# Darboux problem for Caputo-Katugampola fuzzy fractional order differential
equations

Nagwa A. Saeed1,2∗ and Deepak B. Pachpatte1

1 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University,
Aurangabad, India.

2 Department of Mathematics, Taiz University, Taiz, Yemen.

## Abstract

In this paper, we investigate existence and uniqueness of solutions for
Darboux type problem for fuzzy fractional order differential equation. We used
Caputo-Katogampola fuzzy fractional derivative for proving our results.
Schauder’s fixed point theorem is used in proving our results. some
applications are also provided to give the usefulness of our results.
Keywords: Caputo-Katugampola fractional derivative, Fuzzy fractional partial
derivative, Fuzzy valued function, Fixed point theorem, partial differential
equation.
Mathematics Subject Classification (2020) 26A33, 35R11, 35R13.

##  1 Introduction

The significance of fractional calculus and its potential applications has
increased significantly due to its effectiveness in accurately modeling
complex phenomena across a wide range of scientific and engineering fields.
These fields include aerodynamics, control systems, signal processing,
bioengineering and biomedical sciences. Fractional differential equations and
dynamical systems have proven to be valuable tools for modeling various
phenomena in engineering, physics and economics. Moreover, fractional calculus
finds extensive applications in areas such as viscoelasticity, heat conduction
in materials with memory and fluid dynamic traffic modeling (see [1, 2, 3,
4]).

In [5, 6] a novel concept of fractional calculus called Caputo-Katugampola
($\mathcal{CK}$) fractional integral and derivative was introduced. Building
upon this new fractional derivative, the existence and uniqueness of solutions
for fractional differential equations with $\mathcal{CK}$ derivative were
investigated in [7] using Schauder’s second fixed point theorem. In [8] a
discrete version of the $\mathcal{CK}$ derivative was proposed and a numerical
formula was derived to solve a linear fractional differential equation with
$\mathcal{CK}$ fractional derivative. The investigation of chaotic behavior
and stability results of fractional differential equations with $\mathcal{CK}$
derivative was carried out in [9]. A Darboux problem related to a fractional
hyperbolic integro-differential inclusion, characterized by the $\mathcal{CK}$
fractional derivative, is examined by A. Cernea in [10]. In the year 2020,
researchers conducted a study focusing on the existence and uniqueness of
solutions for the Darboux problem of partial differential equations involving
the $\mathcal{CK}$ fractional derivative (see [11]).

Recently, there has been significant development in fuzzy calculus across
various domains. Bede et al. [12, 13, 14] have presented numerous applications
of this calculus. The theory of fuzzy calculus, along with its advantages, has
attracted the attention of many mathematicians, leading to the development of
research studies on existence, uniqueness and stability for fractional
differential equations see [15]-[25]. Recently, Fan Zhang et al. [26] proved
the existence of a weak solutions for a class of new coupled systems of fuzzy
fractional partial differential equations in the sense of Caputo
$g\mathcal{H}$-derivative by using Schauder fixed point theorem.

In [27], the authors put forward the concept of the $\mathcal{CK}$ fuzzy
fractional derivative. They proceeded to investigate the existence and
uniqueness of solutions for the initial value problem of fractional fuzzy
differential equations utilizing this innovative derivative. In reference
[28], the authors conducted an investigation into the existence, uniqueness
and various forms of Ulam-Hyers stability for solutions of an impulsive
coupled system of fractional differential equations. They achieved this by
employing the $\mathcal{CK}$ fuzzy fractional derivative. Furthermore, the
Perov-type fixed point theorem was applied to establish the existence and
uniqueness of the proposed system. Motivated by the previously mentioned
works, the primary objective of our paper is two-fold:

  1. 1.

Extend the concept of the fuzzy $\mathcal{CK}$ derivatives for one-variable
functions, as presented in reference [27], to be applicable for fuzzy-valued
multivariable functions and we examine the existence and uniqueness of
solution for a single initial value problem concerning $\mathcal{CK}$ fuzzy
fractional differential equation of the following form

$\left\\{\begin{array}[]{cc}{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y)),\\\
\\\
\upsilon(\varkappa,0)=\xi_{1}(\varkappa),~{}~{}\upsilon(0,y)=\xi_{2}(y),\end{array}\right.$
(1)

where $\varphi=(\varphi_{1},\varphi_{2})\in(0,1]$, is the fractional order of
$\mathcal{CK}$ $g\mathcal{H}$-fractional derivative operator
${}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}$ defined later in Definition
(2.11), $(\varkappa,y)\in\boldsymbol{\varOmega},$
$\mathcal{F}:\boldsymbol{\varOmega}\times\hat{E_{f}}\rightarrow\hat{E_{f}},$
$\xi_{1}:(0,a]\rightarrow\hat{E_{f}}$, $\xi_{2}:(0,b]\rightarrow\hat{E_{f}}$
and $\boldsymbol{\varOmega}=(0,a]\times(0,b].$

  2. 2.

By using some standard fixed point theorem techniques we establish existence
and uniqueness results of solutions for the following coupled system of fuzzy
fractional differential equations involving the $\mathcal{CK}$
$g\mathcal{H}$-derivative:

$\left\\{\begin{array}[]{cc}{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y)),\\\
\\\
{}^{C}\mathfrak{D}_{0^{+}}^{\psi,\rho}\omega(\varkappa,y,)=\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y)),\end{array}\right.$
(2)

with the following initial conditions:

$\left\\{\begin{array}[]{cc}\upsilon(\varkappa,0)=\xi_{1}(\varkappa),~{}~{}\upsilon(0,y)=\xi_{2}(y),\\\
\omega(\varkappa,0)=\eta_{1}(\varkappa),~{}~{}\omega(0,y)=\eta_{2}(y),\end{array}\right.$
(3)

where
$\varphi=(\varphi_{1},\varphi_{2}),\psi=(\psi_{1},\psi_{2})\in(0,1]\times(0,1]$,
and $\mathcal{CK}$ $g\mathcal{H}$-fractional derivative operators
${}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}$ and
${}^{C}\mathfrak{D}_{0^{+}}^{\psi,\rho}$ are the same as in (1).

##  2 Preliminaries

In this section, we introduce fundamental definitions and concepts within the
realm of fuzzy set theory and fuzzy fractional calculus. These notions will
serve as a foundation for our subsequent discussion.

Let $E_{f}$ be the space of all fuzzy numbers on $\mathbb{R}$, $\hat{E_{f}}$
is a space of fuzzy number $\mathfrak{z}\in E_{f}$, which has the property
that the function $r\mapsto[\mathfrak{z}]^{r}$ is continuous with respect to
Hausdorff metric on [0, 1], $C(\boldsymbol{\varOmega},{\hat{E_{f}}})$ denotes
a space of all continuous fuzzy-valued functions which are on
$\boldsymbol{\varOmega}$, $AC(\boldsymbol{\varOmega},{\hat{E_{f}}})$ denotes
the set of all absolutely continuous fuzzy-valued functions and
$L[\boldsymbol{\varOmega},{\hat{E_{f}}}]$ by the set of Lebesque integrable
for fuzzy-valued functions on $\boldsymbol{\varOmega}$.

The definition of a fuzzy number and its r-level form can be described as
follows:

######  Definition 2.1.

([29]) A fuzzy number can be defined as a mapping
$\mathfrak{z}:\mathbb{R}\rightarrow[0,1]$ with the following characteristics:

(1)

The function $\mathfrak{z}$ is normal, where $\mathfrak{z}(\varkappa_{0})=1$
for a given $\varkappa_{0}\in\mathbb{R}$.

(2)

The function $\mathfrak{z}$ is convex for
$\varkappa_{1},\varkappa_{2},\mathfrak{y}\in\mathbb{R}$ and $t\in[0,1]$,
satisfying the inequality

$\mathfrak{z}(t\varkappa_{1}+(1-t)\varkappa_{2})\geq\min\\{\mathfrak{z}(\varkappa_{1}),\mathfrak{z}(\varkappa_{2})\\}.$

(3)

$\mathfrak{z}$ is semicontinuous.

(4)

$\mathfrak{z}$ is a compactly supported on $\mathbb{R}$.

The set of a fuzzy number $\mathfrak{z}(\varkappa)\in E_{f}$ in the r-level
form is denoted by $[\mathfrak{z}]^{r}$ and defined as:

$[\mathfrak{z}]^{r}=\\{\varkappa\in\mathbb{R}|\mathfrak{z}(\varkappa)\geq
r\\}=[\underline{\mathfrak{z}_{r}},\overline{\mathfrak{\mathfrak{z}}_{r}}],0\leq
r\leq 1).$

It is clear that the set of a fuzzy number $\mathfrak{z}$ in r-level form is a
closed and bounded interval
$[\underline{\mathfrak{z}_{r}},\overline{\mathfrak{\mathfrak{z}}_{r}}]$, where
$\underline{\mathfrak{z}_{r}}$ is the left end point and
$\overline{\mathfrak{\mathfrak{z}}_{r}}$ is the right end point. The diameter
of the r-level set of $\mathfrak{z}$ is denoted by $diam[\mathfrak{z}]^{r}$
and defined as:

$diam[\mathfrak{z}]^{r}=\overline{\mathfrak{\mathfrak{z}}_{r}}-\underline{\mathfrak{z}_{r}}.$

######  Definition 2.2.

([22, 32]) Let ${\mathbb{D}}:E_{f}\times E_{f}\longrightarrow\mathbb{R}$ be
the Hausdorff distance between two fuzzy numbers p,q and defined as

$\begin{split}{\mathbb{D}}(\mathfrak{p},\mathfrak{q})&=\sup_{0\leq r\leq
1}d_{H}\\{[p]^{r},[q]^{r}\\}\\\ &=\sup_{0\leq r\leq
1}max\\{|\underline{\mathfrak{p}_{r}}-\underline{\mathfrak{q}_{r}}|,|\
\overline{\mathfrak{p}_{r}}-\overline{\mathfrak{q}_{r}}|\\}.\end{split}$ (4)

The supremum metric $H^{*}$ on $C(\boldsymbol{\varOmega},E_{f})$ is considered
by

${H^{*}}(p,q)=\sup_{(\varkappa,y)\in I}\\{\
{\mathbb{D}}(p(\varkappa,y),q(\varkappa,y))\\}.$ (5)

According to [29,31,33], fuzzy number spaces $E_{f}$ and $\hat{E_{f}}$ are
semilinear spaces having the cancellation property. Furthermore, it is proved
that fuzzy numbers spaces $E_{f}$ and $\hat{E_{f}}$ with metric $\mathbb{D}$
are complete metric semilinear spaces. Hence, the fuzzy-valued continuous
functions space $C(\boldsymbol{\varOmega},\hat{E_{f}})$ is complete metric
semilinear space, which is a Banach semilinear space with the cancellation
property.

######  Definition 2.3.

[30] Let $\mathfrak{p},\mathfrak{q}\in\hat{E_{f}}$. If there exists
$\boldsymbol{w}\in\hat{E_{f}}$ such that
$\mathfrak{p}=\mathfrak{q}+\boldsymbol{w}$, then $\boldsymbol{w}$ is called
the Hukuhara difference of $\mathfrak{p}$ and $\mathfrak{q}$ and it is denoted
by $\mathfrak{p}\ominus\mathfrak{q}$.

