}(x,0)}\,dx.$

This definition includes the fractional integral considered in [[7],
Definition 3.1]. Observe that, since $1/p_{h}\in L(a,b)$ and $f$ is bounded,
this integral always exists (absolutely). It follows that $I_{p}(f)\in
AC[a,b]$ and consequently $I^{\prime}_{p}(f)$ exists a.e. In this case, the
continuity of $p_{h}$ guarantees that $I_{p}(f)\in C^{1}(a,b)$.

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 4.3.

Let $p$ satisfy (H), and let $f:[a,b]\to\mathbb{R}$ be continuous. Then
$D_{p}(I_{p}(f)(t))=f(t).$

###### Proof.

By Proposition 4.1, since $I_{p}(f)$ is differentiable, we have,

$\displaystyle D_{p}(I_{p}(f)(t))$ $\displaystyle=$ $\displaystyle
p_{h}(t,0)I^{\prime}_{p}(f)(t)=f(t).$

∎

NOTE: The preceding includes [[7]. Theorem 3.2] as a special case.

######  Theorem 4.4.

Let $p$ satisfy (H) and let $F:[a,b]\to\mathbb{R}$ be continuous. If $F$ is
$p$-differentiable on $(a,b)$ and $D_{p}F$ is continuous on $[a,b]$, then
$I_{p}(D_{p}F)(b)=F(b)-F(a).$

###### Proof.

Let $a=x_{0}<x_{1}<x_{2}<....<x_{n}=b$ be a partition of $[a,b]$. Applying
corollary 4.1 to each $[x_{i-1},x_{i}]$ we get, for some $t_{i}$,

$D_{p}F(t_{i})=\frac{F(x_{i})-F(x_{i-1})}{x_{i}-x_{i-1}}p_{h}(t_{i},0)$

or

$F(x_{i})-F(x_{i-1})=(x_{i}-x_{i-1})\frac{D_{p}F(t_{i})}{p_{h}(t_{i},0)}.$

Thus,

$\displaystyle F(b)-F(a)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}F(x_{i})-F(x_{i-1})=\sum_{i=1}^{n}\frac{D_{p}F(t_{i})}{p_{h}(t_{i},0)}\Delta
x_{i}.$

Now since $f$ is continuous on every subinterval $[x_{i-1},x_{i}]$ of $[a,b]$,
we can pass to the limit as $\Delta x_{i}\to 0$. This gives,

$F(b)-F(a)=\int_{a}^{b}\frac{D_{p}F(t)}{p_{h}(t,0)}\,dt.$

This shows that $I_{p}(D_{p}F)(b)=F(b)-F(a)$ and we are done. ∎

Combining Theorem 4.3 and Theorem 4.4 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 4.3.

If $f,g$ are both $p$-differentiable on $(a,b)$ and continuous on $[a,b]$,
then,

$I_{p}(fD_{p}(g))=fg-I_{p}(D_{p}(f)g).$

This is clear on account of the Product Rule in Proposition 3.1(b) above.

##  5 The case where $p_{h}(t,0)=0$.

In this section we consider the exceptional case

$p_{h}(t,0)=0.$ (5.1)

The effect of (5.1) is that it tends to smooth out discontinuities in the
ordinary derivative of functions. A glance at (1.3) would lead one to guess
that whenever (5.1) holds we have $D_{p}f(t)=0$ but that is not the case, in
general.

######  Example 5.1.

Consider the special case $p(t,h)=t+h^{2}$ which satisfies (5.1). Then the
function $f(x)=\sqrt{x}$, $x>0$, although not differentiable at $x=0$, is
clearly right-$p$-differentiable at $x=0$ with $D_{p}^{+}f(0)=1$.

######  Theorem 5.1.

Let (5.1) hold, $p(t,0)=t$, and assume that (3.3), (3.4) are satisfied for
each $t$, as well. If $f$ is continuous on $[a,b]$ and $D_{p}f(t)$ exists,
then $D_{p}f(t)=0$.

###### Proof.

Note that $f$ is continuous on $(a,b)$ on account of the hypothesis and
Theorem 3.2. Using the proofs of Theorem 3.4 and Theorem 3.5 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
$k=p_{h}(c,0)=0$ by (3.5). ∎

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 5.1.

Incidentally, Example 5.1 also shows that (3.3) cannot be waived in the
statement of Theorem 5.1.

