# Existence and Ulam-Hyers Mittag-Leffler stability of $\psi$-Hilfer
fractional functional integrodifferential equation

Mohammed S. Abdo, Abdulkafi M. Saeed, Satish K. Panchal  Mohammed S. Abdo
Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada University,
Aurangabad, (M.S),431001, India<EMAIL_ADDRESS>Abdulkafi M. Saeed
Department of Mathematics, College of Science, Qassim University
Buraydah-51452, Saudi Arabia<EMAIL_ADDRESS>Satish K. Panchal
Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada University,
Aurangabad, (M.S),431001, India<EMAIL_ADDRESS>A PREPRINT

###### Abstract.

This paper is committed to establishing the assumptions essential for the
existence and uniqueness results of a fractional functional
integrodifferential equation (FFIDE) having a derivative of generalized Hilfer
type. Using the Picard operator method, and Banach fixed point theorem, we
obtain the existence and uniqueness solution to the proposed problem. Along
with this, the Ulam-Hyers Mittag-Leffler (UHML) stability is discussed via
Pachpatte’s inequality. For supporting our results, an illustrative example
will be introduced.

###### Key words and phrases:

fractional differential equation; finite delay; Picard operator; Ulam-Hyers-
stability

###### 2010 Mathematics Subject Classification:

34A08, 34A12, 34A60, 37K45

##  1\. Introduction

The theory of fractional differential equations is much significant due to
their nonlocal property is convenient to describe memory phenomena in many
applied fields such as biological sciences, physical sciences, economics,
engineering, and in fluid dynamic traffic model. The existence, stability and
control theory to fractional differential equations have been emerging as an
important area of investigation in the last few decades. For details, we refer
the reader to monographs of Samko [21], Podlubny [19], Hilfer [11], Kilbas
[13], and the papers [1, 4, 5, 9, 27] and the references therein. For the
recent review of the fractional calculus operators, see [7, 6, 5, 23, 13, 28,
29, 30, 31].

On the other hand, the stability results of functional differential equations
have been strongly developed. Very significant contributions about this topic
were introduced by Ulam [25], Hyers [12] and this type of stability called
Ulam-Hyers stability. Thereafter improvement of Ulam-Hyers stability provided
by Rassias [20] in 1978.

Most recently a fractional derivative with kernel of function is introduced by
Almedia in [7], Da Sousa and De Oliveira in [23]. The recent development of
$\psi$-fractional differential equations and the theoretical analysis can be
seen in [2, 3, 10, 14, 16, 22, 27]. For the recent review of fractional
functional differential equations, we will survey some of the works as
follows:

D. Otrocol, V. Ilea in [18] studied the Ulam–Hyers stability and generalized
Ulam–Hyers–Rassias stability for the following delay differential equation

$\left\\{\begin{array}[]{c}u^{\prime}(t)=f(t,u(t),u(h(t)),\text{ \
}t\in[a,b],\\\ u(t)=\psi(t),\qquad t\in[a-h,a].\end{array}\right.$

J. Wang and Y. Zhang [26], proved some results of existence, uniqueness, and
Ulam–Hyers–Mittag-Leffler stable of Caputo-type fractional-order delay
differential equation

$\left\\{\begin{array}[]{c}{}^{C}D_{0^{+}}^{\alpha}u(t)=f(t,u(t),u(h(t)),\text{
\ }t\in[0,d],\\\ u(t)=\psi(t),\qquad t\in[-h,0].\end{array}\right.$ (1.1)

Liu et al. in [16] established the existence, uniqueness, and
Ulam–Hyers–Mittag-Leffler stability of solutions to a class of $\psi$-Hilfer
fractional-order delay differential equations

$\left\\{\begin{array}[]{c}{}^{H}D_{0^{+}}^{\alpha,\beta;\psi}u(t)=f(t,u(t),u(h(t)),\text{
\ }t\in(0,d],\\\ I_{0^{+}}^{1-\gamma;\psi}u(0^{+})=u_{0}\in\mathbb{R},\\\
u(t)=\varphi(t),\qquad t\in[-h,0].\end{array}\right.$ (1.2)

K.D. Kucche, and P.U. Shikhare in [15] studied the existence, uniqueness of a
solution and Ulam type stabilities for Volterra delay integro-differential
equations on a finite interval

$\left\\{\begin{array}[]{c}u^{\prime}(t)=f\left(t,y(t),y(g(t)),\int_{0}^{t}h(t,s,y(s),y(g(s))ds\right),\text{
\ }t\in[0,b],\\\ u(t)=\varphi(t),\qquad t\in[-r,0],\
0<r<\infty,\end{array}\right.$ (1.3)

Motivated by aforesaid works, in this paper, we establish the existence,
uniqueness and UHML stability of solutions for $\psi$-Hilfer fractional-order
functional integrodifferential equations of the form:

$\left\\{\begin{array}[]{c}{}^{H}D_{0^{+}}^{\alpha,\beta;\psi}u(t)=f\left(t,y(t),y(g(t)),\int_{0}^{t}h(t,s,y(s),y(g(s))ds\right),\text{
\ }t\in(0,b],\\\ I_{0^{+}}^{1-\gamma;\psi}u(0^{+})=u_{0},\qquad 0<\gamma\leq
1\\\ u(t)=\varphi(t),\qquad t\in[-r,0],\ 0<r<\infty,\end{array}\right.$ (1.4)

where $0<\alpha<1,$ $0\leq\beta\leq 1,$,
${}^{H}D_{0^{+}}^{\alpha,\beta;\psi}(\cdot)$ and
$I_{0^{+}}^{1-\gamma;\psi}(\cdot)$ are $\psi-$Hilfer fractional derivative of
order$\ \left(\alpha,\beta\right),$ and $\psi-$Riemann–Liouville fractional
integral of order $1-\gamma$ $(\gamma=\alpha+\beta(1-\alpha),$ respectvely,
$\varphi\in C([-r,0],\mathbb{R}),$
$f:[0,b]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$,
$h:[0,b]\times[0,b]\times\mathbb{R}\rightarrow\mathbb{R}$ and
$g:[0,b]\rightarrow[-r,0]$ are continuous functions, and $g(t)\leq t$.

We apply Picard’s operator method, Banach fixed point theorem, and the
Pachpatte’s inequality to achieve our results. The results obtained in this
paper are more general than the known results and include the study of [8, 18,
26, 16, 15] as special cases of (1.4).

