1 Introduction
In this section, we are going to introduce some basic concepts on fractional calculus, a pair-parameter
$(\beta , \gamma )$
matrix Mittag–Leffler function, Babenko’s approach dealing with a fractional differential equation with a nonlocal initial condition, as well as the current work on fractional partial differential equations.
Let
$\omega \in [0, 1]^n \subset {\mathbb R}^n$
and
$\chi \in [0, 1]$
. Then we define for
$\beta _1, \dots , \beta _n \geq 0$
[Reference Kilbas, Srivastava and Trujillo4],

where
$\Lambda $
is a continuous mapping from
$[0, 1] \times [0, 1]^n$
to
$\mathbb R$
.
In particular, we have

from [Reference Li5].
The partial Liouville–Caputo fractional derivative
$_c \partial ^{\alpha }/\partial \chi ^{\alpha }$
of order
$2 < \alpha \leq 3$
with respect to
$\chi $
is defined in [Reference Kilbas, Srivastava and Trujillo4] as

One of the most essential subjects of differential equations is the stability theory of Hyers–Ulam [Reference Mohanapriya, Park, Ganesh and Govindan9]. The idea of such stability for differential equations is the substitution of the equation with a given inequality that acts as a perturbation of the equation.
In this paper, we study the uniqueness and Hyers–Ulam stability for the following new fractional nonlinear partial integro-differential equation (FNPIDE) for
${\alpha _{i j} \geq 0\; (i = 1, 2, \dots , n, j = 1, 2, \dots , l \in \mathbb N)}$
:

where
$(\chi , \omega ) \in [0, 1] \times [0, 1]^n$
,
$a_j, \phi _k \in C([0, 1]^n)$
for
$k = 1, 2, 3$
, and
$\phi : [0, 1] \times [0, 1]^n \times \mathbb R \rightarrow \mathbb R$
satisfies certain conditions to be given later.
In addition, the operator
$I_\chi ^{\alpha }$
is the partial Riemann–Liouville fractional integral of order
$\alpha $
with respect to
$\chi $
, given by

Our main techniques are to derive an equivalent integral equation of equation (1.1) by Babenko’s approach and then to obtain the uniqueness and Hyers–Ulam stability using Banach’s contractive principle and newly established pair-parameter Mittag–Leffler functions below.
Assume
$\alpha _{ij} \geq 0, \alpha _i> 0$
for all
$i = 1, \dots , n, \, j = 1, \dots , l$
, and there is
$1 \leq i_0 \leq n$
such that
$\alpha _{i_0 j}> 0$
for all
$j = 1, \dots , l$
. We define

Definition 1.1 Let
$\beta \geq 0, \; \gamma> 0$
. A pair-parameter
$(\beta , \gamma )$
matrix Mittag–Leffler function is defined by

where
$ \zeta _i \in \mathbb {C} $
for
$i = 1, 2, \dots , l$
, and

It follows that

where
$E_M$
is a matrix Mittag–Leffler function given in [Reference Li, Beaudin, Rahmoune and Remili6].
Since there exists a positive constant
$\theta $
such that

we claim

which implies that
$E_M^{(\beta , \gamma )}(\zeta _1, \dots , \zeta _l) $
is well defined as the multivariate Mittag–Leffler function
$E_{ (\alpha _{i_0 1}, \dots , \alpha _{i_0 l}), \alpha _{i_0}}( |\zeta _l|, \dots , |\zeta _l|)$
converges [Reference Hadid and Luchko3]. Obviously,

where

and

which is the well-known two-parameter Mittag–Leffler function, and

Babenko’s approach (BA) [Reference Babenko1] is a useful tool for dealing with various integral or differential equations (including PDEs) with initial or boundary problems. Let f be a continuous function on
$[0, 1] \times \mathbb R$
with

To demonstrate this method in detail, we convert the following fractional differential equation with a nonlocal initial condition into an equivalent implicit integral equation:

where
$ 0 < \alpha \leq 1$
, a and
$ \beta $
are constants.
Evidently, we get by applying the operator
$I^{\alpha }$
to equation (1.3)

which infers that

and

Treating the factor
$\left (1 + a I^{\alpha }\right )$
as a variable and using BA, we come to

by noting that

and

In summary, equation (1.3) is equivalent to the following integral equation:

The above integral equation, in fact, plays an important role in studying the uniqueness of equation (1.3) in the Banach space
$C[0, 1]$
with the norm

We further assume there is a constant
$\mathcal L> 0$
such that f satisfies the following Lipschitz condition:

and

Then equation (1.3) has a unique solution in
$C[0, 1]$
.
To prove this, we define a nonlinear mapping M over
$C[0, 1]$
as

It follows from the above that
$ (M \Phi ) (x) \in C[0, 1]$
. We are going to show that M is contractive. For
$\Phi _1, \Phi _2 \in C[0, 1]$
, we have