In [22] authors have given some properties of the metric ${\mathbb{D}}$ in
$\hat{E_{f}}$ and Hukuhara difference as:

######  Lemma 2.1.

[22] For all
$\mathfrak{p},\mathfrak{q},\boldsymbol{l},\boldsymbol{k},\boldsymbol{g}\in\hat{E_{f}}$
we have

  * $(1)$

${\mathbb{D}}(\mathfrak{p}+\boldsymbol{l},\mathfrak{q}+\boldsymbol{l})={\mathbb{D}}(\mathfrak{p},\mathfrak{q}),$

  * $(2)$

${\mathbb{D}}(\mathfrak{p}+\mathfrak{q},\boldsymbol{k}+\boldsymbol{g})\leq{\mathbb{D}}(\mathfrak{p},\boldsymbol{k})+{\mathbb{D}}(q,\boldsymbol{g}),$

  * $(3)$

${\mathbb{D}}(\mathfrak{p}+\mathfrak{q},0)={\mathbb{D}}(\mathfrak{u},0)+{\mathbb{D}}(\mathfrak{q},0),$

  * $(4)$

If $\mathfrak{p}\ominus\mathfrak{q}$ exists then
$(-1)\mathfrak{p}\ominus(-1)\mathfrak{q}$ exists and
$(-1)(\mathfrak{p}\ominus\mathfrak{q})=(-1)\mathfrak{p}\ominus(-1)\mathfrak{q},$

  * $(5)$

If $\mathfrak{p}\ominus\mathfrak{q}$ and $\boldsymbol{k}\ominus\boldsymbol{g}$
exist then
${\mathbb{D}}(\mathfrak{p}\ominus\mathfrak{q},\boldsymbol{k}\ominus\boldsymbol{g})\leq{\mathbb{D}}(\mathfrak{p},\boldsymbol{k})+{\mathbb{D}}(\mathfrak{q},\boldsymbol{g})$.

######  Definition 2.4.

([13]) The generalized Hukuhara difference of two fuzzy numbers
$\mathfrak{p},\mathfrak{q}\in\hat{E_{f}}$ ($g\mathcal{H}$-difference for
short) is defined as follows: given by

$\displaystyle\mathfrak{p}\ominus_{g\mathcal{H}}\mathfrak{q}=\boldsymbol{w}\Leftrightarrow(i)~{}\mathfrak{p}=\mathfrak{q}+\boldsymbol{w}~{}~{}or~{}~{}(ii)~{}\mathfrak{q}=\mathfrak{p}+(-1)\boldsymbol{w}.$

A function $\upsilon:\boldsymbol{\varOmega}\longrightarrow\hat{E_{f}}$ is
called d-increasing (d-decreasing) on $\boldsymbol{\varOmega}$ if for every
$r\in[0,1]$ the function $(\varkappa,y)\mapsto diam[\upsilon(\varkappa,y,r)]$
is nondecreasing (nonincreasing) on $\boldsymbol{\varOmega}$. If $\upsilon$ is
d-increasing or d-decreasing on $\boldsymbol{\varOmega}$, then we say that
$\upsilon$ is d-monotone on $\boldsymbol{\varOmega}$.

The definitions of integration and partial derivative incorporating fuzzy
concepts were introduced by authors in [14, 22, 31] as follows:

######  Definition 2.5.

([14]) Let $(\varkappa_{0},\mathfrak{y}_{0})\in J\subset\mathbb{R}^{2}$, then
the g$\mathcal{H}$-partial derivative in first order of a fuzzy mapping
$\mathcal{U}:J\longrightarrow\hat{E_{f}}$ at
$(\varkappa_{0},\mathfrak{y}_{0})$ with respect to $\varkappa$ is the function
$\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}$
given by

$\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}=\lim_{h\to
0}\frac{\mathcal{U}(\varkappa_{0}+h,\mathfrak{y}_{0})\ominus_{g\mathcal{H}}\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{h},$

and with respect to $\mathfrak{y}$ is the function
$\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\mathfrak{y}}$
given by

$\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\mathfrak{y}}=\lim_{k\to
0}\frac{\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0}+k)\ominus_{g\mathcal{H}}\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{k},$

provided that the g$\mathcal{H}$-differences
$f(\varkappa_{0}+h,\mathfrak{y}_{0})\ominus_{g\mathcal{H}}\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0}),\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0}+k)\ominus_{g\mathcal{H}}\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})$
exist and
$\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}$,
$\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\mathfrak{y}}$
in $\hat{E_{f}}$.
The g$\mathcal{H}$-partial derivatives of higher order of $\mathcal{U}$ at the
point $(\varkappa_{0},\mathfrak{y}_{0})\in J$ are defined similarly.

######  Definition 2.6.

([14, 31]) Let $\mathcal{U}:J\longrightarrow\hat{E_{f}}$ be partial
g$\mathcal{H}$-differentiable with respect to $\varkappa$ at
$(\varkappa_{0},\mathfrak{y}_{0})\in J$. Then

  1. 1.

$\mathcal{U}$ is $(i)$-g$\mathcal{H}$ differentiable with respect to
$\varkappa$ at $(\varkappa_{0},\mathfrak{y}_{0})\in J$. If

$\Bigl{[}\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}\Bigl{]}^{r}=\Bigl{[}\frac{\partial\underline{\mathcal{U}}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa},\frac{\partial\overline{\mathcal{U}}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}\Bigl{]},~{}~{}~{}~{}~{}\forall
r\in[0,1],$

  2. 2.

$\mathcal{U}$ is $(ii)$-g$\mathcal{H}$ differentiable with respect to
$\varkappa$ at $(\varkappa_{0},\mathfrak{y}_{0})\in J$. If

$\Bigl{[}\frac{\partial\mathcal{U}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}\Bigl{]}^{r}=\Bigl{[}\frac{\partial\overline{\mathcal{U}}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa},\frac{\partial\underline{\mathcal{U}}(\varkappa_{0},\mathfrak{y}_{0})}{\partial\varkappa}\Bigl{]},~{}~{}~{}~{}~{}\forall
r\in[0,1].$

Similarly, we can defined the types of fuzzy g$\mathcal{H}$-derivatives of
$\mathcal{U}$ with respect to $\mathfrak{y}$ and higher order of $\mathcal{U}$
at the point $(\varkappa_{0},\mathfrak{y}_{0})\in J$.

######  Definition 2.7.

([22, 31]) For any fixed point $\varkappa_{0}$ and
$(\varkappa_{0},\mathfrak{y})\in J$, if in any neighborhood V of
$(\varkappa_{0},\mathfrak{y})\in J$, there exist points
${A_{1}}(\varkappa_{1},\mathfrak{y}),{A_{2}}(\varkappa_{2},\mathfrak{y})$
whereas $\varkappa_{1}<\varkappa_{0}<\varkappa_{2}$ such that

  1. 1.

$\mathcal{F}$ is $(i)$-g$\mathcal{H}$ differentiable at ${A_{1}}$ whereas
$\mathcal{F}$ is $(ii)$-g$\mathcal{H}$ differentiable at ${A_{2}}$ for all
$\mathfrak{y}$, or

  2. 2.

$\mathcal{F}$ is $(i)$-g$\mathcal{H}$ differentiable at ${A_{2}}$ while
$\mathcal{F}$ is $(ii)$-g$\mathcal{H}$ differentiable at ${A_{1}}$ for all
$\mathfrak{y}$. Then,$(\varkappa_{0},\mathfrak{y})\in J$ is known as a
switching point for the differentiability of $\mathcal{F}$ with respect to
$\varkappa$.

######  Definition 2.8.

[22] Let $\mathcal{U}(.,.)\in C(J,\hat{E_{f}})$ such that
$\frac{\partial\mathcal{U}(x,y)}{\partial x}\in C(J,\hat{E_{f}})$ and they
don’t have any switching point on $J$. Denote
$\frac{\partial^{2}\mathcal{U}(x_{0},y_{0})}{\partial x\partial y}$ by the
mixed second order gH-partial derivative of f with respect to x and y at
$(x_{0},y_{0})$ $\in I$. We say that

  * $(1)$

$\frac{\partial^{2}\mathcal{U}(.,.)}{\partial x\partial y}$ is a
$(\hat{i})$-g$\mathcal{H}$ differentiable at $(x_{0},y_{0})$, if the type of
g$\mathcal{H}$-differentiability at $(x_{0},y_{0})$of both $\mathcal{U}(.,.)$
and $\frac{\partial\mathcal{U}(x,y)}{\partial x}$ are the same.

  * $(2)$

$\frac{\partial^{2}\mathcal{U}(.,.)}{\partial x\partial y}$ is a
$(\widehat{ii})$-g$\mathcal{H}$ differentiable at $(x_{0},y_{0})\in I$, if the
type of g$\mathcal{H}$-differentiability at $(x_{0},y_{0})$ of both
$\mathcal{U}(.,.)$ and $\frac{\partial\mathcal{U}(x,y)}{\partial x}$ are
different.

######  Lemma 2.2.

[22] Let $\mathcal{F}\in C(\mathbb{R}^{2},\hat{E_{f}}).$

  * $(i)$

If $\mathcal{F}$ is $(i)$-g$\mathcal{H}$ differentiable with respect to y,
with no switching point on $\mathbb{R}\times[b,\mathfrak{y}]$ then
$\frac{\partial\mathcal{F}(\varkappa,\mathfrak{y})}{\partial\mathfrak{y}}$ is
integrable on $[b,\mathfrak{y}]$ and

$\int_{b}^{\mathfrak{y}}\ \frac{\partial\mathcal{F}(\varkappa,s)}{\partial
s}\,ds=\mathcal{F}(\varkappa,\mathfrak{y})\ominus\mathcal{F}(\varkappa,b).$

  * $(ii)$

If $\mathcal{F}$ is $(ii)$-g$\mathcal{H}$ differentiable with respect to y,
with no switching point on $\mathbb{R}\times[b,\mathfrak{y}]$ then
$\frac{\partial\mathcal{F}(\varkappa,\mathfrak{y})}{\partial\mathfrak{y}}$ is
integrable on $[b,\mathfrak{y}]$ and

$\int_{b}^{\mathfrak{y}}\ \frac{\partial\mathcal{F}(\varkappa,s)}{\partial
s}\,ds=(-1)\mathcal{F}(\varkappa,b)\ominus(-1)\mathcal{F}(\varkappa,\mathfrak{y}).$

######  Remark 2.1.

If the diameter function of $\upsilon(\varkappa,y,r)$
$\big{(}diam[\upsilon(\varkappa,y)]^{r}\big{)}$ is d-increasing (d-decreasing)
for all $r$, then we have $(i)$-g$\mathcal{H}$ differentiable
($(ii)$-g$\mathcal{H}$ differentiable).

The authors in [11] introduced the concept of mixed Katugampola fractional
integral of order $\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ for real-
valued functions $\mu\in L(\boldsymbol{\varOmega},\mathbb{R})$ as follows.

$\mathfrak{I}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)=\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\mu(s,t)dtds,$
(6)

for $(\varkappa,y)\in\boldsymbol{\varOmega}$, $\rho=(\rho_{1},\rho_{2})>0.$
Also they defined the mixed Katugampola fractional derivative of order
$\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ for a real function $\mu\in
AC(\boldsymbol{\varOmega},\mathbb{R})$ by

$\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)=\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}(\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}\mu(\varkappa,y)),$

where
$\mathfrak{D}^{2}_{\varkappa,y}=\frac{\partial^{2}}{\partial\varkappa\partial
y}$ and

$\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}\mu(\varkappa,y)=\frac{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{\varphi_{2}}}\mu(s,t)dtds.$

If $\mu\in L(\boldsymbol{\varOmega},\mathbb{R})$ is a real function such that
$\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)$ exists on
$\boldsymbol{\varOmega}$, then the $\mathcal{CK}$ fractional derivative
${}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)$ of order
$\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ is defined by

${}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)=\mathfrak{D}_{0^{+}}^{\varphi,\rho}(\mu(\varkappa,y)-\mu(\varkappa,0)-\mu(0,y)+\mu(0,0)).$

Therefore, we have

$^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)=\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)-\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,0)-\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(0,y)+\frac{\upsilon(0,0)\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}.$
(7)

If $\mu\in AC[a,b],$ then we have (see lemma (2.1) in [11])

${}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mu(\varkappa,y)=\frac{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{\mathfrak{D}^{2}_{s,t}\mu(s,t)}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{\varphi_{2}}}dtds.$

Now we proceed to establish the definition of the Katugampola fractional
integral and derivative for the fuzzy-valued multivariable function
$\upsilon:\boldsymbol{\varOmega}\rightarrow E_{f}$ as folllows.