The main contributions are as follows: In section 2, some preliminary results
and notations are provided which are useful in the sequel. In Section 3, we
study the existence and uniqueness results on the problem (1.4) by means of
Banach fixed point theorem and Picard operator method. Section 4 is devoted to
discussing the UHML stability result via Pachpatte’s inequality. Finally, an
illustrative example is provided in the last section.

##  2\. Preliminaries

In this section, we will present some preliminaries and lemmas of fractional
calculus theory and nonlinear analysis which are used in this paper. Let
$\left[a,b\right]\subset\mathbb{R}^{+}$ with $(0<a<b<\infty)$ and let
$C\left[a,b\right]$ be the space of continuous function,
$\omega:\left[a,b\right]\rightarrow\mathbb{R}$ with the norm
$\left\|\omega\right\|_{C}=\max\\{\left|\omega(t)\right|:a\leq t\leq b\\}.$ We
consider the weighted spaces $C_{1-\gamma;\psi}[a,b]$ as follows

$C_{1-\gamma;\psi}\left[a,b\right]=\left\\{\omega:(a,b]\rightarrow\mathbb{R};\text{
}\left[\psi(t)-\psi(a)\right]^{1-\gamma}\omega(t)\in
C\left[a,b\right]\right\\},$

where $0<\gamma<1,$ $n\in\mathbb{N},$ with the norm

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

for $0<\gamma<1,$ $\delta\geq 0.$ Denote $E_{\alpha}(\cdot)\ $and
$E_{\alpha,\beta}(\cdot)$ by the Mittag-Leffler functions defined by

$E_{\alpha}(\omega)=\sum\limits_{k=0}^{\infty}\frac{\omega^{k}}{\Gamma(\alpha
k+1)},\ \ \omega\in\mathbb{C},\ \ \Re(\alpha)>0.$
$E_{\alpha,\beta}(\omega)=\sum\limits_{k=0}^{\infty}\frac{\omega^{k}}{\Gamma(\alpha
k+\beta)},\ \ \omega\in\mathbb{C},\ \ \Re(\alpha),\Re(\beta)>0.$

######  Definition 2.1.

[13, 7] Let $\alpha>0$ be a real number and
$\omega:[a,b]\rightarrow\mathbb{R}$ a function. Given another function
$\psi\in C^{1}\left[a,b\right]$ be an increasing having a continuous
derivative $\psi^{\prime}$ on $(a,b)$. Then

The left-sided $\psi-$Riemann-Liouville fractional integral of $\omega,$ of
order $\alpha$ is defined by

$I_{a^{+}}^{\alpha,\psi}\omega(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1}\omega(s)ds.$

The left-sided $\psi-$Riemann-Liouville fractional derivative of $\omega\in
C^{n}[a,b]$ of order $\alpha$ is defined by

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

The left-sided $\psi-$Caputo fractional derivative of $\omega,$ of order
$\alpha$ is defined by

${}^{C}D_{a^{+}}^{\alpha,\psi}\omega(t)=D_{a^{+}}^{\alpha,\psi}\left(\omega(t)-\sum_{k=0}^{n-1}\frac{\omega_{\psi}^{[k]}(a)}{k!}(\psi(t)-\psi(s))^{k}\right),$

where $n=[\alpha]+1$ for $\alpha\notin\mathbb{N},$ and
$\omega_{\psi}^{[k]}(t)=\left(\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right)^{k}\omega(t).$
In particular, if $n=\alpha$, we have
${}^{C}D_{a^{+}}^{\alpha,\psi}\omega(t)=\omega_{\psi}^{[n]}(t).$

######  Lemma 2.2.

[13] Let $\alpha>0$ and $\beta>0$. Then, we have the following semigroup
property given by

$I_{a^{+}}^{\alpha,\psi}I_{a^{+}}^{\beta,\psi}(\cdot)=I_{a^{+}}^{\alpha+\beta,\psi}(\cdot)$

######  Definition 2.3.

[23] Let $n-1<\alpha<n$ $(n\in\mathbb{N})$, and $\omega,\psi\in C^{n}[a,b]$
such that $\psi$ is an increasing with $\psi^{\prime}(t)\neq 0$ for all
$t\in[a,b].$ Then the left-sided $\psi$-Hilfer fractional derivative of
$\omega$ of order $\alpha$ and type $0\leq\beta\leq 1$ is defined by

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

One has,

${}^{H}D_{a^{+}}^{\alpha,\beta,\psi}\omega(t)=I_{a^{+}}^{\beta(n-\alpha);\psi}D_{a^{+}}^{\gamma;\psi}\omega(t),$

where

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

######  Remark 2.4.

From Definition 2.3, we observe that, if $0<\alpha<1,$ $0\leq\beta\leq 1$ and
$\gamma=\alpha+\beta(1-\alpha),$ then

${}^{H}D_{a^{+}}^{\alpha,\beta,\psi}\omega(t)=I_{a^{+}}^{\beta(1-\alpha);\psi}\left(\frac{1}{\psi^{\prime}(t)}\frac{d}{dt}\right)I_{a^{+}}^{(1-\beta)(1-\alpha);\psi}\omega(t).$

One has,

${}^{H}D_{a^{+}}^{\alpha,\beta,\psi}\omega(t)=I_{a^{+}}^{\beta(1-\alpha);\psi}D_{a^{+}}^{\gamma;\psi}\omega(t)=I_{a^{+}}^{\gamma-\alpha;\psi}D_{a^{+}}^{\gamma;\psi}\omega(t),$

where

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

Now, we introduce the weighted spaces

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

and

$C_{1-\gamma;\psi}^{\gamma}[a,b]=\\{\omega\in
C_{1-\gamma;\psi}[a,b],D_{a^{+}}^{\gamma;\psi}\omega\in
C_{1-\gamma;\psi}[a,b]\\},$ (2.1)

where $0<\gamma<1.$ Since
$D_{a^{+}}^{\alpha,\beta;\psi}\omega=I_{a^{+}}^{\beta(1-\alpha);\psi}D_{a^{+}}^{\gamma;\psi}\omega,$
it is obvious that, $C_{1-\gamma;\psi}^{\gamma}[a,b]\subset
C_{1-\gamma;\psi}^{\alpha,\beta}[a,b].$

######  Lemma 2.5.

[2] Let $\alpha>0$, $\beta>0$ and $\gamma=\alpha+\beta(1-\alpha).$ If
$\omega\in C_{1-\gamma;\psi}^{\gamma}[J,\mathbb{R}]$, then

