On existence of solution to nonlinear $\psi-$Hilfer Cauchy-type problem

Mohammed S Abdo1 S K Panchal2 and Sandeep P Bhairat3***author for
correspondence<EMAIL_ADDRESS>
1Department of Mathematics, Hodeidah University, Al-Hodeidah-3114, Yemen.
2Department of Mathematics, Dr Babasaheb Ambedkar Marathwada University,
Aurangabad, (M.S.) India.
3Institute of Chemical Technology Mumbai, Marathwada Campus, Jalna - 431 203
(M.S.) India.

Abstract
The aim of this paper is to obtain the existence of unique solution to
nonlinear Cauchy-type problem. We consider the implicit nonlinear Cauchy-type
problem with $\psi-$Hilfer fractional derivative. The Banach fixed point
theorem is used to obtain the existence of a unique solution whereas the
generalized Gronwall inequality is used to discuss continuous data dependence
of the solution. The results obtained herein are supported with illustrative
example.

Keywords: $\psi$-Hilfer derivative, Cauchy-type problem, existence of
solution, fixed point theory, generalized Gronwall inequality.
Mathematics Subject Classification:26A33; 26D10; 34A08; 40A30.

##  1 Introduction

Fractional Calculus (FC) has glorious history of three decades and has been
developing itself in almost all branches of science and engineering. It has
been emerged and spread wings as a new field of applied mathematics research
in twenty first century due to applicability in many real world applications,
for instance see [14, 15, 22, 25, 29, 30]. Since the beginning of FC, many
fractional differential and integral operators defined and used by timely
mathematicians for serving their own purposes. During the theoretical
development of this arbitrary order calculus, Grunwald-Letnikov, Wyel, Riesz,
Liouville-Caputo, Riemann-Liouville, Hadamard, Hilfer became more famous in
physics, mechanics, material science, signal and image processing, chemical,
biological and electrical engineering, economics and mathematical modelling to
name few. For details on theory and application of FC, see [4, 11, 13, 14, 15,
17, 22, 25, 29, 30] and references therein.

In 2006, Kilbas et al [17] introduced the concept of fractional
differentiation of a function with respect to another function in the Riemann-
Liouville sense. They further defined suitable weighted spaces and studied
some of its properties by using corresponding fractional integral. Using this
idea for Caputo fractional derivative, Almaida [3] defined fractional
derivative of a function with respect to anther function called $\psi$-Caputo
derivative. Many researchers used this $\psi$-Caputo fractional derivative and
studied some of the qualitative properties of fractional differential
equations (FDEs). Recently in 2017, Sousa and Oliviera [26] proposed
interpolator of $\psi$-Riemann-Liouville and $\psi$-Caputo fractional
derivatives in Hilfer’s [14] sense of definition, and named $\psi$-Hilfer
fractional derivative. They generalized the Gronwall inequality and discussed
the data dependence of Cauchy-type problem in suitable weighted space [28],
also see [2, 16, 20, 27].

Recently, Kuchhe et al [20] studied the existence, uniqueness and continuous
dependence of solution to Cauchy-type problem using Weissinger fixed point
theorem and Picard approximation technique. They further studied linear
Cauchy-type problem for $\psi$-Hilfer differential equations with constant as
well as variable coefficients in suitable waited space of functions. For some
recent works on qualitative properties of Riemann-Liouville, Liouville-Caputo,
Hilfer and $\psi$-Hilfer FDEs, see [1, 5, 6, 7, 8, 9, 10, 12, 16, 17, 18, 19,
20, 21, 23, 24].

Motivated by above contributions and the works [20, 21], in this paper, we
consider the following implicit nonlinear Cauchy-type problem:

$D_{a^{+}}^{\alpha,\beta;\psi}u(t)=f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t)),\text{
\ }0<\alpha<1,0\leq\beta\leq 1,t>a,$ (1)
$I_{a^{+}}^{1-\gamma;\psi}u(a)=u_{a},\qquad\qquad\
u_{a}\in\mathbb{R},\quad\gamma=\alpha+\beta-\alpha\beta,$ (2)

where $D_{a^{+}}^{\alpha,\beta;\psi}$ is the $\psi$-Hilfer fractional
derivative [27], $I_{a^{+}}^{1-\gamma;\psi}$ is $\psi-$Riemann-Liouville
fractional integral,
$f:(a,b]\times\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$ is given
function satisfying some assumptions that will be specified in Section 3 and
$u_{a}$ is a constant. In this paper, we use Banach fixed point theorem to
prove the existence, uniqueness results and generalized Gronwall inequality to
discuss the continuous data dependence of Cauchy-type problem (1)-(2).

The rest of the paper is outlined as follows: In Section 2, we list some basic
definitions, preliminary facts and lemmas useful in the subsequent sections.
In section 3, we prove the equivalence of the $\psi-$Hilfer Cauchy-type
problem with Volterra integral equation. Further, we prove the existence of a
uniqueness solution to Cauchy-type problem 1)–(2. The continuous dependence on
order of differentiation and initial data of considered problem will be
discussed in Section 4. Finally, an example illustrating our main results will
be provided in last section.

##  2 Preliminaries

In this section, we list some notations, basic definitions and preliminary
facts which can be found in [17, 27]. Let
$[a,b]\subset{\mathbb{R}^{+}},(0<a<b<\infty),$ and $C[a,b],$
$AC^{n}[a,b],C^{n}[a,b]$ be the spaces of all continuous real functions,
$n$-times absolutely continuous functions, $n$-times continuously
differentiable functions on $[a,b]$, respectively. Let $L^{p}(a,b),$ $(1\leq
p<\infty)$ be a space of Lebesgue measurable functions defined on $(a,b)$. Let
us define the following norms

$\displaystyle\left\|h\right\|_{L^{p}(a,b)}$
$\displaystyle=\left[\int_{a}^{b}\left|h(t)\right|^{p}dt\right]^{\frac{1}{p}}<\infty,\text{
for any }h\in L^{p}(a,b).$ $\displaystyle\left\|h\right\|_{C[a,b]}$
$\displaystyle=\max\\{\left|h(t)\right|:t\in[[a,b]\\},\text{ for any }h\in
C[a,b],$ and $\displaystyle
AC^{n}[a,b]=\\{h:[a,b]\rightarrow\mathbb{R}|h^{(n-1)}\in AC[a,b]\\}.$

We need the following weighted spaces of continuous functions:

$C_{\gamma;\psi}[a,b]=\\{h:(a,b]\rightarrow\mathbb{R}:(\psi(t)-\psi(a))^{\gamma}h(t)\in
C[a,b]\\},\text{ \ }0\leq\gamma<1,$
$C_{\gamma;\psi}^{n}[a,b]=\\{h:(a,b]\rightarrow\mathbb{R}:h(t)\in
C^{n-1}[a,b];h^{(n)}(t)\in C_{\gamma;\psi}[a,b]\\},\text{ \
}0\leq\gamma<1,\text{ }n\in\mathbb{N}$

and

$C_{\gamma;\psi}^{\alpha,\beta}[a,b]=\\{h\in C_{\gamma;\psi}[a,b]:\text{
}D_{a^{+}}^{\alpha,\beta;\psi}h\in C_{\gamma;\psi}[a,b]\\},\text{ \
}\gamma=\alpha+\beta-\alpha\beta.$

In particular, if $n=0$, we have

$C_{\gamma;\psi}^{0}[a,b]=C_{\gamma;\psi}[a,b]$

with the norms

$\left\|h\right\|_{C_{\gamma;\psi}[a,b]}=\left\|(\psi(t)-\psi(a))^{\gamma}h(t)\right\|_{C[a,b]}=\max\\{\left|(\psi(t)-\psi(a))^{\gamma}h(t)\right|:t\in[a,b]\\},$

and

$\left\|h\right\|_{C_{\gamma;\psi}^{n}[a,b]}=\sum_{k=0}^{n-1}\left\|h^{(k)}\right\|_{C[a,b]}+\left\|h^{(n)}\right\|_{C_{\gamma;\psi}[a,b]}.$

######  Definition 1.

The familiar Mittag–Leffler functions $E_{\nu}(z)$ and $E_{\nu,\mu}(z)$ are
defined by the series:

$E_{\nu}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\nu{k}+1)}=:E_{\nu,1}(z)\text{
and }E_{\nu,\mu}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\nu{k}+\mu)},$ (3)

where $\nu,\mu\in\mathbb{C},Re(\nu)>0$ and $\Gamma(\zeta),\zeta>0,$ is Euler
gamma function given by

$\Gamma(\zeta)=\int_{0}^{\infty}e^{-t}t^{\zeta-1}dt.$ (4)