######  Definition 2.9.

Let $\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ and $\upsilon\in
L(\boldsymbol{\varOmega},\hat{E_{f}}).$ The mixed Katugampola fuzzy fractional
integral is defined by

$\mathfrak{I}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\upsilon(s,t)dtds,$

for $(\varkappa,y)\in\boldsymbol{\varOmega}$, $\rho=(\rho_{1},\rho_{2})>0.$
Since
$[\upsilon(\varkappa,y)]^{r}=[\underline{\upsilon}(\varkappa,y,r),\overline{\upsilon}(\varkappa,y,r)]$
and $r\in[0,1],$ by r-level form

$[\mathfrak{I}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)]^{r}=[\mathfrak{I}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r),\mathfrak{I}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r)],$

where
$\mathfrak{I}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r),\mathfrak{I}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r)$
are defined as (6).

######  Definition 2.10.

Let $\upsilon:\boldsymbol{\varOmega}\rightarrow\hat{E_{f}}$,
$\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ and
$\rho=(\rho_{1},\rho_{2})>0.$. The mixed fuzzy Katugampola
$g\mathcal{H}$-fractional derivative of order
$\varphi=(\varphi_{1},\varphi_{2})$ is defined by

$\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}(\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}\upsilon(\varkappa,y)),$

if the g$\mathcal{H}$-derivative
$\mathfrak{D}^{2}_{\varkappa,y}(\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}\upsilon(\varkappa,y))$
exists for $(\varkappa,y)\in\boldsymbol{\varOmega}$, where

$\displaystyle\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}\upsilon(\varkappa,y)$
$\displaystyle=\frac{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{\varphi_{2}}}\mu(s,t)dtds.$

######  Remark 2.2.

Let $\upsilon\in AC(\boldsymbol{\varOmega},\hat{E_{f}})$. Then

  * •

If $\mathfrak{D}^{2}_{\varkappa,y}$ is a $(\hat{i})$-g$\mathcal{H}$ derivative
then $\upsilon$ is (1)-Katugampola g$\mathcal{H}$ differentiable and

$[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)]^{r}=[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r)].$

  * •

If $\mathfrak{D}^{2}_{\varkappa,y}$ is a $(\widehat{ii})$-g$\mathcal{H}$
derivative then $\upsilon$ is (2)-Katugampola g$\mathcal{H}$ differentiable
and

$[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)]^{r}=[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r)].$

######  Proposition 2.1.

Let $\varphi=(\varphi_{1},\varphi_{2}),\psi=(\psi_{1},\psi_{2})\in(0,1)^{2},$
$\upsilon\in C(\boldsymbol{\varOmega},\hat{E_{f}})\cap
L(\boldsymbol{\varOmega},\hat{E_{f}})$ and $\rho=(\rho_{1},\rho_{2})>0.$ Then

$\mathfrak{I}_{0^{+}}^{\varphi,\rho}\mathfrak{I}_{0^{+}}^{\psi,\rho}\upsilon(\varkappa,y)=\mathfrak{I}_{0^{+}}^{\varphi+\psi,\rho}\upsilon(\varkappa,y),$
(8)

and

$\mathfrak{D}_{0^{+}}^{\varphi,\rho}\mathfrak{I}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\upsilon(\varkappa,y).$
(9)

###### Proof.

For $(\varkappa,y)\in\boldsymbol{\varOmega}$ we have

$\displaystyle\mathfrak{I}_{0^{+}}^{\varphi,\rho}\mathfrak{I}_{0^{+}}^{\psi,\rho}\upsilon(\varkappa,y)$
$\displaystyle=\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathfrak{I}_{0^{+}}^{\psi,\rho}\upsilon(\varkappa,y)}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds.$

Now by using Fuzzy Fubini’s Theorem (see [33]) we have

$\displaystyle\mathfrak{I}_{0^{+}}^{\varphi,\rho}\mathfrak{I}_{0^{+}}^{\psi,\rho}\upsilon(\varkappa,y)=$
$\displaystyle\frac{\rho_{1}^{1-\psi_{1}}\rho_{2}^{1-\psi_{2}}\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\psi_{1})\Gamma(\psi_{2})\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s_{1}^{\rho_{1}-1}s_{2}^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-_{1}^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-s_{2}^{\rho_{2}})^{1-\varphi_{2}}}\times$
(10)
$\displaystyle\int_{0}^{s_{1}}\int_{0}^{s_{2}}\frac{t_{1}^{\rho_{1}-1}t_{2}^{\rho_{2}-1}}{(s_{1}^{\rho_{1}}-t_{1}^{\rho_{1}})^{1-\psi_{1}}(s_{2}^{\rho_{2}}-t_{2}^{\rho_{2}})^{1-\psi_{2}}}\upsilon(t_{1},t_{2})dt_{2}dt_{1}ds_{2}ds_{1}$
$\displaystyle=$
$\displaystyle\frac{\rho_{1}^{1-\psi_{1}}\rho_{2}^{1-\psi_{2}}\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\psi_{1})\Gamma(\psi_{2})\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}t_{1}^{\rho_{1}-1}t_{2}^{\rho_{2}-1}\upsilon(t_{1},t_{2})\times$
$\displaystyle\int_{t_{1}}^{\varkappa}\int_{t_{2}}^{y}\frac{s_{1}^{\rho_{1}-1}s_{2}^{\rho_{2}-1}ds_{2}ds_{1}dt_{2}dt_{1}}{(\varkappa^{\rho_{1}}-s_{1}^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-s_{2}^{\rho_{2}})^{1-\varphi_{2}}(s_{1}^{\rho_{1}}-t_{1}^{\rho_{1}})^{1-\psi_{1}}(s_{2}^{\rho_{2}}-t_{2}^{\rho_{2}})^{1-\psi_{2}}}.$

Now we set

$A=\int_{t_{1}}^{\varkappa}\int_{t_{2}}^{y}\frac{s_{1}^{\rho_{1}-1}s_{2}^{\rho_{2}-1}ds_{2}ds_{1}}{(\varkappa^{\rho_{1}}-s_{1}^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-s_{2}^{\rho_{2}})^{1-\varphi_{2}}(s_{1}^{\rho_{1}}-t_{1}^{\rho_{1}})^{1-\psi_{1}}(s_{2}^{\rho_{2}}-t_{2}^{\rho_{2}})^{1-\psi_{2}}}.$

Let

$X=\frac{(s_{1}^{\rho_{1}}-t_{1}^{\rho_{1}})}{(\varkappa^{\rho_{1}}-s_{1}^{\rho_{1}})}~{}\text{and}~{}Y=\frac{(s_{2}^{\rho_{2}}-t_{2}^{\rho_{2}})}{(y^{\rho_{2}}-s_{2}^{\rho_{2}})}.$

By substituting X and Y in $A$ we get

$\displaystyle A$
$\displaystyle=\int_{t_{1}}^{\varkappa}\frac{s_{1}^{\rho_{1}-1}}{(\varkappa^{\rho_{1}}-s_{1}^{\rho_{1}})^{1-\varphi_{1}}(s_{1}^{\rho_{1}}-t_{1}^{\rho_{1}})^{1-\psi_{1}}}ds_{1}\times\int_{t_{2}}^{y}\frac{s_{2}^{\rho_{2}-1}}{(y^{\rho_{2}}-s_{2}^{\rho_{2}})^{1-\varphi_{2}}(s_{2}^{\rho_{2}}-t_{2}^{\rho_{2}})^{1-\psi_{2}}}ds_{1}$
(11)
$\displaystyle=\frac{(\varkappa^{\rho_{1}}-t^{\rho_{1}})^{\varphi_{1}+\psi_{1}-1}(y^{\rho_{2}}-t^{\rho_{2}})^{\varphi_{2}+\psi_{2}-1}}{\rho_{1}\rho_{2}}\int_{0}^{1}(1-X)^{\varphi_{1}-1}X^{\psi_{1}}dX\int_{0}^{1}(1-Y)^{\varphi_{2}-1}Y^{\psi_{2}}dY$
$\displaystyle=\frac{(\varkappa^{\rho_{1}}-t^{\rho_{1}})^{\varphi_{1}+\psi_{1}-1}(y^{\rho_{2}}-t^{\rho_{2}})^{\varphi_{2}+\psi_{2}-1}}{\rho_{1}\rho_{2}}\frac{\Gamma(\varphi_{1})\Gamma(\psi_{1})}{\Gamma(\varphi_{1}+\psi_{1})}\frac{\Gamma(\varphi_{2})\Gamma(\psi_{2})}{\Gamma(\varphi_{2}+\psi_{2})}.$

The result of (8) is obtained by merging equations (10) to (11) as follows.

$\displaystyle\mathfrak{I}_{0^{+}}^{\varphi,\rho}\mathfrak{I}_{0^{+}}^{\psi,\rho}\upsilon(\varkappa,y)=$
$\displaystyle\frac{\rho_{1}^{1-(\varphi_{1}+\psi_{1})}\rho_{2}^{1-(\varphi_{2}+\psi_{2})}}{\Gamma(\varphi_{1}+\psi_{1})\Gamma(\varphi_{2}+\psi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{t_{1}^{\rho_{1}-1}t_{2}^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-t^{\rho_{1}})^{1-(\varphi_{1}+\psi_{1})}(y^{\rho_{2}}-t^{\rho_{2}})^{1-(\varphi_{2}+\psi_{2})}}\times$
$\displaystyle\upsilon(t_{1},t_{2})dt_{2}dt_{1}$ $\displaystyle=$
$\displaystyle\mathfrak{I}_{0^{+}}^{\varphi+\psi,\rho}\upsilon(\varkappa,y).$

The proof of (9) is similar to the proof Lemma (2.3) in [11]. ∎

######  Definition 2.11.

Let $\upsilon:\boldsymbol{\varOmega}\rightarrow\hat{E_{f}}$ be a fuzzy valued
function, $\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ and
$\rho=(\rho_{1},\rho_{2})>0.$ The mixed fuzzy $\mathcal{CK}$
g$\mathcal{H}$-fractional derivative of order
$\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ is defined by

${}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0)\big{)},$

such that $\mathfrak{D}_{0^{+}}^{\varphi,\rho}$ exists on
$\boldsymbol{\varOmega}.$

######  Lemma 2.3.

If $\upsilon\in AC(\boldsymbol{\varOmega},\hat{E_{f}})$ is a d-monotone fuzzy
function. Then the mixed fuzzy $\mathcal{CK}$ g$\mathcal{H}$-fractional
derivative of order $\varphi=(\varphi_{1},\varphi_{2})\in(0,1)^{2}$ is given
by

$\displaystyle{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)$
$\displaystyle=\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\upsilon(\varkappa,y))$
$\displaystyle=\frac{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{\mathfrak{D}^{2}_{\varkappa,y}\upsilon(\varkappa,y)}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{\varphi_{2}}}dtds.$

###### Proof.

Since $\upsilon\in AC(\boldsymbol{\varOmega},\hat{E_{f}})$ Therefore
$\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)$ exists for
$(\varkappa,y)\in\boldsymbol{\varOmega}.$ Now we show the proof in two cases:
First if $\upsilon$ is (1)-Katugampola g$\mathcal{H}$ differentiable then by
Remark (2.6-(i)) in [27] we have

$\begin{split}{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0)\big{)}\\\
=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\big{)}\ominus\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(0,y)\ominus\upsilon(0,0)\big{)}\\\
=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)\ominus\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,0)\ominus\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(0,y)\\\
&+\frac{\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\upsilon(0,0).\end{split}$
(12)

By Remark (2.2) and (7) for $r\in[0,1]$ we have

$\begin{split}[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)]^{r}=&[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r)]\\\
=&[^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r),^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r)]\\\
&+[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,0,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(0,y,r)]\\\
&+[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,0,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(0,y,r)]\\\
&-\frac{\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}[\underline{\upsilon}(0,0,r),\overline{\upsilon}(0,0,r)].\end{split}$
(13)

For every $(\varkappa,y)\in\boldsymbol{\varOmega}$ and $r\in[0,1]$ we get

$\begin{split}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=&\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\upsilon(\varkappa,y))+\mathfrak{D}_{0^{+}}^{\varphi,\rho}{\upsilon}(\varkappa,0)+\mathfrak{D}_{0^{+}}^{\varphi,\rho}{\upsilon}(0,y)\\\
&\ominus\frac{\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}{\upsilon}(0,0).\end{split}$
(14)

Substituting(14) in (12) we get the result.

