On the Nonlinear $\Psi$-Hilfer Fractional Differential Equations

Kishor D. Kucche 1

<EMAIL_ADDRESS>
Ashwini D. Mali 2

<EMAIL_ADDRESS>
J. Vanterler da C. Sousa3

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

3 Department of Applied Mathematics, Imecc-Unicamp, 13083-859, Campinas, SP,
Brazil.

###### Abstract

We consider the nonlinear Cauchy problem for $\Psi$\- Hilfer fractional
differential equations and investigate the existence, interval of existence
and uniqueness of solution in the weighted space of functions. The continuous
dependence of solutions on initial conditions is proved via Weissinger fixed
point theorem. Picard’s successive approximation method has been developed to
solve nonlinear Cauchy problem for differential equations with $\Psi$\- Hilfer
fractional derivative and an estimation have been obtained for the error
bound. Further, by Picard’s successive approximation, we derive the
representation formula for the solution of linear Cauchy problem for
$\Psi$-Hilfer fractional differential equation with constant coefficient and
variable coefficient in terms of Mittag-Leffler function and Generalized
(Kilbas–Saigo) Mittag-Leffler function.

Key words: $\Psi$–Hilfer fractional derivative; Existence and uniqueness;
Continuous dependence of solution; Successive approximations; Mittag-Leffler
function, Generalized Mittag-Leffler function.
2010 Mathematics Subject Classification: 26A33, 34A12, 33E12

##  1 Introduction

During the last decade, fractional differential equations(FDEs) [1, 2, 3]
appeared as rich and beautiful field of research due to their applications to
the physical and life sciences.The theoretical development of FDEs in the
classical Riemann-Liouville’s or Caputo sense have been excellently given in
[1, 2, 3, 4, 5, 6]. For important and interesting works on existence,
uniqueness, data dependence and heterogeneous qualitative properties of
solution of classical FDEs, we propose the work of Lakshmikantham et al.[6, 7,
8], Gejji et al.[9, 10], Kai et al.[11], Trujillo et al.[12], Benchora et
al.[13], and the references cited therein. Other interesting works on these
aspect can be found in [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26].

The fundamental concept and properties of integrals and derivatives of
fractional order of a function with respect to the another function viz.
$\Psi$-Riemann-Liouville integral and derivative have been introduced in [2,
Chapter 2]. Following the similar approach, Almeida [27] introduced
$\Psi$-Caputo fractional derivative and investigated the interesting
properties of this operator and extended few preceding study concerned with
the Caputo and the Caputo–Hadamard derivative operators.

On the other hand, Hilfer [28] introduced a fractional derivative
$\mathcal{D}_{a+}^{\eta,\nu}(\cdot)$ having two parameters
$\eta\in(n-1,n),n\in\mathbb{N}$ and $\nu~{}(0\leq\nu\leq 1)$ which in specific
gives the Riemann–Liouville and the Caputo derivative operator. Furati et al.
[29] analyzed nonlinear FDEs with Hilfer derivative operator. Sousa and
Oliveira [30, 31] introduced a new definition of fractional derivative viz.
$\Psi$–Hilfer fractional derivative and investigated its important properties.
Further, it is proved that the $\Psi$–Hilfer derivative is the generalization
of many existing fractional derivative operators. In [31], Sousa and Oliveira
established generalized Gronwall inequality through the fractional integral
with respect to another function and studied the existence, uniqueness and
data dependence of solution of Cauchy problem with $\Psi$–Hilfer fractional
derivative. For the study relating to existence, uniqueness and stability of
different sorts of FDEs involving $\psi$-Hilfer fractional derivative
operator, we refer to the work of Sousa et al.[32, 33, 34] and the references
given there in.

Motivated by the work of [11, 30, 31] in this paper, existence along with the
interval of existence, uniqueness and continuous dependence of solutions on
initial condition have examined for the nonlinear $\Psi$-Hilfer FDEs
structured as:

${}^{H}\mathcal{D}^{\eta,\,\nu\,;\,\Psi}_{a+}y(t)$
$\displaystyle=f(t,y(t)),~{}~{}~{}t\in\Lambda=[a,a+\xi],~{}~{}\xi>0,~{}0<\eta<1,~{}0\leq\nu\leq
1,~{}$ (1.1) $\displaystyle\mathcal{I}_{a+}^{1-\zeta\,;\,\Psi}y(a)$
$\displaystyle=y_{a}\in\mbox{\Bbb R},\qquad\zeta=\eta+\nu\left(1-\eta\right),$
(1.2)

where ${}^{H}\mathcal{D}^{\eta,\nu;\,\Psi}_{a+}(\cdot)$ is the $\Psi$-Hilfer
derivative of order $\eta$ and type $\nu$,
$\mathcal{I}_{a+}^{1-\zeta;\,\Psi}(\cdot)$ is $\Psi$-Riemann–Liouville
integral of order $1-\zeta$ and $f:\Lambda\times\mbox{\Bbb R}\to\mbox{\Bbb R}$
is an appropriate function.

Further, by Picard’s successive approximation method we derive the existence
along with uniqueness of solution for Cauchy type problem (1.1)-(1.2). We have
established bound for the error between approximated solution $y_{n}$ and
exact solution $y$ and proved that the difference is approaches to zero when
$n$ is very large. The development of Picard’s successive approximation
technique is utilized to obtain representation formula for the solution of
linear Cauchy problem with constant coefficient

$\displaystyle{}^{H}\mathcal{D}_{a+}^{\eta,\,\nu\,;\,\Psi}y(t)-\lambda y(t)$
$\displaystyle=f(t),~{}\lambda\in\mbox{\Bbb R},~{}0<\eta<1,~{}0\leq\nu\leq
1,~{}t\in\Delta=[a,b].$ (1.3)
$\displaystyle\mathcal{I}_{a+}^{1-\zeta\,;\,\Psi}y(a)$
$\displaystyle=y_{a}\in\mbox{\Bbb R},~{}~{}\zeta=\eta+\nu(1-\eta)$ (1.4)

in form of Mittag–Leffler function, where as the representation formula for
the solution of linear Cauchy problem with variable coefficient

$\displaystyle{}^{H}\mathcal{D}_{a+}^{\eta,\,\nu\,;\,\Psi}y(t)-\lambda[\Psi(t)-\Psi(a)]^{\mu-1}y(t)$
$\displaystyle=0,~{}0<\eta<1,~{}0\leq\nu\leq
1,~{}~{}\mu>1-\eta,~{}t\in\Delta,$ (1.5)
$\displaystyle\mathcal{I}_{a+}^{1-\zeta\,;\,\Psi}y(a)$
$\displaystyle=y_{a}\in\mbox{\Bbb R},~{}~{}\zeta=\eta+\nu(1-\eta),$ (1.6)

is obtained in form of generalized (Kilbas–Saigo) Mittag–Leffler function.

This paper is divided in five sections: In the section 2, we provide some
definitions, theorems of $\Psi$-Hilfer fractional derivative and the results
which will be utilized throughout this paper. In Section 3, we establish the
results pertaining to existence, uniqueness and continuous dependence of
solution of (1.1)-(1.2). Section 4, discuss the convergence of Picard’s type
successive approximations to the solution of (1.1)-(1.2). In Section 5,
representation formulas have been obtained for the solution of linear Cauchy
problem with constant coefficient and variable coefficient.

##  2 Preliminaries

We review a few definitions, notations and results of $\Psi$-Hilfer fractional
derivative [30, 31]. Let $\Delta=[a,b]$ $(0<a<b<\infty)$ be a finite interval.
Consider the space $C_{1-\zeta;\,\Psi}(\Delta,\,\mbox{\Bbb R})$ of weighted
functions $h$ defined on $\Delta$ given by

$C_{1-\zeta;\,\Psi}(\Delta,\,\mbox{\Bbb
R})=\left\\{h:\left(a,b\right]\rightarrow\mathbb{R}~{}\big{|}~{}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\zeta}h\left(t\right)\in
C(\Delta,\,\mbox{\Bbb R})\right\\},\text{ }0<\zeta\leq 1$

endowed with the norm

$\left\|h\right\|_{C_{1-\zeta;\,\Psi}\left(\Delta,\mbox{\Bbb
R}\right)}=\underset{t\in\Delta}{\max}\left|\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\zeta}h\left(t\right)\right|.$
(2.1)

###### Definition 2.1

Let $\eta>0~{}(\eta\in\mbox{\Bbb R})$, $h\in L_{1}(\Delta,\,\mbox{\Bbb R})$
and $\Psi\in C^{1}(\Delta,\,\mbox{\Bbb R})$ be an increasing function wth
$\Psi^{\prime}(t)\neq 0$, for all $\,t\in\Delta$. Then, the
$\Psi$-Riemann–Liouville fractional integral of a function $h$ with respect to
$\Psi$ is defined by

$\mathcal{I}_{a+}^{\eta\,;\,\Psi}h\left(t\right)=\frac{1}{\Gamma\left(\eta\right)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)h(\sigma)\,d\sigma.$
(2.2)

where,$\mathrm{Q}^{\eta}_{\Psi}(t,s)=\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\eta-1},~{}t,s\in\Delta.$

###### Definition 2.2

Let $n-1<\eta<n\in\mbox{\Bbb N}$ and $h,\Psi\in C^{n}(\Delta,\,\mbox{\Bbb R})$
two functions such that $\Psi$ is increasing with $\Psi^{\prime}(t)\neq 0,$
for all $\,t\in\Delta$. Then, the $(\mbox{left-sided})$ $\Psi$-Hilfer
fractional derivative ${}^{H}\mathcal{D}^{\eta,\,\nu\,;\,\Psi}_{a+}(\cdot)$ of
a function $h$ of order $\eta$ and type $0\leq\nu\leq 1$, is defined by

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

Following results from [30, 31] play key role in proving our main results.