######  Definition 2.

[17] The left-sided $\psi$-Riemann-Liouville fractional integral and
fractional derivative of order $\alpha$ $(n-1<\alpha<n)$ for an integrable
function $h:[a,b]\rightarrow\mathbb{R}$ with respect to another function
$\psi:[a,b]\rightarrow\mathbb{R},$ that is an increasing differentiable
function such that $\psi^{\prime}(t)\neq 0$, for all $t\in[a,b],$
$(-\infty\leq a<b\leq+\infty),$ are respectively defined as follows:

$I_{a^{+}}^{\alpha;\psi}h(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}h(s)ds$
(5)

and

$\displaystyle D_{a^{+}}^{\alpha;\psi}h(t)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right)^{n}\text{
}I_{a^{+}}^{n-\alpha;\psi}h(t)$ (6) $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(n-\alpha)}\left(\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right)^{n}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{n-\alpha-1}h(s)ds.$

######  Definition 3.

[3] The left-sided $\psi$-Caputo fractional derivative of function $h\in
C^{n}[a,b],$ $(n-1<\alpha<n),$ $n=[\alpha]+1$ with respect to another function
$\psi$ is defined by

$\displaystyle\text{ }^{c}D_{a^{+}}^{\alpha;\psi}h(t)$ $\displaystyle=$
$\displaystyle\text{
}I_{a^{+}}^{n-\alpha;\psi}\left(\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right)^{n}h(t)$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(n-\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{n-\alpha-1}h_{\psi}^{[n]}(s)ds,$

where
$h_{\psi}^{[n]}(t)=\left(\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right)^{n}h(t)$
and $\psi$ defined as in Definition 2.

Moreover, the $\psi-$Caputo fractional derivative of function $h\in
AC^{n}[a,b]$ is determined as

${}^{c}D_{a^{+}}^{\alpha;\psi}h(t)=D_{a^{+}}^{\alpha;\psi}\left[h(t)-\sum_{k=0}^{n-1}\frac{\left[\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right]^{k}h(a)}{k!}(\psi(t)-\psi(a))^{k}\right].$

######  Definition 4.

[27] Let $n-1<\alpha<n,\,\,n\in\mathbb{N}$, with $[a,b],$ $-\infty\leq
a<b\leq+\infty,$ and $\psi\in C^{n}([a,b],\mathbb{R})$ a function such that
$\psi(t)$ is increasing and $\psi^{\prime}(t)\neq 0$, for all $t\in[a,b].$ The
$\psi$-Hilfer fractional derivative (left-sided) of function $h\in
C^{n}([a,b],\mathbb{R})$ of order $\alpha$ and type $\beta\in[0,1]$ is
determined as

$D_{a^{+}}^{\alpha,\beta;\psi}h(t)=I_{a^{+}}^{\beta(n-\alpha);\psi}\left[\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right]^{n}I_{a^{+}}^{(1-\beta)(n-\alpha);\psi}h(t),\text{
}t>a.$

In other way

$D_{a^{+}}^{\alpha,\beta;\psi}h(t)=I_{a^{+}}^{\beta(n-\alpha);\psi}D_{a^{+}}^{\gamma;\psi}h(t),\text{
}t>a,$ (7)

where

$D_{a^{+}}^{\gamma;\psi}h(t)=\left[\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right]^{n}I_{a^{+}}^{(1-\beta)(n-\alpha);\psi}h(t).$

In particular, the $\psi$-Hilfer fractional derivative of order $\alpha(0,1)$
and type $\alpha\in[0,1]$, can be written in the following form

$\displaystyle D_{a^{+}}^{\alpha,\beta;\psi}h(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\gamma-\alpha)}\int_{a}^{t}(\psi(t)-\psi(s))^{\gamma-\alpha-1}D_{a^{+}}^{\gamma;\psi}h(s)ds$
(8) $\displaystyle=$ $\displaystyle
I_{a^{+}}^{\gamma-\alpha;\psi}D_{a^{+}}^{\gamma;\psi}h(t),$

where $\gamma=\alpha+\beta-\alpha\beta$, and
$I_{a^{+}}^{\gamma-\alpha;\psi}(\cdot)$ is defined by (5) and
$D_{a^{+}}^{\gamma;\psi}h(t)=\left[\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right]I_{a^{+}}^{1-\gamma;\psi}h(t).$

######  Lemma 1.

[12] Let $\alpha>0,$ $0\leq\gamma<1$ and $h\in L^{1}(a,b)$. Then

$I_{a^{+}}^{\alpha;\psi}I_{a^{+}}^{\beta;\psi}h(t)=I_{a^{+}}^{\alpha+\beta;\psi}h(t),\text{
}a.e.\text{ }t\in[a,b].$

In particular,

(i)

if $h\in C_{\gamma;\psi}[a,b]$. Then
$I_{a^{+}}^{\alpha;\psi}I_{a^{+}}^{\beta;\psi}h(t)=I_{a^{+}}^{\alpha+\beta;\psi}h(t),$
$t\in(a,b].$

(ii)

If $h\in C[a,b].$ Then
$I_{a^{+}}^{\alpha;\psi}I_{a^{+}}^{\beta;\psi}h(t)=I_{a^{+}}^{\alpha+\beta;\psi}h(t),$
$t\in[a,b].$

######  Lemma 2.

[27] Let $\alpha>0,$ $0\leq\beta\leq 1$ and $0\leq\gamma<1.$ If $h\in
C_{\gamma;\psi}[a,b]$ then

$D_{a^{+}}^{\alpha,\beta;\psi}I_{a^{+}}^{\alpha;\psi}h(t)=h(t),\text{
}t\in(a,b].$

If $h\in C^{1}[a,b]$ then

$D_{a^{+}}^{\alpha,\beta;\psi}I_{a^{+}}^{\alpha;\psi}h(t)=h(t),\text{
}t\in[a,b].$

######  Lemma 3.

[2] Let $0<\alpha<1,$ $0\leq\beta\leq 1$ and
$\gamma=\alpha+\beta-\alpha\beta.$ If $h(t)\in
C_{1-\gamma;\psi}^{\gamma}[a,b]$ then

$I_{a^{+}}^{\gamma;\psi}D_{a^{+}}^{\gamma;\psi}h(t)=I_{a^{+}}^{\alpha;\psi}\text{
}D_{a^{+}}^{\alpha,\beta;\psi}h(t)$ (9)

and

$D_{a^{+}}^{\gamma;\psi}I_{a^{+}}^{\alpha;\psi}h(t)=D_{a^{+}}^{\beta(1-\alpha);\psi}h(t).$
(10)

######  Lemma 4.

[17] Let $t>a,\ \alpha\geq 0,$ and $\delta>0.$ Then

$I_{a^{+}}^{\alpha;\psi}(\psi(t)-\psi(a))^{\delta-1}=\frac{\Gamma(\delta)}{\Gamma(\delta+\alpha)}(\psi(t)-\psi(a))^{\alpha+\delta-1},$

and if $0<\alpha<1,$ we have

$D_{a^{+}}^{\alpha;\psi}(\psi(t)-\psi(a))^{\alpha-1}=0.$

######  Lemma 5.

Let $0<\alpha<1$, $0\leq\beta\leq 1,$ $\gamma=\alpha+\beta-\alpha\beta$ and
let $\psi\in C^{1}([a,b],\mathbb{R})$ be an increasing function such that
$\psi(t)\neq 0$, for all $t\in[a,b].$ Then

(i)

$I_{a^{+}}^{\alpha;\psi}(\cdot)$ maps $C[a,b]$ into $C[a,b].$

(ii)

$I_{a^{+}}^{\alpha;\psi}(\cdot)$ is bounded from $C_{1-\gamma;\psi}[a,b]$ into
$C_{1-\gamma;\psi}[a,b].$

(iii)

If $\gamma\leq\alpha,$ then, $I_{a^{+}}^{\alpha;\psi}(\cdot)$ is bounded from
$C_{1-\gamma;\psi}[a,b]$ into $C[a,b]$.