$I_{a^{+}}^{\gamma;\psi}D_{a^{+}}^{\gamma;\psi}\omega=I_{a^{+}}^{\alpha;\psi}D_{a^{+}}^{\alpha,\beta;\psi}\omega,$

and

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

######  Theorem 2.6.

[23] Let $\omega\in C^{1}[a,b],$ $0<\alpha<1$, and $0\leq\beta\leq 1$. Then

${}^{H}D_{a^{+}}^{\alpha,\beta,\psi}I_{a^{+}}^{\alpha,\psi}\omega(t)=\omega(t).$

######  Theorem 2.7.

[13, 23] Let $\alpha,\sigma>0$, and $0\leq\beta\leq 1.$ Then

$I_{a^{+}}^{\alpha,\psi}\left[\psi(t)-\psi(a)\right]^{\sigma-1}=\frac{\Gamma(\sigma)}{\Gamma(\alpha+\sigma)}(\psi(t)-\psi(a))^{\alpha+\sigma-1}$

and

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

######  Theorem 2.8.

[23] If $0<\alpha<1$, $0\leq\beta\leq 1$, $0<\gamma<1\ $and that $\ \omega\in
C_{1-\gamma}[a,b],$ $I_{a^{+}}^{1-\gamma;\psi}\omega\in
C_{1-\gamma}^{1}[a,b],$ then

$I_{a^{+}}^{\alpha;\psi}\
{}^{H}D_{a^{+}}^{\alpha,\beta,\psi}\omega(t)=\omega(t)-\frac{I_{0^{+}}^{1-\gamma;\psi}\omega(a)}{\Gamma(\gamma)}(\psi(t)-\psi(a))^{\gamma-1}.$

######  Theorem 2.9.

[23] Let $\omega\in C_{\gamma}[a,b],$ $0<\gamma<\alpha<1$. Then we have

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

######  Definition 2.10.

[26] Let $(X,d)$ be a metric space. Now $T:X\longrightarrow X$ is a Picard
operator if there exists $u^{\ast}\in X$ such that $F_{T}=$ $u^{\ast}$ where
$F_{T}$ $=\\{u\in X:T(u)=u\\}$ is the fixed point set of $T$, and the sequence
$(T^{n}(u_{0}))_{n\in\mathbb{N}}$ converges to $u^{\ast}$ for all $u_{0}\in
X$.

######  Lemma 2.11.

[26] Let $(X,d,\leq)$ be an ordered metric space, and let $T:X\longrightarrow
X$ be an increasing Picard operator with $F_{T}$ = $\\{u_{T}^{\ast}\\}$. Then
for $u\in X,$ $u\leq T(u)$ implies$\ u\leq u_{T}^{\ast}.$

######  Lemma 2.12.

([24]) (Pachpatte’s inequality). Let $x(t),$ $p(t)$ and $q(t)$ be nonnegative
continuous functions defined on $\mathbb{R}^{+}$,and $\eta(t)$ be a positive
and nondecreasing continuous function defined on $\mathbb{R}^{+}$ for which
the inequality

$x(t)\leq\eta(t)+\int_{0}^{t}p(s)\left[x(s)+\int_{0}^{s}q(\sigma)x(\sigma)d\sigma\right]ds,$

holds for $t\in\mathbb{R}^{+}$. Then

$x(t)\leq\eta(t)\left[1+\int_{0}^{t}p(s)\exp\int_{0}^{s}\left[p(\sigma)+q(\sigma)\right]d\sigma\right]ds,$
(2.2)

for $t\in\mathbb{R}^{+}.$

Lemma 2.7 Let
$f:(0,b]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
be a continuous function. Then the problem

$\begin{array}[]{c}{}^{H}D_{0^{+}}^{\alpha,\beta;\psi}u(t)=F_{u,g,h}(s),\text{
}t\in(0,b]\\\ I_{0^{+}}^{1-\gamma;\psi}u(0)=u_{0},\end{array}$

is equivalent to integral equation

$u(t)=\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{u,g,h}(s)ds,$
(2.3)

where
$\mathcal{H}_{\psi}^{\gamma}(t,0):=\frac{(\psi(t)-\psi(0))^{\gamma-1}}{\Gamma(\gamma)}$,
$\mathcal{N}_{\psi}^{\alpha}(t,s):=\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1},$and

$F_{u,g,h}(s):=f\left(s,u(s),u(g(s)),\int_{0}^{s}h(s,\tau,u(\tau),u(g(\tau))d\tau\right).$

##  3\. Main results

In this section, we present results on the existence, uniqueness, and UHML
stability of solutions to the problem (1.4). First, we introduce the following
hypotheses:

(H1):

$g:(0,b]\rightarrow[-r,0]$ is continuous function with $g(t)\leq t$.

(H2):

$f:(0,b]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$,
$h:(0,b]\times(0,b]\times\mathbb{R}\rightarrow\mathbb{R}$ are two continuous
functions, and there exist $L_{F},L_{h}>0$ such that

$\left|f(t,u_{1},u_{2},u_{3})-f(t,v_{1},v_{2},v_{3})\right|\leq L_{f}\text{
}\left[\left|u_{1}-v_{1}\right|\text{
}+\left|u_{2}-v_{2}\right|+\left|u_{3}-v_{3}\right|\right],$
$\left|h(t,u_{1},u_{2})-h(t,v_{1},v_{2})\right|\leq L_{h}\text{
}\left[\left|u_{1}-v_{1}\right|\text{ }+\left|u_{2}-v_{2}\right|\right],$

for all $t\in(0,b],$ $u_{i},v_{i}\in\mathbb{R},i=1,2,3.$

$($H${}_{3})$:

The following inequality holds

$\Theta:=2L_{f}\left(\frac{\mathcal{B}(\gamma,\alpha)}{\Gamma(\alpha)}+\frac{L_{h}}{\zeta\gamma}\frac{\mathcal{B}(\gamma+1,\alpha)}{\Gamma(\alpha)}\right)\left[\psi(b)-\psi(0)\right]^{\alpha+1}<1,$

where $\mathcal{B}(\cdot,\cdot)$ is a beta function and
$\zeta=\sup_{s\in(0,b]}\left|\psi^{\prime}(s)\right|.$

Next, before starting and proving our results, we need to the following
remarks.