Next if $\upsilon$ is (2)-Katugampola g$\mathcal{H}$ differentiable then by
Remark (2.6-(ii)) in [27] we have

$\begin{split}{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0)\big{)}\\\
=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\big{)}+(-1)\mathfrak{D}_{0^{+}}^{\varphi,\rho}\big{(}\upsilon(0,y)\ominus\upsilon(0,0)\big{)}\\\
=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)+(-1)\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,0)+(-1)\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(0,y)\\\
&+\frac{\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}\upsilon(0,0).\end{split}$
(15)

By Remark (2.2) and (7) for $r\in[0,1]$ we have

$\begin{split}[^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)]^{r}=&[\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\overline{\upsilon}(\varkappa,y)),\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\underline{\upsilon}(\varkappa,y))]\\\
=&[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(\varkappa,y,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,y,r)]\\\
&+(-1)[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,0,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(0,y,r)]\\\
&+(-1)[\mathfrak{D}_{0^{+}}^{\varphi,\rho}\underline{\upsilon}(\varkappa,0,r),\mathfrak{D}_{0^{+}}^{\varphi,\rho}\overline{\upsilon}(0,y,r)]\\\
&+\frac{\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}[\overline{\upsilon}(0,0,r),\underline{\upsilon}(0,0,r)].\end{split}$
(16)

For every $(\varkappa,y)\in\boldsymbol{\varOmega}$ and $r\in[0,1]$ we get

$\begin{split}\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\upsilon(\varkappa,y))=&\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)+\mathfrak{D}_{0^{+}}^{\varphi,\rho}(-1)\upsilon(\varkappa,0)+\mathfrak{D}_{0^{+}}^{\varphi,\rho}(-1)\upsilon(0,y)\\\
&+\frac{\varkappa^{-\varphi_{1}}y^{-\varphi_{2}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}}{\Gamma(1-\varphi_{1})\Gamma(1-\varphi_{2})}{\upsilon}(0,0).\end{split}$
(17)

Substituting (17) in (15) we get the result.
The proof is complete. ∎

Note:

  * •

If $\mathfrak{D}^{2}_{\varkappa,y}$is a $(\hat{i})$-g$\mathcal{H}$ derivative
then $\upsilon$ is (1)-Caputo Katugampola g$\mathcal{H}$ differentiable.

  * •

If $\mathfrak{D}^{2}_{\varkappa,y}$is a $(\widehat{ii})$-g$\mathcal{H}$
derivative then $\upsilon$ is (2)-Caputo Katugampola g$\mathcal{H}$
differentiable.

######  Proposition 2.2.

Let $\upsilon\in AC(\boldsymbol{\varOmega},\hat{E_{f}})$ be an integrable
function and $\varphi=(\varphi_{1}\varphi_{2})\in(0,1)^{2}$, then

  1. 1.

if $\upsilon$ is (1)-Caputo Katugampola g$\mathcal{H}$ differentiable we have

${\mathfrak{I}_{0^{+}}^{\varphi,\rho}}~{}{{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}}\upsilon(\varkappa,y)=\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0),$
(18)

  2. 2.

if $\upsilon$ is (2)-Caputo Katugampola g$\mathcal{H}$ differentiable we have

${\mathfrak{I}_{0^{+}}^{\varphi,\rho}}~{}{{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}}\upsilon(\varkappa,y)=\ominus(-1)\big{[}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0)\big{]}.$
(19)

###### Proof.

By using (8) and Lemma (2.3) we have

$\begin{split}{\mathfrak{I}_{0^{+}}^{\varphi,\rho}}~{}{{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}}\upsilon(\varkappa,y)&={\mathfrak{I}_{0^{+}}^{\varphi,\rho}}[\mathfrak{I}_{0^{+}}^{1-\varphi,\rho}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\upsilon(\varkappa,y))]\\\
&={\mathfrak{I}_{0^{+}}^{1,\rho}}(\varkappa^{1-\rho_{1}}y^{1-\rho_{2}}\mathfrak{D}^{2}_{\varkappa,y}\upsilon(\varkappa,y))\\\
&=\int_{0}^{\varkappa}\int_{0}^{y}\frac{\partial^{2}}{\partial s\partial
t}\upsilon(s,t))dtds.\end{split}$ (20)

Now, if $\upsilon$ is (1)-Caputo Katugampola g$\mathcal{H}$ differentiable
that is $\upsilon$ and $\frac{\partial\upsilon}{\partial s}$ are
$(i)$-g$\mathcal{H}$ derivative $\boldsymbol{\varOmega}$ or
$(ii)$-g$\mathcal{H}$ derivative. If $\upsilon$ and
$\frac{\partial\upsilon}{\partial s}$ are $(i)$-g$\mathcal{H}$ derivative we
have

$\begin{split}\int_{0}^{\varkappa}\int_{0}^{y}\frac{\partial^{2}}{\partial
s\partial t}\upsilon(s,t))dtds&=\int_{o}^{x}\big{(}\frac{\partial}{\partial
s}\upsilon(s,y)\ominus\frac{\partial}{\partial s}\upsilon(s,0)\big{)}ds\\\
&=\int_{o}^{x}\frac{\partial}{\partial
s}\upsilon(s,y)ds\ominus\int_{o}^{x}\frac{\partial}{\partial
s}\upsilon(s,0)ds.\end{split}$ (21)

Therefore

$\int_{0}^{\varkappa}\int_{0}^{y}\frac{\partial^{2}}{\partial s\partial
t}\upsilon(s,t))dtds=\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0).$

Similarly we get the same result for first item if $\upsilon$ and
$\frac{\partial\upsilon}{\partial s}$ are $(ii)$-g$\mathcal{H}$ derivative.
Now, if $\upsilon$ is (2)-Caputo Katugampola g$\mathcal{H}$ differentiable
that is $\upsilon$ is $(i)$-g$\mathcal{H}$ derivative and
$\frac{\partial\upsilon}{\partial s}$ is $(ii)$-g$\mathcal{H}$ derivative or
$\upsilon$ is $(ii)$-g$\mathcal{H}$ derivative and
$\frac{\partial\upsilon}{\partial s}$ is $(i)$-g$\mathcal{H}$ derivative.
If $\upsilon$ is $(ii)$-g$\mathcal{H}$ derivative and
$\frac{\partial\upsilon}{\partial s}$ is $(i)$-g$\mathcal{H}$ derivative we
get

$\begin{split}\int_{0}^{\varkappa}\int_{0}^{y}\frac{\partial^{2}}{\partial
s\partial t}\upsilon(s,t))dtds&=\int_{o}^{x}\big{(}\frac{\partial}{\partial
s}\upsilon(s,y)\ominus\frac{\partial}{\partial s}\upsilon(s,0)\big{)}ds\\\
&=\int_{o}^{x}\frac{\partial}{\partial
s}\upsilon(s,y)ds\ominus\int_{o}^{x}\frac{\partial}{\partial
s}\upsilon(s,0)ds\\\
&=(-1)\upsilon(0,y)\ominus(-1)\upsilon(x,y)\ominus\big{(}(-1)\upsilon(0,0)\ominus(-1)\upsilon(x,0)\big{)}.\end{split}$
(22)

Therefore

$\int_{0}^{\varkappa}\int_{0}^{y}\frac{\partial^{2}}{\partial s\partial
t}\upsilon(s,t))dtds=\ominus(-1)\big{[}\upsilon(\varkappa,y)\ominus\upsilon(\varkappa,0)\ominus\upsilon(0,y)+\upsilon(0,0)\big{]}.$

Similarly we can get the same result for second item if $\upsilon$ is
$(ii)$-g$\mathcal{H}$ derivative and $\frac{\partial\upsilon}{\partial s}$ is
$(i)$-g$\mathcal{H}$ derivative. ∎

##  3  Darboux problem for Fuzzy Fractional PDE with Caputo Katogumpola
g$\mathcal{H}$ derivative

In this section, we discuss the existence and uniqueness of a solution to the
system (1).

Now we give some concepts and theorems which are needed to solve (1) and (2).

######  Definition 3.1.

[22, 26] Let $B\subseteq\hat{E_{f}}$. For every $z\in B$, a subset B is called
compact-supported if there exists a compact set $L\subseteq\mathbb{R}$ such
that $[z]^{0}\subseteq L$.

######  Definition 3.2.

[22, 26] A subset $B\subseteq\hat{E_{f}}$ is called level-equicontinuous at
$r_{0}\in[0,1]$, if for all $\varepsilon>0$ there exists $\delta>0$ such that
$|r-r_{0}|<\delta$ $\Rightarrow$ $d_{H}([u]^{r},[u]^{r_{0}})<\varepsilon$ for
each $u\in B$.

######  Theorem 3.1.

[22, 26] Suppose B is a subset of $\hat{E_{f}}$ with compact support. Then the
following statements are equivalent

  * •

B is a relatively compact subset of $(\hat{E_{f}},\mathbb{D})$.

  * •

B is level-equicontinuous on $[0,1].$

######  Theorem 3.2.

[22] Let Z be a nonempty, closed, bounded and convex subset of a Banach space
$\mathcal{A}$ , and suppose that $N:Z\rightarrow Z$ is a relatively compact
operator in $\mathcal{A}$. Then N has at least one fixed point in Z.

######  Theorem 3.3.

[34] Let $(\mathcal{A},\mathbb{D})$ be a complete metric space, then each
contraction mapping $N:\mathcal{A}\rightarrow\mathcal{A}$ has a unique fixed
point x of $N$ in $\mathcal{A}$.

For $(\varkappa,y)\in\boldsymbol{\varOmega}$ and $\xi_{1}$, $\xi_{2}$ are
known functions we define the function $h$ by

$h(\varkappa,y)=\xi_{1}(\varkappa)+\xi_{2}(y)-\xi_{1}(0).$

The following lemma gives the equivalent of integral equations for (1).

######  Lemma 3.1.

Let $\upsilon\in C(\boldsymbol{\varOmega},\hat{E_{f}})$ be a fuzzy valued
function satisfying (1),
$\mathcal{F}:\boldsymbol{\varOmega}\times\hat{E_{f}}\rightarrow\hat{E_{f}}$
and $\varphi=(\varphi_{1},\varphi_{2})$ where $0<\varphi_{1},\varphi_{2}<1$
and $\rho=(\rho_{1},\rho_{2})>0$, then one of the following integral equations
is a solution of (1):

$\upsilon(\varkappa,y)=h(\varkappa,y)+\mathfrak{I}^{\varphi,\rho}_{0^{+}}\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))\big{]},$
(23)

or

$\upsilon(\varkappa,y)=h(\varkappa,y)\ominus(-1)\mathfrak{I}^{\varphi,\rho}_{0^{+}}\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))\big{]}.$
(24)

###### Proof.

Let $\upsilon(\varkappa,y)$ be a solution of equation (1). Applying the mixed
Katugampola fuzzy fractional integral ($\mathfrak{I}_{0^{+}}^{\varphi,\rho}$)
to both sides of equation (1), we get

${\mathfrak{I}_{0^{+}}^{\varphi,\rho}}~{}{{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}}\upsilon(\varkappa,y)=\mathfrak{I}^{\varphi,\rho}_{0^{+}}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y)),$

and from the Proposition (2.2) we get (23) and (24). ∎

In this paper we study our results for solution (23) while the outcomes
corresponding to (24) can be demonstrated in a similar manner.

