Resolvent Approach to Atangana--Baleanu Evolution Equations: Laplace Symbols, Mild Solutions, and Regularity

Abstract: Fractional evolution equations with memory terms are widely used to model anomalous diffusion, viscoelastic response, and hereditary dynamics in physics, biology, and engineering. Among the recently introduced operators, the Atangana--Baleanu (AB) derivatives have attracted considerable attention due to their non-singular Mittag--Leffler kernels. However, their analytic treatment remains limited, as the AB kernel does not fall within the classical Volterra or Bernstein-function frameworks. This paper develops a unified resolvent approach for AB-type evolution equations in Banach spaces. Using a Laplace-domain formulation inspired by Hille--Phillips theory, we introduce a fractional resolvent associated with the AB kernel and establish optimal bounds on sectorial contours. Under the natural condition $β<1+α$, we construct an AB--Mittag--Leffler resolvent family and obtain a complete representation of mild solutions to the AB Cauchy problem. Sharp stability and regularity estimates of Mittag--Leffler type are derived, including fractional-domain bounds. Numerical illustrations confirm the predicted decay, and connections with non-autonomous operators, maximal $L^p$-regularity, and weighted AB kernels are outlined. The results place AB-type equations within a functional-analytic framework comparable to the classical theory for Caputo and Volterra models.