######  Remark 3.1.

A function $v\in C_{1-\gamma,\psi}[0,b]$ is a solution of the inequality

$\left|{}^{H}D_{0^{+}}^{\alpha,\beta,\psi}v(t)-F_{v,g,h}(t)\right|\leq\varepsilon
E_{\alpha}(\psi(t)-\psi(0))^{\alpha},\ t\in(0,b],$ (3.1)

if and only if there exists a function $\eta_{v}\in C_{1-\gamma,\psi}[0,b]$
such that

(i):

$\left|\eta_{v}(t)\right|\leq\varepsilon
E_{\alpha}((\psi(t)-\psi(0))^{\alpha}),\ \ t\in(0,b];$

(ii):

${}^{H}D_{0^{+}}^{\alpha,\beta,\psi}v(t)=F_{v,g,h}(t)+\eta_{v}(t),$ $\
t\in(0,b],$ where

$F_{v,g,h}(t):=f\left(t,v(t),v(g(t)),\int_{0}^{t}h(t,s,v(s),v(g(s))ds\right).$

######  Definition 3.2.

Problem (1.4) is UHML stable with respect to
$E_{\alpha}((\psi(t)-\psi(0))^{\alpha})$ if there exists
$C_{{}_{E_{\alpha}}}>0$ such that, for each $\varepsilon>0$ and for each
solution $v\in C[-r,b]$ to the inequality (3.1), there exists a solution
$u\in$ $C[-r,b]$ to first equation of (1.4) with

$\left|v(t)-u(t)\right|\leq C_{{}_{E_{\alpha}}}\varepsilon
E_{\alpha}((\psi(t)-\psi(0))^{\alpha}),\ \ t\in\left[-r,b\right].$

By Lemma 2.12 and above remark, for $t\in(0,b]$ we have

$\displaystyle v(t)$ $\displaystyle=$
$\displaystyle\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{v,g,h}(s)ds$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\eta_{v}(s)ds.$

######  Remark 3.3.

Let $v\in C_{1-\gamma,\psi}[0,b]$ be a solution of the inequality (3.1). Then
$v$ is a solution of the following integral inequality

$\displaystyle\left|v(t)-\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{v,g,h}(s)ds\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\left|\eta_{v}(s)\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)E_{\alpha}((\psi(s)-\psi(0))^{\alpha})ds$
$\displaystyle=$
$\displaystyle\varepsilon\sum\limits_{k=0}^{\infty}\frac{1}{\Gamma((k+1)\alpha+1)}I_{0^{+}}^{\alpha;\psi}\left[\psi(s)-\psi(0)\right]^{\alpha
k}$ $\displaystyle=$
$\displaystyle\varepsilon\sum\limits_{k=0}^{\infty}\frac{\left[\psi(t)-\psi(0)\right]^{\alpha(k+1)}}{\Gamma((k+1)\alpha+1)}$
$\displaystyle\leq$
$\displaystyle\varepsilon\sum\limits_{k=0}^{\infty}\frac{\left(\left[\psi(t)-\psi(0)\right]^{\alpha}\right)^{n}}{\Gamma(n\alpha+1)}$
$\displaystyle=$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha}).$

Now, we are ready to prove our results on the problem (1.4).

######  Theorem 3.4.

Assume that (H1)-(H3) are fulfilled. Then

  1. (1)

The $\psi-$Hilfer problem (1.4) has a unique solution in $C[-r,b]\cap
C_{1-\gamma;\psi}[0,b]$.

  2. (2)

The first equation of (1.4) is UHML stable.

###### Proof.

(1) In view of Lemma 2.12, we get that (1.4) is equivalent to the following
system

$u(t)=\left\\{\begin{array}[]{c}\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{u,g,h}(s)ds,\
\ t\in[0,b],\\\ \varphi(t)\qquad\ \qquad\qquad\qquad\qquad\qquad\ \ \qquad
t\in[-r,0].\end{array}\right.$ (3.2)

where
$\mathcal{H}_{\psi}^{\gamma}(t,0):=\frac{(\psi(t)-\psi(0))^{\gamma-1}}{\Gamma(\gamma)}$,
$\mathcal{N}_{\psi}^{\alpha}(t,s):=\psi^{\prime}(s)(\psi(t)-\psi(s))^{\alpha-1},$and

$F_{u,g,h}(s):=f\left(s,u(s),u(g(s)),\int_{0}^{s}h(s,\tau,u(\tau),u(g(\tau))d\tau\right).$
(3.3)

The existence of a solution for the problem (1.4) can be transformed into a
fixed point problem in $C[-r,b]$ for the operator
$\mathcal{G}_{f}:C[-r,b]\longrightarrow C[-r,b]$ defined by

$\mathcal{G}_{f}u(t)=\left\\{\begin{array}[]{c}\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{u,g,h}(s)ds,\
\ t\in[0,b],\\\ \varphi(t)\qquad\ \qquad\qquad\qquad\qquad\qquad\ \ \qquad
t\in[-r,0].\end{array}\right.$ (3.4)

We remark that for any continuous function $F_{u,g,h},$ the operator
$\mathcal{G}_{f}\mathcal{\ }$is also continuous. Indeed,

Case 1. For all $t,t+\epsilon\in(0,b],$ we have

$\displaystyle\left|\mathcal{G}_{f}u(t+\epsilon)-\mathcal{G}_{f}u(t)\right|$
$\displaystyle=$
$\displaystyle\left|\mathcal{H}_{\psi}^{\gamma}(t+\epsilon,0)u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t+\epsilon}\mathcal{N}_{\psi}^{\alpha}(t+\epsilon,s)F_{u,g,h}(s)ds\right.$
$\displaystyle\left.-\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{u,g,h}(s)ds\right|$
$\displaystyle\rightarrow$ $\displaystyle 0\text{ as }\epsilon\longrightarrow
0.$

Case 2. For all $t,t+\epsilon\in C[-r,0],$ we have

$\left|\mathcal{G}_{f}u(t+\epsilon)-\mathcal{G}_{f}u(t)\right|=\left|\varphi(t+\epsilon)-\varphi(t)\right|\longrightarrow
0,\text{ as }\epsilon\longrightarrow 0.$

Next, we show that $\mathcal{G}_{f}:C[-r,b]\rightarrow C[-r,b]$ defined by
(3.4) is a contraction mapping on $C[-r,b]$ with respect to the weighted norm
$\left\|\cdot\right\|_{C_{1-\gamma;\psi}}.$