Now, in our next theorem we prove the existence of the solution (23) for the
problem (1) by employing Schauder’s fixed point theorem. Let $k>0$ be a
positive constant, we define
$\mathcal{B}(0,k)=\\{\upsilon\in\hat{E_{f}}:\mathbb{D}[\upsilon,0]\leq k\\}.$

######  Theorem 3.4.

Let $k>0$ and $\varkappa_{1},y_{1}>0$ such that
$\varkappa_{1}^{\rho_{1}}y_{1}^{\rho_{2}}\leq\big{[}\frac{k\Gamma(1+\varphi_{1})\Gamma(1+\varphi_{2})}{2M}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\big{]}^{\frac{1}{\varphi_{1}\varphi_{2}}}$,
$h\in C[{\boldsymbol{\varOmega}},\mathcal{B}(0,\frac{k}{2})]$ and Let
$S=min\\{\varkappa_{1},a\\}$, $T=min\\{y_{1},b\\}$. Define

$M=\sup_{(\varkappa,y,\upsilon)\in{\boldsymbol{\varOmega}},\times\mathcal{B}(0,k)}H^{*}[\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y)),0],$

where $M$ is a positive constant. Then for all
$(\varkappa,y)\in\widetilde{\boldsymbol{\varOmega}}=[0,S]\times[0,T]$ there
exists at least a function $\upsilon\in
C[\widetilde{\boldsymbol{\varOmega}},\hat{E_{f}}]$ that solve the problem (1).

###### Proof.

We define the operator $\mathbb{A}$ by

$\mathbb{A}\upsilon(\varkappa,y)=h(\varkappa,y)+\mathfrak{I}^{\varphi,\rho}_{0^{+}}\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))\big{]}.$

Now we show that the operator $\mathbb{A}$ has a fixed point by the following
steps
Step 1. $\mathbb{A}$ is continuous. For
$(\varkappa,y)\in\widetilde{\boldsymbol{\varOmega}}$ we define
$\widetilde{W}=\\{\upsilon\in
C(\widetilde{\boldsymbol{\varOmega}},\hat{E_{f}}):\mathbb{D}(\upsilon,h)\leq
k\\}$.
Firstly, we show that $\mathbb{A}$ maps the set $\widetilde{W}$ in to itself,
for every $\upsilon\in\widetilde{W}$ and
$(\varkappa,y)\in\widetilde{\boldsymbol{\varOmega}}$ we have

$\begin{split}{\mathbb{D}}(\mathbb{A}\upsilon(\varkappa,y),0)\leq&{\mathbb{D}}(h(\varkappa,y),0)+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\times\\\
&{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon(s,t),0)\big{)}dtds\\\ \\\
\leq&{\mathbb{D}}(h(\varkappa,y),0)+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}M}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds\\\
\leq&{\mathbb{D}}(h(\varkappa,y),0)+\frac{M\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{1}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\\\
\leq&k.\end{split}$

Hence $\mathbb{A}\in\widetilde{W}.$

Now we show that $\mathbb{A}$ is continuous. Let $\\{\upsilon_{n}\\}$ be a
sequence such that $\upsilon_{n}\rightarrow\upsilon$ in $\widetilde{W}.$ For
$(\varkappa,y)\in\widetilde{\boldsymbol{\varOmega}}$ we have

$\begin{split}&{\mathbb{D}}(\mathbb{A}\upsilon_{n}(\varkappa,y),\mathbb{A}\upsilon(\varkappa,y))\\\
&\leq\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\times\\\
&\sup_{(s,t)\in{\boldsymbol{\varOmega}}}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon_{n}(s,t),\mathcal{F}(s,t,\upsilon(s,t))\big{)}dtds\\\
\\\
&\leq\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\sup_{(s,t)\in{\boldsymbol{\varOmega}}}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon_{n}(s,t),\mathcal{F}(s,t,\upsilon(s,t))\big{)}\times\\\
&\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds\\\
\\\
&\leq\frac{\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{2}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\alpha_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\sup_{(s,t)\in{\boldsymbol{\varOmega}}}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon_{n}(s,t),\mathcal{F}(s,t,\upsilon(s,t))\big{)}\\\
&\leq\frac{a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}}{\rho_{1}^{\alpha_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\sup_{(s,t)\in{\boldsymbol{\varOmega}}}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon_{n}(s,t),\mathcal{F}(s,t,\upsilon(s,t))\big{)},\end{split}$

since $\mathcal{F}$ is continuous, we have

${\mathbb{D}}(\mathbb{A}\upsilon_{n}(\varkappa,y),\mathbb{A}\upsilon(\varkappa,y))\rightarrow
0~{}\text{as}~{}n\rightarrow\infty.$

Step 2. We show that $\mathbb{A}$ is relatively compact.Let
$(\varkappa_{1},y_{1}),(\varkappa_{2},y_{2})\in\widetilde{\boldsymbol{\varOmega}}$
such that $\varkappa_{1}<\varkappa_{2}$ and $y_{1}<y_{2}$. Then we have

$\begin{split}&{\mathbb{D}}(\mathbb{A}\upsilon(\varkappa_{2},y_{2}),\mathbb{A}\upsilon(\varkappa_{1},y_{1}))\\\
&\leq{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\alpha_{2})}\times\\\
&{\mathbb{D}}\bigg{(}\int_{0}^{\varkappa_{2}}\int_{0}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{1}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{1}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\bigg{)}\end{split}$
$\begin{split}\leq&{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\times\\\
&{\mathbb{D}}\bigg{(}\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\alpha_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\alpha_{2}}},\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{1}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{1}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\bigg{)}\\\
&+{\mathbb{D}}\bigg{(}\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},0\bigg{)}\\\
&+{\mathbb{D}}\bigg{(}\int_{0}^{\varkappa_{1}}\int_{y_{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\alpha_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\alpha_{2}}},0\bigg{)}\\\
&+{\mathbb{D}}\bigg{(}\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{y_{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},0\bigg{)}\end{split}$
$\begin{split}\leq&{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{M\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\times\\\
&\bigg{[}\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\big{[}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}-\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa_{1}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{1}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\big{]}dtds\\\
&+\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\alpha_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\alpha_{2}}}+\int_{0}^{\varkappa_{1}}\int_{y_{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\\\
&+\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{y{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\bigg{]}\\\
\leq&{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{M}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\big{(}\varkappa_{2}^{\rho_{1}\varphi_{1}}y_{2}^{\rho_{2}\varphi_{2}}-\varkappa_{1}^{\rho_{1}\varphi_{1}}y_{1}^{\rho_{2}\varphi_{2}}\big{)}.\end{split}$

As $\varkappa_{1}<\varkappa_{2}$ and $y_{1}<y_{2}$ then
${\mathbb{D}}(\mathbb{A}\upsilon(\varkappa_{2},y_{2}),\mathbb{A}\upsilon(\varkappa_{1},y_{1}))$
tends to zero. This implies that $\mathbb{A}$ is equicontinuous on
$C[\widetilde{\boldsymbol{\varOmega}},\hat{E_{f}}]$.

Now, to prove that $\mathbb{A}$ is relatively compact we must prove
$\mathbb{A}$ is level-equicontinuous and a compact-supported. Since
$\mathcal{F}$ is a compact mapping,
$\mathcal{F}:\widetilde{\boldsymbol{\varOmega}}\times\widetilde{W}\rightarrow\hat{E_{f}}$
is relatively compact and from theorem (3.1) $\mathcal{F}$ is level-
equicontinuous. Then for each $\varepsilon>0$, $q\in[0,1]$ there exists
$\delta>0$ such that from $|r-q|<\delta$ we get

$d_{H}(\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))\big{]}^{r},\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))\big{]}^{q})<\frac{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}{2a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}}\varepsilon$

and
$d_{H}(\big{[}h(\varkappa,y)\big{]}^{r},\big{[}h(\varkappa,y)\big{]}^{q}))<\frac{\varepsilon}{2}$.
Hence, when $|r-q|<\delta$ we have

$\displaystyle d_{H}$
$\displaystyle(\big{[}\mathbb{A}\upsilon(\varkappa,y)\big{]}^{r},\big{[}\mathbb{A}\upsilon(\varkappa,y)\big{]}^{q})\leq
d_{H}(\big{[}h(\varkappa,y)\big{]}^{r},\big{[}h(\varkappa,y)\big{]}^{q}))$
$\displaystyle+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}{d_{H}}(\big{[}\mathcal{F}(s,t,\upsilon)\big{]}^{r},\big{[}\mathcal{F}(s,t,\upsilon)\big{]}^{q})dtds$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon}{2}+\frac{\rho_{1}\rho_{2}\varepsilon\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}{2a^{\alpha_{1}}b^{\alpha_{2}}\Gamma(\alpha_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\alpha_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\alpha_{2}}}dtds$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon}{2}+\frac{\rho_{1}\rho_{2}\varepsilon\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}{2a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\frac{\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{2}\varphi_{2}}}{\rho_{1}\rho_{2}\varphi_{1}\varphi_{2}}\leq\epsilon.$

Thus, $\mathbb{A}$ is level-equicontinuous.

Since $\mathcal{F}$ and $h$ are relative compact, then from theorem (3.1)
$\mathcal{F}$ and $h$ are support-compact and level-equicontinuous. Hence,
there exists a compact subsets $M_{1},M_{2}\subset\mathbb{R}$ such that
$\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))\big{]}^{0}\subseteq
M_{1}$ and $\big{[}h(\varkappa,y)\big{]}^{0}\subseteq M_{2}$ for every
$(\varkappa,y,\upsilon)\in\widetilde{\boldsymbol{\varOmega}}\times\widetilde{W}.$
Thus, we obtain

$\displaystyle\big{[}\mathbb{A}(\upsilon(\varkappa,y))\big{]}^{0}$
$\displaystyle=\bigg{[}h(\varkappa,y)+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\mathcal{F}(s,t,\upsilon)dtds\bigg{]}^{0}$
$\displaystyle=\big{[}h(\varkappa,y)\big{]}^{0}+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\big{[}\mathcal{F}(\varkappa,y,\upsilon)\big{]}^{0}dtds$
$\displaystyle\subseteq
M_{2}+\frac{M_{1}\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{2}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}.$

Since $\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{2}\varphi_{2}}$ is bounded on
$\widetilde{\boldsymbol{\varOmega}}$, then there exists a compact set
$M_{0}\subseteq\mathbb{R}$ such that
$\big{[}\mathbb{A}(\upsilon(\varkappa,y))\big{]}^{0}\subseteq M_{0}$. Hence,
$\mathbb{A}$ is compact-supported. Therefor, by Ascoli-Arzelá theorem
$\mathbb{A}$ is relatively compact.
According to the above steps with Schouder’s theorem, we deduce that
$\mathbb{A}$ has at least one fixed point $\upsilon$ which is a solution to
the $\mathcal{CK}$ system (1). ∎

In the following theorem, we discuss the uniqueness results for the problem
(1).