Hence,

Since
$\mathcal B < 1$
, we claim that equation (1.3) has a unique solution in
$C[0, 1]$
by Banach’s contractive principle (BCP).
We define
$S([0, 1] \times [0, 1]^n)$
as the Banach space of all continuous mappings from
$[0, 1] \times [0, 1]^n$
to
$\mathbb R$
with the norm

Fractional partial differential equations (a generalization of classical PDEs of integer order) are used to model various phenomena in physics, engineering, and other fields. There are intensive studies on fractional PDEs using various approaches, such as integral transforms [Reference Mahor, Mishra and Jain8], analytical and numerical solutions [Reference Momani and Odibat10], homotopy analysis technique [Reference Dehghan and Shakeri2, Reference Singh, Kumar and Swroop11], variational iteration method [Reference Xu, Ling, Zhao and Jassim12] and so on. Very recently, Li et al. [Reference Li, Saadati, O’Regan, Mesiar and Hrytsenko7] investigated the uniqueness of solutions for the following fractional PDE with nonlocal initial value conditions for
$2 < \alpha \leq 3$
,
$0 < \alpha _1 \leq 1$
and
$ \alpha _2> 0$
based on BCP, BA and the multivariate Mittag–Leffler function for a constant
$\eta $
:

where
$(\chi , \omega ) \in [0, 1] \times [0, b]$
,
$\psi \in C[0, 1]$
and
$f: [0, 1] \times [0, b] \times \mathbb R \rightarrow \mathbb R$
satisfies certain conditions.
We will first convert equation (1.1) into an equivalent implicit integral equation in a series by BA in Section 2, and then further study the uniqueness of solutions via BCP in the space
$S([0, 1] \times [0, 1]^n)$
. In Section 3, we derive the Hyers–Ulam stability based on the implicit integral equation and present several examples demonstrating applications of the key results obtained in Section 4. Finally, we summarize the entire work in Section 5.
2 Uniqueness
We begin converting equation (1.1) to an implicit integral equation then derive sufficient conditions for the uniqueness based on Banach’s contractive principle.
Theorem 2.1 Suppose
$a_j, \phi _1, \phi _2, \phi _3 \in C([0, 1]^n)$
for
$j = 1, 2, \dots , j$
,
$\phi $
is a continuous function on
$[0, 1] \times [0, 1]^n \times \mathbb R$
with

$\alpha _{ij} \geq 0 $
for all
$i = 1, \dots , n, \, j = 1, \dots , l$
, and there is
$1 \leq i_0 \leq n$
such that
$\alpha _{i_0 j}> 0$
for all
$j = 1, \dots , l$
. Furthermore, we assume that

and

Then equation (1.1) is equivalent to the following implicit integral equation:

In addition,
$\Lambda $
is a uniformly bounded function satisfying

where

Proof It follows from [Reference Li, Saadati, O’Regan, Mesiar and Hrytsenko7] that

where
$0 < \alpha \leq 3$
.
Applying the integral operator
$ I_\chi ^{\alpha } $
to equation (1.1) and using the condition
$\Lambda (0, \omega ) = \phi _1(\omega )$
, we get

Setting
$\chi = 1$
, we come to

Differentiating equation (2.2) with respect to
$\chi $
, we deduce that for
$\chi = 1$
,

by the given initial condition.
From equations (2.3) and (2.4), we derive that

and

Hence,

Using BA, we deduce that

where

and finally,

Let

Thus,

where

and

Using our assumption

we claim that

which indicates that
$\Lambda $
is a uniformly bounded function. This completes the proof of Theorem 2.1.
Theorem 2.2 Suppose
$a_j, \phi _1, \phi _2, \phi _3 \in C([0, 1]^n)$
for
$j = 1, 2, \dots , j$
,
$\phi $
is a continuous and bounded function on
$[0, 1] \times [0, 1]^n \times \mathbb R$
, satisfying the Lipschitz condition for a positive constant
$\mathcal C$

$\alpha _{ij} \geq 0 $
for all
$i = 1, \dots , n, \, j = 1, \dots , l$
, and there is
$1 \leq i_0 \leq n$
such that
$\alpha _{i_0 j}> 0$
for all
$j = 1, \dots , l$
. Furthermore, we assume that

and

where

Then equation (1.1) has a unique uniformly bounded solution in the space
$S([0, 1] \times [0, 1]^n).$
Proof We define a nonlinear mapping
$\mathcal F$
over
$S([0, 1] \times [0, 1]^n)$
as

It follows from the proof of Theorem 2.1 that
$(\mathcal F \Lambda ) \in S([0, 1] \times [0, 1]^n)$
. We shall show that
$\mathcal F$
is contractive. Indeed, for
$\Lambda _1, \, \Lambda _2 \in S([0, 1] \times [0, 1]^n)$
, we have

