1. Introduction
The idea of stability for a functional equation (FE) arises when we substitute the FE by an inequality which acts as a perturbation of the initial equation. Over the years, stability results of FEs have been developed for obtaining an approximate solution of the perturbed equation which is close to the exact solution (ES). This topic was introduced by Ulam and Hyers in 1940–1941, and this kind of stability is called HU stability [
1,
2]. In 1978, the improvement of HU stability provided by Rassias led to the development of what is now known as HUR stability [
3]. In 1998, Ger and Alsina [
4] established the HU stability of ODEs and many authors defined types of HUR–Mittag–Leffler stability of fractional PDEs to prove that every mapping from this type can be somehow approximated by an ES of the considered equation [
5,
6,
7].
The recent interest in the Mittag–Leffler (ML) function and its various generalizations [
8] is mainly due to their close relations to Fractional Calculus and especially to fractional problems that come from applications. The special functions, along with the ML function, including the functions of Wright type, the functions of hypergeometric type and others [
9] which often appear in solutions of various types of equations with fractional operators, play a prominent role in the theory of the PDEs of fractional order that are applied in modeling of diverse phenomena [
10,
11,
12].
As it is known, the major problem of procuring ES of such equations is very crucial, and the form of the ES (if it exists) is oftentimes so arduous that it is not suitable for numerical calculation [
13,
14,
15]. In view of this, it is imperative to talk about an approximate solution and ask whether it lies close to the ES. Generally, we say that a fractional PDE is stable in the sense of Ulam if, for every solution of the fractional PDE, there exists an approximate solution of the perturbed equation that is near to it.
To clarify the issue, let us introduce the notion of Ulam-type stability of an operator equation [
16,
17]. We consider the Banach space
and an operator
for every
We also consider the operator equation
and the inequality
Equation (
1) is called HU stable if for every solution
of Inequality (
2), there is a solution
v of operator Equation (
1) such that
in which
is a constant dependent on
T.
Now, Equation (
1) is called HUR stable if there is a continuous function
, such that, for every
and for every solution
of the inequality
there is solution
v of operator Equation (
1) with
such that
Equation (
1) has HUR–Mittag–Leffler stability or HUR–Wright stability if the above statement is also true when we replace control function
by the ML function or the Wright function, respectively.
In this paper, we consider a diagonal matrix of special functions as a controller to study a fresh concept of stability, namely multi-stability. The mentioned stability helps us to obtain different approximations depending on the diverse special functions that are initially selected and to evaluate maximal stability with minimal error which enables us to obtain the best approximate solution.
We let the matrix-valued controller
be as follows:
in which
denotes a special function in the the main diagonal of square matrix
, and the natural number
n represents the numbers of special functions that we intend to consider.
We consider normed linear spaces V and Mapping has the multi-stability property if we replace the controller of HUR stability with .
For the special case of multiple stability, i.e., Mittag–Leffler–Gauss-Hypergeometric–Bessel–Maitland–Fox stability, Mittag–Leffler–Supertrigonometric stability, Mittag–Leffler–Superhyperbolic stability and the others, we refer the reader to [
18,
19,
20].
2. Preliminaries
2.1. Some Special Functions
2.1.1. Fox -Function and Related Functions
The
function (sometimes called Fox’s
-function) is a very generally defined special function due to Charles Fox (1928) (see [
21]). We let
be a proper contour of the Mellin–Barnes type in the complex
S-plane. Therefore, the
-function is given by
in which
and
where
and
We now present some special cases of the
-function including the exponential function, the Wright function, the one-parameter Mittag–Leffler function, the Fox–Wright function, the Meijer
–function, the Gauss Hypergeometric function, and the
–function, respectively, as follows:
Note that for
and
we consider
2.1.2. Mittag–Leffler Function and Related Functions
We let
and
and
The m-parameter Mittag–Leffler function is given by [
21]
in which
with
for all
and
Notice that
is given by
Here, we present some examples of (
6) as follows:
2.2. Generalized Triangular Norms (GTNs)
We suppose
with the partial order relation below:
and Symbols
and
are given by
and
Definition 1 ([
18])
. Operation , is called a GTN if for every we have(1)
(2) ,
(3)
(4)
For sequences , converging to , if , ⨀ is continuous.