######  Theorem 3.5.

Let $\upsilon,\hat{\upsilon}\in C[{\boldsymbol{\varOmega}},\hat{E_{f}}]$, for
all $(\varkappa,y)\in{\boldsymbol{\varOmega}}$ there exists a constant $R>0$
such that

${\mathbb{D}}(\mathcal{F}(\varkappa,y,\upsilon),\mathcal{F}(\varkappa,y,\hat{\upsilon}))\leq
R~{}{\mathbb{D}}(\upsilon,\hat{\upsilon}).$

Then the problem(1) has a unique solution if

$\varXi=\frac{R}{\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\varphi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\varphi_{2}}<1.$

###### Proof.

Let $\mathbb{A}$ be an operator defined in Theorem (3.4). We will show that
$\mathbb{A}$ has a unique fuzzy solution. Let $\upsilon,\hat{\upsilon}\in
C[\widetilde{\boldsymbol{\varOmega}},\hat{E_{f}}]$, for all
$(\varkappa,y)\in\widetilde{\boldsymbol{\varOmega}}$. Then

$\displaystyle{\mathbb{D}}(\mathbb{A}\upsilon(\varkappa,y),\mathbb{A}\hat{\upsilon}(\varkappa,y))$
$\displaystyle\leq\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}{\mathbb{D}}(\mathcal{F}(s,t,\upsilon),\mathcal{F}(s,t,\hat{\upsilon}))}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\leq\frac{R\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}H^{*}(\varkappa,y)}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\leq\frac{RH^{*}(\varkappa,y)}{\Gamma(\alpha_{1}+1)\Gamma(\varphi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\varphi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\varphi_{2}}.$

Consequently,

$H^{*}(\mathbb{A}\upsilon(\varkappa,y),\mathbb{A}\hat{\upsilon}(\varkappa,y))\leq\varXi
H^{*}(\varkappa,y).$

Hence $\mathbb{A}$ is a contraction mapping. Thus, by Banach contraction
principle $\mathbb{A}$ has a unique fixed point $\upsilon\in
C[{\boldsymbol{\varOmega}},\hat{E_{f}}]$. ∎

######  Example 3.1.

Consider the following problem:

$\left\\{\begin{array}[]{cc}{}^{C}\mathfrak{D}_{0^{+}}^{\varphi,\rho}\upsilon(\varkappa,y)=\frac{\varkappa
y\upsilon(\varkappa,y)}{2(1+\upsilon(\varkappa,y))},\\\ \\\
\upsilon(\varkappa,0)=\mathcal{K}\varkappa,\upsilon(0,y)=\mathcal{K}y^{2},\upsilon(0,0)=0,\end{array}\right.$
(25)

where $(\varkappa,y)\in(0,\frac{1}{2}]\times(0,1]$,
$\varphi=\frac{1}{2}$,$\rho=\frac{3}{2}$ and $\mathcal{K}=(1,2,3)$ is a
triangle fuzzy number.

We have

$\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y))=\frac{\varkappa
y\upsilon(\varkappa,y)}{2(1+\upsilon(\varkappa,y))}.$

Now for $(\varkappa,y)\in(0,\frac{1}{2}]\times(0,1]$ we can see that the
function $\mathcal{F}$ satisfies the condition of Theorem (3.5) as follows:

$\begin{split}{\mathbb{D}}(\mathcal{F}(\varkappa,y,\upsilon),\mathcal{F}(\varkappa,y,\hat{\upsilon}))&={\mathbb{D}}\big{(}\frac{\varkappa
y\upsilon(\varkappa,y)}{2(1+\upsilon(\varkappa,y))},\frac{\varkappa
y\hat{\upsilon}(\varkappa,y)}{2(1+\hat{\upsilon}(\varkappa,y))}\big{)}\\\
&=\frac{\varkappa
t}{2}{\mathbb{D}}\big{(}\frac{\upsilon}{1+\upsilon},\frac{\hat{\upsilon}}{1+\hat{\upsilon}}\big{)}\\\
&\leq\frac{1}{4}{\mathbb{D}}(\upsilon,\hat{\upsilon}).\end{split}$

Also, for $\varphi_{1}=\varphi_{2}=0.5$ and $\rho_{1}=\rho_{2}=1.5$ we get

$\begin{split}\varXi&=\frac{R}{\Gamma(\alpha_{1}+1)\Gamma(\alpha_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\varphi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\varphi_{2}}\\\
&=\frac{1}{4\Gamma(1.5)\Gamma(1.5)}\bigg{(}\frac{0.5^{1.5}}{1.5}\bigg{)}^{0.5}\bigg{(}\frac{1^{1.5}}{1.5}\bigg{)}^{0.5}\\\
&\backsimeq 0.35689<1.\end{split}$

Since the assumption of Theorem (3.5) are obtained, therefore problem (25) has
a unique solution.

##  4  Fuzzy Fractional Coupled Partial Differential Equations with Caputo-
Katogumpola g$\mathcal{H}$ derivative

In this section, we will prove the existence and uniqueness of a solutions of
the coupled system for fuzzy fractional partial differential equations (2).
For all $(\varkappa,y)\in\boldsymbol{\varOmega}$. let

$h(\varkappa,y)=\xi_{1}(\varkappa)+\xi_{2}(y)\ominus\xi_{1}(0),$ (26)
$\mathfrak{g}(\varkappa,y)=\eta_{1}(\varkappa)+\eta_{2}(y)\ominus\eta_{1}(0),$
(27)

where $\xi_{1}$, $\xi_{2}$, $\eta_{1}$ and $\eta_{2}$ are the given fuzzy
valued functions such that $\xi_{2}(y)\ominus\xi_{1}(0)$ and
$\eta_{2}(y)\ominus\eta_{1}(0)$ exist.

Now we introduce an auxiliary lemma that will have significant importance in
this part of the paper.

######  Lemma 4.1.

Let
$\mathcal{F},\mathcal{G}:\boldsymbol{\varOmega}\times\hat{E_{f}}\rightarrow\hat{E_{f}}$.
For any $\upsilon,\omega\in C(\boldsymbol{\varOmega},\hat{E_{f}})$,
$\varphi=(\varphi_{1},\varphi_{2})\in(0,1)$,
$\psi=(\psi_{1},\psi_{2})\in(0,1)$ and $\rho=(\rho_{1},\rho_{2})>0$, then one
of the following coupled integral equations are equivalent to the coupled
system (2) with (3):

$\left\\{\begin{array}[]{cc}\upsilon(\varkappa,y)=h(\varkappa,y)+\mathfrak{I}^{\varphi,\rho}_{0^{+}}\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]},\\\
\\\
\omega(\varkappa,y)=\mathfrak{g}(\varkappa,y)+\mathfrak{I}^{\psi,\rho}_{0^{+}}\big{[}\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]},\end{array}\right.$
(28)

or

$\left\\{\begin{array}[]{cc}\upsilon(\varkappa,y)=h(\varkappa,y)\ominus(-1)\mathfrak{I}^{\varphi,\rho}_{0^{+}}\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]},\\\
\\\
\omega(\varkappa,y)=\mathfrak{g}(\varkappa,y)\ominus(-1)\mathfrak{I}^{\psi,\rho}_{0^{+}}\big{[}\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}.\end{array}\right.$
(29)

###### Proof.

The equivalent Volterra integral equation for coupled system can be obtained
by applying the same process of single system in Lemma (3.1). ∎

Next, by using Schauder fixed point in Banach spaces, we present existence and
uniqueness theorems only for solutions (28), where we can proved solutions
(28) in similar way.

######  Theorem 4.1.

Let $k$ is a positive constant. For all
$\varphi=(\varphi_{1},\varphi_{2}),\psi=(\psi_{1},\psi_{2})\in(0,1]^{2},$
there exist constants $\varkappa_{1},\varkappa_{2},y_{1},y_{2}>0$ such that
$h,\mathfrak{g}\in C[{\boldsymbol{\varOmega}},\mathcal{B}(0,\frac{k}{2})]$ and

$\varkappa_{1}^{\rho_{1}}y_{1}^{\rho_{2}}\leq\big{[}\frac{k\Gamma(1+\varphi_{1})\Gamma(1+\varphi_{2})}{2\mathcal{M}_{1}}\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\big{]}^{\frac{1}{\varphi_{1}\varphi_{2}}},~{}\varkappa_{2}^{\rho_{1}}y_{2}^{\rho_{2}}\leq\big{[}\frac{k\Gamma(1+\psi_{1})\Gamma(1+\psi_{2})}{2\mathcal{M}_{2}}\rho_{1}^{\psi_{1}}\rho_{2}^{\psi_{2}}\big{]}^{\frac{1}{\psi_{1}\psi_{2}}}.$

Also, let $S_{1}=min\\{\varkappa_{1},\varkappa_{2},a\\}$,
$T_{1}=min\\{y_{1},y_{1},b\\}$. Define

$\mathcal{M}_{1}=\sup_{(\varkappa,y,\upsilon,\omega)\in{\boldsymbol{\varOmega}},\times\mathcal{B}(0,k)\times\mathcal{B}(0,k)}H^{*}[\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y)),0],$
$\mathcal{M}_{2}=\sup_{(\varkappa,y,\upsilon,\omega)\in{\boldsymbol{\varOmega}},\times\mathcal{B}(0,k)\times\mathcal{B}(0,k)}H^{*}[\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y)),0],$

where $\mathcal{M}_{1},\mathcal{M}_{2}$ is a positive constants. Then for all
$(\varkappa,y)\in\Lambda=[0,S_{1}]\times[0,T_{1}]$ there exists a function
$(\upsilon,\omega)\in C[\Lambda,\hat{E_{f}}]\times C[\Lambda,\hat{E_{f}}]$
that solve the problem (2).

###### Proof.

Consider the operator $\Im:\widetilde{W_{1}}\rightarrow\widetilde{W_{1}}$
defined as

$\Im\upsilon(\varkappa,y)=\binom{\Im_{1}\upsilon(\varkappa,y)}{\Im_{2}\omega(\varkappa,y)},$
(30)

where

$\Im_{1}\upsilon(\varkappa,y)=h(\varkappa,y)+\mathfrak{I}^{\varphi,\rho}_{0^{+}}\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]},$
$\Im_{2}\omega(\varkappa,y)=\mathfrak{g}(\varkappa,y)+\mathfrak{I}^{\psi,\rho}_{0^{+}}\big{[}\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}.$

For $(\varkappa,y)\in\Lambda$ we define
$\widetilde{W_{1}}=\\{(\upsilon,\omega)\in C[\Lambda,\hat{E_{f}}]\times
C[\Lambda,\hat{E_{f}}]:max\\{H^{*}(\upsilon,0),H^{*}(\omega,0)\\}]\leq k\\}$.
Also, for each
$\binom{\upsilon_{1}(\varkappa,y)}{\omega_{1}(\varkappa,y)},\binom{\upsilon_{1}(\varkappa,y)}{\omega_{1}(\varkappa,y)}\in\widetilde{W_{1}}$,

$\begin{split}&H^{*}(\Im(\upsilon_{1},\omega_{1}),\Im(\upsilon_{2},\omega_{2}))\\\
&=max\\{H^{*}(\Im_{1}(\upsilon_{1},\omega_{1}),\Im_{1}(\upsilon_{2},\omega_{2})),H^{*}(\Im_{2}(\upsilon_{1},\omega_{1}),\Im_{2}(\upsilon_{2},\omega_{2}))\\}\\\
&=max\big{\\{}\sup_{(\varkappa,y)\in\Lambda}\mathbb{D}(\Im_{1}(\upsilon_{1},\omega_{1}),\Im_{1}(\upsilon_{2},\omega_{2})),\sup_{(\varkappa,y)\in\Lambda}\mathbb{D}(\Im_{2}(\upsilon_{1},\omega_{1}),\Im_{2}(\upsilon_{2},\omega_{2}))\big{\\}}.\end{split}$
(31)