###### Lemma 2.1

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

###### Lemma 2.2

Let $\eta>0$ and $\delta>0$. If
$h(t)=\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\delta-1}$, then

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

###### Lemma 2.3

Let $\Psi\in C^{1}(\Delta,\mathbb{R})$ be increasing function with
$\Psi^{\prime}(t)\neq 0$, for all $t\in\Delta$. If
$\zeta=\eta+\nu\left(1-\eta\right)$ where, $0<\eta<1$ and $0\leq\nu\leq 1,$
then $\Psi$-Riemann-Liouville fractional integral operator
$\mathcal{I}_{a+}^{\eta\,;\,\Psi}\left(\cdot\right):C_{1-\zeta\,;\,\Psi}\left[a,b\right]\rightarrow
C_{1-\zeta\,;\,\Psi}\left[a,b\right]$ is bounded and it is given by :

$\left\|\mathcal{I}_{a+}^{\eta\,;\,\Psi}h\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,b\right]}\leq
M\frac{\Gamma\left(\zeta\right)}{\Gamma\left(\zeta+\eta\right)}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\eta},$
(2.4)

where, $M$ is the bound of a bounded function
$(\Psi(\cdot)-\Psi(a))^{1-\zeta}h(\cdot)$.

###### Theorem 2.4

Let $\tilde{u},$ $\tilde{v}\in L_{1}(\Delta,\,\mathbb{R})$ and $\tilde{g}\in
C(\Delta,\,\mathbb{R}).$ Let $\Psi\in C^{1}(\Delta,\,\mathbb{R})$ be an
increasing function with $\Psi^{\prime}\left(t\right)\neq 0$, for all
$~{}t\in\Delta$. Assume that

  1. 1.

$\tilde{u}$ and $\tilde{v}$ are nonnegative;

  2. 2.

$\tilde{g}$ is nonnegative and nondecreasing.

If

$\tilde{u}(t)\leq\tilde{v}(t)+\tilde{g}(t)\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\,\tilde{u}(\sigma)\,\,d\sigma,$

then

$\tilde{u}(t)\leq\tilde{v}(t)+\int_{a}^{t}\overset{\infty}{\underset{m=1}{\sum}}\frac{\left[\tilde{g}\left(t\right)\Gamma\left(\eta\right)\right]^{m}}{\Gamma\left(m\eta\right)}\,\mathrm{Q}^{m\eta}_{\Psi}(t,s)\,\,\tilde{v}(\sigma)\,d\sigma,~{}t\in\Delta.$
(2.5)

where,
$\mathrm{Q}^{m\eta}_{\Psi}(t,s)=\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{m\eta-1},~{}t,s\in\Delta.$
Further, if $\tilde{v}$ is a nondecreasing function on $\Delta$ then

$\tilde{u}(t)\leq\tilde{v}(t)\,\mathcal{E}_{\eta}\left(\tilde{g}(t)\Gamma(\eta)\left(\Psi(t)-\Psi(a)\right)^{\eta}\right),$

where, $\mathcal{E}_{\eta}(\cdot)$ is the Mittag-Leffler function of one
parameter.

Existence and uniqueness results are proved via following fixed point
theorems.

######  Theorem 2.5 ([11], Schauder)

Let $\mathcal{X}$ be a Banach space, let $\mathcal{U}$ be a nonempty convex
bounded closed subset of $\mathcal{X}$ and let
$\mathcal{A}:\mathcal{U}\rightarrow\mathcal{U}$ be a completely continuous
operator. Then $\mathcal{A}$ has at least one fixed point.

######  Theorem 2.6 ([11], Weissinger)

Assume $(\mathcal{U},d)$ to be a non empty complete metric space and let
$\eta_{j}\geq 0$ for every $j\in\mbox{\Bbb N}_{0}$ such that
$\overset{\infty}{\underset{j=0}{\sum}}\eta_{j}$ converges. Furthermore, let
the mapping $\mathcal{A}:\mathcal{U}\rightarrow{\mathcal{U}}$ satisfy the
inequality

$d(\mathcal{A}^{j}u,\mathcal{A}^{j}v)\leq\eta_{j}\,\,d(u,v)$

for every $j\in\mbox{\Bbb N}$ and every $u,v\in\mathcal{U}.$ Then,
$\mathcal{A}$ has a unique fixed point $u^{*}.$ Moreover, for any
$u_{0}\in\mathcal{U},$ the sequence $(\mathcal{A}^{j}u_{0})_{j=1}^{\infty}$
converges to this fixed point $u^{*}.$

######  Definition 2.3 ([4])

Let $\eta>0,\,\nu>0~{}(~{}\eta,\nu\in\mbox{\Bbb R})$. Then the two parameter
Mittag-Leffler function is defined as

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

######  Definition 2.4 ([4])

Let $\eta,\,m\in\mbox{\Bbb R}$ and $l\in\mbox{\Bbb C}$ such that
$\eta>0,~{}m>0$ and $\eta(jm+l)+1\neq-1,-2,-3,\cdots(j=0,1,2,\cdots)$. Then
generalized (Kilbas–Saigo) Mittag–Leffler type function of three parameters is
defined by

$\mathcal{E}_{\eta,\,m,\,l}\,(z)=\sum_{k=0}^{\infty}c_{k}\,z^{k}$

where

$c_{0}=1,~{}c_{k}=\prod_{j=0}^{k-1}\frac{\Gamma(\eta[jm+l]+1)}{\Gamma(\eta[jm+l+1]+1)}\,\,(k=1,2,\cdots)$

and an empty product is assumed to be equal to one.

##  3 Existence, Uniqueness and Continuous Dependence

The forthcoming theorem establishes the existence of solution along with the
interval of existence of the initial value problem (IVP) (1.1)-(1.2) using its
equivalent fractional Volterra integral equation(VIE) in the weighted space
$C_{1-\zeta;\,\Psi}(\Lambda,\,\mbox{\Bbb R})$.

######  Theorem 3.1 (Existence and interval of existence)

Let $\zeta=\eta+\nu(1-\eta),0<\eta<1$ and $0\leq\nu\leq 1$. Define

$R_{0}=\left\\{(t,y):a\leq t\leq
a+\xi,\left|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right|\leq
k\right\\},~{}\xi>0,~{}k>0.$

where,
$\mathcal{H}^{\Psi}_{\zeta}(t,a):=\dfrac{(\Psi(t)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}$.
Let $f:R_{0}\rightarrow\mbox{\Bbb R}$ is continuous function such that
$f(\cdot\,,y(\cdot))\in C_{1-\zeta\,;\,\Psi}(\Lambda,\,\mbox{\Bbb R})$ for
every $y\in C_{1-\zeta\,;\,\Psi}(\Lambda,\,\mbox{\Bbb R})$. Let $\Psi\in
C^{1}(\Lambda,\,\mbox{\Bbb R})$ be an increasing bijective function with
$\Psi^{\prime}(t)\neq 0,$ for all $\,t\in\Lambda$ . Then the IVP (1.1)-(1.2)
possesses at least one solution $y$ in the space
$C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ of weighted functions, where

$\chi=\min\left\\{\xi,~{}\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\xi\right]}}\right)^{\frac{1}{\eta}}\right]-a\right\\}.$

Proof: The equivalent fractional VIE to the Cauchy problem (1.1)-(1.2) in the
space $C_{1-\zeta\,;\,\Psi}(\Lambda,\,\mbox{\Bbb R})$ is derived in [31] and
it is given by

$y(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f(\sigma,y(\sigma))\,d\sigma,~{}t\in\Lambda.$
(3.1)

Consider the set defined by,

$\mathcal{U}=\left\\{y\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]:\left\|y-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_{a}\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}\leq
k\right\\}.$

Define
$\tilde{y}(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a},~{}t\in[a,a+\chi]$. Then
$(\Psi(t)-\Psi(a))^{1-\zeta}\,\tilde{y}(t)=\dfrac{y_{a}}{\Gamma(\zeta)}\in
C[a,a+\chi]$ and hence we have, $\tilde{y}(t)\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$. Further,

$\left\|\tilde{y}-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_{a}\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}=0\leq
k.$

Thus, $\tilde{y}\in\mathcal{U}$ and hence $\mathcal{U}$ is non-empty set.
Clearly, $\mathcal{U}$ is a convex, bounded and closed subset of space
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right].$ We define an operator
$\mathcal{A}$ on the set $\mathcal{U}$ by

$\mathcal{A}y(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f(\sigma,y(\sigma))\,\,d\sigma.$

By definition of an operator $\mathcal{A}$, the equation (3.1) can be written
as

$y=\mathcal{A}y.$

We analyze the properties of $\mathcal{A}$ so that it admit at least one fixed
point.

First we show that, $\mathcal{A}\,\mathcal{U}\subseteq\mathcal{U}$. In the
view of Lemma 2.1, $\Psi$-Riemann-Liouville fractional integral operator
$\mathcal{I}_{a+}^{\eta\,;\,\Psi}\left(\cdot\right)$ maps
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$ to
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$, and hence
$\mathcal{I}_{a+}^{\eta\,;\,\Psi}f(\cdot,y(\cdot))\in
C_{{1-\zeta};\,\Psi}[a,a+\chi]$ for any $y\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$. Further,

$(\Psi(t)-\Psi(a))^{1-\zeta}\,\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}=\frac{y_{a}}{\Gamma(\zeta)}\in
C[a,a+\chi],$

and hence $\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$.

From the above arguments it follows that $\mathcal{A}y\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$. Next, for any $y\in\mathcal{U}$
and any $x$, $a<x\leq\chi$, we have

$\displaystyle\left\|\mathcal{A}y-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_{a}\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}$
$\displaystyle=\underset{t\in\left[a,a+x\right]}{\max}\left|(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\times\right.$
$\displaystyle\qquad\left.(\Psi(\sigma)-\Psi(a))^{1-\zeta}f(\sigma,y(\sigma))\,d\sigma\right|$
$\displaystyle\leq(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\times$
$\displaystyle\qquad\underset{\varsigma\in\left[a,\,\sigma\right]}{\max}\left|(\Psi(\varsigma)-\Psi(a))^{1-\zeta}f(\varsigma,y(\varsigma))\right|\,d\sigma$
$\displaystyle\leq\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}(\Psi(a+x)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(a+x)-\Psi(a))^{\zeta-1}$
$\displaystyle\leq\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{\eta+\zeta-1}$
$\displaystyle=\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{\eta}.$

Taking $x\rightarrow\chi$ we get,

$\left\|\mathcal{A}y-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_{a}\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}}\leq\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(a+\chi)-\Psi(a))^{\eta}$
(3.2)

Since,

$\chi=\min\left\\{\xi\,\,,\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\xi\right]}}\right)^{\frac{1}{\eta}}\right]-a\right\\}$

we have

$\chi\leq\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\xi\right]}}\right)^{\frac{1}{\eta}}\right]-a.$

Further, since $\Psi\in C^{1}\left(\Lambda,\,\mbox{\Bbb R}\right)$ is a
bijective function, $\Psi^{-1}:\mbox{\Bbb R}\to\Lambda$ exists and from above
inequality, we have

$(\Psi(a+\chi)-\Psi(a))^{\eta}\leq\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\xi\right]}}.$
(3.3)

Using Eq.(3.3) in Eq.(3.2), we get

$\left\|\mathcal{A}y-\mathcal{H}^{\Psi}_{\zeta}(\cdot,a)\,y_{a}\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}}\leq
k.$

We have proved that $\mathcal{A}y\in\mathcal{U}\,\,\text{for
any}\,\,y\in\mathcal{U}$. This proves
$\mathcal{A}\,\,\text{maps}\,\,\mathcal{U}\,\,\text{to itself}$.