For example, we assume the continuous GTNs
given as follows:
and
In this paper, we set
2.3. Matrix-Valued Fuzzy Normed Spaces
We consider vector space We assume to be a set of all matrix-valued fuzzy sets (MVF sets), including the increasing and continuous functions such that, for
In , for every and we assume if
A matrix-valued fuzzy normed space (MVFN space) [
18] is a triple
, such that, for all
, and
we obtain
(i) ,
(ii) ,
(iii) , iff
(iv)
For example,
defines a fuzzy norm (see [
18]) for all
and
Note 1. A matrix-valued fuzzy Banach space (MVFB space) is a complete MVFN space.
2.4. Matrix-Valued Random Normed Spaces
We suppose
is a set of matrix-valued distribution functions (MVDFs) including the non-decreasing and left–continuous functions
such that we have
We let contain such that we obtain In for all we assume if
Note that the maximal element for
is given by
For instance, the function
denotes an MVDF since
and
.
We assume vector space
, DF
and continuous GTN ⨀. A triple
is a matrix-valued random normed space (MVRN space) [
19] if for every
and
we obtain
- (i)
- (ii)
iff
- (iii)
,
in which is the value of at a point
For instance,
is a random norm [
19] for all
and
.
2.5. Multi-Aggregations
We let
and
An n-ary aggregation map [
19] is a mapping,
such that we obtain
In addition, for all
if
then,
Here, we introduce some examples of aggregation maps,
defined as follows:
Geometric mean functions,
Arithmetric mean functions,
Maximum functions,
Minimum functions,
Sum functions,
Product functions,
2.6. Generalized Alternative Fixed Point Theory
We first present vector-valued generalized metric spaces.
Note 2. We suppose and . Thus, for every we have iff and also iff
Definition 2 ([
21])
. Consider the set and A generalized metric d on is a map such that for every , iff
for every , iff
for every ,
Theorem 1 ([
21])
. Consider a complete generalized metric space , with , and a contractive mapping with Lipschitz constant . Hence, for all , either for any or there is an such that;
The fixed point of Γ is a convergence point of the sequence and is unique in the set ;
for every .
3. Application of Multi-Stability for Smoke Transmission Model
In [
22,
23,
24], the authors presented the following basic mathematical model which analyzes the spread of smoking in a population:
where
and
represent smokers, smokers who permanently quit smoking, potential smokers, smokers who temporarily quit smoking, and occasional smokers.
In [
16], teh authors presented an extension of (
7) to a two-age group model: 1-Group including people below 70 years old and 2-Group including people aged above 70 years. Every population consists of
for the 1-Group and
for the 2-Group. For each group, we have the following age-specific parameters:
rate of supply,
effective contact rate between and
natural death rate,
rate of quitting smoking,
rate at which occasional smokers become regular smokers,
the contact rate between smokers and temprorary quitters,
fraction of smokers who temporary quit smoking.
Therefore, the
i -age group transmission model (
) is given as follows:
Now, (
8) under the ABC fractional derivative is given by [
16]
with initial condition
where
is the ABC fractional derivative given by
for any
satisfies property
and
is the one-parameter Mittag–Leffler function given by
Notice that for any
and
the integral of ABC is given by
Reformulating the right side of (
9), we obtain
in which
According to (
10), (
9) can be expressed as follows [
16]:
where
and
for every
Now, we apply the concept of
Z-numbers and we introduce a special matrix of the form
(named the generalized
Z-number) where
is a fuzzy set, time-stamped,
is the probability distribution function and
is a degree of reliability of
that is described as a value of
(see [
25]).
We let the Banach space be
For every
we define
and
Note that every special function
given in the the main diagonal of square matrices
and
is defined as
for example,
Definition 3. Fractional-order Equation (
11)
is multi-stable with respect toif there exist such that, for and all solutions tothere exists solution to (
11)
withfor every and . The diagrams of
are shown separately in
Figure 1. As can be observed,
(brown) and
(yellow) include the highest and the lowest values, and
, are placed between them. Therefore, we can infer that the
proposes a better approximation for (
11) than the others.
As above, we can conclude that
proposes a better approximation for (
11) than
Theorem 2. For every , and we consider the Atangana–Baleanu–Caputo fractional smoke epidemic model (
11)
and the inequalities below:and We let there exist such that, for every , we haveandWe let Then, we can obtain a unique such thatandwhere Proof. According to Lemma 2.4 in [
16], the solution of (
11) is defined as follows:
We consider mapping
defined by
in which
Note that
is a complete generalized metric space (see [
19]).
We consider operator
such that
According to (
21) and (
23), we have
For every
and
we show
is a contraction mapping on
as follows:
and in a similar way,
Thus, we deduce the contractive property of since .