Now we show that the operator $\Im$ has a fixed point by the following steps
Step 1. $\Im$ is continuous.
Firstly, we show that $\Im$ maps the set $\widetilde{W_{1}}$ in to itself, for
every $(\upsilon,\omega)\in\widetilde{W_{1}}$ and $(\varkappa,y)\in\Lambda$ we
have

$\begin{split}&{\mathbb{D}}(\Im_{1}(\upsilon(\varkappa,y),\omega(\varkappa,y)),0)\\\
\leq&{\mathbb{D}}(h(\varkappa,y),0)+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon(s,t),\omega(\varkappa,y),0)\big{)}dtds}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\\\
\\\
\leq&{\mathbb{D}}(h(\varkappa,y),0)+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}\mathcal{M}_{1}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds\\\
\leq&{\mathbb{D}}(h(\varkappa,y),0)+\frac{\mathcal{M}_{1}\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{1}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)},\end{split}$

which implies

${\mathbb{D}}(\Im_{1}(\upsilon(\varkappa,y),\omega(\varkappa,y)),0)\leq\frac{k}{2}+\frac{k}{2}=k.$
(32)

Similarly, we obtain

$\mathbb{D}(\Im_{2}(\upsilon(\varkappa,y),\omega(\varkappa,y)),0)\leq k.$ (33)

For all $(\upsilon,\omega)\in\widetilde{W_{1}}$ and from (31) we get

$\mathbb{D}(\Im(\upsilon_{1},\omega_{1}),0))\\\
=max\\{\mathbb{D}(\Im_{1}(\upsilon_{1},\omega_{1}),0),\mathbb{D}(\Im_{2}(\upsilon_{1},\omega_{1}),0)\\}.$
(34)

Combining (32), (33) with (34), we get
$H^{*}(\Im(\upsilon_{1},\omega_{1}),0)\leq k.$ Hence
$\Im(\upsilon,\omega)\in\widetilde{W_{1}}$.

Now we proof that $\Im$ is continuous. Let $\\{(\upsilon_{n},\omega_{n})\\}$
be a sequence such that
$(\upsilon_{n},\omega_{n})\rightarrow(\upsilon,\omega)$ in
$\widetilde{W_{1}}.$
For $(\varkappa,y)\in\Lambda$ we have

$\begin{split}&{\mathbb{D}}(\Im_{1}(\upsilon_{n},\omega_{n}),\Im_{1}(\upsilon,\omega)\\\
&\leq\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\times\\\
&\sup_{(s,t)\in\Lambda}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon_{n}(s,t),\omega_{n}(s,t)),\mathcal{F}(s,t,\upsilon(s,t),\omega(s,t))\big{)}dtds\\\
&\leq\frac{a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\times\\\
&\sup_{(s,t)\in\Lambda}{\mathbb{D}}\big{(}\mathcal{F}(s,t,\upsilon_{n}(s,t),\omega_{n}(s,t)),\mathcal{F}(s,t,\upsilon(s,t),\omega(s,t))\big{)},\end{split}$
(35)

since $\mathcal{F}$ is continuous. Hence $\Im$ is continuous. Similarly

$\begin{split}{\mathbb{D}}(\Im_{2}(\upsilon_{n},\omega_{n}),\Im_{2}(\upsilon,\omega)&\leq\frac{a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\times\\\
&\sup_{(s,t)\in\Lambda}{\mathbb{D}}\big{(}\mathcal{G}(s,t,\upsilon_{n}(s,t),\omega_{n}(s,t)),\mathcal{G}(s,t,\upsilon(s,t),\omega(s,t))\big{)},\end{split}$
(36)

From (35), (36) and (31)we have

${\mathbb{D}}(\Im(\upsilon_{n},\omega_{n}),\Im(\upsilon,\omega))\rightarrow
0~{}\text{as}~{}n\rightarrow\infty.$

Step 2. We show that $\Im$ is relatively compact. Let
$(\varkappa_{1},y_{1}),(\varkappa_{2},y_{2})\in\Lambda$ such that
$\varkappa_{1}<\varkappa_{2}$ and $y_{1}<y_{2}$. Then we have

$\begin{split}&{\mathbb{D}}(\Im_{1}(\upsilon(\varkappa_{2},y_{2}),\omega(\varkappa_{2},y_{2})),\Im_{1}(\upsilon(\varkappa_{1},y_{1}),\omega(\varkappa_{1},y_{1})))\\\
&\leq{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\times\\\
&{\mathbb{D}}\bigg{(}\int_{0}^{\varkappa_{2}}\int_{0}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{1}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{1}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\bigg{)}\end{split}$
$\begin{split}\leq&{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\times\\\
&{\mathbb{D}}\bigg{(}\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{1}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{1}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\bigg{)}\\\
&+{\mathbb{D}}\bigg{(}\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},0\bigg{)}\\\
&+{\mathbb{D}}\bigg{(}\int_{0}^{\varkappa_{1}}\int_{y_{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},0\bigg{)}\\\
&+{\mathbb{D}}\bigg{(}\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{y_{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}},0\bigg{)}\end{split}$
$\begin{split}\leq&{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{\mathcal{M}_{1}\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\times\\\
&\bigg{[}\int_{0}^{\varkappa_{1}}\int_{0}^{y_{1}}\big{[}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}-\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa_{1}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{1}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\big{]}dtds\\\
&+\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{0}^{y_{1}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}+\int_{0}^{\varkappa_{1}}\int_{y_{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\\\
&+\int_{\varkappa_{1}}^{\varkappa_{2}}\int_{y{1}}^{y_{2}}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}dtds}{(\varkappa_{2}^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y_{2}^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}\bigg{]}\\\
\leq&{\mathbb{D}}(h(\varkappa_{2},y_{2}),h(\varkappa_{1},y_{1}))+\frac{\mathcal{M}_{1}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma\varphi_{2}+1)}\big{(}\varkappa_{2}^{\rho_{1}\varphi_{1}}y_{2}^{\rho_{2}\varphi_{2}}-\varkappa_{1}^{\rho_{1}\varphi_{1}}y_{1}^{\rho_{2}\varphi_{2}}\big{)}.\end{split}$

As $\varkappa_{1}<\varkappa_{2}$ and $y_{1}<y_{2}$ then

${\mathbb{D}}(\Im_{1}(\upsilon(\varkappa_{2},y_{2}),\omega(\varkappa_{2},y_{2})),\Im_{1}(\upsilon(\varkappa_{1},y_{1}),\omega(\varkappa_{1},y_{1})))\longrightarrow
0.$ (37)

In the same way, for $\varkappa_{1}<\varkappa_{2}$ and $y_{1}<y_{2}$

${\mathbb{D}}(\Im_{2}(\upsilon(\varkappa_{2},y_{2}),\omega(\varkappa_{2},y_{2})),\Im_{2}(\upsilon(\varkappa_{1},y_{1}),\omega(\varkappa_{1},y_{1})))\longrightarrow
0.$ (38)

From (37), (38) with (31), we have $\Im$ is equicontinuous on
$\widetilde{W}_{1}$.

Now, to prove that $\Im$ is relatively compact we must prove $\Im$ is level-
equicontinuous and a compact-supported. Since $\mathcal{F}$ and $\mathcal{G}$
are compact mapping,
$\mathcal{F},\mathcal{G}:\Lambda\times\widetilde{W}_{1}\rightarrow\hat{E_{f}}$
are relatively compact and from theorem (3.1) $\mathcal{F}$ and $\mathcal{G}$
are level-equicontinuous. Then for each $\varepsilon>0$, $q\in[0,1]$ there
exists $\delta>0$ such that from $|r-q|<\delta$ we get

$d_{H}(\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}^{r},\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}^{q})<\frac{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}{2a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}}\varepsilon,$
$d_{H}(\big{[}\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}^{r},\big{[}\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}^{q})<\frac{\rho_{1}^{\psi_{1}}\rho_{2}^{\psi_{2}}\Gamma(\psi_{1}+1)\Gamma(\psi_{2}+1)}{2a^{\rho_{1}\psi_{1}}b^{\rho_{2}\psi_{2}}}\varepsilon$

and

$d_{H}(\big{[}h(\varkappa,y)\big{]}^{r},\big{[}h(\varkappa,y)\big{]}^{q}))<\frac{\varepsilon}{2},~{}~{}d_{H}(\big{[}\mathfrak{g}(\varkappa,y)\big{]}^{r},\big{[}\mathfrak{g}(\varkappa,y)\big{]}^{q}))<\frac{\varepsilon}{2}.$

Hence, when $|r-q|<\delta$ we have

$\displaystyle
d_{H}(\big{[}\Im_{1}(\upsilon,\omega)\big{]}^{r},\big{[}\Im_{1}(\upsilon,\omega)\big{]}^{q})$
$\displaystyle\leq
d_{H}(\big{[}h(\varkappa,y)\big{]}^{r},\big{[}h(\varkappa,y)\big{]}^{q}))$
$\displaystyle+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}{d_{H}}(\big{[}\mathcal{F}(s,t,\upsilon,\omega)\big{]}^{r},\big{[}\mathcal{F}(s,t,\upsilon,\omega)\big{]}^{q})}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\leq\frac{\varepsilon}{2}+\frac{\rho_{1}\rho_{2}\varepsilon\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}{2a^{\varphi_{1}}b^{\varphi_{2}}\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\leq\frac{\varepsilon}{2}+\frac{\rho_{1}\rho_{2}\varepsilon\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}{2a^{\rho_{1}\varphi_{1}}b^{\rho_{2}\varphi_{2}}\Gamma(\varphi_{1})\Gamma(\alpha_{2})}\frac{\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{2}\varphi_{2}}}{\rho_{1}\rho_{2}\varphi_{1}\varphi_{2}}.$

This implies

$d_{H}(\big{[}\Im_{1}(\upsilon,\omega)\big{]}^{r},\big{[}\Im_{1}(\upsilon,\omega)\big{]}^{q})\leq\varepsilon.$
(39)

Likewise, we obtain

$d_{H}(\big{[}\Im_{2}(\upsilon,\omega)\big{]}^{r},\big{[}\Im_{2}(\upsilon,\omega)\big{]}^{q})\leq\varepsilon.$
(40)

Based on equations (39), (40) and (31) it can be inferred that for each small
$\varepsilon>0$, there exists $\delta>0$ such that

$d_{H}(\big{[}\Im(\upsilon,\omega)\big{]}^{r},\big{[}\Im(\upsilon,\omega)\big{]}^{q})\leq\varepsilon.$

Thus, $\Im$ is level-equicontinuous.