Next, we prove that $\mathcal{A}$ is a continuous operator. Let any
$\varepsilon>0$. Since $f:R_{0}\rightarrow\mbox{\Bbb R}$ is continuous and
$R_{0}$ is compact set, $f$ is uniformly continuous on $R_{0}$. Thus, there
exists $\tilde{\delta}>0$ such that

$\left|f(t,y)-f(t,z)\right|<\frac{\varepsilon\,\,\Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}},~{}~{}\text{whenever}~{}|y-z|<\tilde{\delta}.$

Since, $\Psi$ is continuous on compact set $\Lambda$, we can choose $\delta>0$
such that $(\Psi(t)-\Psi(a))^{1-\zeta}|y-z|<\delta$. Therefore, we have

$\displaystyle\left|f(t,y)-f(t,z)\right|<\frac{\varepsilon\,\,\Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}},~{}\text{whenever}~{}(\Psi(t)-\Psi(a))^{1-\zeta}|y-z|<\delta.$
(3.4)

Let any $y,z\in\mathcal{U}$ such that
$\left\|y-z\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}<\delta$. Then
in the view of (3.4), we have

$\left|f(t,y(t))-f(t,z(t))\right|<\frac{\varepsilon\,\,\Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}},~{}t\in[a,a+\chi].$

Therefore,

$\displaystyle\left\|\mathcal{A}y-\mathcal{A}z\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]}$
$\displaystyle\leq\underset{t\in\left[a,a+\chi\right]}{\max}(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\left|f(\sigma,y(\sigma))-f(\sigma,z(\sigma))\right|\,d\sigma$
$\displaystyle<\frac{\varepsilon\,\,\Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}\,\,\,\underset{t\in\left[a,a+\chi\right]}{\max}(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\,\,d\sigma$
$\displaystyle=\frac{\varepsilon\,\,\Gamma(\eta+1)}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}\,\,\,\underset{t\in\left[a,a+\chi\right]}{\max}(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta+1)}(\Psi(t)-\Psi(a))^{\eta}$
$\displaystyle\leq\frac{\varepsilon}{(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}}\,\,(\Psi(a+\chi)-\Psi(a))^{\eta-\zeta+1}$
$\displaystyle=\varepsilon.$

Thus, the operator $\mathcal{A}$ is continuous on $\mathcal{U}$. Now, for any
$z\in\mathcal{A}(\mathcal{U})=\\{\mathcal{A}y:y\in\mathcal{U}\\}$ and any $x$
such that $a<x\leq\chi$, we have

$\displaystyle\left\|z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}=\left\|\mathcal{A}y\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}$
$\displaystyle\leq\frac{y_{a}}{\Gamma(\zeta)}+\underset{t\in\left[a,a+x\right]}{\max}\left|\frac{(\Psi(t)-\Psi(a))^{1-\zeta}}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\times\right.$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}(\Psi(\sigma)-\Psi(a))^{1-\zeta}f(\sigma,y(\sigma))\,d\sigma\Big{|}$
$\displaystyle\leq\frac{y_{a}}{\Gamma(\zeta)}+\frac{(\Psi(a+x)-\Psi(a))^{1-\zeta}}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}\underset{\varsigma\in[a,\sigma]}{\max}\left|(\Psi(\varsigma)-\Psi(a))^{1-\zeta}f(\varsigma,y(\varsigma))\right|\,d\sigma$
$\displaystyle\leq\frac{y_{a}}{\Gamma(\zeta)}+{\left\|f\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}(\Psi(a+x)-\Psi(a))^{1-\zeta}}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(a+x)-\Psi(a))^{\zeta-1}$
$\displaystyle=\frac{y_{a}}{\Gamma(\zeta)}+\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}\,\,\left(\Psi(a+x)-\Psi(a)\right)^{1-\zeta}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{\eta+\zeta-1}$
$\displaystyle=\frac{y_{a}}{\Gamma(\zeta)}+\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}\,\,\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{\eta}.$

This shows that $\mathcal{A}(\mathcal{U})$ is uniformly bounded subset of
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$ and hence it is point wise bounded
also. Next, we prove that $\mathcal{A}(\mathcal{U})$ is equicontinuous. For
any $t_{1},t_{2}$ such that $a<t_{1}\leq t_{2}\leq a+\chi$ and any
$y\in\mathcal{U},$ we have

$\displaystyle|\mathcal{A}y(t_{2})-\mathcal{A}y(t_{1})|$
$\displaystyle\leq\left|\frac{y_{a}}{\Gamma(\zeta)}\left\\{\left(\Psi(t_{2})-\Psi(a)\right)^{\zeta-1}-(\Psi(t_{1})-\Psi(a))^{\zeta-1}\right\\}\right|$
$\displaystyle\,\,+\left|\frac{1}{\Gamma(\eta)}\int_{a}^{t_{2}}\mathrm{Q}^{\eta\,;\,\psi}(t_{2},\sigma)\left|f(\sigma,y(\sigma))\right|\,d\sigma-\frac{1}{\Gamma(\eta)}\int_{a}^{t_{1}}\mathrm{Q}^{\eta\,;\,\psi}(t_{1},\sigma)\left|f(\sigma,y(\sigma))\right|\,d\sigma\right|$
$\displaystyle\leq\left|\frac{y_{a}}{\Gamma(\zeta)}\left\\{\left(\Psi(t_{2})-\Psi(a)\right)^{\zeta-1}-(\Psi(t_{1})-\Psi(a))^{\zeta-1}\right\\}\right|$
$\displaystyle\qquad+\left|\frac{1}{\Gamma(\eta)}\int_{a}^{t_{2}}\mathrm{Q}^{\eta\,;\,\psi}(t_{2},\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\underset{\varsigma\in[a,\sigma]}{\max}\left|(\Psi(\varsigma)-\Psi(a))^{1-\zeta}f(\varsigma,y(\varsigma))\right|\,d\sigma\right.$
$\displaystyle\qquad\left.-\frac{1}{\Gamma(\eta)}\int_{a}^{t_{1}}\mathrm{Q}^{\eta\,;\,\psi}(t_{1},\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\underset{\varsigma\in[a,\sigma]}{\max}\left|(\Psi(\varsigma)-\Psi(a))^{1-\zeta}f(\varsigma,y(\varsigma))\right|\,d\sigma\right|$
$\displaystyle\leq\left|\frac{y_{a}}{\Gamma(\zeta)}\left\\{\left(\Psi(t_{2})-\Psi(a)\right)^{\zeta-1}-(\Psi(t_{1})-\Psi(a))^{\zeta-1}\right\\}\right|$
$\displaystyle\qquad+\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\left|\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t_{2})-\Psi(a))^{\zeta-1}-\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t_{1})-\Psi(a))^{\zeta-1}\right|$
$\displaystyle=\frac{y_{a}}{\Gamma(\zeta)}\left|\left(\Psi(t_{2})-\Psi(a)\right)^{\zeta-1}-(\Psi(t_{1})-\Psi(a))^{\zeta-1}\right|$
$\displaystyle\qquad+\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left|(\Psi(t_{2})-\Psi(a))^{\eta+\zeta-1}-(\Psi(t_{1})-\Psi(a))^{\eta+\zeta-1}\right|.$

Observe that, the right hand part in the preceding inequality is free from
$y$. Thus, using the continuity of $\Psi$, $|t_{2}-t_{1}|\rightarrow 0$
implies that $|\mathcal{A}y(t_{2})-\mathcal{A}y(t_{1})|\rightarrow 0$. This
proves that $\mathcal{A}(\mathcal{U})$ is equicontinuous. In the view of
Arzela-Ascoli Theorem [11], it follows that $\mathcal{A}(\mathcal{U})$ is
relatively compact. Therefore, by Schauder’s fixed point Theorem 2.5, operator
$\mathcal{A}$ has at least one fixed point $y\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$ which is the solution of IVP
(1.1)–(1.2). $\Box$

In the next theorem, using Lipschitz condition on $f$ and the Weissinger
theorem, we establish another existence result for the IVP (1.1)–(1.2), which
gives in addition the uniqueness of solution also.

