Existence–Uniqueness and Wright Stability Results of the Riemann–Liouville Fractional Equations by Random Controllers in MB-Spaces

We apply the random controllers to stabilize pseudo Riemann–Liouville fractional equations in MB-spaces and investigate existence and uniqueness of their solutions. Next, we compute the optimum error of the estimate. The mentioned process i.e., stabilization by a control function and finding an approximation for a pseudo functional equation is called random HUR stability. We use a fixed point technique derived from the alternative fixed point theorem (FPT) to investigate random HUR stability of the Riemann–Liouville fractional equations in MB-spaces. As an application, we introduce a class of random Wright control function and investigate existence–uniqueness and Wright stability of these equations in MB-spaces.


Introduction and Preliminaries
We introduce the random control functions that help us to investigate existence, uniqueness, and random Wright stability of integro-differential equations in MB-spaces. Some good references for the theory and application of fractional analysis are [1][2][3].
Here, MB-space represents a complete MN-space [9,10]. In the following, we suppose that * = * M . Theorem 1 ([11,12]). Consider the complete J • -valued metric space (S, δ) and also consider the self-map Λ on S such that Assume that s ∈ S, so there are two options: the sequence Λ m s converges to a fixed point t * of Λ;

Riemann-Liouville Fractional Equations
Let u : [p, q] → R (0 < p < q < ∞) be a continuous function and > 0 a real number. We define the Riemann-Liouville fractional integrals of order , by Using the definition of Riemann-Liouville fractional integrals, we define the Riemann-Liouville derivatives as follows: Let T be a real positive number. Consider the Riemann-Liouville fractional Volterra integro-differential equation, defined by In [13], Goleţ, defined the concept of differentiable functions in an MB-space (W, ζ, * ) and proved that, if the function f : U → (W, ζ, * ) is differentiable in u 0 ∈ U, it is therefore continuous in the point u 0 .
The Wright function [14,15] is one of the special functions defined by the series representation, valid in the whole complex plane where α > −1 and β ∈ C. In this paper, we define a distribution distance mapping (DDM) based on Wright functions.
Consider the DDM-valued W α,β : [0, T] → S + (α, β ∈ J • ), a random control mapping, which is defined as follows: Then, we have • It is left continuous and always increasing for positive values, it means that, for For τ ≤ 0, we have (W α,β ) s τ = 0, And the following conditions also apply to the DDM-valued Wright function . Now, we conclude that Let DDM-valued ψ : [0, T] → S + be a random control mapping. We say that (3) is random HUR stable, when, for a differentiable mapping u(s), satisfying If we replace the control function ψ with the DDM-valued Wright function (W α,β ), we say that (3) is random Wright stable.

Random Stability of Riemann-Liouville Fractional Volterra Integral Equation
Consider the Riemann-Liouville fractional Volterra integral equation In this section, we study random Wright stability of (42).

Applications, Random Wright Stability
Now, as applications, we study the concept of random Wright stability for some fractional equations.