#  neos: End-to-End-Optimised Summary Statistics for High Energy Physics

Nathan Simpson1 and Lukas Heinrich2 1Lund University  2Technical University of
Munich<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The advent of deep learning has yielded powerful tools to automatically
compute gradients of computations. This is because training a neural network
equates to iteratively updating its parameters using gradient descent to find
the minimum of a loss function. Deep learning is then a subset of a broader
paradigm; a workflow with free parameters that is end-to-end optimisable,
provided one can keep track of the gradients all the way through.

This work introduces neos: an example implementation following this paradigm
of a fully differentiable high-energy physics workflow, capable of optimising
a learnable summary statistic with respect to the expected sensitivity of an
analysis. Doing this results in an optimisation process that is aware of the
modelling and treatment of systematic uncertainties.

##  1 Introduction

Summary statistics are typically low-dimensional quantities calculated from
high-dimensional data, often for the purpose of making any following inference
stage easier to compute. They find widespread use in high-energy physics (HEP)
data analysis for this reason, since HEP data, e.g. that produced at the Large
Hadron Collider, is characterised by both its high dimensionality
($\mathcal{O}(10^{8})$ detector read-out channels) as well as the
intractability of computing its probability $p(x|\theta)$ under a given
physics hypothesis $\theta$. A notable example of such a quantity would be the
invariant mass of the Higgs boson, which was used in its discovery in 2012
[1].

Thanks to the presence of high-quality physics simulators, effective machine-
learning based summary statistics can be formed using simulated data,
typically being trained using classification-driven objectives to maximally
distinguish new physics from background processes. While these objectives can
be shown to produce optimal results in simplified hypothesis testing settings,
the presence of systematic uncertainties in the data may cause classification
objectives to become misaligned with physics objectives, e.g. the expected
sensitivity to a signal. This means that algorithms trained in this way may
under-perform when integrated into the actual physics analysis pipeline,
motivating the search for objectives that can better adapt to data with high
systematic uncertainty.

To combat this, we propose to optimise the summary statistic directly on
physics objectives by formulating an end-to-end-differentiable analysis
pipeline, including density estimation of the summary statistic with
histograms, and statistical inference using standard HEP prescriptions. The
training can then based on physics-oriented metrics, such as the expected
sensitivity, which is based on the profile likelihood ratio: a quantity
capable of capturing systematic effects.

###  1.1 Related work

The most similar work to neos is INFERNO [2], which also targets the
optimisation of a summary statistic with respect to an inference-aware loss
function. We compare our approach to INFERNO below both qualitatively and
quantitatively. Other attempts to incorporate robustness to systematic
uncertainties include: directly parameterising the neural network in the
relevant nuisance parameters that model systematic uncertainties [3], and
including an adversarial term in the loss that penalises dependence on these
nuisance parameters [4].

##  2 Making HEP Analysis Differentiable

Given a pre-filtered dataset, an analysis pipeline in HEP involves the
following stages:

  1. 1.

Construction of a learnable 1-D summary statistic from data (with parameters
$\varphi$)

  2. 2.

Binning of the summary statistic, e.g. through a histogram

  3. 3.

Statistical model building, using the summary statistic as a template

  4. 4.

Calculation of a test statistic, used to perform a frequentist hypothesis test
of signal versus background

  5. 5.

A $p$-value (or $\mathrm{CL_{s}}$111$\mathrm{CL_{s}}$ is a modification of the
p-value that protects against rejecting the null hypothesis when the test is
not sensitive to the alternative hypothesis (e.g. through largely overlapping
test statistic distributions). value) resulting from that hypothesis test,
used to characterise the sensitivity of the analysis

We can express this workflow as a direct function of the input dataset
$\mathcal{D}$ and observable parameters $\varphi$:

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}q_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mu}}(x)\
}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\\{}{\color[rgb]{0.31,0.89,0.87}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,0.89,0.87}s},{\color[rgb]{0.96,0.65,0.14}\definecolor[named]{pgfstrokecolor}{rgb}{0.96,0.65,0.14}b},{\color[rgb]{0.73,0,0.09}\definecolor[named]{pgfstrokecolor}{rgb}{0.73,0,0.09}b}\mathrm{{\color[rgb]{0.73,0,0.09}{}_{up}}},{\color[rgb]{0.93,0.86,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.93,0.86,0}b}{\color[rgb]{0.93,0.86,0}{}_{\mathrm{down}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\\}}{\color[rgb]{0,0,0}{}_{i}}\
$$\mathrm{data}\ (d_{i})$$\mathrm{neural\ network}$$\mathrm{binned\ summary\
stat.}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}h_{i}=\mathrm{hist}(f_{{\color[rgb]{0,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.46,1}\varphi}}(d_{i}))}$$\begin{array}[]{l}\mathrm{test\
}\\\ \mathrm{statistic}\end{array}$$\begin{array}[]{l}\mathrm{hypothesis}\\\
\mathrm{test}\end{array}$$\begin{array}[]{l}\mathrm{likelihood\ }\\\
\mathrm{model}\end{array}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}p_{\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}h}}}(x|\mu,\theta)}$$\mathrm{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}CL_{s}}}$$\mathrm{{\color[rgb]{0.29,0.29,0.29}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.29,0.29}update\
}{\color[rgb]{0,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.46,1}\varphi}{\color[rgb]{0.29,0.29,0.29}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.29,0.29}\
with}}{\color[rgb]{0.29,0.29,0.29}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.29,0.29}\
\frac{\partial\mathrm{{\color[rgb]{0.29,0.29,0.29}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.29,0.29}CL_{s}}}}{\partial{\color[rgb]{0,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.46,1}\varphi}}}$$\mathrm{{\color[rgb]{0.29,0.29,0.29}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.29,0.29}backward\
pass}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}q_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mu}}(x_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}obs}})\
}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}f_{{\color[rgb]{0,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.46,1}\varphi}}(d_{i})}$
Figure 1: The pipeline for neos. The dashed line indicating the backward pass
involves updating the weights
${\color[rgb]{0,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.46,1}\varphi}$
of the neural network via gradient descent.

$\mathrm{CL_{s}}=f(\mathcal{D},\varphi)=(f_{\mathrm{sensitivity}}\circ
f_{\mathrm{test\,stat}}\circ f_{\mathrm{likelihood}}\circ
f_{\mathrm{histogram}}\circ f_{\mathrm{observable}})(\mathcal{D},\varphi).$
(1)

In the common case where $f_{\mathrm{observable}}$ is a neural network, it
seems possible to optimise this composition end-to-end, i.e. train the network
to directly optimise the analysis sensitivity. This is exactly the task that
neos sets out to accomplish, with a full workflow detailed in Figure 1. To
train this network by gradient descent, our choice of
$\mathrm{loss}=\mathrm{CL_{s}}$ requires us to be able to calculate
$\partial\,\mathrm{CL_{s}}/\partial\varphi$. However, this is a stronger
condition than it seems, and in fact necessitates the differentiablility of
each individual analysis step via the chain rule applied to Equation 1.

Owing to the fact that neural networks are already differentiable, the last
term $\partial f_{\mathrm{observable}}/{{\partial\varphi}}$ isn’t an issue,
but none of the rest of the steps are differentiable by default. The following
sections detail solutions for calculating the gradient of each intermediate
step.

###  2.1 Binned density estimation (histograms)

Histograms are discontinuous by nature. They are defined for 1-D data as a set
of two quantities: intervals (or bins) over the domain of that data, and
counts of the number of data points that fall into each bin. For small changes
in the underlying data distribution, bin counts will either remain static, or
jump in integer intervals as data migrate between bins, both of which result
in ill-defined gradients. We address this inherent non-differentiability
through implementing a differentiable surrogate: a histogram based on a kernel
density estimate (KDE), which produces a non-parametric density $p(t)$
estimate based on samples $t_{i}$ and a kernel function
$K(t,t_{i})=K(t-t_{i})$, with the full density given as
$p(t)=1/n\sum_{i}K(t,t_{i})$. Normally, a popular kernel function choice is
the standard normal distribution, which comes with a parameter called the
bandwidth that affects the smoothness of the resulting density estimate.

Coming back to gradients: in our case, the data $t_{i}$ we construct the
density estimate over are themselves functions of the summary statistic
parameters, i.e. $t_{i}=f(x_{i}\;\varphi)$. The resulting density estimate
$p(t|\varphi)$ will then be differentiable as long as the kernel $K$ is
differentiable with respect to $t_{i}$, and by extension with respect to
$\varphi$. To extend this differentiability in a binned fashion, we can
accumulate the probability mass of the KDE within the bin edges of the
original histogram – equivalent to evaluations of the Gaussian cumulative
density function – to convert $p(t|\varphi)$ to a binned KDE (bKDE), i.e. a
set of discrete per-bin probabilities $p_{i}(\varphi)$.

In the limit of vanishing bandwidth, the bKDE recovers the standard histogram,
but gradients become susceptible to high variance. Increasing the bandwidth
can alleviate this, but at the cost of introducing a bias. There is then a
trade-off between decreasing the bandwidth enough to minimise this bias, and
increasing it enough to guarantee gradient stability222How low is low enough
when tuning? This relationship between bias and gradient stability is heavily
impacted by the number of data points, and also by the width of the intervals,
with more exploration of this planned.. We can see a demonstration of this
behaviour in Figure 2, where the bandwidth is tuned relative to the bin width.

Figure 2: Illustration of the bias/smoothness tradeoff when tuning the
bandwidth of a bKDE, defined over 200 samples from a bi-modal Gaussian
mixture. All distributions are normalised to unit area. The individual kernels
that make up the KDE are scaled down for visibility.

###  2.2 Likelihood modelling

A popular framework to build statistical models for binned observations based
on “template” histogram data is HistFactory [5], used widely used across HEP.
Thankfully, the resulting log-likelihood function $p_{\mathbf{h}}(x)$ is
differentiable with respect to both the observed data $x$ and histogram data
$\mathbf{h}$ from which the model is constructed. However, these gradients
only recently became readily accessible in software, owing to the development
of pyhf [6, 7]: a Python package for building these likelihoods that leverages
automatic differentiation.

###  2.3 Test statistics (profile likelihood ratio)

A number of test statistics are typically used in HEP333A full list of the
different test statistics and their purposes can be found in [8]., all of
which build on the profile likelihood ratio. For likelihood function $p$,
input data $x$, parameters of interest (POIs) $\mu$, and nuisance parameters
$\psi$, we define the profile likelihood ratio as

$\displaystyle\lambda_{\mu}(x)=\frac{p(x|\mu,\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\psi}}}(\mu))}{p(x|\hat{\mu},\hat{\psi})},$
(2)

where $\hat{\mu},\hat{\psi}$ denote the maximum-likelihood parameters, whilst
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\psi}}}(\mu)$ denotes the
maximum-likelihood estimate of $\psi$ at fixed values of $\mu$. A test
statistic $q_{\mu}$ is then built upon the profile likelihood, following
prescriptions detailed in [8]. Our physics objective, i.e. the expected
$\mathrm{CL}_{s}$, is computed from distributions of $q_{\mu}$ for the null
and alternative hypotheses, which correspond to choosing different values of
$\mu$. In asymptotic theory, the null is distributed as a
$\chi^{2}$-distribution, while the alternative follows a non-central
$\chi^{2}_{\mathrm{nc}}(x,\Lambda^{2})$ distribution, with $\Lambda^{2}$ given
by

$\Lambda^{2}=(\mu-\mu^{\prime})^{2}/\sigma_{\hat{\mu}}^{2},$ (3)

where $\mu$, $\mu^{\prime}$ are the POI values corresponding to the null and
alternative hypothesis respectively, and $\sigma_{\hat{\mu}}$ is the variance
for the maximum likelihood estimate of the POIs. Our goal of minimising the
expected $\mathrm{CL}_{s}$ is then equivalent to both maximising $\Lambda^{2}$
and minimising $\sigma_{\hat{\mu}}$.