######  Theorem 3.2 (Uniqueness)

Let $\zeta=\eta+\nu(1-\eta),0<\eta<1$, $0\leq\nu\leq 1$ and

$R_{0}=\left\\{(t,y):a\leq t\leq
a+\xi,\left|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right|\leq
k\right\\},~{}\xi>0,~{}k>0.$

where,
$\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}$.
Let $f:R_{0}\rightarrow\mbox{\Bbb R}$ be a function such that
$f(\cdot\,,y(\cdot))\in C_{1-\zeta\,;\,\Psi}(\Lambda,\,\mbox{\Bbb R})$ for
every $y\in C_{1-\zeta\,;\,\Psi}(\Lambda,\,\mbox{\Bbb R})$ and satisfies the
Lipschitz condition

$\left|f(t,u)-f(t,v)\right|\leq L|u-v|,\,\,u,v\in\mbox{\Bbb R},~{}L>0.$

Let $\Psi\in C^{1}(\Lambda,\,\mbox{\Bbb R})$ be an increasing bijective
function with $\Psi^{\prime}(t)\neq 0$ for all $t\in\Lambda$. Then, the IVP
(1.1)-(1.2) has a unique solution $y$ in the weighted space
$C_{1-\zeta\,;\,\Psi}[a,a+\chi]$, where

$\chi=\min\left\\{\xi,~{}\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)\left\|f\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+\xi\right]}}\right)^{\frac{1}{\eta}}\right]-a\right\\}.$

Proof: Consider the operator $\mathcal{A}$ used in Theorem 3.1. It is as of
now demonstrated that $\mathcal{A}:\mathcal{U}\rightarrow\mathcal{U}$ is
continuous on closed, convex and bounded subset $\mathcal{U}$ of Banach space
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$. We now prove that, for every
$n\in\mbox{\Bbb N}$ and every $x$ such that $a<x\leq\chi$, we have

$\left\|\mathcal{A}^{n}y-\mathcal{A}^{n}z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}\leq\frac{\Gamma(\zeta)}{\Gamma(n\eta+\zeta)}\left[L\left(\Psi(a+x)-\Psi(a)\right)^{\eta}\right]^{n}\left\|y-z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}.$
(3.5)

We provide the proof of above inequality by using mathematical induction. By
using Lipschitz condition on $f$, for any $x\in(a,\chi]$, we have

$\displaystyle\left\|\mathcal{A}y-\mathcal{A}z\right\|_{{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}$
$\displaystyle\leq(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\left|f(\sigma,y(\sigma))-f(\sigma,z(\sigma))\right|\,d\sigma$
$\displaystyle\leq
L(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}(\Psi(\sigma)-\Psi(a))^{1-\zeta}\left|y(\sigma)-z(\sigma)\right|\,d\sigma$
$\displaystyle\leq
L(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}\underset{\varsigma\in\left[a,\sigma\right]}{\max}\left|(\Psi(\varsigma)-\Psi(a))^{1-\zeta}[y(\varsigma)-z(\varsigma)]\right|\,d\sigma$
$\displaystyle\leq
L\left\|y-z\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}(\Psi(a+x)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(a+x)-\Psi(a))^{\zeta-1}$
$\displaystyle=\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left[L(\Psi(a+x)-\Psi(a))^{\eta}\right]\left\|y-z\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}.$

Hence, the inequality (3.5) is true for $n=1$. Let us assume that, it is true
for $n=k-1$. Now, we prove that inequality (3.5) is also true for $n=k$. Then,
we have

$\displaystyle\left\|\mathcal{A}^{k}y-\mathcal{A}^{k}z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}$
$\displaystyle=\underset{t\in\left[a,a+x\right]}{\max}\left|(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\left[f\left(\sigma,\mathcal{A}^{k-1}y(\sigma)\right)-f\left(\sigma,\mathcal{A}^{k-1}z(\sigma)\right)\right]\,d\sigma\right|$
$\displaystyle\leq
L\,\,(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\left|\mathcal{A}^{k-1}y(\sigma)-\mathcal{A}^{k-1}z(\sigma)\right|\,d\sigma$
$\displaystyle\leq
L\,\,(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}\left\|\mathcal{A}^{k-1}y-\mathcal{A}^{k-1}z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,\sigma\right]}}\,d\sigma$
$\displaystyle\leq
L\,\,(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\times$
$\displaystyle\qquad\frac{\Gamma(\zeta)}{\Gamma((k-1)\eta+\zeta)}\left[L\left(\Psi(\sigma)-\Psi(a)\right)^{\eta}\right]^{k-1}\left\|y-z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,\sigma\right]}}\,d\sigma$
$\displaystyle\leq
L^{k}\frac{\Gamma(\zeta)}{\Gamma((k-1)\eta+\zeta)}\,\,(\Psi(a+x)-\Psi(a))^{1-\zeta}\left\|y-z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}\times$
$\displaystyle\qquad\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(a+x)-\Psi(a))^{\eta(k-1)+\zeta-1}$
$\displaystyle=L^{k}\frac{\Gamma(\zeta)}{\Gamma((k-1)\eta+\zeta)}\,\,(\Psi(a+x)-\Psi(a))^{1-\zeta}\left\|y-z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}\times$
$\displaystyle\qquad\frac{\Gamma((k-1)\eta+\zeta)}{\Gamma(k\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{k\eta+\zeta-1}$
$\displaystyle=\frac{\Gamma(\zeta)}{\Gamma(k\eta+\zeta)}\left[L(\Psi(a+x)-\Psi(a))^{\eta}\right]^{k}\left\|y-z\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,a+x\right]}}.$

Hence, the inequality (3.5) is true for $n=k$. By mathematical induction the
proof of the inequality (3.5) is concluded. Taking $x\to\chi$, we obtain

$\left\|\mathcal{A}^{n}y-\mathcal{A}^{n}z\right\|_{{}_{C_{1-\zeta;\,\psi}\left[a,a+\chi\right]}}\leq\frac{\Gamma(\zeta)}{\Gamma(n\eta+\zeta)}\left[L\left(\psi(a+\chi)-\psi(a)\right)^{\eta}\right]^{n}\left\|y-z\right\|_{{}_{C_{1-\zeta;\,\psi}\left[a,a+\chi\right]}}.$

Note that, the operator $\mathcal{A}$ satisfy all the conditions of the
Weissinger’s theorem 2.6, with

$\eta_{n}=\frac{\left[L\left(\Psi(a+\chi)-\Psi(a)\right)^{\eta}\right]^{n}}{\Gamma(n\eta+\zeta)}.$

Further, by definition of two parameter Mittag–Leffler function, we have

$\sum_{n=0}^{\infty}\,\eta_{n}=\sum_{n=0}^{\infty}\frac{\left[L(\Psi(a+\chi)-\Psi(a))^{\eta}\right]^{n}}{\Gamma(n\eta+\zeta)}=\mathcal{E}_{\eta,\,\zeta}\,\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right),$

which is convergent series. Thus, by using Weissinger’s fixed point theorem
2.6, operator $\mathcal{A}$ has a unique fixed point $y$ in
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$, which is a unique solution of the
IVP (1.1)-(1.2). $\Box$

###### Remark 3.3

  1. 1.

The results obtained by Diethelem and Ford [11] can be regarded as the
particular cases of Theorem 3.1 and Theorem 3.2 and can be derived by putting
$\nu=1,~{}a=0$ and $\Psi(t)=t,~{}t\in\Lambda$.

  2. 2.

We have investigated in Theorem 3.1 and Theorem 3.2, the existence and
uniqueness of the Cauchy problem (1.1)-(1.2) involving the $\Psi$-Hilfer
fractional derivative. One of the fundamental properties of the $\Psi$-Hilfer
fractional derivative is the wide class of fractional derivatives that contain
it as particular cases. Thus, in this sense, the respective results obtained
by Theorem 3.1 and Theorem 3.2, are also valid for these fractional
derivatives. For examples, some particular cases in the items as follows:

    1. (a)

If we take $\nu\rightarrow 1$ on both sides of (1.1)-(1.2),it reduces to the
Cauchy problem with $\Psi$-Caputo fractional derivative and consequently, the
Theorem 3.1 and Theorem 3.2, there are true;

    2. (b)

If we take $\nu\rightarrow 0$ on both sides of the (1.1)-(1.2), we have the
Cauchy problem with $\Psi$-Riemann-Liouville fractional derivative and
consequently, the Theorem 3.1 and Theorem 3.2, there are true;

    3. (c)

Choose $\Psi(t)=t^{\rho}$ and take $\nu\rightarrow 1$ on both sides of
(1.1)-(1.2), we have the Cauchy problem involving the Caputo-Katugampola
fractional derivative and consequently, the Theorem 3.1 and Theorem 3.2, there
are true;

    4. (d)

In cases where they involve the fractional derivatives of Hadamard, Caputo-
Hadamard, Hilfer-Hadamard or any other that is related to Hadamard’s
fractional derivative, we have to impose the condition on the parameter $a>0$,
since in these fractional derivatives they involve the function $\ln t$ and is
not defined when $t=a=0$.

Next, we prove the continuous dependence of solution for the Cauchy type
problem (1.1)-(1.2) via Weissinger’s theorem.

######  Theorem 3.4 (Continuous Dependence)

Let $\zeta=\eta+\nu\left(1-\eta\right)$ where, $0<\eta<1$ and $0\leq\nu\leq
1.$ Let $f:\left[a,a+\chi\right]\times\mathbb{R}\rightarrow\mathbb{R}$ be a
function such that $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ for
every $y\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ and satisfies the Lipschitz
condition

$\left|f(t,u)-f(t,v)\right|\leq L|u-v|,\,\,u,v\in\mbox{\Bbb R},~{}L>0.$

Let $y(t)$ and $z(t)$ be the solutions of, the IVPs,

$\displaystyle{}^{H}\mathcal{D}_{a+}^{\eta,\,\nu;\,\Psi}y(t)=f(t,y(t)),\quad\mathcal{I}_{a+}^{1-\zeta\,;\,\Psi}y(a)=y_{a}$
(3.6)

and

$\displaystyle{}^{H}\mathcal{D}_{a+}^{\eta,\,\nu;\,\Psi}z(t)=f(t,z(t)),\quad\mathcal{I}_{a+}^{1-\zeta\,;\,\Psi}z(a)=z_{a}$
(3.7)

respectively. Then,

$\displaystyle\left\|y-z\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\leq\left\\{1+\Gamma(\zeta)\,\mathcal{E}_{\eta,\,\zeta}\left(L\left(\Psi(a+\chi)-\Psi(a)\right)^{\eta}\right)\right\\}\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]},$
(3.8)

where,
$\tilde{y}_{a}(t)=\dfrac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}y_{a}$
and
$\tilde{z}_{a}(t)=\dfrac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}z_{a}$.