######  Lemma 6.

Let $\alpha>0$, $0\leq\gamma<1$ and $h\in C_{\gamma;\psi}[a,b]$. If
$\alpha>\gamma,\ $then $I_{a^{+}}^{\alpha;\psi}h$ $\in C[a,b]$ and

$I_{a^{+}}^{\alpha;\psi}h(a)=\underset{t\rightarrow
a^{+}}{\lim}I_{a^{+}}^{\alpha;\psi}h(t)=0.$

######  Lemma 7.

[12] Let $0\leq\gamma<1,$ $a<c<b,$ $h\in C_{\gamma;\psi}[a,c],$ $h\in C[c,b]$
and $h$ is continuous at $c$. Then $h\in C_{\gamma;\psi}[a,b].$

######  Theorem 1.

[27] If $h\in C^{n}[a,b],$ $n-1<\alpha<n,$ $0\leq\beta\leq 1,$ and
$\gamma=\alpha+\beta-\alpha\beta.$ Then for all $t\in(a,b],$

$I_{a^{+}}^{\alpha;\psi}D_{a^{+}}^{\alpha,\beta;\psi}h(t)=h(t)-\sum_{k=1}^{n}\frac{\left[\psi(t)-\psi(a)\right]^{\gamma-k}}{\Gamma(\gamma-k+1)}h_{\psi}^{(n-k)}I_{a^{+}}^{(1-\beta)(n-\alpha);\psi}h(a),$

In particular, if $0<\alpha<1,$ we have

$I_{a^{+}}^{\alpha;\psi}D_{a^{+}}^{\alpha,\beta;\psi}h(t)=h(t)-\frac{\left[\psi(t)-\psi(a)\right]^{\gamma-1}}{\Gamma(\gamma)}I_{a^{+}}^{(1-\beta)(1-\alpha);\psi}h(a).$

Moreover, if $h\in C_{1-\gamma;\psi}[a,b]$ and $I_{a^{+}}^{1-\gamma;\psi}h\in
C_{1-\gamma;\psi}^{1}[a,b]$ such that $0<\gamma<1.$ Then for all $t\in(a,b],$

$I_{a^{+}}^{\gamma;\psi}D_{a^{+}}^{\gamma;\psi}h(t)=h(t)-\frac{\left[\psi(t)-\psi(a)\right]^{\gamma-1}}{\Gamma(\gamma)}I_{a^{+}}^{1-\gamma;\psi}h(a).$

######  Lemma 8.

[28] (Gronwall’s lemma) Let $u,v,$ be two integrable functions and $h$ is
continuous on $[a,b]$. Let $\psi\in C[a,b]$ be an increasing function such
that $\psi^{\prime}(t)\neq 0,\forall t\in[a,b].$ Assume that $u$ and $v$ are
nonnegative $h$ is nonnegative and nondecreasing. If

$u(t)\leq
v(t)+h(t)\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}u(s)ds,$

then, for all $t\in[a,b],$we have

$u(t)\leq
v(t)+\int_{a}^{t}\sum_{k=1}^{\infty}\frac{\left[h(t)\Gamma(\alpha)\right]^{k}}{\Gamma(\alpha
k)}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha k-1}v(s)ds.$ (11)

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

$u(t)\leq v(t)E_{\alpha}[h(t)\Gamma(\alpha)(\psi(t)-\psi(a))^{\alpha}],$

where $E_{\alpha}(\cdot)$ is the Mittag-Leffler function.

######  Theorem 2.

(Banach fixed point theorem) Let $(X,d)$ be a nonempty complete metric space
and let $0\leq L<1$. If $T:X\rightarrow X$ is a mapping such that for every
$x,y\in X$

$d(T(x),T(y))\leq Ld(x,y),$ (12)

holds. Then the operator $T$ has a unique fixed point $x^{\ast}\in X.$
Furthermore, for any $k\in\mathbb{N},$ if $T^{k}$ is the sequence of operators
defined by

$T^{1}=T\text{ and }T^{k}=TT^{k-1},\text{ }(k\in\mathbb{N}\backslash\\{1\\}).$

then for any $x_{0}\in X,$
$\underset{k\rightarrow\infty}{\lim}T^{k}(x_{0})=x^{\ast}$(i.e.
$\\{T^{k}x_{0}\\}_{k=1}^{k=\infty}$ converges to the above fixed point
$x^{\ast}$).
The map $T:X\rightarrow X$ satisfying condition (12) is called a contractive
map.

We recall the following weighted spaces:

$C_{1-\gamma;\psi}^{\alpha,\beta}[a,b]=\\{h\in
C_{1-\gamma;\psi}[a,b],D_{a^{+}}^{\alpha,\beta;\psi}h\in
C_{1-\gamma;\psi}[a,b]\\}$

and

$C_{1-\gamma;\psi}^{\gamma}[a,b]=\\{h\in
C_{1-\gamma;\psi}[a,b],D_{a^{+}}^{\gamma;\psi}h\in C_{1-\gamma;\psi}[a,b]\\}.$
(13)

Since
$D_{a^{+}}^{\alpha,\beta;\psi}h=I_{a^{+}}^{\beta(1-\alpha);\psi}D_{a^{+}}^{\gamma;\psi}h$
clearly by Lemma 5, we have $C_{1-\gamma;\psi}^{\gamma}[a,b]\subset
C_{1-\gamma;\psi}^{\alpha,\beta}[a,b].$

##  3 Main results

In this section, we obtain the equivalence between the generalized Cauchy
problem (1)-(2) and the Volterra integral equation

$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}$ (14)
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}f(s,u_{s},D_{a^{+}}^{\alpha,\beta;\psi}u_{s})ds.$

Further we prove the existence and uniqueness of solution to (1)-2 by means of
Banach fixed point theorem. We need the following hypotheses and auxiliary
lemma:

(A1)

$f:(a,b]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a function
such that $f(\cdot,u,v)\in C_{1-\gamma;\psi}[a,b]$ for any $u,v\in
C_{1-\gamma;\psi}[a,b]$.

(A2)

$f(\cdot,u,v)$ satisfies the Lipschitz’s condition with respect to $u,v$ and
is bounded in a region $G\subset\mathbb{R}$ such that

$\displaystyle\left\|f(t,u_{1},v_{1})-f(t,u_{2},v_{2})\right\|_{C_{1-\gamma;\psi}[a,b]}\leq$
$\displaystyle
M\left\|u_{1}-u_{2}\right\|_{C_{1-\gamma;\psi}[a,b]}+M^{\ast}\left\|v_{1}-v_{2}\right\|_{C_{1-\gamma;\psi}[a,b]},$

for all $t\in(a,b]$ and for all $u_{i},v_{i}\in G,$ $(i=1,2)$ where $M>0$ and
$M^{\ast}<1.$

######  Lemma 9.

Let $0<\alpha<1,$ $0\leq\beta\leq 1$ and $\gamma=\alpha+\beta-\alpha\beta,$
and let $f$ satiesfy (A1). If $u\in C_{1-\gamma;\psi}^{\gamma}[a,b]$ then $u$
satisfies the cauchy problem (1)–(2) if and only if $u$ satisfies the Volterra
integral equation (14).

###### Proof.

($\Rightarrow$) Let $u\in C_{1-\gamma;\psi}^{\gamma}[a,b]$ be a solution of
Cauchy-type problem (1))–((2). We show that $u$ is also a solution of (14). By
the definition of $C_{1-\gamma;\psi}^{\gamma}[a,b],$ Lemma 5(ii), and
Definition 4, we have

$I_{a^{+}}^{1-\gamma;\psi}u\in C[a,b]\text{ and
}D_{a^{+}}^{\gamma;\psi}u=\left[\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right]I_{a^{+}}^{1-\gamma;\psi}u\in
C_{1-\gamma;\psi}[a,b],$

Since $\psi\in C^{1}[a,b]$ and by definition of $C_{\gamma;\psi}^{n}[a,b]$,
clearly $I_{a^{+}}^{1-\gamma;\psi}u\in C_{1-\gamma;\psi}^{1}[a,b].$ Hence by
using Lemma 1 and initial condition (2), for $t\in(a,b]$ we have

$\displaystyle I_{a^{+}}^{\alpha;\psi}D_{a^{+}}^{\gamma;\psi}u(t)$
$\displaystyle=$ $\displaystyle
u(t)-\frac{\left[\psi(t)-\psi(a)\right]^{\gamma-1}}{\Gamma(\gamma)}I_{a^{+}}^{1-\gamma;\psi}u(a)$
(15) $\displaystyle=$ $\displaystyle
u(t)-\frac{u_{a}}{\Gamma(\gamma)}\left[\psi(t)-\psi(a)\right]^{\gamma-1},$

From the fact that, $D_{a^{+}}^{\gamma;\psi}u\in C_{1-\gamma;\psi}[a,b],$ (8),
Lemma 3 and equation (1), we have

$\displaystyle I_{a^{+}}^{\gamma;\psi}D_{a^{+}}^{\gamma;\psi}u(t)$
$\displaystyle=$ $\displaystyle
I_{a^{+}}^{\alpha;\psi}D_{a^{+}}^{\alpha,\beta;\psi}u(t)$ (16)
$\displaystyle=$ $\displaystyle
I_{a^{+}}^{\alpha;\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))$

Comparing equations (15) and (16), we obtain the desired integral equation
(14):

$u(t)=\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}+I_{a^{+}}^{\alpha;\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t)).$

($\Leftarrow$) Assume that $u\in C_{1-\gamma;\psi}^{\gamma}[a,b]$ satisfies
the Volterra integral equation (14). We prove that $u$ also satisfies the
Cauchy-type problem (1)–(2). Applying $D_{a^{+}}^{\gamma;\psi}$ on both sides
of equation (4), in view of Lemmas 3 and 4, we have