While the INFERNO approach uses the Fisher Information to estimate and
minimise $\sigma_{\hat{\mu}}$, we take the route of computing $\Lambda^{2}$ as
the test statistic value for the “Asimov dataset” of the alternative
hypothesis. As described in [8], the use of Asimov test statistics is known to
yield better approximations of the true sampling distributions, and therefore
of the final objective $\mathrm{CL_{s}}$. However, the drawback of this
approach is the inclusion of optimisation routines such as the computation of
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\psi}}}$, which are not
a-priori differentiable in an efficient manner. We thus explore methods for
derivatives of optimisation routines that only require gradient information
when optimisation has converged.

####  2.3.1 Implicit differentiation of optimisation routines:

Optimisation algorithms find best-fit values $\hat{\theta}$ for a given
objective function $f$. In the case of the profile likelihood, the objective
$f$ is the likelihood function, which implicitly depends on the summary
statistic parameters $\varphi$; we denote this by $f=f_{\varphi}$, and extend
this dependence to the best-fit parameters $\hat{\theta}(\varphi)$. Iterative
optimisation finds $\hat{\theta}$ by starting at a initial value $\theta_{0}$
and repeatedly updating it through a iteration step
$\hat{\theta}_{i+1}=u(f_{\varphi},\theta_{i})$ until a terminating condition
is satisfied. An explicit expression for $\hat{\theta}(\varphi)$ capturing the
dependence on $\varphi$ is not readily available. Despite this, we can still
compute gradients of $\hat{\theta}$ by noting that at the solution
$\theta_{i}=\hat{\theta}$, the following relation holds for any $\varphi$:

$\hat{\theta}=u(f_{\varphi},\hat{\theta})~{}\Rightarrow~{}g(\varphi,\hat{\theta})=u(f_{\varphi},\hat{\theta})-\hat{\theta}=0,$
(4)

From here, we can leverage a powerful result from the implicit function
theorem that locally guarantees the existence of a continuously differentiable
function $\hat{\theta}(\varphi)$, with gradients given by

$\frac{\partial\hat{\theta}(\varphi)}{\partial\varphi}=-\left[\frac{\partial
g}{\partial\theta}\right]^{-1}\frac{\partial
g}{\partial\varphi}=\left[I-\frac{\partial
u}{\partial\theta}\right]^{-1}\frac{\partial u}{\partial\varphi}$ (5)

This is our saving grace: by evaluating derivatives of just the update step
with respect to the optimised parameters $\theta$ and the parameters
implicitly defining the objective $\varphi$, we can derive gradients of the
resulting optimisation solution without unrolling all iterations. Implicit
differentiation through optimisation procedures like this has been implemented
in software by the authors of the jaxopt package [9], which we use in our
experiments. We refer the reader to Ref. [10] for more detail on the
connection between Equation 5 and automatic differentiation.

This result lets us compute the non-centrality parameter $\Lambda^{2}$ of the
hypothesis test (and as a corollary, the expected $\mathrm{CL_{s}}$) in a
fully differentiable manner, providing the final building block of an end-to-
end differentiable analysis pipeline as described in the beginning of this
section.

##  3 Demonstrating neos: End-to-end Analysis Optimisation

We demonstrate the principle of end-to-end optimisation by training a neural
network optimised directly on the physics sensitivity, expressed as the
expected $\mathrm{CL_{s}}$ value of a `pyhf` model.

###  3.1 Gaussian blob experiment

Figure 3: Analysis metrics evaluated on a test set when running the neos
pipeline with a variety of loss functions, averaged over 7 initialisations of
the neural network weights. Left: the expected $\mathrm{CL_{s}}$ with a null
hypothesis of no signal. Middle: the uncertainty on the signal strength $\mu$.
Right: deviation of the nuisance parameter uncertainty $\gamma$ from its
nominal value of 1.

We consider an example where we generate toy physics data $(x,y)$ from 2-D
Gaussian blobs with covariances equal to the 2-D identity matrix $I_{2}$.

The dataset $\mathcal{D}$ consists of signal
${\color[rgb]{0.31,0.89,0.87}\definecolor[named]{pgfstrokecolor}{rgb}{0.31,0.89,0.87}s}\sim\mathcal{N}(m_{s},I_{2})$,
background
${\color[rgb]{0.96,0.65,0.14}\definecolor[named]{pgfstrokecolor}{rgb}{0.96,0.65,0.14}b}\sim\mathcal{N}(m_{b},I_{2})$,
and two variations of the background data
$\\{{\color[rgb]{0.73,0,0.09}\definecolor[named]{pgfstrokecolor}{rgb}{0.73,0,0.09}b\mathrm{{}_{up}}},{\color[rgb]{0.93,0.86,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.93,0.86,0}b_{\mathrm{down}}}\\}$
that mimic simulating the background process after varying an imagined
physical parameter $\Psi$ up and down by one standard deviation, which is a
common situation encountered in HEP. In practice, $\Psi$ is taken to influence
the mean of the background distribution implicitly through generating blob
data with different means, i.e.
${\color[rgb]{0.73,0,0.09}\definecolor[named]{pgfstrokecolor}{rgb}{0.73,0,0.09}b\mathrm{{}_{up}}}\sim\mathcal{N}(m_{b{\mathrm{up}}},I_{2})$,
${\color[rgb]{0.93,0.86,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.93,0.86,0}b_{\mathrm{down}}}\sim\mathcal{N}(m_{b{\mathrm{down}}},I_{2})$.

After dataset generation, as well as splitting into train and test datasets,
the workflow then follows the steps outlined in Figure 1. Results from running
this experiment can be found in Figure 3, which compares neos to three
baseline loss functions:

  * •

Binary cross-entropy (BCE): signal vs. nominal background

  * •

BCE, but using up/down systematic variations as data augmentation for the
background sample. This is a powerful baseline that maximally isolates the
signal points from all background variations.

  * •

INFERNO: using the signal strength uncertainty $\sigma_{\mu}^{2}$ as the
objective, calculated from the Hessian of the likelihood via automatic
differentiation.

Three metrics are shown: the expected $\mathrm{CL_{s}}$, the parameter of
interest ($\mu$) uncertainty, and the nuisance parameter ($\gamma$)
uncertainty, all evaluated on the test set444During test evaluation and
training with binary cross-entropy, no differentiable approximations are
used.. All metrics are averaged over seven different initialisations of the
weights $\varphi$. Hyperparameters are given in Appendix A.

We can see that neos is able to reach sensitivities lower than any other loss
function (left plot). Moreover, it is able to gradually improve the
uncertainty on the parameter of interest (middle plot) without any additional
tuning or regularisation, far outperforming binary cross-entropy in this
regard. When mixing in the up/down systematic variations as part of the
background class when using BCE, we are able to get similar performance to
neos with lower $\mu$ uncertainty around epoch 4, but neos is able to improve
to match this with more training, all while retaining a lower
$\mathrm{CL_{s}}$. INFERNO is able to achieve the best uncertainty on $\mu$,
but fails to reach the same expected sensitivity as other methods.

We additionally examine the uncertainty on $\gamma$ (right plot), also taken
from the Fisher information matrix. It indicates whether the analysis is able
to determine the systematic effect on the background shape in a way that
differs from the constraint on $\gamma$ provided by the up/down variations.
This is typically undesirable behaviour, as it indicates a strong dependence
of the summary statistic on the nuisance parameters. Here, we see that neos
and BCE (with aug.) are able to keep this metric fairly low, while BCE and
INFERNO show signs of over- or under- constraining $\gamma$.

##  4 Discussion and Future Directions

One of the more immediate concerns for using neos in practice is the issue of
scaling. Training times compared to BCE in the toy problem are a factor of 3
or so higher, since the processing of one batch corresponds to running the all
the downstream parts of a physics analysis. In addition, the batch size needs
to be large enough to faithfully represent the analysis, hopefully avoiding
situations with empty bins. This is an issue that is currently being
investigated; a medium/long term goal would be to scale neos in application to
open data from the major LHC experiments, such as ATLAS and CMS, allowing the
probing of how the performance benefits highlighted in section 3 scale to
realistic problems with many sources of uncertainty.

Despite the focus on summary statistics, the work done in enabling the
differentiability of HEP analysis opens up a variety of new approaches in
optimising any free parameters end-to-end. For example, one could imagine
optimising the pre-selection stage that occurs when filtering data prior to
inference, since one can differentiably approximate a cut with a sigmoid
function. Moreover, one could optimise the binning of an observable for the
best sensitivity, given a fixed number of bins.

An additional, exploratory direction is considering the use of a multi-term
objective function with weighted components. The software released alongside
this work already enables objectives of the form
$a_{0}{\mathrm{CL_{s}}}+a_{1}{\sigma_{\mu}}+a_{2}{(1-\sigma_{\gamma})^{2}}+\dots$,
which could offer regularisation effects to steer away from pathologies that
happen to satisfy one metric only. Work in this direction is also ongoing.

A future comparison to black box methods such as Bayesian optimisation would
also be desirable, as they do not require the use of approximations to enable
optimisation, but still can target the physics goals of choice.

##  5 Conclusion

The typical HEP analysis workflow has been made differentiable, and a use case
for it in the systematic-aware optimisation of a neural network has been
demonstrated. Comparison was made to the more standard approach of using
binary cross-entropy, as well as more competitive baselines like systematic
data augmentation and INFERNO. Improvements were shown in both the expected
analysis sensitivity and the properties of the likelihood function.
Accompanying software to run the experiments detailed in Section 3 can be
found at [11], as well as a toolbox called relaxed [12] that enables the
isolated or combined use of the differentiable components used to assemble
neos.

We would like to thank the authors of INFERNO (Pablo de Castro, Tommaso
Dorigo) for initial productive discussions. We also extend thanks to Kyle
Cranmer, Alexander Held, Giordon Stark, and Matthew Feickert for conceptual
and technical support in the development of this project. Visualisations are
produced using the matplotlib package [13]. LH is supported by the Excellence
Cluster ORIGINS, which is funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy -
EXC-2094-390783311. NS completed this work with financial support from
Insights, which is funded by the European Union’s Horizon 2020 research and
innovation programme, call H2020-MSCA-ITN-2017, under Grant Agreement n.
765710. NS extends gratitude to friends and colleagues that indirectly
facilitated this work, particularly to those in the “noonch” lunch group at
CERN.

## Appendix A

Hyperparameters for the Gaussian blob study in Section 3:

  * •

10000 data points, split evenly between all four blobs,

  * •

3-layer neural network of size (1024, 1024, 1),

  * •

Training with Adam optimiser, learning rate 1e-3,

  * •

Adam optimiser also used in maximum likelihood fits with learning rate 1e-2,

  * •

$m_{\mathrm{s}}=(-1,1)$, $m_{\mathrm{b}}=(2.5,2)$,
$m_{\mathrm{bup}}=(-2.5,-1.5)$, $m_{\mathrm{bdown}}=(1,-1)$,

  * •

Multiplicative histogram scale factors: signal scale=2, background scale=10,
global scale=10,

  * •

ReLU activations, with sigmoid activation on the final layer,

  * •

15 epochs, with a batch size of 2000.