Proof: Consider the sequences $\\{\mathcal{A}^{m}\tilde{y}_{a}\\}$ and
$\\{\mathcal{A}^{m}\tilde{z}_{a}\\}$ defined by

$\displaystyle\mathcal{A}^{0}\tilde{y}_{a}(t)=\tilde{y}_{a}(t),$
$\displaystyle\mathcal{A}^{m}\tilde{y}_{a}(t)=\tilde{y}_{a}(t)+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f\left(\sigma,\mathcal{A}^{m-1}\tilde{y}_{a}(\sigma)\right)\,d\sigma,~{}(m=1,2,\cdots)$

and

$\displaystyle\mathcal{A}^{0}\tilde{z}_{a}(t)=\tilde{z}_{a}(t),$
$\displaystyle\mathcal{A}^{m}\tilde{z}_{a}(t)=\tilde{z}_{a}(t)+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)f\left(\sigma,\mathcal{A}^{m-1}\tilde{z}_{a}(\sigma)\right)\,d\sigma,~{}(m=1,2,\cdots)$
$\displaystyle respectively.$

Then, for each $x$ with $a<x\leq\chi$,

$\displaystyle\left\|\mathcal{A}^{m}\tilde{y}_{a}-\mathcal{A}^{m}\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}$
$\displaystyle=\underset{t\in\left[a,a+x\right]}{\max}\left|(\Psi(t)-\Psi(a))^{1-\zeta}\Big{\\{}\tilde{y}_{a}(t)-\tilde{z}_{a}(t)\right.$
$\displaystyle\left.\left.\qquad+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta\,;\,\psi}(t,\sigma)\left[f\left(\sigma,\mathcal{A}^{m-1}\tilde{y}_{a}(\sigma)\right)-f\left(\sigma,\mathcal{A}^{m-1}\tilde{z}_{a}(\sigma)\right)\right]\,d\sigma\right\\}\right|$
$\displaystyle\leq\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}+(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{L}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}\underset{\varsigma\in\left[a,\sigma\right]}{\max}\left|(\Psi(\varsigma)-\Psi(a))^{1-\zeta}\left\\{\mathcal{A}^{m-1}\tilde{y}_{a}(\varsigma)-\mathcal{A}^{m-1}\tilde{z}_{a}(\varsigma)\right\\}\right|\,d\sigma$
$\displaystyle\leq\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}+L(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}\left\|\mathcal{A}^{m-1}\tilde{y}_{a}-\mathcal{A}^{m-1}\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,\sigma]}\,d\sigma.$

Using the inequality (3.5),

$\displaystyle\left\|\mathcal{A}^{m}\tilde{y}_{a}-\mathcal{A}^{m}\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}$
$\displaystyle\leq\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}+(\Psi(a+x)-\Psi(a))^{1-\zeta}\frac{L}{\Gamma(\eta)}\int_{a}^{a+x}\mathrm{Q}^{\eta\,;\,\psi}(a+x,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}\frac{\Gamma(\zeta)}{\Gamma((m-1)\eta+\zeta)}\left[L\left(\Psi(\sigma)-\Psi(a)\right)^{\eta}\right]^{m-1}\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,\sigma\right]}}\,d\sigma$
$\displaystyle=\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}+L^{m}\frac{\Gamma(\zeta)}{\Gamma((m-1)\eta+\zeta)}\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}\times$
$\displaystyle\qquad(\Psi(a+x)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(a+x)-\Psi(a))^{(m-1)\eta+\zeta-1}$
$\displaystyle=\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}\left\\{1+L^{m}\frac{\Gamma(\zeta)}{\Gamma((m-1)\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{1-\zeta}\times\right.$
$\displaystyle\left.\qquad\frac{\Gamma((m-1)\eta+\zeta)}{\Gamma(m\eta+\zeta)}(\Psi(a+x)-\Psi(a))^{m\eta+\zeta-1}\right\\}$
$\displaystyle=\left\\{1+{\Gamma(\zeta)}\frac{\left[L(\Psi(a+x)-\Psi(a))^{\eta}\right]^{m}}{\Gamma(m\eta+\zeta)}\right\\}\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}$
$\displaystyle\leq\left\\{1+{\Gamma(\zeta)}\sum_{k=0}^{m}\frac{\left[L(\Psi(a+x)-\Psi(a))^{\eta}\right]^{m}}{\Gamma(m\eta+\zeta)}\right\\}\left\|\tilde{y}_{a}-\tilde{z}_{a}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+x]}.$

Taking limit as $m\to\infty$ and $x\to\chi$ and utilizing Theorem 2.6, we get
the desired inequality (3.8). $\Box$

###### Remark 3.5

In the above theorem, in particular if $y_{a}=z_{a}$, then we get uniqueness
of solution.

##  4 Picard’s Successive Approximations: Nonlinear Case

In this section, we define the Picard’s type successive approximations and
prove that it converges to a unique solution of nonlinear Cauchy problem
(1.1)-(1.2). Further, we obtain an estimation for the error bound.

###### Theorem 4.1

Let $\zeta=\eta+\nu(1-\eta),0<\eta<1$ and $0\leq\nu\leq 1$. Define

$R_{0}=\left\\{(t,y):a\leq t\leq
a+\xi,\left|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right|\leq
k\right\\},~{}\xi>0,~{}k>0.$

where,
$\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}$.
Let $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ for every $y\in
C_{1-\zeta\,;\,\Psi}[a,a+\chi]$. Let $\Psi\in
C^{1}\left([a,a+\chi],~{}\mbox{\Bbb R}\right)$ be an increasing bijective
function with $\Psi^{\prime}(t)\neq 0$, for all $\,t\in[a,a+\chi]$, where

$\chi=\min\left\\{\xi,~{}\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)M}\right)^{\frac{1}{\eta}}\right]-a\right\\}$

and $M$ is constant such that

$|\left(\Psi(t)-\Psi(a)\right)^{1-\zeta}f(t)|\leq
M,~{}\forall\,t\in[a,a+\chi].$

Then the successive approximations defined by

$\displaystyle y_{0}(t)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}$ $\displaystyle
y_{n}(t)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\mathrm{I}_{a+}^{\eta\,;\,\Psi}f\left(t,y_{n-1}(t)\right),n=1,2,\cdots.$
(4.1)

satisfies the following conditions:

  1. (a)

$y_{n}\in C_{1-\zeta;\,\Psi}[a,a+\chi],~{}n=0,1,2,\cdots$ ;

  2. (b)

$(\Psi(t)-\Psi(a))^{1-\zeta}|y_{n}(t)-y_{0}(t)|\leq\dfrac{M\,\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta},~{}t\in[a,a+\chi],~{}n=0,1,2,\cdots.$

Proof: (a) Since $y_{0}(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}$, we have
$(\Psi(t)-\Psi(a))^{1-\zeta}y_{0}(t)=\dfrac{y_{a}}{\Gamma(\zeta)}\in
C[a,a+\chi]$ and hence $y_{0}\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]$. Further, by
Lemma 2.1, $\Psi$–Riemann fractional integral operator
$\mathcal{I}_{a+}^{\eta\,;\,\Psi}\left(\cdot\right)$ maps
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$ to
$C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$ and hence
$\mathcal{I}_{a+}^{\eta;\,\Psi}f(\cdot,y_{n-1}(\cdot))\in
C_{{1-\zeta};\,\Psi}[a,a+\chi]$ for any $y_{n-1}\in
C_{1-\zeta\,;\,\Psi}\left[a,a+\chi\right]$ for $n=1,2,\cdots$. This proves,
$y_{n}\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]~{}\forall\,n\in\mbox{\Bbb N}$.

(b) For any $t\in[a,a+\chi]$ and $n\in\mbox{\Bbb N}$, we have

$\displaystyle|y_{n}(t)-y_{0}(t)|$
$\displaystyle=\left|\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,y_{n-1}(\sigma))\,d\sigma\right|$
$\displaystyle\leq\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\times$
$\displaystyle\qquad\left|(\Psi(\sigma)-\Psi(a))^{1-\zeta}f(\sigma,y_{n-1}(\sigma))\right|\,d\sigma$
$\displaystyle\leq
M\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{\zeta-1}$
$\displaystyle=M\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta+\zeta-1}.$

Therefore,

$(\Psi(t)-\Psi(a))^{1-\zeta}|y_{n}(t)-y_{0}(t)|\leq\frac{M\,\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta},~{}t\in[a,a+\chi],~{}n=0,1,2,\cdots.$

$\Box$

###### Theorem 4.2

Let $\zeta=\eta+\nu(1-\eta),0<\eta<1$ and $0\leq\nu\leq 1$. Define

$R_{0}=\left\\{(t,y):a\leq t\leq
a+\xi,\left|y-\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\right|\leq
k\right\\},~{}\xi>0,~{}k>0.$

where,
$\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}$.
Let $f(\cdot\,,y(\cdot))\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ for every $y\in
C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ and satisfies the Lipschitz condition

$\left|f(t,u)-f(t,v)\right|\leq L|u-v|,\,\,u,v\in\mbox{\Bbb R},~{}L>0.$

Let $\Psi\in C^{1}\left([a,a+\chi],~{}\mbox{\Bbb R}\right)$ be an increasing
bijective function with $\Psi^{\prime}(t)\neq 0$,for all $\,t\in[a,a+\chi]$,
where

$\chi=\min\left\\{\xi,~{}\Psi^{-1}\left[\Psi(a)+\left(\frac{k\,\Gamma(\eta+\zeta)}{\Gamma(\zeta)M}\right)^{\frac{1}{\eta}}\right]-a\right\\}$

and $M$ is constant such that

$|\left(\Psi(t)-\Psi(a)\right)^{1-\zeta}f(t)|\leq
M,~{}\forall\,t\in[a,a+\chi].$

Then the successive approximations defined by (4.1) converges to the unique
solution $y$ of the Cauchy type problem (1.1)-(1.2) in the weighted space
$C_{1-\zeta\,;\,\Psi}[a,a+\chi]$.

Proof: We give the proof of the theorem in the following steps.

Step 1: In the above theorem we have already proved that $y_{n}\in
C_{1-\zeta\,;\,\Psi}[a,a+\chi]$, $\forall\,n\in\mbox{\Bbb N}$.