######  Example 5.2.

As before we let $p(t,h)=t+h^{2}$. Then $f(x)=|x|$, and although it is not
differentiable at $x=0$, it is clearly $p$-differentiable at $x=0$ with
$D_{p}f(0)=0$.

Below we study the consequences of this extraordinary assumption (5.1) and its
impact on the study of such $p$-derivatives.

###  5.1 Consequences of (5.1)

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 p-derivative equal to zero, as well. We first look
at some simple special cases of $p$ satisfying (5.1).

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 5.2.

Let $p(t,h)=t+h^{2}$. Then every polygonal function $f$ on $\mathbf{R}$ is
$p$-differentiable everywhere and $D_{p}f(x)=0$ for all $x\in\mathbf{R}$.

###### 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
zero (see Example 3.1 in [10]). ∎

######  Remark 5.2.

In contrast with the case where $p_{h}(t,0)\neq 0$ where an integral can be
defined via Definition 4.1, 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
(5.1) for a class of power functions defining the derivative. Our main result
is Theorem 3.1 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 $p_{h}(t,0)\neq 0$, Proposition 4.1 makes it easy to construct functions
that are nowhere $p$-differentiable on the whole real axis simply by choosing,
in particular, any function with $p_{h}(t,0)=1$. For a fascinating historical
survey of classical nowhere differentiable functions, the reader is encouraged
to look at [12] (available online).

At this point one may think that $p$-differentiability is normal and that most
functions have a zero $p$-derivative if $p_{h}(t,0)=0$. This motivates the
next question. Does there exist a function $p$ satisfying (5.1) such that it
is continuous and nowhere $p$-differentiable on $\mathbf{R}$? 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 (5.2), first considered by
Weierstrass, and one that led to a continuous nowhere differentiable function,
[11], can also serve as the basis for a continuous nowhere $p$-differentiable
function for a large class of functions $p$ satisfying (5.1), namely power
functions. Below we show that for each $\alpha>1$ there is a function $p$
satisfying (5.1) and a function $f$ that is nowhere $p$-differentiable on the
whole line.

######  Theorem 5.3.

Let $p(t,h)=t+h^{\alpha}$ where $\alpha>1$. Then Weierstrass’ continuous and
nowhere differentiable function

$f(x)=\sum_{n=0}^{\infty}\,b^{n}\,\cos(a^{n}\pi x)$ (5.2)

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

$\sqrt[\alpha]{a}\,b>1+\frac{3}{2}\,\pi,$ (5.3)

is nowhere $p$-differentiable on $\mathbf{R}$.

###### Proof.

Observe that the cases where $\alpha\leq 1$ are excluded by (5.1), so we let
$\alpha>1$. We will show, as usual, that there exists a sequence of $h\to 0$
along which $|(f(x+h^{\alpha})-f(x))/h|\to\infty$. Now, for fixed
$x\in\mathbf{R}$,

$\displaystyle\frac{f(x+h^{\alpha})-f(x)}{h}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}b^{n}\,\frac{\cos(a^{n}\pi(x+h^{\alpha}))-\cos(a^{n}\pi
x)}{h}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{m-1}b^{n}\,\frac{\cos(a^{n}\pi(x+h^{\alpha}))-\cos(a^{n}\pi
x)}{h}+\sum_{n=m}^{\infty}b^{n}\,\frac{\cos(a^{n}\pi(x+h^{\alpha}))-\cos(a^{n}\pi
x)}{h}$ $\displaystyle:=$ $\displaystyle S_{m}+R_{m}.$

Estimating $S_{m}$ by the Mean Value Theorem shows that for some $0<\theta<1$,

$|\cos(a^{n}\pi(x+h^{\alpha}))-\cos(a^{n}\pi
x)|=|h^{\alpha}a^{n}\pi\sin(a^{n}\pi(x+\theta h^{\alpha}))|\leq
a^{n}\pi|h|^{\alpha},$ (5.4)

so that

$|S_{m}|\leq\pi|h|^{\alpha-1}\,\sum_{n=0}^{m-1}(ab)^{n}<\frac{\pi\,|h|^{\alpha-1}(ab)^{m}}{ab-1}.$
(5.5)