Therefore,

Since
$q < 1$
, equation (1.1) has a unique uniformly bounded solution in
$S([0, 1] \times [0, 1]^n)$
by BCP. The proof is completed.
3 The Hyers–Ulam stability
In this section, we are going to derive the Hyers–Ulam stability of equation (1.1) using the implicit integral equation from Section 2.
Definition 3.1 We say that the FNPIDE (1.1) is Hyers–Ulam stable if there exists a constant
$\mathcal K> 0$
such that for all
$\epsilon> 0$
and a continuously differentiable function
$\Lambda $
satisfying the three boundary conditions and the inequality

then there exists a solution
$\Lambda _0$
of equation (1.1) such that

where
$\mathcal K$
is a Hyers–Ulam stability constant.
Theorem 3.1 Suppose
$a_j, \phi _1, \phi _2, \phi _3 \in C([0, 1]^n)$
for
$j = 1, 2, \dots , j$
,
$\phi $
is a continuous function on
$[0, 1] \times [0, 1]^n \times \mathbb R$
satisfying the Lipschitz condition for a positive constant
$\mathcal C$

$\alpha _{ij} \geq 0 $
for all
$i = 1, \dots , n, \, j = 1, \dots , l$
, and there is
$1 \leq i_0 \leq n$
such that
$\alpha _{i_0 j}> 0$
for all
$j = 1, \dots , l$
. Furthermore, we assume that

and

where

Then equation (1.1) is Hyers–Ulam stable in the space
$S([0, 1] \times [0, 1]^n).$
Proof Let

Then

and from our assumption

It follows from the proof of Theorem 2.1 that

and

by noting that
$\phi $
is a continuous and

if
$\Lambda \in S([0, 1] \times [0, 1]^n).$
Hence,

which implies that

Finally, we have

where

This completes the proof.
Remark 3.2 We should point out that Theorem 3.1 does not require the condition that
$\phi $
is a bounded function. Moreover,
$\Lambda _0$
is not a uniformly bounded function in general, which is different from Theorem 2.2. Since
$\Omega = [0, 1] \times [0, 1]^n$
is bounded and closed (compact) the Hyers–Ulam stability is guaranteed by noting the fact that all continuous functions reach their maximum and minimum over
$\Omega $
. The Hyers–Ulam stability constant
$\mathcal K$
obtained above is the best possible in our approach. There is a possible lower bound on the Hyers–Ulam stability constant but it would be tough and difficult to find it.
4 Examples
We will present two examples demonstrating applications of key theorems obtained from previous sections.
Example 4.1 The following fractional differential equation with a nonlocal initial condition:

has a unique solution in
$C[0, 1]$
.
Proof Clearly,

is bounded and

if
$x \in [0, 1]$
. It remains to find the value

Hence, equation (4.1) has a unique solution in the Banach space
$C[0, 1]$
.
Example 4.2 The following FNPIDE with a mixed boundary condition:

where

and

has a unique uniformly bounded solution and the Hyers–Ulam stability in the space
$S([0, 1] \times [0, 1]^4)$
.
Proof Clearly,
$a_j$
for
$j = 1, 2, 3, 4$
,
$\phi _1, \, \phi _2, \, \phi _3 \in C([0, 1]^4)$
and

is a continuous and bounded function on
$[0, 1] \times [0, 1]^4 \times \mathbb R$
, satisfying the Lipschitz condition with
$\mathcal C = 1/59$
:

Furthermore,

We need to compute the value

where




and finally

Using the following Python codes to get

Hence, equation (4.2) has a unique uniformly bounded solution in the space
$S([0, 1] \times [0, 1]^4)$
by Theorem 2.2, and it is Hyers–Ulam stable by Theorem 3.1.

Remark 4.3 We have used the Python language to find approximates values of our newly established pair-parameter matrix Mittag–Leffler functions to study the uniqueness of solutions to equation (1.1). Slightly changing the codes we can compute values of the multivariate Mittag–Leffler functions. As far as we know from current research related to computation of the Mittag–Leffler functions, this approach is efficient and simple.
5 Conclusion
We have studied the uniqueness and Hyers–Ulam stability to the new equation (1.1) based on the pair-parameter matrix Mittag–Leffler functions, Banach’s contractive principle as well as Babenko’s approach. A few examples were provided to demonstrate applications of main results derived. The methods used in the current work are also suitable for different types of differential equations with various initial or boundary conditions, as well as integral equations with variable coefficients, which cannot be handled by any existing integral transforms.
Acknowledgments
The author is thankful to the reviewers and editor for giving valuable comments and suggestions.
Competing interests
The author declares no competing interests.
Data availability statement
No data were used to support this study.