Step 2: We show that the sequence $\\{y_{n}\\}$ converges to a function $y\in
C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ with respect to the norm
$\|\cdot\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}$. Observe that $y_{n}$ can be
written as

$y_{n}=y_{0}+\sum_{k=1}^{n}(y_{k}-y_{k-1})$ (4.2)

which is partial sum of series

$y_{0}+\sum_{k=1}^{\infty}(y_{k}-y_{k-1}).$ (4.3)

Therefore, to show that sequence $\\{y_{n}\\}$ is convergent, we prove that
the series (4.3) is convergent. For any $x\in[a,a+\chi]$, consider the space
$C_{1-\zeta\,;\,\Psi}[a,x]$ having norm

$\left\|h\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,x\right]}=\underset{t\in\left[a,x\right]}{\max}\left|\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\zeta}h\left(t\right)\right|.$
(4.4)

By mathematical induction, we now prove , for each $x\in[a,a+\chi]$ and
$y_{j}\in C_{1-\zeta\,;\,\Psi}[a,x]$,

$\left\|y_{n+1}-y_{n}\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,x\right]}}\leq\frac{M\Gamma(\zeta)}{L}\frac{\left[L(\Psi(x)-\Psi(a))^{\eta}\right]^{n+1}}{\Gamma((n+1)\eta+\zeta)},~{}n\in\mbox{\Bbb
N}.$ (4.5)

Using Lemma 2.3, we obtain

$\displaystyle\left\|y_{1}-y_{0}\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,x\right]}}$
$\displaystyle=\underset{t\in[a,x]}{\max}\left|(\Psi(t)-\Psi(a))^{1-\zeta}\left\\{y_{0}(t)+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,y_{0}(\sigma))\,d\sigma-
y_{0}(t)\right\\}\right|$
$\displaystyle=\underset{t\in[a,x]}{\max}\left|(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,y_{0}(\sigma))\,d\sigma\right|$
$\displaystyle=\left\|\mathcal{I}_{a+}^{\eta\,;\,\Psi}f\left(\cdot,y_{0}(\cdot)\right)\right\|_{{}_{C_{1-\zeta\,;\,\Psi}\left[a,x\right]}}$
$\displaystyle\leq
M\,\,\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left(\Psi(t)-\Psi(a)\right)^{\eta}$
$\displaystyle\leq
M\,\,\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}\left(\Psi(x)-\Psi(a)\right)^{\eta},$

which is the inequality (4.5) for $n=0$.

Now, assume that the inequality (4.5) is hold for $n=k$. We prove it is also
hold for $n=k+1$. In fact,

$\displaystyle\left\|y_{k+2}-y_{k+1}\right\|_{C_{1-\zeta\,;\,\Psi}\left[a,x\right]}$
$\displaystyle\leq
L\underset{t\in[a,x]}{\max}(\Psi(t)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\times$
$\displaystyle\qquad(\Psi(\sigma)-\Psi(a))^{\zeta-1}(\Psi(\sigma)-\Psi(a))^{1-\zeta}\left|y_{k+1}(\sigma)-y_{k}(\sigma)\right|\,d\sigma$
$\displaystyle\leq
L(\Psi(x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{x}\mathrm{Q}^{\eta:\Psi}(x,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\left\|y_{k+1}-y_{k}\right\|_{{C_{1-\zeta\,;\,\Psi}\left[a,\sigma\right]}}\,d\sigma$
$\displaystyle\leq
L(\Psi(x)-\Psi(a))^{1-\zeta}\frac{1}{\Gamma(\eta)}\int_{a}^{x}\mathrm{Q}^{\eta:\Psi}(x,\sigma)(\Psi(\sigma)-\Psi(a))^{\zeta-1}\times$
$\displaystyle\qquad\frac{M\Gamma(\zeta)}{L}\frac{\left[L(\Psi(\sigma)-\Psi(a))^{\eta}\right]^{k+1}}{\Gamma((k+1)\eta+\zeta)}\,d\sigma$
$\displaystyle=M\,\,L^{k+1}\,\,\frac{\Gamma(\zeta)}{\Gamma((k+1)\eta+\zeta)}(\Psi(x)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(x)-\Psi(a))^{(k+1)\eta+\zeta-1}$
$\displaystyle=M\,\,L^{k+1}\,\,\frac{\Gamma(\zeta)}{\Gamma((k+1)\eta+\zeta)}(\Psi(x)-\Psi(a))^{1-\zeta}\times\frac{\Gamma((k+1)\eta+\zeta)}{\Gamma((k+2)\eta+\zeta)}(\Psi(x)-\Psi(a))^{(k+2)\eta+\zeta-1}$
$\displaystyle=\frac{M\,\,\Gamma(\zeta)}{L}\frac{\left(L\left(\Psi(x)-\Psi(a)\right)^{\eta}\right)^{k+2}}{\Gamma((k+2)\eta+\zeta)}.$

Thus, inequality (4.5) is true for $n=k+1$. Hence, by using principle of
mathematical induction the inequality (4.5) holds for each $n\in\mbox{\Bbb N}$
and $x\in[a,a+\chi]$. Taking $x\to a+\chi$ in (4.5), we get,

$\left\|y_{k}-y_{k-1}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\leq\frac{M\Gamma(\zeta)}{L}\frac{\left[L(\Psi(a+\chi)-\Psi(a))^{\eta}\right]^{k}}{\Gamma(k\eta+\zeta)}.$

Therefore,

$\displaystyle\sum_{k=1}^{\infty}\left\|y_{k}-y_{k-1}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}$
$\displaystyle\leq\frac{M\Gamma(\zeta)}{L}\sum_{k=1}^{\infty}\frac{\left[L(\Psi(a+\chi)-\Psi(a))^{\eta}\right]^{k}}{\Gamma(k\eta+\zeta)}$
$\displaystyle\leq\frac{M\Gamma(\zeta)}{L}\left[\mathcal{E}_{\eta,\,\zeta}\left(L\left(\Psi(a+\chi)-\Psi(a)\right)^{\eta}\right)-\frac{1}{\Gamma(\zeta)}\right].$

This proves the series $y_{0}+\sum_{k=1}^{\infty}\left(y_{k}-y_{k-1}\right)$
is convergent in the space $C_{1-\zeta\,;\,\Psi}[a,a+\chi]$. Let us suppose
that

$\tilde{y}=y_{0}+\sum_{k=1}^{\infty}(y_{k}-y_{k-1}).$

Therefore,

$\left\|y_{n}-\tilde{y}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\rightarrow
0~{}\text{as}\,\,n\rightarrow\infty.$ (4.6)

Step 3: $\tilde{y}$ is solution of fractional integral equation (3.1). In
fact, we have

$\displaystyle\left\|\mathcal{I}_{a+}^{\eta\,;\,\Psi}f\left(\cdot,y_{n}(\cdot)\right)-\mathcal{I}_{a+}^{\eta\,;\,\Psi}f\left(\cdot,\tilde{y}(\cdot)\right)\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}$
$\displaystyle\leq
L\left\|y_{n}-\tilde{y}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\underset{t\in[a,a+\chi]}{\max}(\Psi(t)-\Psi(a))^{1-\zeta}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{\zeta-1}$
$\displaystyle\leq
L\left\|y_{n}-\tilde{y}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\underset{t\in[a,a+\chi]}{\max}(\Psi(t)-\Psi(a))^{1-\zeta}\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta+\zeta-1}$
$\displaystyle\leq L\,\,\
\frac{\Gamma(\zeta)}{\Gamma(\eta+\zeta)}(\Psi(a+\chi)-\Psi(a))^{\eta}\left\|y_{n}-\tilde{y}\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}.$

Using (4.6), we obtain

$\left\|\mathcal{I}_{a+}^{\eta\,;\,\Psi}f\left(\cdot,y_{n}(\cdot)\right)-\mathcal{I}_{a+}^{\eta\,;\,\Psi}f\left(\cdot,\tilde{y}(\cdot)\right)\right\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\rightarrow
0\,\,\,\text{as}\,\,n\rightarrow\infty.$ (4.7)

Therefore, taking limit as $n\rightarrow\infty$ in (4.1), we get

$\displaystyle\tilde{y}(t)$
$\displaystyle=y_{0}(t)+\mathcal{I}_{a+}^{\eta\,;\,\Psi}f\left(t,\tilde{y}(t)\right)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma,\tilde{y}(\sigma))\,d\sigma.$

This proves, $\tilde{y}\in C_{1-\zeta\,;\,\Psi}[a,a+\chi]$ is solution of
(3.1).

Step 4: Uniqueness of the solution.
Let $y(t)$ and ${y}^{\ast}(t)$ be any two solutions of the IVP (1.1)–(1.2) and
consider the function defined by $z(t)=\left|y(t)-{y}^{\ast}(t)\right|$. Then,
we get

$\displaystyle z(t)$ $\displaystyle=\left|y(t)-{y}^{\ast}(t)\right|$
$\displaystyle=\left|\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left[f(\sigma,y(\sigma))-f(\sigma,{y}^{\ast}(\sigma))\right]\,d\sigma\right|$
$\displaystyle\leq\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left|f(\sigma,y(\sigma))-f(\sigma,{y}^{\ast}(\sigma))\right|\,d\sigma$
$\displaystyle\leq\frac{L}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left|y(\sigma)-{y}^{\ast}(\sigma)\right|\,d\sigma$
$\displaystyle\leq\frac{L}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)z(\sigma)\,d\sigma.$

Using Gronwall inequality given in the Theorem 2.4, we get

$z(t)\leq 0\times\mathcal{E}_{\eta}[L\left(\Psi(t)-\Psi(a)\right)^{\eta}].$

Therefore $z(t)=0$ and we have $y(t)={y}^{\ast}(t)$. $\Box$

###### Remark 4.3

For $\eta=1,\,\nu=1$ and $\Psi(t)=t$ the above theorems includes the results
of Coddington [[40],Chapter 5] for the ordinary differential equations.