$\displaystyle D_{a^{+}}^{\gamma;\psi}u(t)$ $\displaystyle=$ $\displaystyle
D_{a^{+}}^{\gamma;\psi}\left[\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}+I_{a^{+}}^{\alpha;\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))\right]$
(17) $\displaystyle=$ $\displaystyle
D_{a^{+}}^{\gamma;\psi}\left[\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}\right]+D_{a^{+}}^{\gamma;\psi}I_{a^{+}}^{\alpha;\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))$
$\displaystyle=$ $\displaystyle
D_{a^{+}}^{\beta(1-\alpha);\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t)).$

Since $D_{a^{+}}^{\gamma;\psi}u\in C_{1-\gamma;\psi}[a,b],$ equation (17)
implies that

$D_{a^{+}}^{\gamma;\psi}u(t)=\left[\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right]I_{a^{+}}^{1-\beta(1-\alpha);\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))\in
C_{1-\gamma;\psi}[a,b].$

As $f(\cdot,u(\cdot),D_{a^{+}}^{\alpha,\beta;\psi}u(\cdot))\in
C_{1-\gamma;\psi}[a,b],$ and from Lemma 5(ii), follows

$I_{a^{+}}^{1-\beta(1-\alpha);\psi}f(\cdot,u(\cdot),D_{a^{+}}^{\alpha,\beta;\psi}u(\cdot))\in
C_{1-\gamma;\psi}[a,b].$ (18)

By the definition of $C_{\gamma;\psi}^{n}[a,b]$, equation (18) means

$I_{a^{+}}^{1-\beta(1-\alpha);\psi}f(\cdot,u(\cdot),D_{a^{+}}^{\alpha,\beta;\psi}u(\cdot))\in
C_{1-\gamma;\psi}^{1}[a,b].$

Now apply $I_{a^{+}}^{\beta(1-\alpha);\psi}$ on both sides of equation (17),
in view of Lemmas 1 and 6, we have

$\displaystyle I_{a^{+}}^{\beta(1-\alpha);\psi}D_{a^{+}}^{\gamma;\psi}u(t)$
(19) $\displaystyle=$ $\displaystyle
I_{a^{+}}^{\beta(1-\alpha);\psi}D_{a^{+}}^{\beta(1-\alpha);\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))$
$\displaystyle=$ $\displaystyle
f(t,u(t),D_{a^{+}}^{\gamma;\psi}u(t))-\frac{I_{a^{+}}^{1-\beta(1-\alpha);\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))\mid_{t=a}}{\Gamma(\beta(1-\alpha))}\left[\psi(t)-\psi(a)\right]^{\beta(1-\alpha)-1}$
$\displaystyle=$ $\displaystyle f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t)).$

Comparing equation (19) and equation (7), we obtain the differential equtions
(1):

$D_{a^{+}}^{\alpha,\beta;\psi}u(t)=f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t)).$

Now we show that $u\in C_{1-\gamma;\psi}^{\gamma}[a,b]$ given by (14) also
satisfies the initial condition (2). To this end, multiplying both sides of
equation (14) by $I_{a^{+}}^{1-\gamma;\psi},$ use Lemma 5(iii) and Lemma 4, we
have

$\displaystyle I_{a^{+}}^{1-\gamma;\psi}u(t)$ $\displaystyle=$ $\displaystyle
I_{a^{+}}^{1-\gamma;\psi}\left[\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}+I_{a^{+}}^{\alpha;\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))\right]$
(20) $\displaystyle=$ $\displaystyle
I_{a^{+}}^{1-\gamma;\psi}\left[\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}\right]+I_{a^{+}}^{1-\gamma;\psi}I_{a^{+}}^{\alpha;\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))$
$\displaystyle=$ $\displaystyle
u_{a}+I_{a^{+}}^{1-\beta(1-\alpha);\psi}f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t))$

Taking limit $t\rightarrow a$ in equation (20), by Lemma 6, we conclude that
$I_{a^{+}}^{1-\gamma;\psi}u(a)=u_{a}.$ This completes the proof. ∎

Next, we prove the existence and uniqueness of solution for the Cauchy problem
(1)–(2) in the weighted space $C_{1-\gamma;\psi}^{\alpha,\beta}[a,b]$ by using
the Banach fixed point theorem.

######  Theorem 3.

Let $0<\alpha<1,$ $0\leq\beta\leq 1$ and $\gamma=\alpha+\beta-\alpha\beta.$
Assume that the hypotheses (A1) and (A2) are fulfilled. Then there exists a
unique solution $u$ for the Cauchy type problem (1)–(2) in the space
$C_{1-\gamma;\psi}^{\alpha,\beta}[a,b].$

###### Proof.

Let us rewrite Cauchy-type problem (1)-(2) as:

$D_{a^{+}}^{\alpha,\beta;\psi}u(t)=F_{u}(t),\text{ \
}I_{a^{+}}^{1-\gamma;\psi}u(a)=u_{a},\text{ \
}\gamma=\alpha+\beta-\alpha\beta,$

then by Lemma 9,

$u(t)=u_{0}(t)+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}F_{u}(s)ds,\
\ t>a,$ (21)

where

$u_{0}(t)=\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}$ (22)

and

$F_{u}(t)=f(t,u_{0}(t)+I_{a^{+}}^{\alpha;\psi}F_{u}(t),F_{u}(t)).$ (23)

That is

$u(t)=u_{0}(t)+I_{a^{+}}^{\alpha;\psi}F_{u}(t).$ (24)

To start with, we partition the interval $(a,b]$ into $N$ subintervals namely,
$(t_{0},t_{1}],[t_{1},t_{2}],...[t_{N-1},t_{N}]$ on which the operator $T$ is
a contraction mapping on all subintervals, where $a=t_{0}<t_{1}<...<t_{N}=b$.

Let $C_{1-\gamma;\psi}[a,b]$ is a complete metric space with the metric

$d(u_{1},u_{2})=\left\|u_{1}-u_{2}\right\|_{C_{1-\gamma;\psi}[a,b]}=\underset{t\in[a,b]}{\max}\left[(\psi(t)-\psi(a)\right]^{1-\gamma}\left|u_{1}(t)-u_{2}(t)\right|$

for $u_{1},u_{2}\in C_{1-\gamma;\psi}[a,b]$ and consider the operator
$T:C_{1-\gamma;\psi}[a,b]\rightarrow C_{1-\gamma;\psi}[a,b]$ defined by

$(Tu)(t)=u_{0}(t)+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}F_{u}(s)ds,\
\ t\in[a,b].$ (25)

Now, on the interval $(a,t_{1}]$ note that $\ C_{1-\gamma;\psi}([a,t_{1}]$ is
a complete metric space with the metric

$d(u_{1},u_{2})=\left\|u_{1}-u_{2}\right\|_{C_{1-\gamma;\psi}[a,t_{1}]}=\underset{t\in[a,t_{1}]}{\max}\left[(\psi(t)-\psi(a)\right]^{1-\gamma}\left|u_{1}(t)-u_{2}(t)\right|,$

for all $u_{1},u_{2}\in C_{1-\gamma;\psi}[a,t_{1}].$ Choose $t_{1}\in(a,b]$
such that

$\frac{\Gamma(\gamma)\left[(\psi(t_{1})-\psi(a)\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{M}{1-M^{\ast}}<1$
(26)

holds, where $M$ and $M^{\ast}$ are defined as in the condition (A2).

Next, we show that $Tu(t)\in C_{1-\gamma;\psi}[a,t_{1}]$ for all
$t\in(a,t_{1}].$ From equation (22) it follows that $u_{0}(t)\in
C_{1-\gamma;\psi}[a,t_{1}]$ since
$\left[(\psi(t)-\psi(a)\right]^{1-\gamma}u_{0}(t)=\frac{u_{a}}{\Gamma(\gamma)}\in
C[a,t_{1}]$. In view of Lemma 5(ii), then $I^{\alpha;\psi}f$ is bounded from
$C_{1-\gamma;\psi}[a,t_{1}]$ into $C_{1-\gamma;\psi}[a,t_{1}]$ which implies
$Tu\in C_{1-\gamma;\psi}[a,t_{1}],$ i.e.
$T:C_{1-\gamma;\psi}[a,t_{1}]\rightarrow C_{1-\gamma;\psi}[a,t_{1}]$.