## References

## References

  * [1] ATLAS-Collaboration 2012 Physics Letters B 716 1–29 ISSN 0370-2693 URL http://dx.doi.org/10.1016/j.physletb.2012.08.020
  * [2] de Castro P and Dorigo T 2019 Computer Physics Communications 244 170–179 ISSN 0010-4655 URL http://dx.doi.org/10.1016/j.cpc.2019.06.007
  * [3] Ghosh A, Nachman B and Whiteson D 2021 Physical Review D 104 ISSN 2470-0029 URL http://dx.doi.org/10.1103/PhysRevD.104.056026
  * [4] Louppe G, Kagan M and Cranmer K 2017 Learning to pivot with adversarial networks (Preprint 1611.01046)
  * [5] Cranmer K, Lewis G, Moneta L, Shibata A and Verkerke W (ROOT Collaboration) 2012 HistFactory: A tool for creating statistical models for use with RooFit and RooStats Tech. rep. New York U. New York URL https://cds.cern.ch/record/1456844
  * [6] Heinrich L, Feickert M and Stark G pyhf: v0.6.3 https://github.com/scikit-hep/pyhf/releases/tag/v0.6.3 URL https://doi.org/10.5281/zenodo.1169739
  * [7] Heinrich L, Feickert M, Stark G and Cranmer K 2021 Journal of Open Source Software 6 2823 URL https://doi.org/10.21105/joss.02823
  * [8] Cowan G, Cranmer K, Gross E and Vitells O 2011 The European Physical Journal C 71 ISSN 1434-6052 URL http://dx.doi.org/10.1140/epjc/s10052-011-1554-0
  * [9] Blondel M, Berthet Q, Cuturi M, Frostig R, Hoyer S, Llinares-López F, Pedregosa F and Vert J P 2021 arXiv preprint arXiv:2105.15183
  * [10] Duvenaud D, Johnson M and Kolter Z Deep implicit layers - neural odes, deep equilibirum models, and beyond URL http://implicit-layers-tutorial.org/
  * [11] Simpson N and Heinrich L 2021 neos: version 0.2.0 URL https://github.com/gradhep/neos
  * [12] Simpson N 2022 relaxed: version 0.1.3 URL https://github.com/gradhep/relaxed
  * [13] Hunter J D 2007 Computing in Science & Engineering 9 90–95


Nonlocal Boundary Value Problem for Generalized Hilfer Implicit Fractional
Differential Equations

Ashwini D. Mali 1

<EMAIL_ADDRESS>
Kishor D. Kucche 2

<EMAIL_ADDRESS>
1,2 Department of Mathematics, Shivaji University, Kolhapur-416 004,
Maharashtra, India.

###### Abstract

In this paper, we derive the equivalent fractional integral equation to the
nonlinear implicit fractional differential equations involving
$\varphi$-Hilfer fractional derivative subject to nonlocal fractional integral
boundary conditions. The existence of a solution, Ulam-Hyers, and Ulam-Hyers-
Rassias stability has been acquired by means equivalent fractional integral
equation. Our investigations depend on the fixed point theorem due to
Krasnoselskii and the Gronwall inequality involving
$\varphi$-Riemann–Liouville fractional integral. An example is provided to
show the utilization of primary outcomes.

Key words: Implicit nonlinear fractional differential equations;
$\varphi$-Hilfer fractional derivative; Nonlocal integral boundary conditions;
Existence of solution; Ulam-Hyers stability.
2010 Mathematics Subject Classification: 34A08, 34A09, 34D20, 34A40, 35A23.

##  1 Introduction

The fundamental study about initial value problem (IVP) for Caputo implicit
fractional differential equations (FDEs) of the form

$\displaystyle{}^{c}D^{\alpha}x(t)=f(t,x(t),{}^{c}D^{\alpha}x(t)),~{}x(0)=x_{0},$

have been initiated by Nieto et al. [1]. The major investigation relating the
existence and uniqueness of the solution and Ulam-Hyers stabilities for
implicit FDEs with different kinds of initial and boundary conditions can be
found in the work of Benchohra et al. [2, 3, 4, 5, 6, 7, 8] . Kucche et al.
[9] investigated existence, the interval of existence and uniqueness of
solutions along with various qualitative properties of solution for the
implicit FDEs mentioned above.

There have appeared numerous significant works about the nonlinear boundary
value problems( BVPs) for FDEs, out of which we are mentioning here only a few
that are relating to the works of the present paper. Existence and uniqueness
results for nonlocal BVPs of nonlinear FDEs involving Caputo fractional
derivative has been examined by Benchohra et al.[10] and Ahmad and Nieto[11]
through various types of fixed point theorems. Utilizing the methodology of
[11] the study has been extended to nonlocal BVPs for nonlinear
integrodifferential equations [12]. Zhang[13], obtained results relating to
the existence and multiplicity of positive solutions of BVP for nonlinear
Caputo FDEs. In [14], Jiang and Wang investigated results concerning the
existence of solutions of multi-point BVP for Riemann-Liouville FDEs.

On the other hand, Hilfer[15] defined a two parameter fractional derivative
called Hilfer fractional derivative that incorporates both fractional
operators, Riemann-Liouville derivative and Caputo derivative. The fundamental
work on the theory of IVP for FDEs involving Hilfer derivative can be found in
[16]. Asawasamrit et al. [17] initiated the study of BVPs for FDEs involving
Hilfer fractional derivatives subject to nonlocal integral boundary
conditions. The BVP for fractional integrodifferential equations with Hilfer
derivative has been researched in [18] for existence and data dependence of
solutions. The implicit FDEs with a nonlocal condition involving Hilfer
fractional derivative investigated in [19, 20] for existence and Ulam types
stabilities.

The Hilfer version of the fractional derivative with another function called
$\varphi$-Hilfer fractional derivative has been presented by Jose et al. [21].
The basic study about existence and uniqueness of the solution of a nonlinear
$\varphi$-Hilfer FDEs with different kinds of initial conditions and the Ulam-
Hyers and Ulam-Hyers-Rassias stabilities of its solutions have been explored
in [22, 23, 24, 25, 26, 27, 28, 29] . The implicit FDEs involving
$\varphi$-Hilfer derivative has been investigated in [30] for the existence
and uniqueness of the solution and the Ulam-Hyers-Rassias stability.

Thinking about available literature, it is seen that the study of boundary
value problems for nonlinear implicit FDEs involving generalized fractional
derivative is still in the underlying stages and numerous parts of this theory
should be investigated. Inspired by the work of the papers mentioned above and
the work of [10, 17], the primary goal of the present work is to establish the
existence the solutions and to examine the Ulam-Hyers stabilities of the
following implicit FDEs involving $\varphi$-Hilfer fractional derivative
subject to nonlocal integral boundary conditions,

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)$
$\displaystyle=f(t,y(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)),~{}t\in
J^{\prime},~{}1<\mu<2,~{}0\leq\nu\leq 1$ (1.1) $\displaystyle y(a)$
$\displaystyle=0,$ (1.2) $\displaystyle y(b)$
$\displaystyle=\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i}),\qquad$
(1.3)

where $J^{\prime}=(a,\,b],$
${}^{H}\mathcal{D}^{\mu,\nu;\,\varphi}_{a+}(\cdot)$ is the $\varphi$-Hilfer
fractional derivative of order $\mu$ and type $\nu$,
$\mathcal{I}_{a+}^{\delta_{i};\,\varphi}(\cdot)$ is
$\varphi$-Riemann–Liouville fractional integral of order $\delta_{i}>0$,
$\lambda_{i}\in\mbox{\Bbb R},\,i=1,2,\cdots,m,$ $0\leq
a\leq\tau_{1}<\tau_{2}<\tau_{3}<\cdots<\tau_{m}\leq b$ and $f\in
C(J^{\prime}\times\mbox{\Bbb R}\,,\mbox{\Bbb R})$.

The $\varphi$-Hilfer implicit FDEs with nonlocal integral boundary conditions
considered in the present paper is the more broad class of BVP that
incorporates for different values of $\nu$ and $\varphi$ the class of BVP for
implicit FDEs involving fractional derivative operators to be specific
Riemann-Liouville, Caputo, Hadamard, Katugampola and other fractional
derivative operators referenced in [21]. In particular,

  * •

for $\varphi(t)=t$, the outcomes acquired in the present paper incorporates
the results of [17] for non-implicit Hilfer FDEs with nonlocal BVP.

  * •

for $\nu=1,\,\varphi(t)=t,\delta_{i}=0,a=0,b=1$ and
$\tau_{1}=\tau_{2}=\dots=\tau_{m-1}=0$, incorporates the results of [11] for
non-implicit Caputo FDEs with nonlocal BVP.

The paper is structured in five sections as follows: In sections 2, we present
the definitions and the results that are utilized in the paper. Section 3,
provide an equivalent fractional integral equation to the nonlocal implicit
BVP problem (1.1)-(1.3). Section 4, deals with existence of solution for
nonlocal BVP problem (1.1)-(1.3). Ulam-Hyers stability and Ulam-Hyers-Rassias
stability has been examined in section 5. In section 6, we provided an example
to justify our results.

##  2 Preliminaries

Let $\xi=\mu+\nu\left(2-\mu\right)$, $1<\mu<2,~{}0\leq\nu\leq 1$. Then
$1<\xi\leq 2$. Let $\varphi\in C^{1}(J,\,\mbox{\Bbb R})$ be an increasing
function with $\varphi^{\prime}(t)\neq 0$, for all $\,t\in J=[a,b]$. Consider
the space

$C_{2-\xi;\,\varphi}(J,\,\mbox{\Bbb
R})=\left\\{\mathfrak{h}:\left(a,b\right]\rightarrow\mathbb{R}~{}\big{|}~{}\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\mathfrak{h}\left(t\right)\in
C(J,\,\mbox{\Bbb R})\right\\},$

with the norm

$\left\|\mathfrak{h}\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}=\underset{t\in
J}{\max}\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\mathfrak{h}\left(t\right)\right|.$
(2.1)

######  Definition 2.1 ([31])

Let $\mu>0~{}(\mu\in\mbox{\Bbb R})$, $\mathfrak{h}\in L_{1}(J,\,\mbox{\Bbb
R})$. Then, the $\varphi$-Riemann–Liouville fractional integral of a function
$\mathfrak{h}$ with respect to $\varphi$ is defined by

$\mathcal{I}_{a+}^{\mu\,;\,\varphi}\mathfrak{h}\left(t\right)=\frac{1}{\Gamma\left(\mu\right)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\mathfrak{h}(s)\,ds.$
(2.2)

######  Definition 2.2 ([21])

Let $n-1<\mu<n\in\mbox{\Bbb N}$ and $h\in C^{n}(J,\,\mbox{\Bbb R})$. Then, the
$\varphi$-Hilfer fractional derivative
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}(\cdot)$ of a function
$\mathfrak{h}$ of order $\mu$ and type $0\leq\nu\leq 1$, is defined by

$^{H}\mathcal{D}_{a+}^{\mu,\,\nu\,;\,\varphi}\mathfrak{h}\left(t\right)=\mathcal{I}_{a+}^{\nu\left(n-\mu\right)\,;\,\varphi}\left(\frac{1}{\varphi^{\prime}\left(t\right)}\frac{d}{dt}\right)^{n}\mathcal{I}_{a+}^{\left(1-\nu\right)\left(n-\mu\right)\,;\,\varphi}\mathfrak{h}\left(t\right).$
(2.3)

######  Lemma 2.1 ([31, 21])

Let $\mu,\chi>0$ and $\delta>0$. Then

  1. (a)

$\mathcal{I}_{a+}^{\mu\,;\,\varphi}\mathcal{I}_{a+}^{\chi\,;\,\varphi}\mathfrak{h}(t)=\mathcal{I}_{a+}^{\mu+\chi\,;\,\varphi}\mathfrak{h}(t)$

  2. (b)

$\mathcal{I}_{a+}^{\mu\,;\,\varphi}\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\delta-1}=\frac{\Gamma\left(\delta\right)}{\Gamma\left(\mu+\delta\right)}\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\mu+\delta-1}.$

  3. (c)

${}^{H}\mathcal{D}_{a+}^{\mu,\,\nu\,;\,\varphi}\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-1}=0.$

######  Lemma 2.2 ([21])

If $\mu>0$ and $0\leq\alpha<1,$ then $I_{a+}^{\mu\,;\varphi}(\cdot)$ is
bounded from $C_{\alpha\,;\varphi}\left[a,b\right]$ to
$C_{\alpha\,;\varphi}\left[a,b\right].$ In addition, if $\alpha\leq\mu$, then
$I_{a+}^{\mu\,;\varphi}(\cdot)$ is bounded from
$C_{\alpha\,;\varphi}\left[a,b\right]$ to $C\left[a,b\right]$.

