Abstract
We stabilize pseudostochastic (mathcal{G}_{1},mathcal{G}_{2})-random operator inequality using a class of stochastic matrix control functions in matrix Menger Banach algebras. We get an approximation for stochastic (mathcal{G}_{1},mathcal{G}_{2})-random operator inequality by means of both direct and fixed point methods. As an application, we apply both stochastic Mittag-Leffler and mathbb{H}-fox control functions to get a better approximation in a random operator inequality.
Highlights
Introduction and preliminariesThe 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 1903, the Swedish mathematician Gosta Mittag-Leffler presented a generalization of the exponential function and introduced some properties of this function
Mittag-Leffler function naturally appears as the solution of fractional order integro-differential equations and 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
Summary
In 1961, Charles Fox presented a generalization of the Meijer G-function and the Fox– Wright function. We introduce a class of stochastic matrix control functions and apply them to approximate the following pseudostochastic additive (G1, G2)-random operator inequalities in matrix Menger Banach algebras: Q(j,S+R+A)–Q(j,S)–Q(j,R)–Q(j,A). Definition 1.1 ([10]) A generalized t-norm on diag Nn( ) is an operation : diag Nn( ) × diag Nn( ) → diag Nn( ) satisfying the following conditions:. Consider the following examples of a continuous generalized t-norm:. Consider E+, the set of matrix distribution functions, including left continuous and increasing maps : R ∪ {–∞, ∞} → diag Nn( ) such that 0 = 0 and +∞ = 1. Definition 1.2 Let S be a linear space, be a continuous generalized t-norm, and : S →. Definition 1.3 Let (S, , ) be a matrix Menger normed space and , generalized t-norms. For all S, R, A ∈ 1, j ∈ J , and > 0
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