Minimum Superstability of Stochastic Ternary Antiderivations in Symmetric Matrix-Valued FB-Algebras and Symmetric Matrix-Valued FC- (cid:5) -Algebras

: Our main goal in this paper is to investigate stochastic ternary antiderivatives (STAD). First, we will introduce the random ternary antiderivative operator. Then, by introducing the aggregation function using special functions such as the Mittag-Lefﬂer function (MLF), the Wright function (WF), the H -Fox function ( H FF), the Gauss hypergeometric function (GHF), and the exponential function (EXP-F), we will select the optimal control function by performing the necessary calculations. Next, by considering the symmetric matrix-valued FB-algebra (SMV-FB-A) and the symmetric matrix-valued FC- (cid:5) -algebra (SMV-FC- (cid:5) -A), we check the superstability of the desired operator. After stating each result, the superstability of the minimum is obtained by applying the optimal control function.


Introduction
The study of the stability problem for group homomorphisms started with the famous Ulam question in 1940, and in 1941 Hyers established the stability of nonlinear functions in a particular case.In 1978, Rassias extended this problem and now it is known as the Hyers-Ulam-Rasias problem.Various aspects of the Hyers-Ulam stability for various functions and mappings have been investigated and among these equations and mappings, we refer the reader to Euler-Lagrange functional equations (E-L-FE), differential equations (DE), Navier-Stokes equations (N-SE), as well as mappings such as Cauchy-Jensen mappings, k-additive mappings, multiplicative mappings [1][2][3][4]. Most of the investigations have been carried out in Banach spaces and now research is conducted in other spaces as well [5][6][7][8]. Also, the stability of groups, Banach algebra, ternary Banach algebras, and C-ternary algebras have attracted the attention of many researchers [9][10][11]. Among the various applications of ternary algebras, we mention Nambu mechanics and quark in physics and mathematics and we also point to different applications of the additive principle in physics in the field of internal energy and the superposition principle. In addition, many physics problems can be considered as a linear system and can be solved [12,13].
In 2003, Radu-Mihet proposed a new method to obtain the exact solution and error estimation, which was based on the alternative fixed-point method. Note that fixed point theory arises in various fields such as dynamical systems, equilibrium problems, and differential equations because this theory provides basic tools for examining these problems [14,15]. (1) where σ 1 , σ 2 ∈ C and |σ 1 | + |σ 2 | > 2. The general structure of the article is as follows.
In the second section, all the required concepts including special functions and spaces used to prove the desired results are given. In the third section, after stating the necessary lemmas, the stability of the stochastic operator in SMVFB-A is investigated, and then the minimum stability of this operator is proved. In the fourth section, the superstability of stochastic ternary antiderivatives is proved by introducing stochastic ternary antiderivatives and considering T-SMVFB and T-SMVFC--A spaces. Also, in the form of an example, minimum stability is investigated. In the last part, the superstability of continuous stochastic ternary antiderivatives in the introduced spaces is proved.

Preliminaries
We denote the set of all p × p diagonal matrices by D H = diagH([0, 1]), and we consider this set as follows For the above set, we have (II) h g = g h for all h, g ∈ D H ; (III) h (g k) = (h g) k for all h, g, k ∈ D H ; (IV) h g and k s implies that h k g s, for all h, k, g, s ∈ D H ; For convergent sequences {h p } and {g p } with convergence points h and g, if we have (V) lim m (h p g p ) = h g, then the GTN is a continuous GTN (CGTN).
In the following, we define some examples of CGTN:  We refer to [5,6,9] to see more numerical examples of the introduced CGTNs.

Definition 8 ([21]
). We consider an SMVFB-A (M, Ω M , , ). We say that M is an SMVFB--A if the mapping δ → δ on M has the following conditions Moreover, with the following condition Definition 9 ( [11][12][13]). We consider the complex SMVFB-S M (CSMVFB-S). If by using M, which is from M 3 to M, such that it has the following properties then, we say that M is a ternary SMVFB-A (T-SMVFB-A).
We assume that (M, [., ., .]) is T-SMVFB-A. If M has an identity member say such that where u * is an unital algebra. If (M, •) is an unital algebra, then the relation Definition 10 ([3,12,13]). We assume that M and N are SMVFB-As. A C-linear stochastic mapping Y : is called a stochastic ternary homomorphism (STH).
Definition 11 ([12,13]). We assume that M is an SMVFB-A. A C-linear stochastic mapping is called a stochastic ternary derivation (STD).
In the following, we introduce special functions that we need to select the optimal control function. We have also performed the necessary calculations on these functions and have shown a representation of these functions and calculations in the form of graphs.
where u f = exp{f(log |u| + i arg u)} and A ∈ C is a path that is deleted. Also, the symbol Definition 15 ([6,8,9]). The generalized Bessel Maitland function (GBMF) or the Wright function (WF) of order 1/(1 + σ) is represented by using the series In the Figures 1-3 we can see the values of introduced special functions in this paper. Definition 16 ([9]). An p-ary (p ∈ N) generalized aggregation function (p-AGAF) A (p) : R p −→ R has the following property • u ι ≤ v ι =⇒ A (p) (u 1 , · · · , u p ) ≤ A (p) (v 1 , · · · , v p ), for all ι ∈ {1, · · · , p}, and for (u 1 , · · · , u p ), (v 1 , · · · , v p ) ∈ R p . For the sake of simplicity, we can remove the m number, which represents the number of variables of the aggregation function, and denote this function as A. Also, when p = 1, the aggregation function is shown as A (1) (u) = u for all φ ∈ R.
Example 2. The geometric mean function (GMF) GM : R p −→ R is defined by Example 3. The projection function (PF) P β : R p −→ R for β ∈ [p] and βth argument is defined by where u (β) is the βth lowest coordinate of u, i.e., u (1) ≤ · · · ≤ u (β) ≤ · · · u (p) . Also, the following functions show the PF in the first and last coordinates P F (u) = P 1 (u) = u 1 , P L (u) = P p (u) = u p .
Example 6. The median function (MF) is defined as follows for odd and even values of (u 1 , · · · , u 2β−1 ) and (u 1 , · · · , u 2β ), respectively MED(u 1 , · · · , u 2β−1 ) = u (β) , (a) AM and GM functions for u ∈ (−500, 500) (b) The aggregation AM function for u ∈ (10, 200) By referring to [9] and studying the information in the presented table, we choose the minimum function as the control function. Also Figures 1-3 help us to choose this function. Consider the function and we choose the following function as a control function We call this set the set of all contraction mappings that is, every contractive function is located in this set. Next, we present the Diaz-Margolis theorem (FTP) [5,6,8,9]. Theorem 1. We consider GCMS (M, T M ) and assume that u, v ∈ M, and also J ∈ CON (M) such that J < 1. With these assumptions, we assume that for every r, r 0 ∈ N (r ≥ r 0 ) and for u ∈ M, T M J r u, J r+1 v < ∞,. If this condition holds, we have (1) The fixed point j of J is the convergence point of the sequence {J r u}; (2) In the set K = {u ∈ M | T M (J r 0 u, v) < ∞}, j is the unique fixed point of J;