######  Lemma 2.3 ([21])

If $\mathfrak{h}\in C^{n}[a,b]$, $n-1<\mu<n$ and $0\leq\nu\leq 1$, then

  1. 1.

$I_{a+}^{\mu\,;\varphi}\text{
}^{H}\mathbb{D}_{a+}^{\mu,\nu\,;\varphi}\mathfrak{h}\left(t\right)=\mathfrak{h}\left(t\right)-\overset{n}{\underset{k=1}{\sum}}\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-k}}{\Gamma\left(\xi-k+1\right)}h_{\varphi}^{\left[n-k\right]}I_{a+}^{\left(1-\nu\right)\left(n-\mu\right)\,;\varphi}h\left(a\right)$
where,
$\mathfrak{h}_{\varphi}^{\left[n-k\right]}\mathfrak{h}(t)=\left(\frac{1}{\varphi^{\prime}(t)}\frac{d}{dt}\right)^{n-k}\mathfrak{h}(t)$.

  2. 2.

${}^{H}\mathbb{D}_{a+}^{\mu,\nu\,;\varphi}I_{a+}^{\mu\,;\varphi}\mathfrak{h}\left(t\right)=\mathfrak{h}\left(t\right).$

######  Theorem 2.4 ([32])

Let $u,$ $v$ be two integrable functions and $g$ be continuous with domain
$\left[a,b\right].$ Let $\varphi\in C^{1}\left[a,b\right]$ an increasing
function such that $\varphi^{\prime}\left(t\right)\neq 0$,
$\forall~{}t\in\left[a,b\right]$. Assume that

  1. 1.

$u$ and $v$ are nonnegative;

  2. 2.

$g$ is nonnegative and nondecreasing.

If

$u\left(t\right)\leq
v\left(t\right)+g\left(t\right)\int_{a}^{t}\varphi^{\prime}\left(s\right)\left(\varphi\left(t\right)-\varphi\left(s\right)\right)^{\mu-1}u\left(s\right)ds,$

then

$u\left(t\right)\leq
v\left(t\right)+\int_{a}^{t}\overset{\infty}{\underset{k=1}{\sum}}\frac{\left[g\left(t\right)\Gamma\left(\alpha\right)\right]^{k}}{\Gamma\left(k\alpha\right)}\varphi^{\prime}\left(s\right)\left[\varphi\left(t\right)-\varphi\left(s\right)\right]^{k\mu-1}v\left(s\right)ds,~{}t\in\left[a,b\right].$
(2.4)

Further, if $v$ is a nondecreasing function on $[a,b]$ then

$u(t)\leq
v(t)\,E_{\mu}\left(g(t)\Gamma(\mu)\left(\varphi(t)-\varphi(a)\right)^{\mu}\right),$

where $E_{\mu}(\cdot)$ is the Mittag-Leffler function of one parameter[33],
defined as

$\mathcal{E}_{\mu}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(k\mu+1)}.$

######  Theorem 2.5 ([34], Krasnoselskii)

Let $\mathcal{M}$ be a closed, convex, and nonempty subset of a Banach space
$\mathcal{X}$, and $P$, $Q$ the operators such that

  1. 1.

$Px+Qy\in\mathcal{M}$ whenever $x,y\in\mathcal{M}$;

  2. 2.

$P$ is a contraction mapping;

  3. 3.

$Q$ is compact and continuous.

Then, there exists ${y}^{\ast}\in\mathcal{M}$ such that
${y}^{\ast}=P{y}^{\ast}+Q{y}^{\ast}.$

##  3 Equivalent Fractional Integral Equation

In this section, we derive equivalent fractional integral equation to the
nonlocal BVP (1.1)-(1.3).

###### Theorem 3.1

Let $1<\mu<2,~{}0\leq\nu\leq 1$ and $h:J^{\prime}\to\mbox{\Bbb R}$ be a
continuous function. Then the nonlocal BVP for $\varphi$-Hilfer FDEs

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)$
$\displaystyle=h(t),~{}~{}~{}t\in J^{\prime}=(a,\,b],~{}$ (3.1) $\displaystyle
y(a)$ $\displaystyle=0,$ (3.2) $\displaystyle y(b)$
$\displaystyle=\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i})$
(3.3)

is equivalent to

$y(t)=\frac{(\varphi(t)-\varphi(a))^{\xi-1}}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]+\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(t),\,t\in
J,$

where $\lambda_{i}\in\mbox{\Bbb R}(i=1,2,\cdots,m)$ are the constants such
that

$\Lambda=\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Gamma(\xi)}-\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\xi+\delta_{i})}(\varphi(\tau_{i})-\varphi(a))^{\xi+\delta_{i}-1}\neq
0.$ (3.4)

Proof: Assume that $y$ is the solution of the nonlocal BVP for
$\varphi$-Hilfer FDEs (3.1)-(3.3). Operating $\varphi$-fractional integral
$\mathcal{I}_{a+}^{\mu\,;\,\varphi}$ on both sides of equation (3.1) and using
Lemma 2.3, we obtain

$y\left(t\right)-\sum_{k=1}^{2}\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-k}}{\Gamma\left(\xi-k+1\right)}y_{\varphi}^{\left[2-k\right]}\mathcal{I}_{a+}^{\left(1-\nu\right)\left(2-\mu\right)\,;\varphi}y\left(a\right)=\mathcal{I}_{a+}^{\mu\,;\varphi}h(t),\,t\in
J.$

But $\left(1-\nu\right)\left(2-\mu\right)=2-\xi$. Therefore,

$\displaystyle y\left(t\right)$
$\displaystyle=\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-1}}{\Gamma\left(\xi\right)}\left(\frac{1}{\varphi^{\prime}(t)}\frac{d}{dt}\right)\mathcal{I}_{a+}^{2-\xi\,;\varphi}y\left(t\right)|_{t=a}$
$\displaystyle\qquad+\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-2}}{\Gamma\left(\xi-1\right)}\mathcal{I}_{a+}^{2-\xi\,;\varphi}y\left(t\right)|_{t=a}+\mathcal{I}_{a+}^{\mu\,;\varphi}h(t)$
$\displaystyle=\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-1}}{\Gamma\left(\xi\right)}\,^{H}\mathcal{D}^{\xi-1,\,\nu\,;\,\varphi}_{a+}y\left(t\right)|_{t=a}$
$\displaystyle\qquad+\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-2}}{\Gamma\left(\xi-1\right)}\mathcal{I}_{a+}^{2-\xi\,;\varphi}y\left(t\right)|_{t=a}+\mathcal{I}_{a+}^{\mu\,;\varphi}h(t).$

Set

$C_{1}=^{H}\mathcal{D}^{\xi-1,\,\nu\,;\,\varphi}_{a+}y\left(t\right)|_{t=a}\quad\text{and}\quad
C_{2}=\mathcal{I}_{a+}^{2-\xi\,;\varphi}y\left(t\right)|_{t=a},\,t\in J.$

Then,

$y\left(t\right)=C_{1}\,\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-1}}{\Gamma\left(\xi\right)}\,+C_{2}\,\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-2}}{\Gamma\left(\xi-1\right)}+\mathcal{I}_{a+}^{\mu\,;\varphi}h(t),\,t\in
J.$ (3.5)

Since $\lim\limits_{t\rightarrow
a}\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-2}=\infty,$ in
the view of boundary condition (3.2), we must have $C_{2}=0.$ In this case
(3.5) becomes

$y\left(t\right)=C_{1}\,\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi-1}}{\Gamma\left(\xi\right)}\,+\mathcal{I}_{a+}^{\mu\,;\varphi}h(t),\,t\in
J.$ (3.6)

Next, to determine the constant $C_{1}$, we utilize the boundary condition
(3.3). Operating $\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}$ on both sides of
equation (3.6), we obtain

$\displaystyle\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(t)$
$\displaystyle=\frac{C_{1}}{\Gamma\left(\xi+\delta_{i}\right)}\,\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi+\delta_{i}-1}+\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(t).$
(3.7)

From (3.3) and (3.7) we have

$\displaystyle y(b)$
$\displaystyle=\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i})$
$\displaystyle=C_{1}\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma\left(\xi+\delta_{i}\right)}\left(\varphi\left(\tau_{i}\right)-\varphi\left(a\right)\right)^{\xi+\delta_{i}-1}+\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i}).$
(3.8)

But from (3.6) and (3), we have

$\displaystyle
C_{1}\,\frac{\left(\varphi\left(b\right)-\varphi\left(a\right)\right)^{\xi-1}}{\Gamma\left(\xi\right)}\,+\mathcal{I}_{a+}^{\mu\,;\varphi}h(b)$
$\displaystyle\qquad=C_{1}\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma\left(\xi+\delta_{i}\right)}\left(\varphi\left(\tau_{i}\right)-\varphi\left(a\right)\right)^{\xi+\delta_{i}-1}+\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i}),$
(3.9)

Since $\lambda_{i}\in\mbox{\Bbb R}(i=1,2,\cdots,m)$ are the constants such
that

$\Lambda=\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Gamma(\xi)}-\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\xi+\delta_{i})}(\varphi(\tau_{i})-\varphi(a))^{\xi+\delta_{i}-1}\neq
0,$

the equation (3) can be written as

$C_{1}=\frac{1}{\Lambda}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\varphi}h(b)\right].$

Thus, equation (3.6) takes the form

$y(t)=\frac{(\varphi(t)-\varphi(a))^{\xi-1}}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]+\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(t),\,t\in
J.$ (3.10)

Which is the desired equivalent fractional integral equation to the problem
(3.1)-(3.3).

Conversely, suppose that $y$ is the solution of the fractional integral
equation (3.10). Operating fractional derivative
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}$ on both sides of equation
(3.10) and using the Lemma 2.1 and Lemma 2.3, we obtain

$\displaystyle^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)$
$\displaystyle=\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}(\varphi(t)-\varphi(a))^{\xi-1}$
$\displaystyle\qquad+^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(t)$
$\displaystyle=h(t),\,t\in J.$ (3.11)

This proves $y$ satisfies the equation (3.1). Next, we prove that $y$ given by
(3.10) verifies the boundary conditions. From (3.10), clearly

$y(a)=0.$ (3.12)

Now we prove that $y$ satisfy the boundary condition (3.3). For each
$i\,(i=1,2,\cdots,m)$, from equation (3.10) we have

$\displaystyle\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(t)$
$\displaystyle=\frac{1}{\Lambda}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{\xi+\delta_{i}-1}}{\Gamma\left(\xi+\delta_{i}\right)}\,+\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(t).$

Therefore,

$\displaystyle\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i})$
$\displaystyle=\frac{1}{\Lambda}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma\left(\xi+\delta_{i}\right)}\left(\varphi\left(\tau_{i}\right)-\varphi\left(a\right)\right)^{\xi+\delta_{i}-1}$
$\displaystyle\qquad+\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i}).$
(3.13)

But from (3.4), we have

$\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\xi+\delta_{i})}(\varphi(\tau_{i})-\varphi(a))^{\xi+\delta_{i}-1}=\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Gamma(\xi)}-\Lambda.$

Thus, equation (3) reduces to

$\displaystyle\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i})$
$\displaystyle=\frac{1}{\Lambda}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]\left(\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Gamma(\xi)}-\Lambda\right)$
$\displaystyle\qquad+\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})$
$\displaystyle=\frac{1}{\Lambda}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Gamma(\xi)}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b).$
(3.14)