Now, we prove that $T$ has a fixed point in $C_{1-\gamma;\psi}[a,t_{1}]$ which
is the unique solution to (1)-(2) on $(a,t_{1}].$ To this end, it is enough to
show that the operator $T$ is a contraction map. Indeed, from equation (25),
assumption (A2), Lemma’s 4, 5, and for any $u_{1},u_{2}\in
C_{1-\gamma;\psi}[a,t_{1}],$ we have

$\displaystyle|{[\psi(t)-\psi(a)]}^{1-\gamma}(Tu_{1})(t)$
$\displaystyle-{[\psi(t)-\psi(a)]}^{1-\gamma}(Tu_{2})(t)|$
$\displaystyle=\left|\left[(\psi(t)-\psi(a)\right]^{1-\gamma}I_{a^{+}}^{\alpha;\psi}(F_{u_{1}}(t)-F_{u_{2}}(t))\right|$
$\displaystyle=\left[(\psi(t)-\psi(a)\right]^{1-\gamma}\left|I_{a^{+}}^{\alpha;\psi}\left[(\psi(t)-\psi(a)\right]^{\gamma-1}\right|$
$\displaystyle\times\left[(\psi(t)-\psi(a)\right]^{1-\gamma}\left|F_{u_{1}}(t)-F_{u_{2}}(t)\right|$
$\displaystyle\leq\left\|I_{a^{+}}^{\alpha;\psi}\left[(\psi(t)-\psi(a)\right]^{\gamma-1}\right\|_{C_{1-\gamma;\psi}[a,t_{1}]}\left\|F_{u_{1}}(t)-F_{u_{2}}(t)\right\|_{C_{1-\gamma;\psi}[a,t_{1}]}$
$\displaystyle\leq\frac{\Gamma(\gamma)\left[(\psi(t_{1})-\psi(a)\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{\left[(\psi(t_{1})-\psi(a)\right]^{\gamma}}{(\psi(t_{1})-\psi(a)}\left\|F_{u_{1}}(t)-F_{u_{2}}(t)\right\|_{C_{1-\gamma;\psi}[a,t_{1}]}$
(27)

and

$\displaystyle{\|F_{u_{1}}(t)-F_{u_{2}}(t)\|}_{C_{1-\gamma;\psi}[a,t_{1}]}=$
$\displaystyle{\|f(t,u_{1}(t),F_{u_{1}}(t))-f(t,u_{2}(t),F_{u_{2}}(t))\|}_{C_{1-\gamma;\psi}[a,t_{1}]}$
$\displaystyle\leq
M{\|u_{1}-u_{2}\|}_{C_{1-\gamma;\psi}[a,t_{1}]}+M^{\ast}{F_{u_{1}}(t)-F_{u_{2}}(t)\|}_{C_{1-\gamma;\psi}[a,t_{1}]}$
$\displaystyle\leq\frac{M}{1-M^{\ast}}{\|u_{1}-u_{2}\|}_{C_{1-\gamma;\psi}[a,t_{1}]}.$
(28)

Further by (27) and (3), we get

$\displaystyle\left\|Tu_{1}-Tu_{2}\right\|_{{}_{C_{1-\gamma;\psi}[a,t_{1}]}}\leq\frac{\Gamma(\gamma)\left[(\psi(t_{1})-\psi(a)\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{\left[(\psi(t_{1})-\psi(a)\right]^{\gamma}}{(\psi(t_{1})-\psi(a)}\frac{M}{1-M^{\ast}}\left\|u_{1}-u_{2}\right\|_{C_{1-\gamma;\psi}[a,t_{1}]}.$

Since
$\frac{\left[(\psi(t_{1})-\psi(a)\right]^{\gamma}}{(\psi(t_{1})-\psi(a)}<1$
and by condition (26), we conclude that $T$ is a contraction map. By the
Banach fixed point theorem, we can deduce that $T$ has a fixed point
$u^{\ast}\in C_{1-\gamma;\psi}[a,t_{1}]$, which is just the unique solution to
the integral equation (21) on $(a,t_{1}]$. This solution $u^{\ast}$ is
obtained as a limit of a convergent sequence
$(T^{k}u_{0}^{\ast})_{k\in\mathbb{N}}$ as follows:

$\left\|T^{k}u_{0}^{\ast}-u^{\ast}\right\|_{{}_{C_{1-\gamma;\psi}[a,t_{1}]}}\rightarrow
0\text{ as }k\rightarrow\infty.$ (29)

For any function $u_{0}^{\ast}\in C_{1-\gamma;\psi}[a,t_{1}]$ and

$T^{k}u_{0}^{\ast}(t)=TT^{k-1}u_{0}^{\ast}(t)=u_{0}(t)+I_{a^{+}}^{\alpha;\psi}F_{T^{k-1}u_{0}^{\ast}}(t),\text{
}k\in\mathbb{N}.$ (30)

Set $u_{0}^{\ast}(t):=u_{0}(t)$ such that $u_{0}(t)$ is defined by (22) and we
consider

$u_{k}(t)=T^{k}u_{0}^{\ast}(t),\text{ }t\in(a,t_{1}],\text{ }k\in\mathbb{N},$
(31)

then equation (31) leads to

$u_{k}(t)=u_{0}(t)+I_{a^{+}}^{\alpha;\psi}F_{u_{k-1}}(t),\text{
}k\in\mathbb{N}.$ (32)

On the other hand, by equations 30, (31) and (32), limit in (29) can be
rewritten as

$\left\|u_{k}-u^{\ast}\right\|_{{}_{C_{1-\gamma;\psi}[a,t_{1}]}}\rightarrow
0,\text{ as }k\rightarrow\infty.$

If $t_{1}\neq b$ then we consider the interval $[t_{1},b].\ $For
$t\in[t_{1},b],$ we consider $u\in C[t_{1},b],$ and rewrite integral equation
(21) as

$\displaystyle u(t)$ $\displaystyle=$ $\displaystyle
Tu(t)=u_{01}(t)+\frac{1}{\Gamma(\alpha)}\int_{t_{1}}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}F_{u}(s)ds$
(33) $\displaystyle=$ $\displaystyle
u_{01}(t)+I_{t_{1}^{+}}^{\alpha;\psi}F_{u}(t),$

where

$\displaystyle u_{01}(t)$ $\displaystyle=$ $\displaystyle
u_{0}(t)+\frac{1}{\Gamma(\alpha)}\int_{a}^{t_{1}}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}F_{u}(s)ds$
(34) $\displaystyle=$ $\displaystyle
u_{0}(t)+I_{a^{+}}^{\alpha;\psi}F_{u}(t).$

Let us choose $t_{2}\in(t_{1},b]$ such that

$\frac{\Gamma(\gamma)\left[(\psi(t_{2})-\psi(t_{1})\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{M}{1-M^{\ast}}<1.$
(35)

As seen above, $u_{01}(t)$ is uniquely defined on $[a,t_{1}],$ and the
integral in equation (34) can be considered as a known function i.e.
$u_{01}(t)\in C_{1-\gamma;\psi}[t_{1},t_{2}]$. By equation (33), assumption
(A2), Lemma 5(i) and relation (3), for $u_{1},u_{2}\in C[t_{1},t_{2}]$ and
$t\in[t_{1},t_{2}],$ we have

$\displaystyle\left|(Tu_{1})(t)-(Tu_{2})(t)\right|=$
$\displaystyle\left|I_{t_{1}^{+}}^{\alpha;\psi}(F_{u_{1}}(t)-F_{u_{2}}(t))\right|$
$\displaystyle\leq\left\|I_{t_{1}^{+}}^{\alpha;\psi}(F_{u_{1}}(t)-F_{u_{2}}(t))\right\|_{C[t_{1},t_{2}]}$
$\displaystyle\leq\left\|I_{t_{1}^{+}}^{\alpha;\psi}\left|F_{u_{1}}(t)-F_{u_{2}}(t)\right|\right\|_{C[t_{1},t_{2}]}$
$\displaystyle\leq\frac{\Gamma(\gamma)\left[(\psi(t_{2})-\psi(t_{1})\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\left\|F_{u_{1}}(t)-F_{u_{2}}(t)\right\|_{C[t_{1},t_{2}]}$
$\displaystyle\leq\frac{\Gamma(\gamma)\left[(\psi(t_{2})-\psi(t_{1})\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{M}{1-M^{\ast}}\left\|u_{1}-u_{2}\right\|_{C[t_{1},t_{2}]}.$

From condition (35), we obtain

$\left\|Tu_{1}-Tu_{2}\right\|_{C[t_{1},t_{2}]}\leq\eta\left\|u_{1}-u_{2}\right\|_{C[t_{1},t_{2}]},$

where
$\eta=\frac{\Gamma(\gamma)\left[(\psi(t_{2})-\psi(t_{1})\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{M}{1-M^{\ast}}<1.$
This means that $T$ is a contraction. Since $F_{u}(t)\in C[t_{1},t_{2}]$ for
any function $u\in C[t_{1},t_{2}],$ then Lemma 5(i) yields that
$I_{t_{1}^{+}}^{\alpha;\psi}F_{u}(t)\in C[t_{1},t_{2}].$ Thus, $Tu\in
C[t_{1},t_{2}].$ Consequently, the operator $T$ maps $C[t_{1},t_{2}]$ into
$C[t_{1},t_{2}]$. An application of Banach fixed point theorem shows that
there exists a unique solution $u^{\ast}\in C[t_{1},t_{2}]$ to equation (21)
on the interval $[t_{1},t_{2}]$. This solution $u^{\ast}$ which is defined
explicitly as a limit of iterations of the mapping $T$. i.e.