Lemma 1. Assume that M, Ω M , ,
is an SMVFB-S. We consider the stochastic mapping φ : Π × M → M in such a way that the following inequality holds for all u, v, w ∈ M. Then the stochastic mapping φ : Π × M → M is additive.
Proof. First, we assume that u = v = w = 0 and apply this assumption to condition (S2), which is inequality (8). We get according to |σ 1 | + |σ 2 | > 2 and condition (S1) means (7) , we have φ(z, 0) = 0 and Ψ(0, 0, 0, η) = 1. Now, let us assume u = w = ζ 2 and v = ζ. By placing the new assumption in condition (S2), we have for all ζ ∈ M. On the set Υ = {R : Π × M → M : R(z, 0) = 0}, we define the mapping T M : Υ × Υ → Υ as follows It is easy to see that T M is a complete generalized metric space (CGMS) [5,6,8]. In the following, for all u ∈ M, we define a stochastic linear mapping S : Υ → Υ as follows: For Q, R ∈ Υ, we assume T M (Q, R) = κ. As a result, for all u ∈ M, we have Then, for all u ∈ M, we get Ψ u 2 , u, u 2 , η θκ , and this means T M (S(Q(z, u)), S(R(z, u))) ≤ θκ or T M (S(Q(z, u)), S(R (z, u))) ≤ θT M (Q, R). In the sequel, due to (9), for all u ∈ M, we get and this means T M (φ, Sφ) ≤ θ 2 . According to Theorem 1, there exists a unique fixed point such as the stochastic mapping E : Υ × M → M for S, which is defined as Then, considering this fixed point, there is a ς ∈ (0, ∞) such that for all u ∈ M, we have On the other hand, because lim p→∞ Ω M S p φ − E , η = 1, then for all u ∈ M, we have lim p→∞ 2 p φ z, u 2 p = E (z, u).
According to conditions (S1) and (S2), that is, inequalities (7) and (8), for all u, v, w ∈ A , we have Therefore, according to Lemma 1, E is a stochastic additive mapping (SAM).

Example 7.
We consider the stochastic mapping φ : Π × M → M such that for all u, v, w ∈ M the following inequality holds Proof. From the proof of Theorem 2, assuming θ = 1 100 and , and the proof is complete.

Stochastic Ternary Antiderivations in T-SMVFB-A and T-SMVFC--A
In this section, we first define the stochastic ternary antiderivatives in T-SMVFB-A and T-SMVFC--A and we prove the minimum superstability. As we mentioned before, all the results are proved by considering the (α, β)-functional inequality. Definition 17. [12,13] Consider the T-SMVFB-S M. A stochastic ternary antiderivative is a stochastic C-linear mapping ψ : for all u, v, w ∈ M.
(S4) For all ω ∈ T 1 and all x, y, z ∈ A For every ω ∈ B 1 and all u, v, w ∈ M If φ is continuous and φ(z, 2u) = 2φ(z, u) for all u ∈ M, then the stochastic mapping φ : Π × M → M is a stochastic ternary antiderivation.
According to conditions (S3) and (S4), that is, inequalities (10) and (11), for all u, v, w ∈ M, we have We assume that u = ∑ p ι=1 m i u ι for all m i ∈ C, u i ∈ I(M) and u ∈ M. Since ψ in the second variable is C-bilinear, for all u, v, w ∈ M, we get Then, ψ : Π × M → M is a stochastic ternary antiderivation (STAD).

Superstability of Continuous Stochastic Ternary Antiderivations in T-SMVFB-A and T-SMVFC--A
Here, we show the superstability of continuous ternary antiderivatives ternary SMVFB-As and T-SMVFC--As.
Using the continuity Ψ, φ and Ω M (u, η) and considering the limit when p → ∞, we have