Case 1. For all $t\in[-r,0],$and for each $u,v\in C\left[-r,b\right],$ we have

$\left|\mathcal{G}_{f}u(t)-\mathcal{G}_{f}v(t)\right|=0.$

Case 2. From our assumption (3.3), and for each $t\in(0,b],$ $u,v\in
C_{1-\gamma,\psi}\left[0,b\right],$ we have

$\displaystyle\left|F_{u,g,h}(s)-F_{v,g,h}(s)\right|$ $\displaystyle=$
$\displaystyle\left|f\left(s,u(s),u(g(s)),\int_{0}^{s}h(s,\tau,u(\tau),u(g(\tau))d\tau\right)\right.$
(3.8)
$\displaystyle\left.-f\left(s,v(s),v(g(s)),\int_{0}^{s}h(s,\tau,v(\tau),v(g(\tau))d\tau\right)\right|$
$\displaystyle\leq$ $\displaystyle
L_{f}\left[\begin{array}[]{c}\left|u(s)-v(s)\right|\text{
}+\left|u(g(s)-v(g(s)\right|\\\
+\int_{0}^{s}\left|h(s,\tau,u(\tau),u(g(\tau))-h(s,\tau,v(\tau),v(g(\tau))\right|d\tau\end{array}\right]$
$\displaystyle\leq$ $\displaystyle
2L_{f}\left[\left[\psi(s)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]$
$\displaystyle+2L_{f}L_{h}\int_{0}^{s}\text{
}\left[\left[\psi(\tau)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]d\tau.$

Since $\psi\in C^{1}[0,b],$ there exists a constant $\zeta\neq 0$ such that
$\underset{\tau\in(0,b]}{sup}\left|\psi^{\prime}(\tau)\right|\leq\zeta.$
Therefore

$\displaystyle\int_{0}^{s}\text{
}\left[\left[\psi(\tau)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]d\tau$
(3.9) $\displaystyle=$ $\displaystyle\int_{0}^{s}\text{
}\left[\psi^{\prime}(\tau)\left[\psi(\tau)-\psi(0)\right]^{\gamma-1}\left[\psi^{\prime}(\tau)\right]^{-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]d\tau$
$\displaystyle\leq$ $\displaystyle\frac{1}{\zeta}\int_{0}^{s}\text{
}\left[\psi^{\prime}(\tau)\left[\psi(\tau)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]d\tau$
$\displaystyle=$ $\displaystyle\frac{1}{\zeta\gamma}\text{
}\left[\left[\psi(s)-\psi(0)\right]^{\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right].$

The equations (3.8) and (3.9), gives

$\displaystyle\left|F_{u,g,h}(s)-F_{v,g,h}(s)\right|$ $\displaystyle\leq$
$\displaystyle
2L_{f}\left[\left[\psi(s)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]$
$\displaystyle+2L_{f}L_{h}\frac{1}{\zeta\gamma}\text{
}\left[\left[\psi(s)-\psi(0)\right]^{\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right].$

Consequently,

$\displaystyle\left|\mathcal{G}_{f}u(t)-\mathcal{G}_{f}v(t)\right|$
$\displaystyle\leq$
$\displaystyle\frac{2L_{f}}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\left[\left[\psi(s)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]ds$
$\displaystyle+\frac{2L_{f}L_{h}}{\zeta\gamma}\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\text{
}\left[\left[\psi(s)-\psi(0)\right]^{\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]ds$
$\displaystyle=$ $\displaystyle
2L_{f}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}I_{0^{+}}^{\alpha;\psi}\left[\psi(t)-\psi(0)\right]^{\gamma-1}$
$\displaystyle+\frac{2L_{f}L_{h}}{\zeta\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}I_{0^{+}}^{\alpha;\psi}\left[\psi(t)-\psi(0)\right]^{\gamma}$
$\displaystyle=$ $\displaystyle
2L_{f}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\frac{\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}\left[\psi(t)-\psi(0)\right]^{\alpha+\gamma-1}$
$\displaystyle+\frac{2L_{f}L_{h}}{\zeta\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+\alpha+1)}\left[\psi(t)-\psi(0)\right]^{\alpha+\gamma}.$

From the definition of beta function, it follows that

$\displaystyle\left\|\mathcal{G}_{f}u-\mathcal{G}_{f}v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}$
$\displaystyle\leq$ $\displaystyle
2L_{f}\left(\frac{\mathcal{B}(\gamma,\alpha)}{\Gamma(\alpha)}+\frac{L_{h}}{b\gamma}\frac{\mathcal{B}(\gamma+1,\alpha)}{\Gamma(\alpha)}\right)$
(3.10)
$\displaystyle\times\left[\psi(b)-\psi(0)\right]^{\alpha+1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}.$

The condition (H3) shows that $\mathcal{G}_{f}$ is a contraction mapping on
$C[-r,b],$ via the norm
$\left\|\cdot\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}$. An application the
Banach contraction principle shows that the problem (1.4) has a unique
solution in $C[-r,b]\cap C_{1-\gamma;\psi}[0,b].$

Now we prove our second claim (2). Let $\varepsilon>0,$ and let $v\in
C[-r,b]\cap C_{1-\gamma;\psi}\left[0,b\right]$ be a function which satisfies
the inequality (3.1). We denote by $u\in C[-r,b]\cap
C_{1-\gamma;\psi}\left[0,b\right]$ the unique solution to the problem

$\left\\{\begin{array}[]{c}{}^{H}D_{0^{+}}^{\alpha,\beta;\psi}u(t)=F_{u,g,h}(t),\text{
\ }t\in(0,b],\\\
I_{0^{+}}^{1-\gamma;\psi}u(0^{+})=I_{0^{+}}^{1-\gamma;\psi}v(0^{+}),\\\
u(t)=v(t),\qquad t\in[-r,0],\ 0<r<\infty,\end{array}\right.$

Now, by using our first claim (1),

$u(t)=\left\\{\begin{array}[]{c}v(t)\ ,\qquad\qquad\qquad\qquad\qquad\qquad\
\qquad\ t\in\left[-r,0\right],\\\
\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\mathcal{F}_{u}(s)ds,\text{
\ }t\in\left(0,b\right],\end{array}\right.$

Obviously, for $t\in(0,b]$ the Remark 3.3 gives,

$\displaystyle\left|v(t)-\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{v,g,h}(s)ds\right|$
(3.11) $\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha}).$

Note that, for all$\ t\in[-r,0],$ $\left|v(t)-u(t)\right|=0$.