###### Theorem 4.4

Let $\\{y_{n}\\}$ be the sequence of Picard’s successive approximation defined
by (4.1) and $y$ is the solution of the IVP (1.1)-(1.2). Then the error
$y-y_{n}$ satisfies the condition

$\|y-y_{n}\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}\leq\frac{M\Gamma(\zeta)}{L}\left(\mathcal{E}_{\eta,\,\zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\sum_{k=0}^{n}\frac{(L(\Psi(a+\chi)-\Psi(a))^{\eta})^{k}}{\Gamma(k\eta+\zeta)}\right).$
(4.8)

Proof: From the proof of the Theorem 4.2, we have

$y=y_{0}+\sum_{k=1}^{\infty}(y_{k}-y_{k-1})$

and

$y_{n}=y_{0}+\sum_{k=1}^{n}(y_{k}-y_{k-1}).$

Hence

$y-y_{n}=\sum_{k=n+1}^{\infty}(y_{k}-y_{k-1}).$

From the above relations, in view of (4.5), we have

$\displaystyle\|y-y_{n}\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}$
$\displaystyle\leq\sum_{k=n+1}^{\infty}\|y_{k}-y_{k-1}\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}$
$\displaystyle\leq\sum_{k=n+1}^{\infty}\frac{M\Gamma(\zeta)}{L}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}$
$\displaystyle\leq\frac{M\Gamma(\zeta)}{L}\left(\sum_{k=0}^{\infty}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}-\sum_{k=0}^{n}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}\right)$
$\displaystyle\leq\frac{M\Gamma(\zeta)}{L}\left(\mathcal{E}_{\eta,\,\zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\sum_{k=0}^{n}\frac{(L(\Psi(a+\chi)-\Psi(a))^{\eta})^{k}}{\Gamma(k\eta+\zeta)}\right).$

$\Box$

###### Remark 4.5

From the inequality (4.8), it follows that

$\displaystyle\lim\limits_{n\rightarrow\infty}\|y-y_{n}\|_{C_{1-\zeta\,;\,\Psi}[a,a+\chi]}$
$\displaystyle\leq\frac{M\Gamma(\zeta)}{L}\lim\limits_{n\rightarrow\infty}\left(\sum_{k=0}^{\infty}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}-\sum_{k=0}^{n}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}\right)$
$\displaystyle=\frac{M\Gamma(\zeta)}{L}\left(\mathcal{E}_{\eta,\,\zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\sum_{k=0}^{\infty}\frac{\left(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)^{k}}{\Gamma(k\eta+\zeta)}\right)$
$\displaystyle=\frac{M\Gamma(\zeta)}{L}\left(\mathcal{E}_{\eta,\,\zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta})-\mathcal{E}_{\eta,\,\zeta}(L(\Psi(a+\chi)-\Psi(a))^{\eta}\right)$
$\displaystyle=0.$

This implies the sequence $\\{y_{n}\\}$ of successive approximation converges
to the solution $y$ of the problem (1.1)-(1.2) as $n\rightarrow\infty.$

##  5 Picard’s Successive Approximations: Linear Case

Here we derive the representation formula for the solution of linear Cauchy
problem with constant coefficient and variable coefficients which extend the
results of [4, Chapter 7].

###  5.1 Linear Cauchy Type problem with Constant Coefficient

###### Theorem 5.1

Let $f\in C_{1-\zeta\,;\,\Psi}\left(\Delta,\,\mbox{\Bbb R}\right)]$ and
$\lambda\in\mbox{\Bbb R}$. Then, the solution of the Cauchy problem for FDE
with constant coefficient involving $\Psi$-Hilfer fractional derivative
(1.3)-(1.4) is given by

$\displaystyle y\left(t\right)$
$\displaystyle=y_{a}(\Psi(t)-\Psi(a))^{\zeta-1}\,\mathcal{E}_{\eta,\,\zeta}\left(\lambda(\Psi(t)-\Psi(a))^{\eta}\right)$
$\displaystyle\qquad+\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\,\mathcal{E}_{\eta,\,\eta}\left(\lambda(\Psi(t)-\Psi(\sigma))^{\eta}\right)\,f(\sigma)\,\,d\sigma$

where,
$\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)=\Psi^{\prime}(\sigma)(\Psi(t)-\Psi(\sigma))^{\eta-1}$

Proof: The linear Cauchy problem (1.3)-(1.4) is equivalent to

$\displaystyle y(t)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y(\sigma)\,d\sigma+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma.$
(5.1)

Employing the method of successive approximation we derive the solution of
(5.1). Consider the sequences $\\{y_{m}\\}$ defined by

$\displaystyle y_{0}(t)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}$ $\displaystyle
y_{m}(t)$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y_{m-1}(\sigma)\,d\sigma$
$\displaystyle\qquad+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma,~{}(m=1,2,\cdots).$

By method of induction, we prove that

$\displaystyle
y_{m}(t)=y_{a}\sum_{k=0}^{m}\frac{\lambda^{k}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}}{\Gamma(k\eta+\zeta)}+\int_{a}^{t}\Psi^{\prime}(\sigma)\sum_{k=0}^{m-1}\frac{\lambda^{k}(\Psi(t)-\Psi(\sigma))^{(k+1)\eta-1}}{\Gamma((k+1)\eta)}f(\sigma)\,d\sigma.$
(5.2)

For the case $m=1$, we have

$\displaystyle y_{1}(t)$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y_{0}(\sigma)\,d\sigma+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\frac{(\Psi(\sigma)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}y_{a}\,\,d\sigma$
$\displaystyle\qquad+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma$
$\displaystyle=y_{0}(t)+y_{a}\frac{\lambda}{\Gamma(\zeta)}\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{\zeta-1}+\mathcal{I}_{a+}^{\eta\,;\,\Psi}f(t)$
$\displaystyle=y_{a}\,\frac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}+y_{a}\,\frac{\lambda}{\Gamma(\eta+\zeta)}(\Psi(t)-\Psi(a))^{\eta+\zeta-1}$
$\displaystyle\qquad+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma.$

This is the equation (5.2) for $m=1$. Now, we assume that the equation (5.2)
is hold for $m=j$ and prove that it is also true for $m=j+1$. In fact,

$\displaystyle y_{j+1}(t)$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)y_{j}(\sigma)\,d\sigma+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\left\\{y_{a}\sum_{k=0}^{j}\frac{\lambda^{k}(\Psi(\sigma)-\Psi(a))^{k\eta+\zeta-1}}{\Gamma(k\eta+\zeta)}\right.$
$\displaystyle\left.\qquad+\int_{a}^{\sigma}\Psi^{\prime}(\tau)\sum_{k=0}^{j-1}\frac{\lambda^{k}}{\Gamma((k+1)\eta)}(\Psi(\sigma)-\Psi(\tau))^{(k+1)\eta-1}f(\tau)d\tau)\right\\}\,d\sigma+\mathcal{I}_{a+}^{\eta\,;\,\Psi}f(t)$
$\displaystyle=y_{0}(t)+y_{a}\sum_{k=0}^{j}\frac{\lambda^{k+1}}{\Gamma(k\eta+\zeta)}\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}$
$\displaystyle\qquad+\sum_{k=0}^{j-1}\frac{\lambda^{k+1}}{\Gamma((k+1)\eta)}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\times$
$\displaystyle\qquad\qquad\left\\{\int_{a}^{\sigma}\Psi^{\prime}(\tau)(\Psi(\sigma)-\Psi(\tau))^{(k+1)\eta-1}f(\tau)d\tau\right\\}\,d\sigma+\mathcal{I}_{a+}^{\eta\,;\,\Psi}f(t).$

Changing the order of integration in the second last term,

$\displaystyle y_{j+1}(t)$
$\displaystyle=y_{a}\frac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}+y_{a}\sum_{k=0}^{j}\frac{\lambda^{k+1}}{\Gamma((k+1)\eta+\zeta)}(\Psi(t)-\Psi(a))^{(k+1)\eta+\zeta-1}$
$\displaystyle\qquad+\sum_{k=0}^{j-1}\frac{\lambda^{k+1}}{\Gamma((k+1)\eta)}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\Psi^{\prime}(\tau)\left\\{\int_{\tau}^{t}\Psi^{\prime}(\sigma)(\Psi(\sigma)-\Psi(\tau))^{(k+1)\eta-1}(\Psi(t)-\Psi(\sigma))^{\eta-1}f(\sigma)\,d\sigma\right\\}f(\tau)d\tau$
$\displaystyle\qquad+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma$
$\displaystyle=y_{a}\sum_{k=0}^{j+1}\frac{\lambda^{k}}{\Gamma(k\eta+\zeta)}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}$
$\displaystyle\qquad+\sum_{k=0}^{j-1}\frac{\lambda^{k+1}}{\Gamma((k+1)\eta)}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\frac{\Gamma(\eta)\Gamma((k+1)\eta)}{\Gamma((k+2)\eta)}(\Psi(t)-\Psi(\tau))^{(k+2)\eta-1}\Psi^{\prime}(\tau)f(\tau)d\tau$
$\displaystyle\qquad+\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)f(\sigma)\,d\sigma$
$\displaystyle=y_{a}\sum_{k=0}^{j+1}\frac{\lambda^{k}}{\Gamma(k\eta+\zeta)}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}$
$\displaystyle\qquad+\int_{a}^{t}\left\\{\sum_{k=0}^{j-1}\frac{\lambda^{k+1}}{\Gamma((k+2)\eta)}(\Psi(t)-\Psi(\sigma))^{(k+2)\eta-1}+\frac{(\Psi(t)-\Psi(\sigma))^{\eta-1}}{\Gamma(\eta)}\right\\}\Psi^{\prime}(\sigma)f(\sigma)\,d\sigma$
$\displaystyle=y_{a}\sum_{k=0}^{j+1}\frac{\lambda^{k}}{\Gamma(k\eta+\zeta)}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}+\int_{a}^{t}\Psi^{\prime}(\sigma)\sum_{k=0}^{j}\frac{\lambda^{k}}{\Gamma((k+1)\eta)}(\Psi(t)-\Psi(\sigma))^{(k+1)\eta-1}f(\sigma)\,d\sigma.$

This proves equation (5.2) is true for $m=j+1$. By mathematical induction the
equation (5.2) is true for all $m\in\mbox{\Bbb N}$. Taking limit as
$m\rightarrow\infty$, on both sides of (5.2), we get the solution of the
Cauchy problem (1.3)-(1.4) as