Recall that $x$ is fixed at the outset. Now, for any positive integer $m$, we
can write $a^{m}x$ in the form $a^{m}x=\alpha_{m}+t_{m}$ where $\alpha_{m}$ is
an integer and $|t_{m}|\leq 1/2$. Define a sequence $h_{m}>0$ by

$h_{m}=\sqrt[\alpha]{\frac{1-t_{m}}{a^{m}}}.$

Then $0<h_{m}^{\alpha}\leq 3/(2a^{m}).$ From this choice of a sequence and
(5.5) we get the estimate,

$|S_{m}|<\pi\,\left(\frac{1-t_{m}}{a^{m}}\right)^{\frac{\alpha-1}{\alpha}}\,\frac{(ab)^{m}}{ab-1}\leq\pi\,\left(\frac{3}{2a^{m}}\right)^{\frac{\alpha-1}{\alpha}}\,\frac{(ab)^{m}}{ab-1}=\pi\,\left(\frac{3}{2}\right)^{\frac{\alpha-1}{\alpha}}\frac{{a^{\frac{m}{\alpha}}}b^{m}}{ab-1}.$
(5.6)

The next step is to show that the remainder term, $R_{m}$, remains bounded
away from $0$. To this end note that
$a^{n}\,\pi\,(x+h_{m}^{\alpha})=a^{n-m}a^{m}\pi\,(x+h_{m}^{\alpha})=a^{n-m}\pi(\alpha_{m}+1).$
It follows that since $a$ is odd, then for $n\geq m$, we have

$\cos(a^{n}\,\pi\,(x+h_{m}^{\alpha}))=(-1)^{\alpha_{m}+1}.$ (5.7)

A similar calculation shows that

$\cos(a^{n}\,\pi\,x)=\cos(a^{n-m}\,\pi\,(\alpha_{m}+t_{m}))=(-1)^{\alpha_{m}}\cos(a^{n-m}\pi
t_{m}).$ (5.8)

Combining (5.7) and (5.8) we see that

$\displaystyle R_{m}$ $\displaystyle=$
$\displaystyle\sum_{n=m}^{\infty}b^{n}\frac{(-1)^{\alpha_{m}+1}-(-1)^{\alpha_{m}}\cos(a^{n-m}\pi
t_{m})}{h_{m}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\alpha_{m}+1}}{h_{m}}\sum_{n=m}^{\infty}b^{n}(1+\cos(a^{n-m}\pi
t_{m}))$ $\displaystyle{\rm i.e.,}\ |R_{m}|$ $\displaystyle=$
$\displaystyle\frac{1}{|h_{m}|}\sum_{n=m}^{\infty}b^{n}(1+\cos(a^{n-m}\pi
t_{m})).$

Since the previous series is a series of non-negative terms we can drop all
terms except the first. In this case note that $\cos(\pi t_{m})\geq 0$ since
$|t_{m}|\leq 1/2$. So,

$|R_{m}|>\frac{b^{m}}{|h_{m}|}>\sqrt[\alpha]{\frac{2}{3}}a^{m/\alpha}\,b^{m}.$
(5.9)

Finally, using (5.9) and (5.6) we get

$\displaystyle\bigg{|}\frac{f(x+h_{m}^{\alpha})-f(x)}{h_{m}}\bigg{|}\geq|R_{m}|-|S_{m}|>\sqrt[\alpha]{\frac{2}{3}}a^{\frac{m}{\alpha}}\,b^{m}-\pi\,\left(\frac{3}{2}\right)^{\frac{\alpha-1}{\alpha}}\frac{{a^{\frac{m}{\alpha}}}b^{m}}{ab-1}=\left(\sqrt[\alpha]{\frac{2}{3}}-\frac{\pi}{ab-1}\,\left(\frac{3}{2}\right)^{\frac{\alpha-1}{\alpha}}\right)a^{\frac{m}{\alpha}}b^{m}$
(5.10)

Since $b<1$ we must have $a\geq 3$ so that $ab>\sqrt[\alpha]{a}b$. The
stronger hypothesis (5.3) forces both $\sqrt[\alpha]{a}b>1$ and the term in
the parentheses in (5.10) to be positive. Since $h_{m}\to 0$ as $m\to\infty$,
the left hand side of (5.10) 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 6.1.

