On the α -norm in neutral fractional equations with nonlocal terms defined by probability densities

This paper investigates a class of fractional differential equations formulated on the α-norm. Central to our approach is the Wright probability density, whose heavy-tailed profile naturally captures the long-memory effects encoded by Caputo-type derivatives. By embedding this kernel into the problem, we derive sharp, easily verifiable criteria for the existence and uniqueness of mild solutions. The proof combines fixed-point and the Wright kernel’s Laplace transform. To illustrate the practical scope of the theory, we present a detailed numerical example driven by the explicit densityhighlighting how the interplay between the fractional index β and the α-norm shapes solution behavior.