On the Darboux Problem for Partial Fractional Random Differential Equations Involving Unbounded Delay in Fréchet Spaces

This paper investigates the qualitative and topological behavior of random solutions for a class of partial fractional random differential equations governed by the Darboux problem. Unlike classical configurations that rely on bounded or finite delays, our theoretical framework explicitly addresses systems involving unbounded infinite delay. The dynamics of the state transitions are formulated using left-sided mixed Riemann–Liouville fractional integrals and joint Caputo fractional derivatives of order ε=(ε1,ε2)∈(0,1]×(0,1]. Because of the infinite historical horizon, the underlying model is constructed and analyzed within abstract, semi-normed axiomatic phase spaces defined over topological Fréchet spaces. By avoiding restrictive compactness assumptions on the nonlinear operational bounds, we establish novel random mild existence theorems. The structural proofs are achieved through a combination of a regular, sublinear family of axiomatic measures of noncompactness and an advanced generalization of the classical Darbo fixed-point theorem tailored for Fréchet domains. Finally, a concrete mathematical example is systematically analyzed to confirm the validity, consistency, and practical applicability of the established theoretical bounds.