Now from (3.10), we have

$y(b)=\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}h(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b)\right]+\mathcal{I}_{a+}^{\mu\,;\,\varphi}h(b).$
(3.15)

From equations (3) and (3.15) we obtain

$y(b)=\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i}).$
(3.16)

From (3), (3.12) and (3.16), it follows that the $y$ defined by (3.10)
satisfies the problem (3.1)-(3.3). $\Box$

###### Theorem 3.2

Let $f:J^{\prime}\times\mbox{\Bbb R}\to\mbox{\Bbb R}$ be a continuous function
such that
$f\left(\cdot,y(\cdot),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(\cdot)\right)\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ for each $y\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$. Then, the nonlocal BVP for
$\varphi$-Hilfer implicit FDEs (1.1)-(1.3) is equivalent to the fractional
integral equation

$y(t)=(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t),$
(3.17)

where $g_{y}(\cdot)\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ satisfies the
functional equation

$g_{y}(t)=f\left(t,(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t),g_{y}(t)\right),\quad
t\in J$ (3.18)

and

$\tilde{A}_{y}=\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}g_{y}(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(b)\right].$
(3.19)

Proof: Assume that $y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ is the solution
of integral equation (3.17). Operating
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}$ on both sides of (3.17) and
using the Lemma 2.1 and Lemma 2.3, we obtain

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)$
$\displaystyle=\tilde{A}_{y}\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}(\varphi(t)-\varphi(a))^{\xi-1}+^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}\,\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t)$
$\displaystyle=g_{y}(t)$
$\displaystyle=f\left(t,(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t),g_{y}(t)\right)$
$\displaystyle=f\left(t,y(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)\right).$

Thus, $y$ satisfies the equation (1.1). The proof of the function $y$ given by
(3.17) satisfies the boundary conditions (1.2) and (1.3) can be completed
similarly as in the proof of Theorem 3.1 with $h(t)$ replaced by $g_{y}(t)$.
$\Box$

##  4 Existence Result

In this section we derive existence result for nonlocal implicit BVP
(1.1)-(1.3).

###### Theorem 4.1

Assume the following hypothesis hold:
$(H1)$ $f:J^{\prime}\times\mbox{\Bbb R}\to\mbox{\Bbb R}$ be a continuous
function such that
$f\left(\cdot,y(\cdot),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(\cdot)\right)\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ for each $y\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ that satisfy the following Lipschitz
type condition

$\left|f(t,x_{1},y_{1})-f(t,x_{2},y_{2})\right|\leq
K\left|x_{1}-x_{2}\right|+L\left|y_{1}-y_{2}\right|,\,t\in J,$

where $x_{1},x_{2},y_{1},y_{2}\in\mbox{\Bbb R},\,K>0$ and $0<L<1.$ Then,
nonlocal BVP for $\varphi$-Hilfer implicit FDEs (1.1)-(1.3) has at least one
solution, provided

$\displaystyle\sigma$
$\displaystyle=\frac{K\Gamma(\xi-1)(\varphi(b)-\varphi(a))^{\mu}}{1-L}\left\\{\frac{(\varphi(b)-\varphi(a))^{\xi-1}}{\Gamma(\xi)\Lambda}\Omega+\frac{1}{\Gamma(\xi+\mu-1)}\right\\}<1,$
(4.1)

where

$\displaystyle\Omega$
$\displaystyle=\sum_{i=1}^{m}\frac{\lambda_{i}(\varphi(b)-\varphi(a))^{\delta_{i}}}{\Gamma(\xi+\mu+\delta_{i}-1)}+\frac{1}{\Gamma(\xi+\mu-1)}$
(4.2)

and $\Lambda$ is defined in equation (3.4).

Proof: In the view of Theorem 3.2, the equivalent fractional integral equation
to the nonlocal BVP for $\varphi$-Hilfer implicit FDEs (1.1)-(1.3) can be
written as operator equation as follows

$y=Py+Qy,\,y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R}),$ (4.3)

where $P$ and $Q$ are the operators defined on
$C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ by

$\displaystyle Py(t)$
$\displaystyle=(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y},\,t\in J\quad\
\text{and}$ $\displaystyle Qy(t)$
$\displaystyle=\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t)=\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}g_{y}(s)ds,\,t\in
J,$

where $g_{y}\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ satisfies the functional
equation

$g_{y}(t)=f\left(t,(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t),g_{y}(t)\right),\quad
t\in J,$

and

$\tilde{A}_{y}=\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}g_{y}(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(b)\right].$

To prove the nonlocal implicit BVP (1.1)-(1.3) has a solution in
$C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ is equivalent to show that the operator
equation (4.3) has a fixed point. Define,

$\displaystyle\zeta$
$\displaystyle=\frac{M\Gamma(\xi-1)(\varphi(t)-\varphi(a))^{\mu}}{1-L}\left\\{\frac{(\varphi(t)-\varphi(a))^{\xi-1}}{\Gamma(\xi)}\Omega+\frac{1}{\Gamma(\xi+\mu-1)}\right\\},$

where

$M=\underset{t\in
J}{\max}\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}f\left(t,\,0,\,0\right)\right|.$

Choose $r$ such that

$r\geq\frac{\zeta}{1-\sigma}$

and consider the closed ball

$B_{r}=\left\\{y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb
R}):\left\|y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb R}\right)}\leq
r\right\\}.$

To obtain the fixed point of the operator equation (4.3), we prove that the
operators $P$ and $Q$ satisfies the conditions of Theorem 2.5 (Krasnoselskii
fixed point theorem). We give the proof in the following steps:
Step 1: $Px+Qy\in B_{r}$ for all $x,y\in B_{r}.$
Let any $x,y\in B_{r}$. Then,
$\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}Px(t)=(\varphi(t)-\varphi(a))\tilde{A}_{x}\in
C(J,\mbox{\Bbb R}).$ This implies, $Px\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb
R})$.

Since $g_{y}\left(\cdot\right)=f(\cdot,y(\cdot),g_{y}(\cdot))\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ for any $y\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R}),0\leq 2-\xi<1$ and
$Qy(t)=\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t)$, by Lemma 2.2 we have
$Qy\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$. Using triangle inequality in the
space $C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ we have

$\left\|Px+Qy\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\left\|Px\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}+\left\|Qy\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}.$ (4.4)

Now, by using hypothesis ($H1$), we have

$\displaystyle\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}Px(t)\right|$
$\displaystyle=\left|(\varphi(t)-\varphi(a))\tilde{A}_{x}\right|$
$\displaystyle=\left|\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}g_{x}(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{x}(b)\right]\right|$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\left|f(s,x(s),g_{x}(s))\right|ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left|f(s,x(s),g_{x}(s))\right|ds\right]$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\times\right.$
$\displaystyle\left.\qquad\qquad\left\\{\left|f(s,x(s),g_{x}(s))-f(s,0,0)\right|+\left|f(s,0,0)\right|\right\\}ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left\\{\left|f(s,x(s),g_{x}(s))-f(s,0,0)\right|+\left|f(s,0,0)\right|\right\\}ds\right]$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\left\\{K\left|x(s)\right|+L\left|g_{x}(s)\right|+\left|f(s,0,0)\right|\right\\}ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left\\{K\left|x(s)\right|+L\left|g_{x}(s)\right|+\left|f(s,0,0)\right|\right\\}ds\right]$
(4.5)

But, by hypothesis ($H1$), we have

$\displaystyle\left|g_{x}(t)\right|$
$\displaystyle\leq\left|f(t,x(t),g_{x}(t))-f(t,0,0)\right|+\left|f(t,0,0)\right|$
$\displaystyle\leq
K\left|x(t)\right|+L\left|g_{x}(t)\right|+\left|f(t,0,0)\right|,\,t\in J.$

This gives

$\left|g_{x}(t)\right|\leq\frac{K}{1-L}\left|x(t)\right|+\frac{1}{1-L}\left|f(t,0,0)\right|,\,t\in
J.$ (4.6)

Using equation (4.6) in equation (4), we get

$\displaystyle\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}Px(t)\right|$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\right.$
$\displaystyle\qquad\left.\times\left\\{K\left|x(s)\right|+L\left\\{\frac{K}{1-L}\left|x(s)\right|+\frac{1}{1-L}\left|f(s,0,0)\right|\right\\}+\left|f(s,0,0)\right|\right\\}ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left\\{K\left|x(s)\right|+L\left\\{\frac{K}{1-L}\left|x(s)\right|+\frac{1}{1-L}\left|f(s,0,0)\right|\right\\}+\left|f(s,0,0)\right|\right\\}ds\right]$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\left\\{\frac{K}{1-L}\left|x(s)\right|+\frac{1}{1-L}\left|f(s,0,0)\right|\right\\}ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left\\{\frac{K}{1-L}\left|x(s)\right|+\frac{1}{1-L}\left|f(s,0,0)\right|\right\\}ds\right]$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\frac{K}{1-L}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}(\varphi(s)-\varphi(a))^{\xi-2}\right.\right.$
$\displaystyle\qquad\left.\left.\times\left|(\varphi(s)-\varphi(a))^{2-\xi}x(s)\right|\right.\right.ds$
$\displaystyle\left.\left.\qquad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}x(s)\right|ds\right\\}\right.$
$\displaystyle\left.\qquad+\frac{1}{1-L}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}f(s,0,0)\right|\right.\right.ds$
$\displaystyle\left.\left.\qquad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}f(s,0,0)\right|ds\right\\}\right]$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\frac{K}{1-L}\left\|x\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu+\delta_{i}-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right.\right.$
$\displaystyle\left.\left.\qquad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right\\}\right.$
$\displaystyle\left.\qquad+\frac{M}{1-L}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu+\delta_{i}-1}(\varphi(s)-\varphi(a))^{\xi-2}\right.\right.ds$
$\displaystyle\left.\left.\qquad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right\\}\right]$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\frac{K}{1-L}\left\|x\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}\Gamma(\xi-1)}{\Gamma(\mu+\delta_{i}+\xi-1)}(\varphi(b)-\varphi(a))^{\mu+\delta_{i}+\xi-2}\right.\right.$
$\displaystyle\left.\left.\qquad+\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(b)-\varphi(a))^{\mu+\xi-2}\right\\}\right.$
$\displaystyle\left.\qquad+\frac{M}{1-L}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}\Gamma(\xi-1)}{\Gamma(\mu+\delta_{i}+\xi-1)}(\varphi(b)-\varphi(a))^{\mu+\delta_{i}+\xi-2}+\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(b)-\varphi(a))^{\mu+\xi-2}\right\\}\right].$

Therefore,

$\displaystyle\left\|Px\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}$ $\displaystyle=\underset{t\in
J}{\max}\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}Px\left(t\right)\right|$
$\displaystyle\leq\frac{(\varphi(b)-\varphi(a))^{\mu+\xi-1}\Gamma(\xi-1)}{\Lambda\Gamma(\xi)}\left[\frac{K\,r}{1-L}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i}+\xi-1)}(\varphi(b)-\varphi(a))^{\delta_{i}}+\frac{1}{\Gamma(\mu+\xi-1)}\right\\}\right.$
$\displaystyle\left.\qquad+\frac{M}{1-L}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i}+\xi-1)}(\varphi(b)-\varphi(a))^{\delta_{i}}+\frac{1}{\Gamma(\mu+\xi-1)}\right\\}\right]$
$\displaystyle\leq\frac{(\varphi(b)-\varphi(a))^{\mu+\xi-1}\Gamma(\xi-1)}{\Lambda\Gamma(\xi)}\frac{(K\,r+M)}{1-L}\Omega=A.$
(4.7)