$\left\|T^{k}u_{1}^{\ast}-u^{\ast}\right\|_{{}_{C[t_{1},t_{2}]}}\rightarrow
0,\text{ as }k\rightarrow\infty.$ (36)

Therefore, for any function $u_{1}^{\ast}\in C[t_{1},t_{2}]$ we have

$\displaystyle T^{k}u_{1}^{\ast}(t)$ $\displaystyle=$ $\displaystyle
TT^{k-1}u_{1}^{\ast}(t)$ (37) $\displaystyle=$ $\displaystyle
u_{01}(t)+I_{t_{1}^{+}}^{\alpha;\psi}F_{T^{k-1}u_{1}^{\ast}}(t),\ \
t>t_{1},\text{ }k\in\mathbb{N}.$

Set $u_{1}^{\ast}(t):=u_{01}(t)$ such that $u_{01}(t)$ is defined by equation
(34) and we consider

$u_{k}(t)=T^{k}u_{1}^{\ast}(t),\text{ }t\in[t_{1},t_{2}],\text{
}k\in\mathbb{N},$ (38)

then equation (38) leads to

$u_{k}(t)=u_{01}(t)+I_{t_{1}^{+}}^{\alpha;\psi}F_{u_{k-1}}(t),\ \
t>t_{1},\text{ }k\in\mathbb{N}.$ (39)

On the other hand, in the light of (37), (38) and (39), equation (36) can be
rewritten as

$\left\|u_{k}-u^{\ast}\right\|_{C[t_{1},t_{2}]}\rightarrow 0,\text{ as
}k\rightarrow\infty.$

Since $u_{0}^{\ast}\in C_{1-\gamma;\psi}[a,t_{1}]$ and $u_{1}^{\ast}\in
C[t_{1},t_{2}]$ implies $u_{0}^{\ast}(t_{1})=u_{1}^{\ast}(t_{1})$. Therefore,
if

$u^{\ast}(t)=\left\\{\begin{array}[]{c}u_{0}^{\ast}(t),\qquad a<t\leq
t_{1},\\\ u_{1}^{\ast}(t),\qquad t_{1}\leq t\leq t_{2},\end{array}\right.$

then by Lemma 7, $u^{\ast}\in C_{1-\gamma;\psi}[a,t_{2}].$ So $u^{\ast}$ is
the unique solution of equation (21) in $C_{1-\gamma;\psi}[a,t_{2}]$ on the
interval $(a,t_{2}].$

If $t_{2}\neq b,$ we repeat the process $N-2$ times to obtain the unique
solution $u_{k}^{\ast}\in C_{1-\gamma;\psi}[t_{k},t_{k+1}]$ for equation (21)
on the interval $[t_{k},t_{k+1}],$ where $a=t_{0}<t_{1}<....<t_{N}=b$ such
that

$\frac{\Gamma(\gamma)\left[(\psi(t_{k+1})-\psi(t_{k})\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{M}{1-M^{\ast}}<1.$

Consequently, we have the unique solution $u^{\ast}\in C_{1-\gamma;\psi}[a,b]$
of integral equation (21) given by

$u^{\ast}(t)=u_{k}^{\ast}(t),\qquad t\in(t_{k},t_{k+1}],\text{ }k=0,1,...N-1.$

Finally, we must show that the obtained unique solution $u^{\ast}\in
C_{1-\gamma;\psi}^{\alpha,\beta}[a,b].$ So, we need to show that
$D_{a^{+}}^{\alpha,\beta;\psi}u^{\ast}(\cdot)\in C_{1-\gamma;\psi}[a,b].$ We
know that $u^{\ast}\in C_{1-\gamma;\psi}[a,b]$ is the limit of the sequence
$u_{k}\in C_{1-\gamma;\psi}[a,b],$ $k\in\mathbb{N},$ i.e.

$\left\|u_{k}-u^{\ast}\right\|_{{}_{C_{1-\gamma;\psi}[a,b]}}\rightarrow
0,\text{ as }k\rightarrow\infty.$ (40)

Therefore, we use fractional differential equation (1), Lipschitz condition
(A2) and equation (40) to obtain

$\displaystyle\|D_{a^{+}}^{\alpha,\beta;\psi}u_{k}-D_{a^{+}}^{\alpha,\beta;\psi}u^{\ast}\|_{{}_{C_{1-\gamma;\psi}[a,b]}}$
$\displaystyle=\|f(t,u_{k},F_{u_{k}}(t))-f(t,u^{\ast},F_{u^{\ast}}(t))\|_{{}_{C_{1-\gamma;\psi}[a,b]}}$
$\displaystyle\leq
M\|u_{k}-u^{\ast}\|_{C_{1-\gamma;\psi}[a,b]}+M^{\ast}\|F_{u_{k}}(t)-F_{u^{\ast}}(t)\|_{C_{1-\gamma;\psi}[a,b]}$
$\displaystyle\leq[M+\frac{M}{1-M^{\ast}}]\|u_{k}-u^{\ast}\|_{C_{1-\gamma;\psi}[a,b]}.$

Taking limit as $k\rightarrow\infty,$ the right-hand side of the above
inequality tends to zero independently. i.e.

$\left\|D_{a^{+}}^{\alpha,\beta;\psi}u_{k}-D_{a^{+}}^{\alpha,\beta;\psi}u^{\ast}\right\|_{{}_{C_{1-\gamma;\psi}[a,b]}}\rightarrow
0,\text{ as }k\rightarrow\infty.$

This last expression gives us to $D_{a^{+}}^{\alpha,\beta;\psi}u^{\ast}\in
C_{1-\gamma;\psi}[a,b],$ if $D_{a^{+}}^{\alpha,\beta;\psi}u_{k}\in
C_{1-\gamma;\psi}[a,b]$ $k\in\mathbb{N}.$

Since $D_{a^{+}}^{\alpha,\beta;\psi}u_{k}(t)=f(t,u_{k-1}(t),F_{u_{k-1}}(t)),$
then by the preceding argument
$f(\cdot,u^{\ast}(\cdot),F_{u^{\ast}}(\cdot))\in C_{1-\gamma;\psi}[a,b]$ for
any $u^{\ast}\in C_{1-\gamma;\psi}[a,b].$ Consequently, $u^{\ast}\in
C_{1-\gamma;\psi}^{\alpha,\beta}[a,b]$ and the proof is complete. ∎

##  4 Continuous dependence

In this section, we discuss the continuous dependence of solution of an
implicit fractional differential equation involving Hilfer derivative with
respect to another function via the generalized Gronwall inequality.

######  Theorem 4.

Let $\psi\in C([a,b],\mathbb{R})$ a function such that it is increasing and
$\psi^{\prime}(t)\neq 0$, for all $t\in[a,b]$, and $f\in C([a,b],\mathbb{R})$
satisfying Lipschitz condition ((A2)) on $\mathbb{R}$. Let $\alpha>0,$
$\epsilon>0$ such that $0<\alpha-\epsilon<\alpha\leq 1$ and $0\leq\beta\leq
1$. For any $a\leq t\leq b$, assume that $u$ is the solution of the Cauchy-
type problem (1)-(2) and $u^{\ast}$ is the solution of the following problem:

$D_{a^{+}}^{\alpha-\epsilon,\beta;\psi}u^{\ast}(t)=f(t,u^{\ast}(t),D_{a^{+}}^{\alpha-\epsilon,\beta;\psi}u^{\ast}(t)),\text{
\ }0<\alpha<1,0\leq\beta\leq 1,t>a,$ (41)
$I_{a^{+}}^{1-\gamma-\epsilon(\beta-1);\psi}u^{\ast}(a)=u_{a}^{\ast}.\qquad\qquad\
\ \qquad\ \gamma=\alpha+\beta-\alpha\beta.$ (42)

