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

TL;DR

This work develops a resolvent-based functional-analytic framework for AB-type fractional evolution equations on Banach spaces by leveraging the AB kernel’s Laplace symbol $\widehat{k}_{\alpha,\beta}(s)=\dfrac{s^{\alpha-\beta}}{s^{\alpha}+c}$. It introduces the operator-valued AB–Mittag–Leffler resolvent $R_{\alpha,\beta}(s;A)$ and the associated resolvent family $V_{\alpha,\beta}(t)$, proving existence under the natural condition $\beta<1+\alpha$ and establishing a representation of mild solutions via $u(t)=V_{\alpha,\beta}(t)u_0$. The paper develops sharp two-regime resolvent bounds, Mittag–Leffler stability, and fractional-domain regularity results including domain regularisation, time-derivative regularity, and smoothing with fractional powers of $A$, while providing numerical demonstrations that confirm the predicted decay and mode behavior. By embedding AB-type equations into the Hille–Phillips framework, this work places non-singular memory models on par with classical Caputo/Volterra theory and outlines extensions to non-autonomous problems, weighted kernels, and matrix-valued generalizations with potential applications in diffusion, viscoelasticity, and related memory-driven dynamics.

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.

Figures (3)

• Figure 1: Decay of the fundamental Dirichlet mode. The curve matches the Mittag--Leffler prediction $E_{\alpha,\beta}(-t^\alpha)$.
• Figure 2: Normalized decay of the first two modes for $(\alpha,\beta)=(0.8,1.2)$. The dependence on $\lambda_k$ matches $E_{\alpha,\beta}(-\lambda_k t^\alpha)$.
• Figure 3: Heatmap of the reconstructed two-mode solution $u(t,x)$. Fractional smoothing and Mittag--Leffler decay are apparent.

Theorems & Definitions (21)

• Remark 2.1: No additional angular constraints
• Theorem 3.1: Fractional resolvent bounds
• proof : Sketch of proof
• Proposition 4.1
• proof
• Lemma 4.2: Laplace transform
• proof
• Proposition 4.3: Operator-valued Mittag--Leffler decomposition
• proof : Sketch
• Theorem 4.4: Representation formula for mild solutions