Now, for all $t\in(0,b]$, it follows from (H2) and (3.11) that

$\displaystyle\left|v(t)-u(t)\right|$ $\displaystyle\leq$
$\displaystyle\left|v(t)-\mathcal{H}_{\psi}^{\gamma}(t,0)u_{0}-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{v,g,h}(s)ds\right|$
(3.12)
$\displaystyle+\left|\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{v,g,h}(s)ds-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)F_{u,g,h}(s)ds\right|$
$\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})$
$\displaystyle+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\left|F_{v,g,h}(s)-F_{u,g,h}(s)\right|ds$
$\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})$
$\displaystyle+L_{f}\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\bigg{\\{}\left|u(s)-v(s)\right|+\left|u(g(s)-v(g(s)\right|$
$\displaystyle+L_{h}\int_{0}^{s}\big{(}\left|u(\tau)-v(\tau)\right|+\left|u(g(\tau)-v(g(\tau)\right|\big{)}d\tau\bigg{\\}}ds.$

In view of (3.12), for $z\in C([-r,b],\mathbb{R}^{+})$ we consider the
operator $\mathcal{T}:C([-r,b],\mathbb{R}^{+})\rightarrow
C([-r,b],\mathbb{R}^{+})$ defined by

$\mathcal{T}z(t)=0,\ \ \ t\in\left[-r,0\right],$
$\displaystyle\mathcal{T}z(t)$ $\displaystyle=$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})+\frac{L_{f}}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\bigg{\\{}z(s)+z(g(s))$
$\displaystyle+L_{h}\int_{0}^{s}z(\tau)+z(g(\tau))d\tau\bigg{\\}}ds,$

for $t\in\left(0,b\right].$ We prove that $\mathcal{T}$ is a Picard operator.

Case 1. Observe first that for any $z,w\in C([-r,b],\mathbb{R}^{+}),$

$\left|\mathcal{T}z(t)-\mathcal{T}w(t)\right|=0,\ \ \ t\in\left[-r,0\right],$

Case 2. For each $t\in\left(0,b\right]$ and $z,w\in
C_{1-\gamma;\psi}\left[0,b\right],$ it follows from ($H_{2}$) that

$\displaystyle\left|\mathcal{T}z(t)-\mathcal{T}w(t)\right|$
$\displaystyle\leq$
$\displaystyle\frac{L_{f}}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\bigg{\\{}\left|z(s)-w(s)\right|+\left|z(g(s))-w(g(s))\right|$
$\displaystyle+L_{h}\int_{0}^{s}\big{(}\left|z(\tau)-w(\tau)\right|+\left|z(g(\tau))-w(g(\tau))\right|\big{)}d\tau\bigg{\\}}ds$
$\displaystyle\leq$
$\displaystyle\frac{2L_{f}}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\left[\left[\psi(s)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]ds$
$\displaystyle+\frac{2L_{f}L_{h}}{\zeta\gamma}\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\text{
}\left[\left[\psi(s)-\psi(0)\right]^{\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}\right]ds$
$\displaystyle=$ $\displaystyle
2L_{f}\left(\frac{\mathcal{B}(\gamma,\alpha)}{\Gamma(\alpha)}+\frac{L_{h}}{\zeta\gamma}\frac{\mathcal{B}(\gamma+1,\alpha)}{\Gamma(\alpha)}\right)\left[\psi(t)-\psi(0)\right]^{\alpha+\gamma}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}.$

Then we obtain

$\displaystyle\left\|\mathcal{T}z-\mathcal{T}w\right\|_{C_{1-\gamma,\psi[0,b]}}$
$\displaystyle\leq$ $\displaystyle
2L_{f}\left(\frac{\mathcal{B}(\gamma,\alpha)}{\Gamma(\alpha)}+\frac{L_{h}}{b\gamma}\frac{\mathcal{B}(\gamma+1,\alpha)}{\Gamma(\alpha)}\right)$
$\displaystyle\times\left[\psi(b)-\psi(0)\right]^{\alpha+1}\left\|u-v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]},$

By (H3), $\mathcal{T}$ is a contraction mapping on $C([-r,b],\mathbb{R}^{+})$
via the wieghted norm $\left\|.\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}.$
Applying the Banach contraction principle to $\mathcal{T}$, we see that
$\mathcal{T}$ is a Picard operator and $F_{\mathcal{T}}=\\{z^{\ast}\\}$. Then,
for all$\ t\in(0,b]$, we have

$\displaystyle z^{\ast}(t)$ $\displaystyle=$
$\displaystyle\mathcal{T}z^{\ast}(t))$ $\displaystyle=$
$\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})+\frac{L_{f}}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\bigg{\\{}z^{\ast}(s)+z^{\ast}(g(s))$
$\displaystyle+L_{h}\int_{0}^{s}z^{\ast}(\tau)+z^{\ast}(g(\tau))d\tau\bigg{\\}}ds,$

Next, we prove that the solution $z^{\ast}$ is increasing. Let
$\sigma:=\min_{s\in[0,b]}[z^{\ast}(s)+z^{\ast}(h(s))]\in\mathbb{R}_{+}.$Then
for all $0<t_{1}<t_{2}\leq b$, we have