Then, for $a<t\leq b,$

$\displaystyle\left|u^{\ast}(t)-u(t)\right|\leq
A(t)+\int_{a}^{t}\left[\sum_{k=1}^{\infty}\left(\frac{M\Gamma(\alpha-\epsilon)}{\Gamma(\alpha)(1-M^{\ast})}\right)^{k}\frac{\psi^{\prime}(s)(\psi(t)-\psi(s))^{k(\alpha-\epsilon)-1}}{\Gamma(k(\alpha-\epsilon))}A(s)\right]ds,$

where

$\displaystyle A(t)=$
$\displaystyle\left|\frac{u_{a}^{\ast}(\psi(t)-\psi(a))^{\gamma+\epsilon(\beta-1)-1}}{\Gamma(\gamma+\epsilon(\beta-1))}-\frac{u_{a}(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}\right|$
(43)
$\displaystyle+\left\|f\right\|\left|\frac{(\psi(t)-\psi(a))^{\alpha-\epsilon}}{\Gamma(\alpha-\epsilon+1)}-\frac{(\psi(t)-\psi(a))^{\alpha-\epsilon}}{\Gamma(\alpha-\epsilon)\Gamma(\alpha)}\right|+\left\|f\right\|\left|\frac{(\psi(t)-\psi(a))^{\alpha-\epsilon}}{\Gamma(\alpha-\epsilon)\Gamma(\alpha)}-\frac{(\psi(t)-\psi(a))^{\alpha}}{\Gamma(\alpha+1)}\right|,$

$\left\|f\right\|=\underset{t\in[a,b]}{\max}\left|f(t,u(t),F_{u}(t))\right|$
and $F_{u}(t)=D_{a^{+}}^{\alpha,\beta;\psi}u(t).$

###### Proof.

The Cauchy-type problems (1)-(2) and (41)-(42), have integral equations which
are given by

$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}f(s,u(s),F_{u}(s))ds,\text{
}t>a,$

and

$\displaystyle u^{\ast}(t)$ $\displaystyle=$
$\displaystyle\frac{u_{a}^{\ast}}{\Gamma(\gamma+\epsilon(\beta-1))}(\psi(t)-\psi(a))^{\gamma+\epsilon(\beta-1)-1}$
$\displaystyle+\frac{1}{\Gamma(\alpha-\epsilon)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-\epsilon-1}f(s,u^{\ast}(s),F_{u^{\ast}}(s))ds,\text{
}t>a,$

respectively. It follows that

$\displaystyle|u^{\ast}(t)-u(t)|\leq$
$\displaystyle\bigg{|}\frac{u_{a}^{\ast}(\psi(t)-\psi(a))^{\gamma+\epsilon(\beta-1)-1}}{\Gamma(\gamma+\epsilon(\beta-1))}-\frac{u_{a}(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}\bigg{|}$
$\displaystyle+\bigg{|}\frac{1}{\Gamma(\alpha-\epsilon)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-\epsilon-1}f(s,u^{\ast}(s),F_{u^{\ast}}(s))ds$
$\displaystyle-\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}f(s,u(s),F_{u}(s))ds\bigg{|}$
$\displaystyle\leq$
$\displaystyle\bigg{|}\frac{u_{a}^{\ast}(\psi(t)-\psi(a))^{\gamma+\epsilon(\beta-1)-1}}{\Gamma(\gamma+\epsilon(\beta-1))}-\frac{u_{a}(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}\bigg{|}$
$\displaystyle+\bigg{|}\int_{a}^{t}\psi^{\prime}(s)\left[\frac{(\psi(t)-\psi(s))^{\alpha-\epsilon-1}}{\Gamma(\alpha-\epsilon)}-\frac{(\psi(t)-\psi(s))^{\alpha-\epsilon-1}}{\Gamma(\alpha)}\right]f(s,u^{\ast}(s),F_{u^{\ast}}(s))ds$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-\epsilon-1}\left[f(s,u^{\ast}(s),F_{u^{\ast}}(s))-f(s,u(s),F_{u}(s))\right]ds$
$\displaystyle+\int_{a}^{t}\psi^{\prime}(s)\left[\frac{(\psi(t)-\psi(s))^{\alpha-\epsilon-1}}{\Gamma(\alpha)}-\frac{(\psi(t)-\psi(s))^{\alpha-1}}{\Gamma(\alpha)}\right]f(s,u(s),F_{u}(s))ds\bigg{|}.$

Since

$\displaystyle|F_{u^{\ast}}(t)-F_{u}(t)|=$
$\displaystyle|f(t,u^{\ast}(t),F_{u^{\ast}}(t))-f(t,u(t),F_{u}(t))|$
$\displaystyle\leq M|u^{\ast}(t)-u(t)|+M^{\ast}|F_{u^{\ast}}(t)-F_{u}(t)|$
$\displaystyle\leq\frac{M}{1-M^{\ast}}|u^{\ast}(t)-u(t)|.$

Then

$\displaystyle\left|u^{\ast}(t)-u(t)\right|$ $\displaystyle\leq$
$\displaystyle\left|\frac{u_{a}^{\ast}(\psi(t)-\psi(a))^{\gamma+\epsilon(\beta-1)-1}}{\Gamma(\gamma+\epsilon(\beta-1))}-\frac{u_{a}(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}\right|$
$\displaystyle+\left\|f\right\|\left|\frac{(\psi(t)-\psi(a))^{\alpha-\epsilon}}{\Gamma(\alpha-\epsilon+1)}-\frac{(\psi(t)-\psi(a))^{\alpha-\epsilon}}{\Gamma(\alpha)\Gamma(\alpha-\epsilon)}\right|$
$\displaystyle+\frac{M}{1-M^{\ast}}\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-\epsilon-1}\left|u^{\ast}(s)-u(s)\right|ds$
$\displaystyle+\left\|f\right\|\left[\frac{(\psi(t)-\psi(s))^{\alpha-\epsilon}}{\Gamma(\alpha)\Gamma(\alpha-\epsilon)}-\frac{(\psi(t)-\psi(s))^{\alpha}}{\Gamma(\alpha+1)}\right]$
$\displaystyle=$ $\displaystyle
A(t)+\frac{M}{1-M^{\ast}}\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-\epsilon-1}\left|u^{\ast}(s)-u(s)\right|ds,$

where $A(t)$ is defined as in (43).By applying Lemma 8, we conclude that

$|u^{\ast}(t)-u(t)|\leq
A(t)+\int_{a}^{t}\left[\sum_{k=1}^{\infty}\left(\frac{M\Gamma(\alpha-\epsilon)}{\Gamma(\alpha)(1-M^{\ast})}\right)^{k}\frac{\psi^{\prime}(s)(\psi(t)-\psi(s))^{k(\alpha-\epsilon)-1}}{\Gamma(k(\alpha-\epsilon))}A(s)\right]ds.$

∎

Next, we consider the following fractional differential equation

$D_{a^{+}}^{\alpha,\beta;\psi}u(t)=f(t,u(t),D_{a^{+}}^{\alpha,\beta;\psi}u(t)),\text{
\ }0<\alpha<1,0\leq\beta\leq 1,t>a.$ (44)

with initial condition

$I_{a^{+}}^{1-\gamma;\psi}u(a)=u_{a}+\delta.\qquad\qquad\ \ \qquad\
\gamma=\alpha+\beta-\alpha\beta,$ (45)

######  Theorem 5.

Assume that hypotheses of Theorem 3 hold. Let $u$ and $u^{\ast}$ are solutions
of the Cauchy-type problems (1)-(2) and (44)-(45) respectively. Then

$\left|u(t)-u^{\ast}(t)\right|\leq\left|\delta\right|(\psi(t)-\psi(a))^{\gamma-1}E_{\alpha,\gamma}\left[\frac{M}{(1-M^{\ast})}(\psi(t)-\psi(a))^{\alpha}\right],\text{
}t\in[a,b].$

###### Proof.

In view of Theorem 3, we have
$u(t)=\underset{k\rightarrow\infty}{\lim}u_{k}(t)$ with

$u_{0}(t)=\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}$ (46)

and

$\displaystyle u_{k}(t)$
$\displaystyle=u_{0}(t)+I_{a^{+}}^{\alpha;\psi}F_{u_{k-1}}(t)$
$\displaystyle=\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}f(s,u_{k-1}(s),F_{u_{k-1}}(s))ds.$
(47)

Clearly, we can write
$u^{\ast}(t)=\underset{k\rightarrow\infty}{\lim}u_{k}^{\ast}(t)$ with

$u_{0}^{\ast}(t)=\frac{(u_{a}+\delta)}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1},$
(48)

and

$\displaystyle u_{k}^{\ast}(t)$
$\displaystyle=u_{0}^{\ast}(t)+I_{a^{+}}^{\alpha;\psi}F_{u_{k-1}^{\ast}}(t)$
$\displaystyle=\frac{(u_{a}+\delta)}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}f(s,u_{k-1}^{\ast}(s),F_{u_{k-1}^{\ast}}(s))ds.$
(49)

