Best approximation of (G1,G2)$(\mathcal{G}_{1},\mathcal{G}_{2})$-random operator inequality in matrix Menger Banach algebras with application of stochastic Mittag-Leffler and H$\mathbb{H}$-Fox control functions

*Correspondence: rsaadati@iust.ac.ir; rsaadati@eml.cc 1School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran Full list of author information is available at the end of the article Abstract We stabilize pseudostochastic (G1,G2)-random operator inequality using a class of stochastic matrix control functions in matrix Menger Banach algebras. We get an approximation for stochastic (G1,G2)-random operator inequality by means of both direct and fixed point methods. As an application, we apply both stochastic Mittag-Leffler andH-fox control functions to get a better approximation in a random operator inequality.


Introduction and preliminaries
The theory of special functions, such as Mittag-Leffler function, hypergeometric function, Wright function, H-Fox function, and so on, encircles a significant segment of mathematics. In recent centuries, the necessity of solving problems taking place in various fields of science motivated the advancement of the theory of special functions. These functions have extensive applications in a variety of different fields, together with material science and engineering science, biology, chemistry, mathematical physics, and both applied and pure mathematics. The interested readers can review the literature [1][2][3][4].
In 1903, the Swedish mathematician Gosta Mittag-Leffler presented a generalization of the exponential function and introduced some properties of this function. In 1905, Wiman introduced its general form. Mittag-Leffler function naturally appears as the solution of fractional order integro-differential equations and particularly in the investigations of electric networks, random walks, fluid flow, superdiffusive transport, the fractional generalization of the kinetic equation, diffusive transport akin to diffusion, and in the study of complex systems. During the last 20 years, the interest in Mittag-Leffler function has remarkably increased among scientists and engineers, owing to it wide potential in applications. We suggest the readers to consult the literature [5,6].
In 1961, Charles Fox presented a generalization of the Meijer G-function and the Fox-Wright function. H-Fox function is defined by a Mellin-Barnes integral in the context of symmetrical Fourier kernels. It has a number of great applications, most notably in fractional calculus and statistics. Also, it plays a significant role in a wide range of responserelated topics, like reaction-diffusion, theoretical physics, and mathematical probability theory. For more details, see [7,8].
Here, we introduce a class of stochastic matrix control functions and apply them to approximate the following pseudostochastic additive (G 1 , G 2 )-random operator inequalities in matrix Menger Banach algebras: Next, we define a generalized t-norm on diag N n ( ).  then on diag N n ( ) is continuous, see [11,12]. Consider the following examples of a continuous generalized t-norm: Then P is a continuous generalized t-norm.
Then M is a continuous generalized t-norm.
Then L is a continuous generalized t-norm. Furthermore, we present some numerical examples and compare the results: Also, since Consider E + , the set of matrix distribution functions, including left continuous and increasing maps : R ∪ {-∞, ∞} → diag N n ( ) such that 0 = 0 and +∞ = 1. Now + ⊆ E + are all (proper) mappings ∈ E + for which -= lim σ →σ = 1. Notice that proper matrix distribution functions are the matrix distribution functions of real random variables q such that P(|q| = ∞) = 0.

4)
for all S, R, A ∈ 1 , j ∈ J , and > 0. for all S ∈ 1 , j ∈ J , and > 0. Thus for all S ∈ 1 , j ∈ J , and > 0. Replacing S by S 2 n in (2.7), we get (2.8) It follows from for all S ∈ 1 , j ∈ J , > 0. That is, (2.9) Replacing S with S 2 m in (2.9), we get is Cauchy for all S ∈ 1 , j ∈ J . Since 2 is a matrix Menger Banach algebra, the sequence for all S ∈ 1 , j ∈ J , and > 0. Now, (2.4) implies that for all S, R, A ∈ 1 , j ∈ J , > 0, since φ for all S, R, A ∈ 1 , j ∈ J , > 0. Lemma 2.1 implies that the random operator V : J × 1 → 2 is stochastic additive. Now, to prove the uniqueness of the random operator V, suppose that there exists a stochastic additive random operator V : J × 1 → 2 which satisfies (2.5). Then tends to ∇ 0 as n → ∞.
Thus we conclude that 2 n V(j, S 2 n )-2 n V (j, S 2 n ) = 1 for all S ∈ 1 , j ∈ J , > 0. So V(j, S) = V (j, S) for all S ∈ 1 , and j ∈ J . for all S ∈ 1 , j ∈ J , and > 0. Thus for all S ∈ 1 , j ∈ J , and > 0. Replacing S by 2 n S in (2.14), we have for each S ∈ 1 , j ∈ J , and > 0. That is, (2.16) Replacing S by 2 m S in (2.16), we get tends to ∇ 0 . It implies that the sequence { 1 2 n Q(j, 2 n S)} is Cauchy for any S ∈ 1 and j ∈ J . Since 2 is a matrix Menger Banach algebra, the sequence { 1 2 n Q(j, 2 n S)} converges. Now, we determine the random operator V : J × 1 → 2 as follows: for all S ∈ 1 , j ∈ J , and > 0. Using Theorem 2.2 completes the proof.

Fixed point method for approximating inequality (1.1)
We use the fixed point technique to get an approximation of the additive (G 1 , G 2 )-random operator inequality (1.1) in matrix Menger Banach algebras.
In [29], Miheţ and Radu showed that (ξ , ζ ) is complete. We define the linear function L : ξ → ξ as almost everywhere for any S ∈ 1 and > 0, and also almost everywhere for each S ∈ 1 and > 0. Thus, from ζ (H, K) = ε, we conclude that ζ (LH, LK) ≤ Pε, and so for each H, K ∈ ξ . By (3.1), we have that almost everywhere for each S ∈ 1 and > 0, which implies that ζ (Q, LQ) ≤ P 2 . Theorem 1.4 implies that there is a random operator V : J × 1 → 2 such that: (1) A fixed point for the function L is V, almost everywhere for each x ∈ 1 , which is unique in the set (2) ζ (L p Q, V) → 0 as p → ∞, which implies that almost everywhere for each S ∈ 1 ; almost everywhere for each S ∈ 1 and > 0. Using (2.4) and (3.3), we have almost everywhere for any S, R, A ∈ 1 and > 0. Thus

Application with stochastic Mittag-Leffler and Fox's H-control functions
In this section, we apply stochastic Mittag-Leffler control functions and stochastic Fox Hcontrol functions to get a better approximation in the random operator inequality (1.1). Now, we introduce the concepts of the above stochastic control functions.
Suppose T is a vector space and • > 0.
We will present an example of a stochastic normed space by means of Mittag-Leffler function, but before that we introduce Mittag-Leffler function itself.
The special function , σ ∈ C, (σ ) > 0, z ∈ C (4.1) is called Mittag-Leffler function [3], where C and are respectively the set of complex numbers and the gamma function.
Consider the one-parameter Mittag-Leffler function σ - Now we want to show in the following four steps that (T, σ (-T • • ), min) is a random normed space.
). Then we have which implies that σ - Hence we have for all T • , T • ∈ T and • , • > 0. Therefore, σ (-T • • ) is a stochastic Mittag-Leffler control function. Now, we introduce the Fox H-function [30] as follows: H m,n p,q z| (a j ,A j ) 1,p (b j ,B j ) 1,q := 1 2πi L θ (ξ )z ξ dξ , As for the future research directions, we can replace the above control functions with hypergeometric function, Wright function, Fox-Wright function, and so on. Also, we can use matrix-valued fuzzy control functions instead of a class of stochastic matrix control functions.