$\displaystyle z^{\ast}(t_{2})-z^{\ast}(t_{1})$ $\displaystyle=$
$\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t_{2})-\psi(0)\right]^{\alpha})-\varepsilon
E_{\alpha}(\left[\psi(t_{1})-\psi(0)\right]^{\alpha})$
$\displaystyle+\frac{L_{f}}{\Gamma(\alpha)}\int_{0}^{t_{1}}\big{(}\mathcal{N}_{\psi}^{\alpha}(t_{2},s)-\mathcal{N}_{\psi}^{\alpha}(t_{1},s)\big{)}\bigg{\\{}z^{\ast}(s)+z^{\ast}(g(s))$
$\displaystyle+L_{h}\int_{0}^{s}z^{\ast}(\tau)+z^{\ast}(g(\tau))d\tau\bigg{\\}}ds,$
$\displaystyle+\frac{L_{f}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}\mathcal{N}_{\psi}^{\alpha}(t_{2},s)\bigg{\\{}z^{\ast}(s)+z^{\ast}(g(s))$
$\displaystyle+L_{h}\int_{0}^{s}z^{\ast}(\tau)+z^{\ast}(g(\tau))d\tau\bigg{\\}}ds$
$\displaystyle\geq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t_{2})-\psi(0)\right]^{\alpha})-\varepsilon
E_{\alpha}(\left[\psi(t_{1})-\psi(0)\right]^{\alpha})$
$\displaystyle+\frac{L_{f}}{\Gamma(\alpha)}\int_{0}^{t_{1}}\big{(}\mathcal{N}_{\psi}^{\alpha}(t_{2},s)-\mathcal{N}_{\psi}^{\alpha}(t_{1},s)\big{)}\sigma\big{(}1+L_{h}s\big{)}ds$
$\displaystyle+\frac{L_{f}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}\mathcal{N}_{\psi}^{\alpha}(t_{2},s)\big{)}\sigma\big{(}1+L_{h}s\big{)}ds$
$\displaystyle=$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t_{2})-\psi(0)\right]^{\alpha})-\varepsilon
E_{\alpha}(\left[\psi(t_{1})-\psi(0)\right]^{\alpha})$
$\displaystyle+\frac{\sigma
L_{f}}{\Gamma(\alpha+1)}\left[(\psi(t_{2})-\psi(0))^{\alpha}-\psi(t_{1})-\psi(0))^{\alpha}\right]$
$\displaystyle+\frac{\sigma
L_{f}L_{h}}{\Gamma(\alpha+2)\zeta}\left[(\psi(t_{2})-\psi(0))^{\alpha+1}-(\psi(t_{1})-\psi(0))^{\alpha+1}\right]$
$\displaystyle>$ $\displaystyle 0,$

Therefore, $u^{\ast}$ is increasing, so $z^{\ast}(g(t))\leq z^{\ast}(t)\ $due
to $g(t)\leq t$ and

$\displaystyle z^{\ast}(t)$ $\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})$
$\displaystyle+\int_{0}^{t}\frac{2L_{f}}{\Gamma(\alpha)}\mathcal{N}_{\psi}^{\alpha}(t,s)\bigg{\\{}z^{\ast}(s)+\int_{0}^{s}L_{h}z^{\ast}(\tau)d\tau\bigg{\\}}ds.$

Applying Pachpatte’s inequality given in the Lemma 2.12 to the inequality
(2.2) with $x(t)=z^{\ast}(t),$ $\eta(t)=\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha}),$
$p(s)=\frac{2L_{f}}{\Gamma(\alpha)}\mathcal{N}_{\psi}^{\alpha}(t,s)$ and
$q(\sigma)=L_{h},$ we obtain

$\displaystyle z^{\ast}(t)$ $\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})$
$\displaystyle\times\bigg{(}1+\int_{0}^{t}\frac{2L_{f}}{\Gamma(\alpha)}\mathcal{N}_{\psi}^{\alpha}(t,s)\exp\left\\{\int_{0}^{s}\bigg{(}\frac{2L_{f}}{\Gamma(\alpha)}\mathcal{N}_{\psi}^{\alpha}(s,\tau)+L_{h}\bigg{)}d\tau\right\\}ds\bigg{)}$
$\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})$
$\displaystyle\times\bigg{(}1+\int_{0}^{t}\frac{2L_{f}}{\Gamma(\alpha)}\mathcal{N}_{\psi}^{\alpha}(t,s)\exp\left\\{\frac{2L_{f}}{\Gamma(\alpha+1)}(\psi(s)-\psi(0))^{\alpha}+L_{h}s\right\\}ds\bigg{)}.$

By Lagrange Mean value theorem, there exist $c\in(0,s]$ such that
$(\psi(s)-\psi(0))=s\psi^{\prime}(c),$ it follows from fact that $\psi\in
C^{1}[0,b],$ there exists a constant $\kappa\neq 0$ such that
$\sup_{\tau\in(0,c]}\left|\psi^{\prime}(\tau)\right|=\kappa,$ and we have from
$0<\alpha<1,$ that

$\left[\psi(s)-\psi(0)\right]^{\alpha}<\left[\psi(s)-\psi(0)\right],\text{ and
}s=\frac{1}{\kappa}\left[\psi(s)-\psi(0)\right].$

This gives

$\displaystyle z^{\ast}(t)$ $\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})$
$\displaystyle\times\bigg{(}1+\int_{0}^{t}\frac{2L_{f}}{\Gamma(\alpha)}\mathcal{N}_{\psi}^{\alpha}(t,s)\exp\left\\{\left(\frac{2L_{f}}{\Gamma(\alpha+1)}+\frac{L_{h}}{\kappa}\right)\left[\psi(s)-\psi(0)\right]\right\\}ds\bigg{)}$
$\displaystyle=$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})\bigg{(}1+2L_{f}I_{0^{+}}^{\alpha;\psi}e^{A\left[\psi(t)-\psi(0)\right]}\bigg{)}$
$\displaystyle\leq$ $\displaystyle\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha})\bigg{(}1+2L_{f}\left[\psi(b)-\psi(0)\right]^{\alpha}E_{1,\alpha+1}(A\left[\psi(b)-\psi(0)\right])\bigg{)}$

where $A:=\left(\frac{2L_{f}}{\Gamma(\alpha+1)}+\frac{L_{h}}{\kappa}\right).$
Take

$C_{E_{\alpha}}=\bigg{(}1+2L_{f}\left[\psi(b)-\psi(0)\right]^{\alpha}E_{1,\alpha+1}(A\left[\psi(b)-\psi(0)\right])\bigg{)},$

we get

$z^{\ast}(t)\leq C_{E_{\alpha}}\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha}),$

In particular, if $z=|v-u|$, from (3.12), $z\leq\mathcal{T}z$ and applying the
Lemma 2.11, we obtain $z\leq z^{\ast}$, where $\mathcal{T}$ is an increasing
Picard operator. As a result, we get

$\left|v(t)-u(t)\right|\leq C_{E_{\alpha}}\varepsilon
E_{\alpha}(\left[\psi(t)-\psi(0)\right]^{\alpha}),\ \ \
t\in\left[-r,b\right].$