By (46) and (48) we get

$\left|u_{0}(t)-u_{0}^{\ast}(t)\right|=\left|\frac{u_{a}}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}-\frac{(u_{a}+\delta)}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}\right|\leq\left|\delta\right|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}.$
(50)

Using relations (46), (4), (48), (4), the Lipschitz condition ((A2)) and the
inequality (50), we get

$\displaystyle|u_{1}(t)-u_{1}^{\ast}(t)|\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}|f(s,u_{0}(s),F_{u_{0}}(s))-f(s,u_{0}^{\ast}(s),F_{u_{0}^{\ast}}(s)|ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}M|u_{0}(t)-u_{0}^{\ast}(t)|ds$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}M^{\ast}|F_{u_{0}}(s)-F_{u_{0}^{\ast}}(s)|ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M}{1-M^{\ast}}\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}|u_{0}(t)-u_{0}^{\ast}(t)|ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M|\delta|}{1-M^{\ast}}\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}ds$
$\displaystyle=$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M|\delta|}{1-M^{\ast}}\frac{(\psi(t)-\psi(a))^{\gamma+\alpha-1}}{\Gamma(\gamma+\alpha)}.$

Hence,

$\left|u_{1}(t)-u_{1}^{\ast}(t)\right|\leq\left|\delta\right|(\psi(t)-\psi(a))^{\gamma-1}\sum_{i=0}^{1}\left(\frac{M}{1-M^{\ast}}\right)^{i}\frac{(\psi(t)-\psi(a))^{\alpha
i}}{\Gamma(\gamma+\alpha i)}.$ (51)

On the other hand, we have

$\displaystyle|u_{2}(t)-u_{2}^{\ast}(t)|\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}|f(s,u_{1}(s),F_{u_{1}}(s))-f(s,u_{1}^{\ast}(s),F_{u_{1}^{\ast}}(s)|ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}M|u_{1}(t)-u_{1}^{\ast}(t)|ds$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}M^{\ast}|F_{u_{1}}(s)-F_{u_{1}^{\ast}}(s)|ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M}{1-M^{\ast}}\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}|u_{1}(t)-u_{1}^{\ast}(t)|ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M|\delta|}{1-M^{\ast}}\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}(\psi(t)-\psi(a))^{\gamma-1}$
$\displaystyle\times\sum_{i=0}^{1}\left(\frac{M}{1-M^{\ast}}\right)^{i}\frac{(\psi(t)-\psi(a))^{\alpha
i}}{\Gamma(\gamma+\alpha i)}ds$ $\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M|\delta|}{1-M^{\ast}}\frac{1}{\Gamma(\alpha)}\sum_{i=0}^{1}\left(\frac{M}{1-M^{\ast}}\right)^{i}\frac{(\psi(t)-\psi(a))^{\alpha
i}}{\Gamma(\gamma+\alpha i)}$
$\displaystyle\times\left(\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}(\psi(t)-\psi(a))^{\gamma+\alpha
i-1}ds\right)$ $\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M|\delta|}{1-M^{\ast}}\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\alpha)\Gamma(\gamma)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}ds$
$\displaystyle+\left(\frac{M}{1-M^{\ast}}\right)^{2}|\delta|\frac{(\psi(t)-\psi(a))^{\gamma+\alpha-1}}{\Gamma(\alpha)\Gamma(\gamma+\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}ds$
$\displaystyle\leq$
$\displaystyle|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}}{\Gamma(\gamma)}+\frac{M|\delta|}{1-M^{\ast}}\frac{(\psi(t)-\psi(a))^{\gamma-1}(\psi(t)-\psi(a))^{\alpha}}{\Gamma(\gamma+\alpha)}$
$\displaystyle+\left(\frac{M}{1-M^{\ast}}\right)^{2}|\delta|\frac{(\psi(t)-\psi(a))^{\gamma-1}(\psi(t)-\psi(a))^{2\alpha}}{\Gamma(\gamma+2\alpha)}$
$\displaystyle=$
$\displaystyle|\delta|(\psi(t)-\psi(a))^{\gamma-1}\sum_{i=0}^{2}\left(\frac{M}{1-M^{\ast}}\right)^{i}\frac{(\psi(t)-\psi(a))^{\alpha
i}}{\Gamma(\gamma+\alpha i)}.$

Using the mathematical induction, we get

$\left|u_{k}(t)-u_{k}^{\ast}(t)\right|\leq\left|\delta\right|(\psi(t)-\psi(a))^{\gamma-1}\sum_{i=0}^{k}\left(\frac{M}{1-M^{\ast}}\right)^{i}\frac{(\psi(t)-\psi(a))^{\alpha
i}}{\Gamma(\gamma+\alpha i)}.$ (52)

Taking the limit $m\rightarrow\infty$ in inequation (52), we obtain

$\left|u_{k}(t)-u_{k}^{\ast}(t)\right|\leq\left|\delta\right|(\psi(t)-\psi(a))^{\gamma-1}E_{\gamma,\alpha}\left(\frac{M}{1-M^{\ast}}(\psi(t)-\psi(a))^{\alpha}\right).$

∎

##  5 An example.

Fix a kernel function $\psi:[0,1]\rightarrow\mathbb{R}$ such that
$\psi(t)=\sqrt{t+1}$. Let $\alpha=\frac{1}{2},\beta=\frac{1}{3}$ and
$\gamma=\frac{2}{3}$. Consider the following implicit Cauchy-type problem:

$D_{0^{+}}^{\frac{1}{2},\frac{1}{3},\psi}u(t)=f(t,u(t),D_{0^{+}}^{\frac{1}{2},\frac{1}{3},\psi}u(t)),\text{
\ },t>0,$ (53) $I_{0^{+}}^{\frac{1}{3},\psi}u(0)=u_{0}.\qquad
u_{0}\in\mathbb{R},$ (54)

where

$f(t,u(t),D_{0^{+}}^{\frac{1}{2},\frac{1}{3},\psi}u(t))=\frac{1}{(1+9e^{t})\left(1+\left|u(t)\right|+\left|D_{0^{+}}^{\frac{1}{2},\frac{1}{3},\psi}u(t)\right|\right)}.$

It is easy to see that
$f(t,u(t),D_{0^{+}}^{\frac{1}{2},\frac{1}{3},\psi}u(t)\in
C_{1-\frac{2}{3},\psi}[0,1].$ Moreover, for any
$x,y,x^{\ast},y^{\ast}\in\mathcal{\mathbb{R^{+}}}$ and $t\in(0,1],$ we have

$\displaystyle|f(t,x,y)-f(t,x^{\ast},$ $\displaystyle
y^{\ast})|_{C_{1-\frac{2}{3},\psi}[0,1]}=\max_{t\in[0,1]}[\psi(t)-\psi(0)]^{\frac{1}{3}}|f(t,x,y)-f(t,x^{\ast},y^{\ast})|$
$\displaystyle=\max_{t\in[0,1]}[\psi(t)-\psi(0)]^{\frac{1}{3}}\bigg{|}\frac{1}{(1+9e^{t})(1+|x|+|y|)}-\frac{1}{(1+9e^{t})(1+|x^{\ast}|+|y^{\ast}|)}\bigg{|}$
$\displaystyle\leq\frac{1}{10}\max_{t\in[0,1]}[\psi(t)-\psi(0)]^{\frac{1}{3}}\bigg{|}\frac{|x^{\ast}|+|y^{\ast}|-|x|-|y|}{(1+|x|+|y|)(1+|x^{\ast}|+|y^{\ast}|)}\bigg{|}$
$\displaystyle\leq\frac{1}{10}\max_{t\in[0,1]}[\psi(t)-\psi(0)]^{\frac{1}{3}}[|x-x^{\ast}|+|y-y^{\ast}|]$
$\displaystyle=\frac{1}{10}|x-x^{\ast}|_{C_{1-\frac{2}{3},\psi}[0,1]}+\frac{1}{10}|y-y^{\ast}|_{C_{1-\frac{2}{3},\psi}[0,1]}.$

Therefore, all the conditions of Theorem 3 are satisfied with
$M=M^{\ast}=\frac{1}{10}$. It is easy to check that the condition (26) holds,
i.e.

$\left(\frac{\Gamma(\gamma)\left[(\psi(t)-\psi(a)\right]^{\alpha}}{\Gamma(\gamma+\alpha)}\frac{M}{1-M^{\ast}}\right)=\left(\frac{\Gamma(\frac{2}{3})\left[\sqrt{(t+1}-1\right]^{\frac{1}{2}}}{\Gamma(\frac{2}{3}+\frac{1}{2})}\frac{\frac{1}{10}}{1-\frac{1}{10}}\right)<1$

for all $t\in[0,1].$ Now we can apply Theorem 3 and conclude that Cauchy-type
problem (53)-(54) has a unique solution in
${}_{C_{\frac{1}{3},\sqrt{t+1}}[0,1]}.$

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