Further, by using ($H1$) and inequality (4.6), we have

$\displaystyle\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}Qy(t)\right|$
$\displaystyle=\left|\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}g_{y}(s)ds\right|$
$\displaystyle\leq\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left|f(s,y(s),g_{y}(s))\right|ds$
$\displaystyle\leq\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left\\{\left|f(s,y(s),g_{y}(s))-f(s,0,0)\right|+\left|f(s,0,0)\right|\right\\}ds$
$\displaystyle\leq\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left\\{K\left|y(s)\right|+L\left|g_{y}(s)\right|+\left|f(s,0,0)\right|\right\\}ds$
$\displaystyle\leq\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}$
$\displaystyle\qquad\times\left\\{K\left|y(s)\right|+L\left\\{\frac{K}{1-L}\left|y(s)\right|+\frac{1}{1-L}\left|f(s,0,0)\right|\right\\}+\left|f(s,0,0)\right|\right\\}ds$
$\displaystyle\leq\frac{\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left\\{\frac{K}{1-L}\left|y(s)\right|+\frac{1}{1-L}\left|f(s,0,0)\right|\right\\}ds$
$\displaystyle\leq\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\left\\{\frac{K}{(1-L)\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}y(s)\right|ds\right.$
$\displaystyle\left.\qquad+\frac{1}{(1-L)\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}f(s,0,0)\right|ds\right\\}$
$\displaystyle\leq\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\left\\{\frac{K}{1-L}\left\|y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right.$
$\displaystyle\left.\qquad+\frac{M}{1-L}\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right\\}$
$\displaystyle\leq\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\left\\{\frac{K}{1-L}\left\|y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(t)-\varphi(a))^{\mu+\xi-2}\right.$
$\displaystyle\left.\qquad+\frac{M}{1-L}\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(t)-\varphi(a))^{\mu+\xi-2}\right\\}$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))^{\mu}\Gamma(\xi-1)}{(1-L)\Gamma(\mu+\xi-1)}\left\\{K\left\|y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}+M\right\\}.$

Therefore,

$\displaystyle\left\|Qy\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}$ $\displaystyle=\underset{t\in
J}{\max}\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}Qy\left(t\right)\right|$
$\displaystyle\leq\frac{(\varphi(b)-\varphi(a))^{\mu}\Gamma(\xi-1)}{(1-L)\Gamma(\mu+\xi-1)}\left\\{Kr+M\right\\}=B.$
(4.8)

From inequalities (4) and (4) and the definition of $r$ given in (4.4), we
obtain

$\left\|Px+Qy\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb R}\right)}\leq
A+B=r\sigma+\zeta\leq r.$

This proves, $Px+Qy\in B_{r}$ for all $x,y\in B_{r}.$
Step 2: Operator $P$ is contraction on $B_{r}$.
Let any $y_{1},y_{2}\in B_{r}$ and $t\in J$. Then

$\displaystyle\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\left(Py_{1}(t)-Py_{2}(t)\right)\right|$
$\displaystyle\leq\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\left|g_{y_{1}}(s)-g_{y_{2}}(s)\right|ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left|g_{y_{1}}(s)-g_{y_{2}}(s)\right|ds\right].$
(4.9)

But by hypothesis ($H1$), we have

$\displaystyle\left|g_{y_{1}}(t)-g_{y_{2}}(t)\right|$
$\displaystyle=\left|f(t,y_{1}(t),g_{y_{1}}(t))-f(t,y_{2}(t),g_{y_{2}}(t))\right|$
$\displaystyle\leq
K\left|y_{1}(t)-y_{2}(t)\right|+L\left|g_{y_{1}}(t)-g_{y_{2}}(t)\right|,\,t\in
J.$

Thus,

$\left|g_{y_{1}}(t)-g_{y_{2}}(t)\right|\leq\frac{K}{1-L}\left|y_{1}(t)-y_{2}(t)\right|,\,t\in
J.$ (4.10)

Using equation (4.10) in equation (4) we have

$\displaystyle\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\left(Py_{1}(t)-Py_{2}(t)\right)\right|$
$\displaystyle\leq\frac{K}{1-L}\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\right.$
$\displaystyle\left.\qquad\times\left(\varphi\left(s\right)-\varphi\left(a\right)\right)^{\xi-2}\left|\left(\varphi\left(s\right)-\varphi\left(a\right)\right)^{2-\xi}(y_{1}(s)-y_{2}(s))\right|ds\right.$
$\displaystyle\left.\quad+\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left(\varphi\left(s\right)-\varphi\left(a\right)\right)^{\xi-2}\left|\left(\varphi\left(s\right)-\varphi\left(a\right)\right)^{2-\xi}(y_{1}(s)-y_{2}(s))\right|ds\right]$
$\displaystyle\leq\frac{K}{1-L}\frac{(\varphi(t)-\varphi(a))}{\Lambda\Gamma(\xi)}\left\|y_{1}-y_{2}\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\times$
$\displaystyle\qquad\left[\sum_{i=1}^{m}\frac{\lambda_{i}\Gamma(\xi-1)}{\Gamma(\mu+\delta_{i}+\xi-1)}(\varphi(b)-\varphi(a))^{\mu+\delta_{i}+\xi-2}+\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(b)-\varphi(a))^{\mu+\xi-2}\right].$

Therefore,

$\displaystyle\left\|Py_{1}-Py_{2}\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}$ $\displaystyle=\underset{t\in
J}{\max}\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\left(Py_{1}(t)-Py_{2}(t)\right)\right|$
$\displaystyle\leq\left\|y_{1}-y_{2}\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\frac{K\Gamma(\xi-1)}{1-L}\frac{(\varphi(t)-\varphi(a))^{\mu+\xi-1}}{\Lambda\Gamma(\xi)}\Omega$
$\displaystyle\leq\sigma\left\|y_{1}-y_{2}\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}.$

Since $\sigma<1$, $P$ is contraction on $B_{r}.$

Step 3: The operator $Q$ is compact and continuous.
We have already proved that, $Q$ is self mapping on
$C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb R}\right).$ Further, from Step 1 we
have

$\left\|Qy\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\frac{(\varphi(b)-\varphi(a))^{\mu}\Gamma(\xi-1)}{(1-L)\Gamma(\mu+\xi-1)}\left\\{Kr+M\right\\},\,y\in
B_{r}.$

Therefore, $Q(B_{r})=\left\\{Qy:y\in B_{r}\right\\}$ is uniformly bounded.
Next, we prove that $Q(B_{r})$ is equicontinuous. Let any $y\in B_{r}$ and
$t_{1},\,t_{2}\in J$ with $t_{1}>t_{2}$. Then

$\displaystyle\left|Qy(t_{1})-Qy(t_{2})\right|$
$\displaystyle=\left|\frac{1}{\Gamma(\mu)}\int_{a}^{t_{1}}\varphi^{\prime}(s)(\varphi(t_{1})-\varphi(s))^{\mu-1}g_{y}(s)ds-\frac{1}{\Gamma(\mu)}\int_{a}^{t_{2}}\varphi^{\prime}(s)(\varphi(t_{2})-\varphi(s))^{\mu-1}g_{y}(s)ds\right|$
$\displaystyle\leq\left|\frac{1}{\Gamma(\mu)}\int_{a}^{t_{1}}\varphi^{\prime}(s)(\varphi(t_{1})-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}f(s,y(s),g_{y}(s))\right|ds\right.$
$\displaystyle\left.~{}-\frac{1}{\Gamma(\mu)}\int_{a}^{t_{2}}\varphi^{\prime}(s)(\varphi(t_{2})-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}\left|(\varphi(s)-\varphi(a))^{2-\xi}f(s,y(s),g_{y}(s))\right|ds\right|.$

Since $g_{y}\left(\cdot\right)=f(\cdot,y(\cdot),g_{y}(\cdot))\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ for any $y\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$,
$(\varphi(\cdot)-\varphi(a))^{2-\xi}f(\cdot,y(\cdot),g_{y}(\cdot))$ is
continuous on $J=[a,\,b]$. Therefore, there exists $\mathfrak{K}\in\mbox{\Bbb
R}$ such that

$\left|(\varphi(t)-\varphi(a))^{2-\xi}f(t,y(t),g_{y}(t))\right|\leq\mathfrak{K},~{}\text{for
all}~{}t\in J.$

Thus,

$\displaystyle\left|Qy(t_{1})-Qy(t_{2})\right|$
$\displaystyle\leq\left|\frac{\mathfrak{K}}{\Gamma(\mu)}\int_{a}^{t_{1}}\varphi^{\prime}(s)(\varphi(t_{1})-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right.$
$\displaystyle\left.~{}-\frac{\mathfrak{K}}{\Gamma(\mu)}\int_{a}^{t_{2}}\varphi^{\prime}(s)(\varphi(t_{2})-\varphi(s))^{\mu-1}(\varphi(s)-\varphi(a))^{\xi-2}ds\right|$
$\displaystyle\leq\left|\mathfrak{K}\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(t_{1})-\varphi(a))^{\mu+\xi-2}-\mathfrak{K}\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}(\varphi(t_{2})-\varphi(a))^{\mu+\xi-2}\right|.$

Hence,

$\left|Qy(t_{1})-Qy(t_{2})\right|\leq\mathfrak{K}\frac{\Gamma(\xi-1)}{\Gamma(\mu+\xi-1)}\left|(\varphi(t_{1})-\varphi(a))^{\mu+\xi-2}-(\varphi(t_{2})-\varphi(a))^{\mu+\xi-2}\right|.$

Using continuity of $\varphi$, we observe that
$\left|Qy(t_{1})-Qy(t_{2})\right|\to 0$ as $\left|t_{1}-t_{2}\right|\to 0$.
This proves $Q(B_{r})$ is equicontinuous. Thus, by Arzel$\grave{a}$-Ascoli
Theorem, $Q(B_{r})$ is relatively compact. We proved that, operator $Q$ is
compact.

Since $P$ and $Q$ satisfy all the conditions of Krasnoselskii fixed point
theorem (Theorem 2.5), the operator equation defined in the equation (4.3),
has at least one fixed point in the space
$C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb R}\right)$ which is the solution of
the nonlocal BVP for implicit FDEs (1.1)-(1.3).

$\Box$

##  5 Ulam Stability Results

We begin with definitions regarding Ulam stabilities. To discuss the these
stabilities for the problem (1.1), we follow the approach of [30].

###### Definition 5.1

The equation (1.1) is Ulam-Hyers stable if there exists a real number
$C_{f}>0$, such that for each $\epsilon>0$ and for each solution $z\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ of the inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)-f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))\right|\leq\epsilon,\,\,t\in
J,$ (5.1)

there exists solution $y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ of the
equation (1.1) with

$\left\|z-y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb R}\right)}\leq
C_{f}\,\epsilon,\quad t\in J.$

###### Definition 5.2

The equation (1.1) is generalized Ulam-Hyers stable if there exists a function
$\phi_{f}\in C(\mbox{\Bbb R}_{+},\mbox{\Bbb R}_{+})$, $\phi_{f}(0)=0$, such
that for each solution $z\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ of the
inequality (5.1) there exists solution $y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb
R})$ of the equation (1.1) with

$\left\|z-y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\phi_{f}(\epsilon),\quad t\in J.$

###### Definition 5.3

The equation (1.1) is Ulam-Hyers-Rassias stable, with respect to the $\chi\in
C(J,\,\mbox{\Bbb R}_{+})$, if there exists a $C_{f,\chi}>0$ such that for each
$\epsilon>0$ and for each solution $z\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$
of the inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)-f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))\right|\leq\epsilon\,\chi(t),\quad
t\in J,$ (5.2)

there exists solution $y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ of the
equation (1.1) with

$\left|\left(\varphi(t)-\varphi(a)\right)^{2-\xi}\left(z(t)-y(t)\right)\right|\leq\,\epsilon\,C_{f,\chi}\,\chi(t),\quad
t\in J.$

###### Definition 5.4

The equation (1.1) is generalized Ulam-Hyers-Rassias stable, with respect to
the $\chi\in C(J,\,\mbox{\Bbb R}_{+})$, if there exists a $C_{f,\chi}>0$ such
that for each solution $z\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ of the
inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)-f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))\right|\leq\,\chi(t),\quad
t\in J,$ (5.3)

there exists solution $y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ of the
equation (1.1) with

$\left|\left(\varphi(t)-\varphi(a)\right)^{2-\xi}\left(z(t)-y(t)\right)\right|\leq\,C_{f,\chi}\,\chi(t),\quad
t\in J.$

###### Remark 5.1

A function $z\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ is solution of the
inequality (5.1) if and only if there exists a function $w\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ (which depend on $z$) such that

  1. 1.