We remark that (see [
18]) function
is a solution of (
16)–(
17) if there exists function
(which depends on
) such that, for every
and
(1) , and ; and
Now, the solution of (
35) is given by
Making use of (
26) and (
30), we have
and in a similar way,
for every
By means of fixed point theory, we can obtain such that
(1)
is a fixed point of
,
which is unique in the set
(2) as .
(3) We have
which infers
where
The plots of special functions given in the the main diagonal of square matrix
are displayed in
Figure 2a, where the diagrams of the Wright function and the Fox
-function are displayed in green and yellow colors, and the rest are in between. Similarly, the plots of special functions given in the the main diagonal of square matrix
are displayed in
Figure 2b, where the diagrams of one-parameter Mittag–Leffler function and eight-parameter Mittag–Leffler function are displayed in brown and yellow colors, and the rest are in between.
The plots of
(the bottom chart) and
(the above chart) are displayed in
Figure 3. Now, we obtain
□
4. A New Optimal Method for a Smoke Epidemic Model in Banach Spaces
In [
16], the authors applied the Krasnoselskii-type fixed point theorem to study the existence, uniqueness, and UHR stability of (
11) in Banach spaces. We study the existence, uniqueness, and UHR stability of the governing model using the Cadariu–Radu method derived from an alternative fixed point theorem. By comparing methods, we conclude that our method provides an optimal solution with the same error value obtained through the Krasnoselskii-type fixed point theorem.
We let the Banach space be
and let
Now, we assume the following for :
- (ı)
There is and such that
- (ıı)
There is such that
We consider operator
such that
According to (
21) and (
29), we have
4.1. The Krassnoselskii Fixed Point Theorem
Lemma 1 ([
16])
. Consider Banach space with norm for any Assume is a closed convex subset of ⊤ and there are operators and with- (1)
- (2)
is compact and continuous and is a contraction operator.
Thus, there is solution such that
Theorem 3. Consider assumptions and with Then, (
11)
has at least one solution. Proof. First, we prove
is a contraction. We assume
and
which is a closed convex set. Now, we have
Thus,
Therefore, is a contraction since
We now prove
is compact and continuous for any
then,
Thus,
is bounded. We assume
then, we have
As
Thus, according to the Arzela–Ascoli theorem,
is compact and equicontinuous. Therefore, based on Lemma 1, System (
11) has at least one solution. □
Theorem 4. Consider constant with Then, operator has a unique fixed point.
Proof. We assume
. Then,
Therefore, based on the contraction principle, the operator has a unique fixed point. □
Definition 4 ([
26])
. System (
11)
is UHR stable with respect to if there exists such that for all and any solution to the following inequality,there exists solution to (
11)
with Remark 1. Sunction is a solution of (
33)
if and only if there exists function (which depends on Ξ
) such that (1)
(2)
Lemma 2. We consider the fractional-order system below,Then, for we obtain Proof. According to Lemma 2.4 in [
16], the solution of (
35) is given by
Now, based on (
30), we have
□
Theorem 5. System (
11)
has UHR stability if Proof. We suppose
is any solution and
is the unique solution of (
11) (see Theorem 4); then, for
, we obtain
Now, making use of Definition 4, we conclude that (
11) is UHR stable. □
4.2. The Cădariu–Radu Method
Here, we study the existence, uniqueness, and UHR stability of System (
11) using the Cădariu–Radu method derived from an alternative fixed point Theorem 1 (see [
26]).
Theorem 6. Consider the following inequality:where is a control function. Assume where Then, there exists a unique withand Proof. We consider mapping
defined by
Now, we define
as
According to the previous subsection, we see that
is contractive on
i.e., for any
we have
Then, we deduce the contraction property of
, since
Again using the previous subsection, we obtain
Thus, the assumptions of Theorem 1 are satisfied, and we obtain
where
is the unique solution in
□
4.3. Discussion
The concept of fractional derivative is more than three thousand years old. The role of fractional calculus has been increasing due to its application zone in diverse domains like laser propagation, energy quantization, semiconductor industry, wave propagation, biology, optical communication, quantum chemistry, etc. The fractional derivative of ABC uses the Mittag–Leffler function to consider random fuzzy models with uncertain constraint conditions [
27,
28,
29,
30,
31]. An analysis method is used to consider the best decision on the stability of the smoking epidemic model by using a new class of controllers powered by special functions. Under some special conditions, we compared the findings of our study with the obtained results in [
16] and we concluded that our method provides an optimal solution with the same error value obtained through the Krasnoselskii-type fixed point theorem [
16]. Our technique offers a constructive process to obtain our fixed points.