Since $\mathcal{F}$ and $\mathcal{G}$ are relatively compact, it can be seen
from theorem (3.1) that $\mathcal{F}$ and $\mathcal{G}$ are compact-supported
and level-equicontinuous on $[0,1]$. Hence, there exists a compact subsets
$M_{1}^{*},M_{2}^{*}\subset\mathbb{R}$ such that

$\big{[}\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y)\big{]}^{0}\subseteq
M_{1}^{*}~{}~{}and~{}~{}\big{[}\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y)\big{]}^{0}\subseteq
M_{2}^{*}.$

Also, since $h,\mathfrak{g}$ are compact-supported, then from (3.1) there
exists a compact subsets $M_{11}^{*},M_{22}^{*}\subset\mathbb{R}$ such that
$\big{[}h(\varkappa,y)\big{]}^{0}\subseteq M_{11}^{*}$ and
$\big{[}\mathfrak{g}(\varkappa,y)\big{]}^{0}\subseteq M_{22}^{*}$ for every
$(\varkappa,y,\upsilon,\omega)\in\Lambda\times\widetilde{W}_{1}.$ Thus, we
obtain

$\displaystyle\big{[}\Im_{1}(\upsilon,\omega)\big{]}^{0}$
$\displaystyle=\bigg{[}h(\varkappa,y)+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\mathcal{F}(s,t,\upsilon,\omega)}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds\bigg{]}^{0}$
$\displaystyle=\big{[}h(\varkappa,y)\big{]}^{0}+\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}\big{[}\mathcal{F}(\varkappa,y,\upsilon,\omega)\big{]}^{0}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\subseteq
M_{11}^{*}+\frac{M_{1}^{*}\varkappa^{\rho_{1}\varphi_{1}}y^{\rho_{2}\varphi_{2}}}{\rho_{1}^{\varphi_{1}}\rho_{2}^{\varphi_{2}}\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}.$

Since $\varkappa^{\rho_{1}\alpha_{1}}y^{\rho_{2}\alpha_{2}}$ is bounded on
$\Lambda$, then there exists a compact set
$\mathcal{N}_{1}^{*}\subseteq\mathbb{R}$ such that
$\big{[}\Im_{1}(\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}^{0}\subseteq\mathcal{N}_{1}^{*}$
Hence, $\Im_{1}$ is compact-supported.
In a like manner, $\big{[}\Im_{2}(\upsilon,\omega)\big{]}^{0}\subseteq
M_{22}^{*}+\frac{M_{2}^{*}\varkappa^{\rho_{1}\beta_{1}}y^{\rho_{2}\beta_{2}}}{\rho_{1}^{\beta_{1}}\rho_{2}^{\beta_{2}}\Gamma(\beta_{1}+1)\Gamma(\beta_{2}+1)}$
hold, then there exists a compact set $\mathcal{N}_{2}^{*}\subseteq\mathbb{R}$
such that
$\big{[}\Im_{2}(\upsilon(\varkappa,y),\omega(\varkappa,y))\big{]}^{0}\subseteq\mathcal{N}_{2}^{*}$.
Hence $\Im_{2}$ is compact-supported. From (30), it follows that

$\big{[}\Im(\upsilon,\omega)\big{]}^{0}\subseteq\\{\mathcal{N}_{i}^{*}\subseteq\mathbb{R}|\mathcal{N}_{i}^{*}\subseteq\mathcal{N}_{j}^{*},i,j=1,2\\},$

which shows that $\Im$ is compact-supported. And by Ascoli-Arzelá theorem
$\Im$ is relatively compact.
According to the above steps with Schouder’s theorem , we deduce that $\Im$
has at least one fixed point, which is a solution to the $\mathcal{CK}$
coupled system (2). ∎

Now in our next theorem, we discuss the uniqueness results for the problem
(2).

######  Theorem 4.2.

Let $(\upsilon,\omega),(\hat{\upsilon},\hat{\omega})\in
C[{\boldsymbol{\varOmega}},\hat{E_{f}}]\times
C[{\boldsymbol{\varOmega}},\hat{E_{f}}]$, for all
$(\varkappa,y)\in{\boldsymbol{\varOmega}}$ there exists a constants
$R_{i}>0(i=1,2)$ such that

${\mathbb{D}}(\mathcal{F}(\varkappa,y,\upsilon,\omega),\mathcal{F}(\varkappa,y,\hat{\upsilon},\hat{\omega}))\leq
R_{1}\big{(}~{}{\mathbb{D}}(\upsilon,\hat{\upsilon})+~{}{\mathbb{D}}(\omega,\hat{\omega})\big{)},$
${\mathbb{D}}(\mathcal{G}(\varkappa,y,\upsilon,\omega),\mathcal{G}(\varkappa,y,\hat{\upsilon},\hat{\omega}))\leq
R_{2}\big{(}~{}{\mathbb{D}}(\upsilon,\hat{\upsilon})+~{}{\mathbb{D}}(\omega,\hat{\omega})\big{)}.$

Then the problem(2) has a unique solution if

$\varXi_{1}=\frac{R_{1}}{\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\varphi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\varphi_{2}}<1,$
$\varXi_{2}=\frac{R_{2}}{\Gamma(\psi_{1}+1)\Gamma(\psi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\beta_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\psi_{2}}<1,$

and

$\varXi^{*}=\binom{\varXi_{1}}{\varXi_{2}}<1.$

###### Proof.

Let $\Im$ be an operator defined in Theorem (4.1). We will prove that $\Im$
has a unique fuzzy solution. Let
$(\upsilon,\omega),(\hat{\upsilon},\hat{\omega})\in C[\Lambda]\times
C[\Lambda]$, for all $(\varkappa,y)\in C[\Lambda]$. Then

$\displaystyle{\mathbb{D}}(\Im_{1}((\upsilon,\omega)(\varkappa,y)),\Im_{1}((\hat{\upsilon},\hat{\omega})(\varkappa,y)))$
$\displaystyle\leq\frac{\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}{\mathbb{D}}(\mathcal{F}(s,t,\upsilon,\omega),\mathcal{F}(s,t,\hat{\upsilon},\hat{\omega}))}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\leq\frac{R_{1}\rho_{1}^{1-\varphi_{1}}\rho_{2}^{1-\varphi_{2}}\big{(}{H^{*}}(\upsilon,\hat{\upsilon})+{H^{*}}(\omega,\hat{\omega})\big{)}}{\Gamma(\varphi_{1})\Gamma(\varphi_{2})}\int_{0}^{\varkappa}\int_{0}^{y}\frac{s^{\rho_{1}-1}t^{\rho_{2}-1}}{(\varkappa^{\rho_{1}}-s^{\rho_{1}})^{1-\varphi_{1}}(y^{\rho_{2}}-t^{\rho_{2}})^{1-\varphi_{2}}}dtds$
$\displaystyle\leq\frac{R_{1}}{\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\varphi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\varphi_{2}}\big{(}{H^{*}}(\upsilon,\hat{\upsilon})+{H^{*}}(\omega,\hat{\omega})\big{)}.$

Consequently,

$H^{*}(\Im_{1}((\upsilon,\omega)(\varkappa,y)),\Im_{1}((\hat{\upsilon},\hat{\omega})(\varkappa,y)))\leq\varXi_{1}\big{(}{H^{*}}(\upsilon,\hat{\upsilon})+{H^{*}}(\omega,\hat{\omega})\big{)}.$
(41)

Similarly, we get

$H^{*}(\Im_{2}((\upsilon,\omega)(\varkappa,y)),\Im_{2}((\hat{\upsilon},\hat{\omega})(\varkappa,y)))\leq\varXi_{2}\big{(}{H^{*}}(\upsilon,\hat{\upsilon})+{H^{*}}(\omega,\hat{\omega})\big{)}.$
(42)

Therefore,

$H^{*}(\Im((\upsilon,\omega)(\varkappa,y)),\Im((\hat{\upsilon},\hat{\omega})(\varkappa,y)))\leq\varXi^{*}\big{(}{H^{*}}(\upsilon,\hat{\upsilon})+{H^{*}}(\omega,\hat{\omega})\big{)},$

for all $(\upsilon,\omega),(\hat{\upsilon},\hat{\omega})\in C[\Lambda]\times
C[\Lambda]$. Hence, $\Im$ is a contraction mapping. Thus, the $\mathcal{CK}$
system (2) has a unique solution. ∎

######  Example 4.1.

Consider the following coupled system of fuzzy fractional PDEs

$\left\\{\begin{array}[]{cc}{}^{C}\mathfrak{D}_{0^{+}}^{\frac{1}{2},\frac{3}{2}}\upsilon(\varkappa,y)=\frac{\varkappa
y[\upsilon(\varkappa,y)+\omega(\varkappa,y)]}{2(1+\upsilon(\varkappa,y)+\omega(\varkappa,y))},\\\
\\\
{}^{C}\mathfrak{D}_{0^{+}}^{\frac{2}{3},\frac{3}{2}}\omega(\varkappa,y,)=\frac{8(\Gamma(\frac{2}{3}))^{2}}{9e^{\varkappa+y+2}}[\upsilon(\varkappa,y)+\omega(\varkappa,y)],\end{array}\right.$
(43)

with the following initial conditions:

$\left\\{\begin{array}[]{cc}\upsilon(\varkappa,0)=\mathcal{K}\varkappa,~{}~{}\upsilon(0,y)=\mathcal{K}y^{2},\\\
\omega(\varkappa,0)=2\mathcal{K}\varkappa,~{}~{}\omega(0,y)=\mathcal{K}y,\end{array}\right.$
(44)

where $(\varkappa,\zeta)\in(0,\frac{1}{2}]\times(0,1]$, $\mathcal{K}=(1,2,3)$
is a triangle fuzzy number,
$\mathcal{F}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))=\frac{\varkappa
y[\upsilon(\varkappa,y)+\omega(\varkappa,y)]}{2(1+\upsilon(\varkappa,y)+\omega(\varkappa,y))}$
and
$\mathcal{G}(\varkappa,y,\upsilon(\varkappa,y),\omega(\varkappa,y))=\frac{8(\Gamma(\frac{2}{3}))^{2}}{9e^{\varkappa+y+2}}[\upsilon(\varkappa,y)+\omega(\varkappa,y)]$,
For any $\upsilon,\omega,\hat{\upsilon},\hat{\omega}\in E_{f}$ and
$(\varkappa,y)\in(0,\frac{1}{2}]\times(0,1],$ we have

${\mathbb{D}}(\mathcal{F}(\varkappa,y,\upsilon,\omega),\mathcal{F}(\varkappa,y,\hat{\upsilon},\hat{\omega}))\leq\frac{1}{4}\big{(}~{}{\mathbb{D}}(\upsilon,\hat{\upsilon})+~{}{\mathbb{D}}(\omega,\hat{\omega})\big{)},$

and

${\mathbb{D}}(\mathcal{G}(\varkappa,y,\upsilon,\omega),\mathcal{G}(\varkappa,y,\hat{\upsilon},\hat{\omega}))\leq\frac{8(\Gamma(\frac{2}{3}))^{2}}{9e^{2}}\big{(}~{}{\mathbb{D}}(\upsilon,\hat{\upsilon})+~{}{\mathbb{D}}(\omega,\hat{\omega})\big{)}.$

Moreover, for
$\rho=(\rho_{1},\rho_{2})=\frac{3}{2},\varphi=(\varphi_{1},\varphi_{2})=\frac{1}{2}$
$\psi=(\psi_{1},\psi_{2})=\frac{2}{3}$, $R_{1}=\frac{1}{4}$ and
$R_{2}=\frac{8(\Gamma(\frac{2}{3}))^{2}}{9e^{2}}$ we have

$\varXi_{1}=\frac{R_{1}}{\Gamma(\varphi_{1}+1)\Gamma(\varphi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\varphi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\varphi_{2}}\backsimeq
0.35689<1,$
$\varXi_{2}=\frac{R_{2}}{\Gamma(\psi_{1}+1)\Gamma(\psi_{2}+1)}\bigg{(}\frac{a^{\rho_{1}}}{\rho_{1}}\bigg{)}^{\psi_{1}}\bigg{(}\frac{b^{\rho_{2}}}{\rho_{2}}\bigg{)}^{\psi_{2}}\backsimeq
0.07882<1.$

Hence the conditions of theorem (4.2) are satisfied.Thus the coupled system
given by (43)-(44) has a unique solution on
$\boldsymbol{\varOmega}=(0,\frac{1}{2}]\times(0,1]$.

##  5 Conclusion

Study of properties of fractional partial differential equations has important
implications for various fields, including physics, biology, finance, and
control engineering. The incorporation of fuzzy logic and fractional calculus
in the modeling of real-world problems with uncertainty and memory effects is
crucial for accurately representing and understanding complex systems. In this
paper we have extended the concept of fuzzy Caputo Katugampola calculus for
one-variable functions to be applicable for fuzzy-valued multivariable
functions. By utilizing Schauder’s fixed point theorem and certain of notions
of relative compactness in fuzzy number spaces, we have studied the existence
and uniqueness of solutions for both single and coupled systems of fuzzy
partial differential equations under Caputo Katugampola derivatives.

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