Let $p$ satisfy (H) and $q:[0,T]\to\mathbb{R},T<\infty$ be continuous. Assume
that for some $b>0,$ we have

$\|\frac{1}{p_{h}}\|_{L^{1}[0,T]}<\min\left\\{\frac{b}{\|q\|_{\infty}+b^{2}},\frac{1}{2b}\right\\}.$
(6.1)

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

$D_{p}u(t)+u^{2}(t)=q(t),\quad u(0)=u_{0},$ (6.2)

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

###### Proof.

Let $B=\\{u\in C[0,T],\|u\|_{\infty}\leq b\\}$. Then $B$ is a complete metric
space. Define an operator $F$ on $B$ by $F(u)=I_{p}(q(t)-u^{2}(t))+u_{0}.$
Then for every $u,v\in B$.

$\displaystyle|Fu-Fv|$ $\displaystyle=$
$\displaystyle|I_{p}(q(t)-u^{2}(t))-I_{p}(q(t)-v^{2}(t))|$ $\displaystyle=$
$\displaystyle\bigg{|}\int_{0}^{t}\frac{q(s)-u^{2}(s)}{p_{h}(s,0)}-\frac{q(s)-v^{2}(s)}{p_{h}(s,0)}\,ds\bigg{|}$
$\displaystyle=$
$\displaystyle\bigg{|}\int_{0}^{t}\frac{(v(s)-u(s))(v(s)+u(s))}{p_{h}(s,0)}\,ds\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2b\|u-v\|_{\infty}\,\int_{0}^{t}\bigg{|}\frac{1}{p_{h}(s,0)}\bigg{|}\,ds$

It follows that $\|Fu-Fv\|_{\infty}<k\|u-v\|$ is a contraction on $B$, with
$k=2b\|\frac{1}{p_{h}}\|_{L^{1}[0,T]}<1$, by hypothesis.

Next, we show that $F:B\to B$. Clearly, for $u\in B$, $Fu$ is continuous on
$[0,T]$. Next, observe that

$\displaystyle\|Fu\|_{\infty}\leq\int_{0}^{T}\frac{\|q(s)-u^{2}(s)\|_{\infty}}{|p_{h}(s,0)|}\,ds\leq(\|q\|_{\infty}+b^{2})\|\frac{1}{p_{h}}\|_{L^{1}[0,T]}\leq
b,$

by hypothesis. Hence $F$ maps $B$ into itself. Applying the contraction
principle we get that $F$ has a unique fixed point $u\in C[0,T]$ such that
$Fu=u.$ Theorem 4.3 gives us the final result. ∎

######  Remark 6.1.

Observe that there are no sign restrictions on $p_{h}(t,0)$. Note that $D_{p}$
may in fact depend on a parameter $\alpha$, subject only to the
$L^{1}$-condition on $1/p_{h}$ at the outset. For example, if we choose
$p(t,h)=t+ht^{1-\alpha}$ as in [8], the hypothesis (6.1) above becomes,

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

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

Similarly, if
$p(t,h)=t+\frac{\Gamma(\beta)}{\Gamma(\beta-\alpha+1)}ht^{1-\alpha}$ with
$\beta>-1$, $\beta\in\mathbb{R^{+}}$ and $0<\alpha\leq 1$ as in [1], the
generalized Rolle’s Theorem, Mean Value Theorem and Riccati differential
equation studied here include the corresponding theorems in [1]. In addition,
this existence theorem clarifies the purely numerical results obtained in [1]
when solving a special Riccati equation of the form (6.2) using the fractional
derivative (1.6), which, as we have shown, is contained in our theory.

##  7 Open question

  1. 1.

Is there a function $p$, satisfying (5.1), and a function $f$ such that $f$ is
$p$-differentiable and such that $D_{p}f(t)\neq 0$ for all t in some interval
(or, more generally, some set of positive measure)?

##  8 Conclusion

In this paper we have extended the theory of $p$-derivatives in [10] to
include results such as the mean-value theorem, Rolle’s theorem and
integration by parts. In so doing we pointed out that the conformable
fractional derivative of a given function, as considered by [8] and [7], is
actually an ordinary derivative except possibly at one point. We expanded on
the cases where the partial derivative $p_{h}(t,0)$ either vanishes or doesn’t
and in so doing showed that in the former case there exists, for each
$\alpha>1$, a fractional derivative and a function whose fractional derivative
exists nowhere on the real line. In the case where $p_{h}(t,0)=0$ 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.

## References

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