$\left|w(t)\right|\leq\epsilon$.

  2. 2.

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)=f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))+w(t)$.

######  Theorem 5.2 (Ulam-Hyers stability)

Assume that the hypothesis $(H1)$ hold. Then, the equation (1.1) is Ulam-Hyers
stable provided the condition (4.1) hold.

Proof: Let any $\epsilon>0$. Let $z\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$
be any solution of the inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)-f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))\right|\leq\epsilon,\quad
t\in J.$ (5.4)

Then, there exists $w\in C(J,\,\mbox{\Bbb R})$ such that

$^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)=f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))+w(t),$
(5.5)

and $\left|w(t)\right|\leq\epsilon,\,t\in J.$ In the view of the Theorem 3.2.

$z(t)=(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{z}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{z}(t)+\mathcal{I}_{a+}^{\mu\,;\,\varphi}w(t)$
(5.6)

is the solution of the equation (5.5), where $g_{z}(\cdot)\in
C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ satisfies the functional equation

$g_{z}(t)=f\left(t,(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{z}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{z}(t)+\mathcal{I}_{a+}^{\mu\,;\,\varphi}w(t),g_{z}(t)\right),\quad
t\in J,$

and

$\tilde{A}_{z}=\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}g_{z}(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{z}(b)\right].$

From (5.6), we have

$\left|z(t)-(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{z}-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{z}(t)\right|\leq\mathcal{I}_{a+}^{\mu\,;\,\varphi}\left|w(t)\right|\leq\epsilon\,\,\frac{(\varphi(t)-\varphi(a))^{\mu}}{\Gamma(\mu+1)}.$
(5.7)

Let $y\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb R})$ be solution of the problem

$\begin{cases}{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)=f(t,y(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}y(t)),\\\
y(a)=z(a),\qquad y(b)=z(b),\end{cases}$ (5.8)

where
$y(b)=\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}y(\tau_{i})$
and
$z(b)=\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\delta_{i}\,;\,\varphi}z(\tau_{i})$.
By Theorem 3.2, the equivalent fractional integral equation to (5.8) is

$y(t)=(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t),$
(5.9)

where, $g_{y}$ satisfies functional equation

$g_{y}(t)=f\left(t,(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(t),g_{y}(t)\right),\quad
t\in J,$

and

$\tilde{A}_{y}=\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}g_{y}(\tau_{i})-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{y}(b)\right].$

Now,

$\displaystyle\left|\tilde{A}_{y}-\tilde{A}_{z}\right|$
$\displaystyle=\left|\left\\{\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}g_{y}(s)ds\right.\right.\right.$
$\displaystyle\left.\left.\left.\quad-\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}g_{y}(s)ds\right]\right.\right\\}$
$\displaystyle-\left.\left\\{\frac{1}{\Lambda\Gamma(\xi)}\left[\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}g_{z}(s)ds\right.\right.\right.$
$\displaystyle\left.\left.\left.\quad-\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}g_{z}(s)ds\right]\right\\}\right|$
$\displaystyle\leq\frac{1}{\Lambda\Gamma(\xi)}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\left|g_{y}(s)-g_{z}(s)\right|ds\right.$
$\displaystyle\left.\quad-\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left|g_{y}(s)-g_{z}(s)\right|ds\right\\}.$
(5.10)

By hypothesis ($H1$), we obtain

$\displaystyle\left|g_{y}(t)-g_{z}(t)\right|$
$\displaystyle=\left|f(t,y(t),g_{y}(t))-f(t,z(t),g_{z}(t))\right|$
$\displaystyle\leq K\left|y(t)-z(t)\right|+L\left|g_{y}(t)-g_{z}(t)\right|.$

This gives,

$\left|g_{y}(t)-g_{z}(t)\right|\leq\frac{K}{1-L}\left|y(t)-z(t)\right|.$
(5.11)

Using equation (5.11) in equation (5) we get

$\displaystyle\left|\tilde{A}_{y}-\tilde{A}_{z}\right|$
$\displaystyle\leq\frac{K}{1-L}\frac{1}{\Lambda\Gamma(\xi)}\left\\{\sum_{i=1}^{m}\frac{\lambda_{i}}{\Gamma(\mu+\delta_{i})}\int_{a}^{\tau_{i}}\varphi^{\prime}(s)(\varphi(\tau_{i})-\varphi(s))^{\mu+\delta_{i}-1}\left|y(s)-z(s)\right|ds\right.$
$\displaystyle\left.\quad-\frac{1}{\Gamma(\mu)}\int_{a}^{b}\varphi^{\prime}(s)(\varphi(b)-\varphi(s))^{\mu-1}\left|y(s)-z(s)\right|ds\right\\}$
$\displaystyle\leq\frac{K}{1-L}\frac{1}{\Lambda\Gamma(\xi)}\left\\{\sum_{i=1}^{m}\lambda_{i}\,\mathcal{I}_{a+}^{\mu+\delta_{i}\,;\,\varphi}\left|y(\tau_{i})-z(\tau_{i})\right|+\mathcal{I}_{a+}^{\mu\,;\,\varphi}\left|y(b)-z(b)\right|\right\\}.$
(5.12)

Since $y(b)=z(b)$, we must have $y(\tau_{i})=z(\tau_{i})(i=1,2,\cdots,m)$.
Therefore, from inequality (5), we obtain $\tilde{A}_{y}=\tilde{A}_{z}$.

Using equations (5.7) and (5.11), we have

$\displaystyle\left|z(t)-y(t)\right|$
$\displaystyle=\left|z(t)-\left[(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{y}+\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}g_{y}(s)ds\right]\right|$
$\displaystyle\leq\left|z(t)-(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{z}-\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}g_{z}(s)ds\right|$
$\displaystyle~{}~{}+\left|\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}g_{z}(s)ds-\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}g_{y}(s)ds\right|$
$\displaystyle\leq\epsilon\,\,\frac{(\varphi(b)-\varphi(a))^{\mu}}{\Gamma(\mu+1)}+\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left|g_{z}(s)-g_{y}(s)\right|ds$
$\displaystyle\leq\epsilon\,\,\frac{(\varphi(b)-\varphi(a))^{\mu}}{\Gamma(\mu+1)}+\frac{K}{1-L}\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left|z(s)-y(s)\right|ds.$

Applying Gronwall inequality(Theorem 2.4) with
$u(t)=\left|z(t)-y(t)\right|,\,v(t)=\epsilon\,\,\frac{(\varphi(b)-\varphi(a))^{\mu}}{\Gamma(\mu+1)}$
and $g(t)=\frac{K}{(1-L)\Gamma(\mu)}$, we obtain

$\left|z(t)-y(t)\right|\leq\epsilon\,\,\frac{(\varphi(b)-\varphi(a))^{\mu}}{\Gamma(\mu+1)}E_{\mu}\left(\frac{K}{1-L}\left(\varphi(t)-\varphi(a)\right)^{\mu}\right),\quad
t\in J.$

Therefore,

$\displaystyle\left\|z-y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}$ $\displaystyle=\underset{t\in
J}{\max}\left|\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}(z(t)-y(t))\right|$
$\displaystyle\leq\left(\varphi\left(t\right)-\varphi\left(a\right)\right)^{2-\xi}\epsilon\,\,\frac{(\varphi(b)-\varphi(a))^{\mu}}{\Gamma(\mu+1)}E_{\mu}\left(\frac{K}{1-L}\left(\varphi(t)-\varphi(a)\right)^{\mu}\right)$
$\displaystyle=C_{f}\,\epsilon,$ (5.13)

where
$C_{f}:=\frac{(\varphi(b)-\varphi(a))^{\mu+2-\xi}}{\Gamma(\mu+1)}E_{\mu}\left(\frac{K}{1-L}\left(\varphi(b)-\varphi(a)\right)^{\mu}\right)$.
Thus, the equation (1.1) is Ulam-Hyers stable. $\Box$

###### Remark 5.3

Define $\phi_{f}:\mbox{\Bbb R}_{+}\to\mbox{\Bbb R}_{+}$ by
$\phi_{f}(\epsilon)=C_{f}\,\epsilon$. Then, $\phi_{f}\in C(\mbox{\Bbb
R}_{+},\,\mbox{\Bbb R}_{+})$ and $\phi_{f}(0)=0$. Then (5) can be written as

$\left\|z-y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\phi_{f}(\epsilon).$

Thus, the equation (1.1) is generalized Ulam-Hyers stable.

######  Theorem 5.4 (Ulam-Hyers-Rassias stability)

Assume that the hypothesis $(H1)$ hold. Let there exists non decreasing
function such $\chi\in C(J,\,\mbox{\Bbb R}_{+})$ and $K^{*}>0$ such that

$\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\chi(s)ds\leq
K^{*}\,\chi(t),\quad\text{for all}\,\,t\in J.$ (5.14)

Then, the implicit FDE (1.1) is Ulam-Hyers-Rassias stable provided the
condition (4.1) hold.

Proof: Let any $\epsilon>0$ and let $z\in C_{2-\xi;\,\varphi}(J,\mbox{\Bbb
R})$ be any solution of the inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t)-f(t,z(t),^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{a+}z(t))\right|\leq\epsilon\,\chi(t),\,\,t\in
J.$ (5.15)

Then, proceeding as in the proof of Theorem 5.2, we obtain

$\left|z(t)-(\varphi(t)-\varphi(a))^{\xi-1}\tilde{A}_{z}-\mathcal{I}_{a+}^{\mu\,;\,\varphi}g_{z}(t)\right|\leq\mathcal{I}_{a+}^{\mu\,;\,\varphi}\left|w(t)\right|\leq\epsilon\,\,\mathcal{I}_{a+}^{\mu\,;\,\varphi}\chi(t)\leq\epsilon\,K^{*}\chi(t),\,\,t\in
J.$ (5.16)

Taking $y\in C_{2-\xi}(J,\,\mbox{\Bbb R})$ be any solution of the problem
(5.8), and following similar steps as in the proof of Theorem 5.2, we obtain

$\left|z(t)-y(t)\right|\leq\epsilon\,K^{*}\chi(t)+\frac{K}{1-L}\frac{1}{\Gamma(\mu)}\int_{a}^{t}\varphi^{\prime}(s)(\varphi(t)-\varphi(s))^{\mu-1}\left|z(s)-y(s)\right|ds,\,\,t\in
J.$

By applying Gronwall inequality (Theorem 2.4), we get

$\left|z(t)-y(t)\right|\leq\epsilon\,K^{*}\chi(t)E_{\mu}\left(\frac{K}{1-L}\left(\varphi(t)-\varphi(a)\right)^{\mu}\right),\,\,t\in
J,$

which gives,

$\left|\left(\varphi(t)-\varphi(t)\right)^{2-\xi}\left(z(t)-y(t)\right)\right|\leq
C_{f,\chi}\,\epsilon\,\chi(t),\,\,t\in J,$ (5.17)

where
$C_{f,\chi}:=K^{*}(\varphi(b)-\varphi(a))^{2-\xi}E_{\mu}\left(\frac{K}{1-L}\left(\varphi(b)-\varphi(a)\right)^{\mu}\right)$.
This proves, equation (1.1) is Ulam-Hyers-Rassias stable. $\Box$

###### Remark 5.5

The proof of the equation (1.1) is generalized Ulam-Hyers-Rassias stable,
follows by taking $\epsilon=1$, in the inequality (5.17).

