Fractional Evolution Equations With Infinite Delay Gain a Sharp Regularity Criterion

A new paper in Boundary Value Problems establishes an exact criterion for maximal regularity of a class of time-fractional evolution equations posed on the torus, extending results previously known only for first-order equations. The work, authored by Rachid Bahloul and Houssame Rachad and published open access on 28 September 2026, treats a linear problem in which the time derivative is fractional of order between zero and one, the leading term is a closed operator on a complex Banach space, and an infinite-delay convolution acts on the operator part of the solution.

The equation is formulated as a periodic boundary value problem and analysed on the scale of vector-valued periodic Triebel–Lizorkin spaces of functions with vanishing mean. The fractional time derivative is taken to be the periodic Caputo–Weyl derivative, placing the problem within a functional-analytic framework designed for periodic-in-time evolution equations.

A central feature of the paper is how the delay term is defined. Rather than relying on integral representations suited to Laplace transform techniques, the authors define the delay spectrally, through the sequence of Fourier symbols of its kernel. This choice admits kernels that are not integrable at infinity — the Abel kernel in particular — which lie outside the usual Laplace transform framework that constrains much of the existing literature on equations with infinite delay.

The main theorem is a characterization. Under a hypothesis on the symbol sequence combining a uniform gap condition with discrete smoothness up to order three, the authors prove that the periodic problem has maximal regularity precisely when an explicit sequence of complex numbers, built from the fractional symbol and from the kernel symbol, lies in the resolvent set of the operator, and when the associated resolvent family is uniformly bounded. The criterion is thus necessary and sufficient, not merely sufficient.

A notable structural point is that the underlying operator-valued multiplier theorem is valid on an arbitrary Banach space. As a consequence, the characterization requires neither the UMD property nor R-boundedness, two assumptions that are standard in vector-valued harmonic analysis and maximal regularity theory but that exclude many Banach spaces of practical interest.

The paper also verifies the symbol hypothesis explicitly for the regularized Abel kernel, showing that the abstract condition is checkable in a concrete case involving a non-integrable kernel.

The result is consistent with earlier work in a precise limiting sense: when the order of the fractional derivative equals one, the symbol appearing in the criterion reduces to the one that arises in earlier studies of first-order integro-differential equations with infinite delay. The new theorem therefore recovers the first-order case while extending the analysis to fractional orders strictly between zero and one.

Establishing an exact resolvent-based criterion for such regularity, in a setting that tolerates non-integrable delay kernels and arbitrary Banach spaces, is the paper’s principal contribution to the theory of abstract fractional evolution equations.

The article is published open access, and the version shared early is peer-reviewed and accepted research carrying a permanent DOI, subject to further edits before the final Version of Record.

Subject of Research: Maximal regularity for time-fractional integro-differential equations with infinite delay on periodic Triebel–Lizorkin spaces

Article Title: Maximal regularity for time-fractional integro-differential equations with infinite delay on periodic Triebel–Lizorkin spaces

Article References: Bahloul, R., & Rachad, H. (2026). Maximal regularity for time-fractional integro-differential equations with infinite delay on periodic Triebel–Lizorkin spaces. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02358-x

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DOI: 10.1186/s13661-026-02358-x

Keywords: maximal regularity, time-fractional equations, infinite delay, Triebel–Lizorkin spaces, Caputo–Weyl derivative, Abel kernel, resolvent set, sectorial operators, Banach spaces, periodic boundary value problems, Maximal, regularity