Rational-Kernel Fractional Evolution Equations with Almost Sectorial Operators: A Resolvent Framework Unifying ABC and W Dynamics

TL;DR

This paper develops a unified resolvent framework for fractional evolution equations driven by rational-kernel operators, including ABC and W types, extending classical theory to non-singular memory operators and analyzing their long-term behavior.

Contribution

It introduces a resolvent-based approach for fractional evolution equations with almost sectorial operators, encompassing non-Bernstein class kernels like ABC and W, and compares their dynamics.

Findings

Established existence and uniqueness of mild solutions.

Analyzed decay and smoothing properties of solutions.

Provided examples illustrating long-time behavior.

Abstract

We study fractional evolution equations driven by rational-kernel time operators with non-singular memory, including the Atangana-Baleanu-Caputo operator and a generalized W-operator. These operators are characterized by Laplace symbols that do not necessarily belong to the classical Bernstein class. The analysis is carried out in the framework of almost sectorial operators, which allows resolvent estimates beyond standard analytic semigroup theory. Existence, uniqueness, and temporal regularity of mild solutions are established by Laplace transform techniques and contour integration, leading to the construction of associated resolvent families. A unified resolvent framework is developed, enabling a precise comparison between ABC and W dynamics and clarifying the influence of rational memory kernels on decay and smoothing properties. Several examples, including fractional diffusion-type equations, illustrate the abstract theory and highlight the impact of non-singular memory on long-time behavior.