##  6 Example

###### Example 6.1

Consider the following nonlocal BVP for implicit FDE involving
$\varphi$-Hilfer fractional derivative

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{0+}y(t)$ $\displaystyle=\frac{\cos
t}{10\,e^{t+1}}\left[\sin
y(t)+^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{0+}y(t)\right],~{}t\in(0,\,1],~{}1<\mu<2,~{}0\leq\nu\leq
1,~{}$ (6.1) $\displaystyle y(0)$ $\displaystyle=0,$ (6.2) $\displaystyle
y(1)$
$\displaystyle=\frac{10}{7}\,\mathcal{I}_{0+}^{\frac{4}{5}\,;\,\varphi}y\left(\frac{1}{3}\right)+\frac{13}{6}\,\mathcal{I}_{0+}^{\frac{8}{3}\,;\,\varphi}y\left(\frac{1}{2}\right).\qquad$
(6.3)

Define $f:(0,\,1]\times\mbox{\Bbb R}^{2}\rightarrow\mbox{\Bbb R}$ by

$f(t,\,u,\,v)=\frac{\cos t}{10\,e^{t+1}}\left[\sin u+v\right].$

Then, for any $u_{i},v_{i}\in\mbox{\Bbb R}(i=1,2)$ and $t\in(0,\,1]$ we have

$\displaystyle\left|f(t,\,u_{1},\,v_{1})-f(t,\,u_{2},\,v_{2})\right|$
$\displaystyle\leq\frac{1}{10\,e}\left\\{\left|\sin u_{1}-\sin
u_{2}\right|+\left|v_{1}-v_{2}\right|\right\\}.$

Without loss of generality we may take $u_{1}<u_{2}$. Then, by Mean value
theorem, $\exists\,\gamma\in(u_{1},u_{2})$ such that, $\sin u_{1}-\sin
u_{2}=(u_{1}-u_{2})\cos\gamma.$ This gives, $\left|\sin u_{1}-\sin
u_{2}\right|\leq\left|u_{1}-u_{2}\right|.$ Thus,

$\left|f(t,\,u_{1},\,v_{1})-f(t,\,u_{2},\,v_{2})\right|\leq\frac{1}{10\,e}\left\\{\left|u_{1}-u_{2}\right|+\left|v_{1}-v_{2}\right|\right\\}.$

This proves $f$ satisfies the Lipschitz type condition in hypothesis $(H1)$
with $K=L=\frac{1}{10\,e}=0.0368.$ Further compering the problem (6.1)-(6.3)
to the problem (1.1)-(1.3) we have
$a=0,\,b=1,\,\lambda_{1}=\frac{10}{7},\,\lambda_{2}=\frac{13}{6},\,\delta_{1}=\frac{4}{5},\,\delta_{2}=\frac{8}{3},\,\tau_{1}=\frac{1}{3},\,\tau_{2}=\frac{1}{2}$.
With these constants the equations (3.4), (4.1) and (4.2) becomes

$\displaystyle\sigma$
$\displaystyle=\frac{0.0368\,\Gamma(\xi-1)(\varphi(1)-\varphi(0))^{\mu}}{1-0.0368}\left\\{\frac{(\varphi(1)-\varphi(0))^{\xi-1}}{\Lambda\Gamma(\xi)}\Omega+\frac{1}{\Gamma(\xi+\mu-1)}\right\\},$
(6.4)

where

$\displaystyle\Lambda$
$\displaystyle=\frac{(\varphi(1)-\varphi(0))^{\xi-1}}{\Gamma(\xi)}-\left[\frac{10}{7}\frac{(\varphi(\frac{1}{3})-\varphi(0))^{\xi-1+\frac{4}{5}}}{\Gamma(\xi+\frac{4}{5})}+\frac{13}{6}\frac{(\varphi(\frac{1}{2})-\varphi(0))^{\xi-1+\frac{8}{3}}}{\Gamma(\xi+\frac{8}{3})}\right]$
(6.5)

and

$\displaystyle\Omega=\frac{10}{7}\frac{(\varphi(1)-\varphi(0))^{\frac{4}{5}}}{\Gamma(\xi-1+\mu+\frac{4}{5})}+\frac{13}{6}\frac{(\varphi(1)-\varphi(0))^{\frac{8}{3}}}{\Gamma(\xi-1+\mu+\frac{8}{3})}+\frac{1}{\Gamma(\xi+\mu-1)}.$
(6.6)

Since all the assumptions of the Theorem 4.1 satisfied, the problem
(6.1)-(6.3) has unique solution provided $\sigma<1.$ Further, by Theorem 5.2,
for any solution $z\in C_{2-\xi;\,\varphi}([0,1],\mbox{\Bbb R})$ of the
inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{0+}z(t)-\frac{\cos
t}{10\,e^{t+1}}\left[\sin
z(t)+^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{0+}z(t)\right]\right|\leq\epsilon,\,\,t\in[0,1],$

there exists a unique solution $y\in C_{2-\xi;\,\varphi}([0,1],\mbox{\Bbb R})$
of the problem (6.1)-(6.3) such that

$\left\|z-y\right\|_{C_{2-\xi;\,\varphi}\left(J,\,\mbox{\Bbb R}\right)}\leq
C_{f}\,\epsilon,$

where
$C_{f}=\frac{(\varphi(1)-\varphi(0))^{\mu+2-\xi}}{\Gamma(\mu+1)}E_{\mu}\left(\frac{0.0368}{1-0.0368}\left(\varphi(1)-\varphi(0)\right)^{\mu}\right)$.

Define

$\chi(t)=E_{\mu}\left(\frac{1}{9}\left(\varphi(t)-\varphi(0)\right)^{\mu}\right),\,t\in[0,1].$
(6.7)

Then, $\chi:[0,\,1]\rightarrow\mbox{\Bbb R}$ is continuous non-decreasing
function such that

$\mathcal{I}_{0+}^{\mu\,;\,\varphi}E_{\mu}\left(\frac{1}{9}\left(\varphi(t)-\varphi(0)\right)^{\mu}\right)=\frac{1}{9}\left[E_{\mu}\left(\frac{1}{9}\left(\varphi(t)-\varphi(0)\right)^{\mu}\right)-1\right]\leq\frac{1}{9}E_{\mu}\left(\frac{1}{9}\left(\varphi(t)-\varphi(0)\right)^{\mu}\right).$

This proves $\chi$ satisfies condition (5.14) with $K^{*}=\frac{1}{9}$.
Therefore, by Theorem 5.4, for each solution $z\in
C_{2-\xi;\,\varphi}([0,1],\mbox{\Bbb R})$ of the inequality

$\left|{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{0+}z(t)-\frac{\cos
t}{10\,e^{t+1}}\left[\sin
z(t)+^{H}\mathcal{D}^{\mu,\,\nu\,;\,\varphi}_{0+}z(t)\right]\right|\leq\epsilon
E_{\mu}\left(\frac{1}{9}\left(\varphi(t)-\varphi(0)\right)^{\mu}\right),\,\,t\in[0,1],$

there exists a unique solution $y\in C_{2-\xi;\,\varphi}([0,1],\mbox{\Bbb R})$
of the problem (6.1)-(6.3) such that

$\left|\left(\varphi(t)-\varphi(a)\right)^{2-\xi}\left(z(t)-y(t)\right)\right|\leq\epsilon
C_{f,\,\chi}\,E_{\mu}\left(\frac{1}{9}\left(\varphi(t)-\varphi(0)\right)^{\mu}\right),$

where
$C_{f,\,\chi}=\frac{1}{9}(\varphi(1)-\varphi(0))^{2-\xi}E_{\mu}\left(\frac{0.0368}{1-0.0368}\left(\varphi(1)-\varphi(0)\right)^{\mu}\right)$.

A particular case: For $\mu=\frac{3}{2},\,\nu=1$ and $\varphi(t)=t$, the
problem (6.1)-(6.3) reduces to the following problem

${}^{C}\mathcal{D}^{\frac{3}{2}}_{0+}y(t)$ $\displaystyle=\frac{\cos
t}{10\,e^{t+1}}\left[\sin
y(t)+^{C}\mathcal{D}^{\frac{3}{2}}_{0+}y(t)\right],~{}$ (6.8) $\displaystyle
y(0)$ $\displaystyle=0,$ (6.9) $\displaystyle y(1)$
$\displaystyle=\frac{10}{7}\,\mathcal{I}_{0+}^{\frac{4}{5}}y\left(\frac{1}{3}\right)+\frac{13}{6}\,\mathcal{I}_{0+}^{\frac{8}{3}}y\left(\frac{1}{2}\right),\qquad$
(6.10)

which is the nonlocal BVP for implicit FDEs involving Caputo fractional
derivative. In this case $\xi=\mu+\nu(2-\mu)=2$. Further putting the values of
the constants $\xi,\mu$ with $\varphi(t)=t$ in (6.4), (6.5) and (6.6) has the
following values
$\Lambda=0.87045,\Omega=1.35464\,\,\text{and}\,\,\sigma=0.0881987<1.$ It can
be seen that all the assumptions of the Theorem 4.1 and Theorem 5.2 are
satisfied. Therefore by Theorem 4.1, the Caputo nonlocal implicit BVP
(6.8)-(6.10) has at least one solution. Further, for any solution $z\in
C\left([0,\,1],\mbox{\Bbb R}\right)=\mathcal{C}$ of the inequality

$\left|{}^{C}\mathcal{D}^{\frac{3}{2}}_{0+}z(t)-\frac{\cos
t}{10\,e^{t+1}}\left[\sin
z(t)+^{C}\mathcal{D}^{\frac{3}{2}}_{0+}z(t)\right]\right|\leq\epsilon,\,\,t\in[0,1],$

there is solution $y\in C\left([0,\,1],\mbox{\Bbb R}\right)=\mathcal{C}$ of
problem (6.8)-(6.10) such that

$\left\|z-y\right\|_{\mathcal{C}}\leq C_{f}\,\epsilon,$

where $\left\|\cdot\right\|_{\mathcal{C}}$ is supremum norm on $\mathcal{C}$
and the problem (6.8) is Ulam-Hyers stable. Further, for
$\varphi(t)=t,\mu=\frac{3}{2}$ from equation (6.7) we have
$\chi(t)=E_{\frac{3}{2}}\left(\frac{1}{9}t^{\frac{3}{2}}\right)$ which satisfy

$\mathcal{I}_{0+}^{\frac{3}{2}}E_{\frac{3}{2}}\left(\frac{1}{9}t^{\frac{3}{2}}\right)\leq\frac{1}{9}E_{\frac{3}{2}}\left(\frac{1}{9}t^{\frac{3}{2}}\right).$

Therefore, by Theorem 5.4, for each solution $z\in\mathcal{C}$ of the
inequality

$\left|{}^{C}\mathcal{D}^{\frac{3}{2}}_{0+}z(t)-\frac{\cos
t}{10\,e^{t+1}}\left[\sin
z(t)+^{C}\mathcal{D}^{\frac{3}{2}}_{0+}z(t)\right]\right|\leq\epsilon\chi(t),\,\,t\in[0,1],$

there is solution $y\in\mathcal{C}$ of problem (6.8)-(6.10) such that

$\left|z(t)-y(t)\right|\leq\epsilon\,C_{f,\,\chi}\,\chi(t),$

and hence (6.8) is Ulam-Hyers-Rassias stable.

## Acknowledgment

The second author acknowledges the Science and Engineering Research Board
(SERB), New Delhi, India for the Research Grant (Ref: File no.
EEQ/2018/000407).

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