$\displaystyle y(t)$ $\displaystyle=\lim\limits_{m\rightarrow\infty}y_{m}(t)$
$\displaystyle=y_{a}\sum_{k=0}^{\infty}\frac{\lambda^{k}(\Psi(t)-\Psi(a))^{k\eta+\zeta-1}}{\Gamma(k\eta+\zeta)}+\int_{a}^{t}\Psi^{\prime}(\sigma)\sum_{k=0}^{\infty}\frac{\lambda^{k}(\Psi(t)-\Psi(\sigma))^{(k+1)\eta-1}}{\Gamma((k+1)\eta)}f(\sigma)\,d\sigma$
$\displaystyle=y_{a}(\Psi(t)-\Psi(a))^{\zeta-1}\mathcal{E}_{\eta,\,\zeta}\left(\lambda(\Psi(t)-\Psi(a))^{\eta}\right)$
$\displaystyle\qquad+\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)\mathcal{E}_{\eta,\,\eta}\left(\lambda(\Psi(t)-\Psi(\sigma))^{\eta}\right)f(\sigma)\,d\sigma.$

$\Box$

###  5.2 Linear Cauchy Type problem with Variable Coefficient

###### Theorem 5.2

Let $f\in C_{1-\zeta\,;\,\Psi}\left(\Delta,\mbox{\Bbb
R}\right),~{}\lambda\in\mbox{\Bbb R}~{}\text{and}~{}\mu>1-\eta$. Then, the
solution of Cauchy problem for homogeneous FDE with variable coefficient
involving $\Psi$-Hilfer fractional derivative (1.5)-(1.6) is given by

$\displaystyle
y\left(t\right)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\,\mathcal{E}_{\eta,\,1+\frac{\mu-1}{\eta},\,\frac{\mu+\zeta-2}{2}}\left(\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right).$

where,
$\mathcal{H}^{\Psi}_{\zeta}(t,a)=\frac{\left(\Psi(t)-\Psi(a)\right)^{\zeta-1}}{\Gamma(\zeta)}$

Proof: The equivalent fractional integral of (1.5)-(1.6) is

$y(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu-1}y(\sigma)\,d\sigma.$
(5.3)

We consider the sequences $\\{y_{m}\\}$ of successive approximation defined by

$\displaystyle y_{0}(t)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}$ $\displaystyle
y_{m}(t)$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu-1}y_{m-1}(\sigma)\,d\sigma,~{}(m=1,2,\cdots),$

and derive the solution of (5.3). By mathematical induction, we prove that

$\displaystyle
y_{m}(t)=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\\{1+\sum_{k=1}^{m}c_{k}\,\left(\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right)^{k}\right\\},$
(5.4)

where

$\displaystyle c_{k}$
$\displaystyle=\prod_{j=0}^{k-1}\frac{\Gamma(j(\eta+\mu-1)+\mu+\zeta-1)}{\Gamma(j(\eta+\mu-1)+\eta+\mu+\zeta-1)}$
$\displaystyle=\prod_{j=0}^{k-1}\frac{\Gamma(\eta[j(1+\frac{\mu-1}{\eta})+\frac{\mu+\zeta-2}{\eta}]+1)}{\Gamma(\eta[j(1+\frac{\mu-1}{\eta})+\frac{\mu+\zeta-2}{\eta}+1]+1)}.$

For the case $m=1$, we have

$\displaystyle y_{1}(t)$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu-1}y_{0}(\sigma)\,d\sigma$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu-1}y_{a}\frac{(\Psi(\sigma)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}\,d\sigma$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}+y_{a}\frac{\lambda}{\Gamma(\zeta)}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{\mu+\zeta-2}$
$\displaystyle=y_{a}\frac{(\Psi(t)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}+y_{a}\,\,\frac{\lambda}{\Gamma(\zeta)}\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}(\Psi(t)-\Psi(a))^{\eta+\mu+\zeta-2}$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\\{1+\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}\left(\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right)\right\\}.$

This is the equation (5.4) for $m=1$. Now, we assume that the equation (5.4)
is hold for $m=j$. we prove it is also hold for $m=j+1$. In fact,

$\displaystyle y_{j+1}(t)$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu-1}y_{j}(\sigma)\,d\sigma$
$\displaystyle=y_{0}(t)+\frac{\lambda}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu-1}\times$
$\displaystyle\qquad\frac{(\Psi(\sigma)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}y_{a}\,\left\\{1+\sum_{k=1}^{j}c_{k}\left(\lambda(\Psi(\sigma)-\Psi(a))^{\eta+\mu-1}\right)^{k}\right\\}\,d\sigma$
$\displaystyle=y_{0}(t)+y_{a}\frac{\lambda}{\Gamma(\zeta)}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{\mu+\zeta-2}\,d\sigma$
$\displaystyle\qquad+y_{a}\sum_{k=1}^{j}\frac{c_{k}\,\,\lambda^{k+1}}{\Gamma(\zeta)}\frac{1}{\Gamma(\eta)}\int_{a}^{t}\mathrm{Q}^{\eta}_{\Psi}(t,\sigma)(\Psi(\sigma)-\Psi(a))^{(k+1)\mu+k\eta+\zeta-k-2}\,d\sigma$
$\displaystyle=y_{0}(t)+y_{a}\frac{\lambda}{{\Gamma(\zeta)}}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{\mu+\zeta-2}$
$\displaystyle\qquad+y_{a}\sum_{k=1}^{j}\frac{c_{k}\,\,\lambda^{k+1}}{\Gamma(\zeta)}\,\,\mathcal{I}_{a+}^{\eta\,;\,\Psi}(\Psi(t)-\Psi(a))^{(k+1)\mu+k\eta+\zeta-k-2}$
$\displaystyle=y_{a}\frac{(\Psi(t)-\Psi(a))^{\zeta-1}}{\Gamma(\zeta)}+y_{a}\frac{\lambda\,\Gamma(\mu+\zeta-1)}{\Gamma(\zeta)\Gamma(\eta+\mu+\zeta-1)}(\Psi(t)-\Psi(a))^{\eta+\mu+\zeta-2}$
$\displaystyle\qquad+y_{a}\sum_{k=1}^{j}\frac{c_{k}\,\,\lambda^{k+1}}{\Gamma(\zeta)}\frac{\Gamma((k+1)\mu+k\eta+\zeta-(k+1))}{\Gamma((k+1)(\mu+\eta)+\zeta-(k+1))}(\Psi(t)-\Psi(a))^{(k+1)(\eta+\mu)+\zeta-k-2}$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\\{1+\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right.$
$\displaystyle\qquad~{}\left.+\sum_{k=1}^{j}c_{k}\,\,\frac{\Gamma((k+1)\mu+k\eta+\zeta-(k+1)}{\Gamma((k+1)(\mu+\eta)+\zeta-(k+1)}\lambda^{k+1}(\Psi(t)-\Psi(a))^{(k+1)(\eta+\mu-1)}\right\\}$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\\{1+\frac{\Gamma(\mu+\zeta-1)}{\Gamma(\eta+\mu+\zeta-1)}\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right.$
$\displaystyle\qquad~{}+\left.\sum_{k=1}^{j}c_{k}\frac{\Gamma(k(\eta+\mu-1)+\mu+\zeta-1)}{\Gamma(k(\eta+\mu-1)+\eta+\mu+\zeta-1)}\lambda^{k+1}(\Psi(t)-\Psi(a))^{(k+1)(\eta+\mu-1)}\right\\}$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\\{1+\sum_{k=1}^{j+1}c_{k}{\left(\lambda(\Psi(t)-\Psi(a))^{(\eta+\mu-1)}\right)}^{k}\right\\}$

where,

$c_{k}=\prod_{j=0}^{k-1}\frac{\Gamma(j(\eta+\mu-1)+\mu+\zeta-1)}{\Gamma(j(\eta+\mu-1)+\eta+\mu+\zeta-1)}.$

This proves equation (5.4) is true for $m=j+1$. By mathematical induction the
equation (5.4) is true for all $m\in\mbox{\Bbb N}$. Taking limit as
$m\rightarrow\infty$, on both sides of (5.4), we get the solution of the
Cauchy problem (1.5)-(1.6), given by

$\displaystyle y(t)$ $\displaystyle=\lim\limits_{m\rightarrow\infty}y_{m}(t)$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\left\\{1+\sum_{k=1}^{\infty}c_{k}[\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}]^{k}\right\\}$
$\displaystyle=\mathcal{H}^{\Psi}_{\zeta}(t,a)\,y_{a}\,\mathcal{E}_{\eta,\,1+\frac{\mu-1}{\eta},\,\frac{\mu+\zeta-2}{2}}\left(\lambda(\Psi(t)-\Psi(a))^{\eta+\mu-1}\right).$

where
$\mathcal{E}_{\eta,\,1+\frac{\mu-1}{\eta},\,\frac{\mu+\zeta-2}{2}}(\cdot)$ is
generalized (Kilbas–Saigo) Mittag–Leffler type function of three parameters.
$\Box$

##  6 Concluding remarks

In the present paper, we are able to provide a brief study of the theory of
FDEs by means of the $\Psi$-Hilfer fractional derivative. We examined the
existence along with the interval of existence, uniqueness, dependence of
solutions and Picard’s successive approximations in cases: nonlinear and
linear.

The existence results pertaining to Cauchy problem (1.1)-(1.2), was obtained
through the theorems of Schauder and Arzela-Ascoli. On the other hand, we can
prove the uniqueness and continuous dependence of the problem (1.1)-(1.2) by
making use of mathematical induction and Weissinger’s fixed point theorem. In
addition, we investigate Picard’s successive approximation in the nonlinear
case for the Cauchy problem (1.1)-(1.2) and in the linear cases: (1.3)-(1.4)
with constant coefficients and (1.5)-(1.6) with variable coefficients, making
use of the Gronwall’s inequality and mathematical induction. It should be
noted that the results obtained in the space of the weighted functions
$C_{1-\zeta;\,\Psi}\left(\Delta,\mbox{\Bbb R}\right)$ are contributions to the
fractional calculus field, in particular, the fractional analysis.

The question that arises is: will it be possible to perform the same study in
space $L_{p}([0,1],\mathbb{R})$ with the norm $||(\cdot)||_{p,\,\eta}$,
involving $\Psi$-Hilfer fractional derivative and Banach’s fixed point theorem
[39]? If the answer is yes. What are the restrictions for such results, if
any? These issues and others, are studies that are in progress and will be
published in the future.

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