Thus, the first equation of (1.4) is UHML stable. ∎

Next, we use the Bielecki’s norm

$\left\|\omega\right\|_{B}:=\underset{t\in\left[a,b\right]}{\max}e^{-\delta\left[\psi(t)-\psi(a)\right]}\left|\left[\psi(t)-\psi(a)\right]^{1-\gamma}\omega(t)\right|,\text{
}\delta\geq 0,$

where

$B=\left\\{\omega:(a,b]\rightarrow\mathbb{R};\text{
}e^{-\delta\left[\psi(t)-\psi(a)\right]}\left[\psi(t)-\psi(a)\right]^{1-\gamma}\omega(t)\in
C\left[a,b\right]\right\\}.$

######  Theorem 3.5.

Assume that $(H_{1})$\- $(H_{3})$ are satisfied. If we have the inequality

$2L_{f}e^{\delta\left[\psi(b)-\psi(0)\right]}\frac{\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}\left(1+\frac{L_{h}}{\zeta(\gamma+\alpha)}\right)\left[\psi(b)-\psi(0)\right]^{\alpha+1}<1.$
(3.13)

Then

  1. (1)

The $\psi-$Hilfer problem (1.4) has a unique solution in $C[-r,b]\cap
C_{1-\gamma;\psi}[0,b]$.

  2. (2)

The first equation of (1.4) is UHML stable.

###### Proof.

Just like the debate in Theorem 3.4, we only show that we show that
$\mathcal{G}_{f}:C[-r,b]\rightarrow C[-r,b]$ defined by (3.4) is a contracting
mapping on $C[-r,b]$ with respect to the Bielecki’s norm
$\left\|\cdot\right\|_{B}.$ Since the procedure is standard, we only present
the main variation in the proof as follows:

For each $t\in[-r,0]$ and for each $u,v\in C\left[-r,b\right].$we have

$\left|\mathcal{G}_{f}(u)(t)-\mathcal{G}_{f}(v)(t)\right|=0.$

On the other hand, for each $u,v\in B$ and for all $t\in(0,b]$, we have

$\displaystyle\left|\mathcal{G}_{f}(u)(t)-\mathcal{G}_{f}(v)(t)\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)\left|F_{u,g,h}(s)-F_{u,g,h}(s)\right|ds$
(3.14) $\displaystyle\leq$
$\displaystyle\frac{2L_{f}}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)e^{\delta\left[\psi(s)-\psi(0)\right]}\left[\left[\psi(s)-\psi(0)\right]^{\gamma-1}\left\|u-v\right\|_{B}\right]ds$
$\displaystyle+\frac{2L_{f}L_{h}}{\zeta\gamma\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)e^{\delta\left[\psi(s)-\psi(0)\right]}\left[\left[\psi(s)-\psi(0)\right]^{\gamma}\left\|u-v\right\|_{B}\right]ds$
$\displaystyle=$
$\displaystyle\left(2L_{f}J_{1}+\frac{2L_{f}L_{h}}{\zeta\gamma}J_{2}\right)\left\|u-v\right\|_{B},$

where

$\displaystyle J_{1}$ $\displaystyle:$
$\displaystyle=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)e^{\delta\left[\psi(s)-\psi(0)\right]}\left[\psi(s)-\psi(0)\right]^{\gamma-1}ds,$
$\displaystyle J_{2}$ $\displaystyle:$
$\displaystyle=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\mathcal{N}_{\psi}^{\alpha}(t,s)e^{\delta\left[\psi(s)-\psi(0)\right]}\left[\psi(s)-\psi(0)\right]^{\gamma}ds.$

We also estimate $J_{1}$, $J_{2}$ terms separately. By Theorem (2.7) we have

$J_{1}\leq
e^{\delta\left[\psi(b)-\psi(0)\right]}\frac{\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}\left[\psi(t)-\psi(0)\right]^{\alpha+\gamma-1},$
(3.15)

and

$J_{2}\leq
e^{\delta\left[\psi(b)-\psi(0)\right]}\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+\alpha+1)}\left[\psi(t)-\psi(0)\right]^{\alpha+\gamma}.$
(3.16)

Equation (3.14) with (3.15) and (3.16), lead to

$\displaystyle\left\|\mathcal{G}_{f}u-\mathcal{G}_{f}v\right\|_{C_{1-\gamma;\psi}\left[0,b\right]}$
$\displaystyle\leq$ $\displaystyle
2L_{f}e^{\delta\left[\psi(b)-\psi(0)\right]}\frac{\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}\left[\psi(b)-\psi(0)\right]^{\alpha}\left\|u-v\right\|_{B}$
$\displaystyle+\frac{2L_{f}L_{h}}{\zeta\gamma}e^{\delta\left[\psi(b)-\psi(0)\right]}\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+\alpha+1)}\left[\psi(b)-\psi(0)\right]^{\alpha+1}\left\|u-v\right\|_{B}$
$\displaystyle\leq$ $\displaystyle
2L_{f}e^{\delta\left[\psi(b)-\psi(0)\right]}\frac{\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}\left(1+\frac{L_{h}}{\zeta(\gamma+\alpha)}\right)\left[\psi(b)-\psi(0)\right]^{\alpha+1}\left\|u-v\right\|_{B}.$

By the inequality (3.13), $\mathcal{G}_{f}$ is a contraction mapping on
$C[-r,b],$ via the Bielecki’s norm $B$. An application the Banach contraction
principle shows that $\psi-$ Hilfer problem (1.4) has a unique solution in
$C\left([-r,b],\mathbb{R}\right)\cap
C_{1-\gamma;\psi}\left([0,b],\mathbb{R}\right).$

The proof of UHML stability is just like in Theorem 3.4 so we omit it here. ∎

######  Remark 3.6.

  1. (1)

If $\int_{0}^{t}h(t,s,y(s),y(g(s))ds=0,$ then problem (1.4) reduces to the
problem (1.2) in [16].

  2. (2)

If $\int_{0}^{t}h(t,s,y(s),y(g(s))ds=0,$ and $\beta=1,$ then problem (1.4)
reduces to the problem (1.1) in [26].

  3. (3)

If $\alpha=\beta=1,$ then problem (1.4) reduces to the problem (1.3) in [15].

##  4\. An example

Will be provided in the revised submission.

##  5\. Conclusion

We have obtained some existence, uniqueness and Ulam–Hyers–Mittag-Leffler
(UHML) stability results for the solution of Cauch type problem for
$\psi-$Hilfer FFIDEs based on the reduction of fractional differential
equations (FDEs) to integral equations. The employed techniques the Picard
operator method and generalized Pachpatte’s inequality are quite general and
effective. We trust the reported results here will have a positive impact on
the development of further applications in engineering and applied sciences.

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