Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space

In this article, we introduce a class of stochastic matrix control functions to stabilize a nonlinear fractional Volterra integro-differential equation with Ψ-Hilfer fractional derivative. Next, using the fixed-point method, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability of the nonlinear fractional Volterra integro-differential equation in matrix MB-space.


Introduction
Fractional calculus is considered as a branch of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. Therefore, fractional calculus is an extension of the integer-order calculus that considers integrals and derivatives of any real or complex order [1,2], i.e., unifies and generalizes the notions of integer-order differentiation and n-fold integration.
Different forms of fractional operators have been introduced, like the Riemann-Liouville, Grinwald-Letnikov, Weyl, Caputo, Marchaud, and Hadamard fractional derivatives. The first approach is that Riemann-Liouville, which is based on iterating the classical integral operator n times and then considering the Cauchy's formula where n! is replaced by the Gamma function, and hence the fractional integral of noninteger order is defined.
Fractional calculus has attracted the attention of many mathematicians, but also of some researchers in other areas like physics, chemistry, and engineering. As it is well known, several physical phenomena are often better described by fractional derivatives. This is mainly due to the fact that fractional operators take into consideration the evolution of the system, by taking the global correlation, and not only local characteristics. Moreover, integer-order calculus sometimes contradicts the experimental results, and therefore, derivatives of fractional order may be more suitable [3][4][5].
Very useful physical applications have given birth to the variable-order fractional calculus, for example, in modeling mechanical behaviors [6]. Nowadays, variable-order frac-tional calculus is particularly recognized as a useful and promising approach in the modeling of diffusion processes, in order to characterize time-or concentration-dependent anomalous diffusion, or diffusion processes in inhomogeneous porous media [7].
Results on existence and stability of solutions of implicit fractional differential equations can be found in [8][9][10][11].
By proposing the study of solution stability via fractional integrals and fractional derivatives, we can generalize the results and obtain the usual ones as particular cases. In this article, we study distribution functions with the ranges in a class of matrix algebras with the generalized triangular norms, to define MB-space and introduce a new class of matrix control functions. Also, we will use two recent fractional operators, that is, of general differentiation and integration [12].
These concepts help us study the Hyers-Ulam (in short HU) and Hyers-Ulam-Rassias (in short HUR) stability of fractional nonlinear Volterra integro-differential equation (in short VIDE),
Let : U → U be given by for all α ∈ 1 and ς ∈ 1 .