A mathematical preprint proves that its semilinear problem has a unique mild solution when the stated hypotheses are met, within a class of abstract differential equations whose delays change over time and extend indefinitely into the past. Here, “mild solution” is the paper’s abstract way of saying that the equation has a mathematically valid solution in the framework it sets up.
The same work reports existence results under several operator assumptions, then constructs an evolution family for the homogeneous problem and identifies a variation-of-constants formula for the inhomogeneous one. Those are linked pieces of solution theory: one addresses existence, while the others organize how solution histories are carried through time and how external terms enter the representation.
A problem built around an unlimited past
The analyzed object is a class of first-order abstract retarded functional differential equations in Banach spaces, with a time-dependent infinite delay governed by a regulated function. It is a mathematical equation class rather than a dataset.
The central question is whether these equations admit mild solutions. The paper also examines whether the solution maps of the associated linear problem can support an evolution family and a variation-of-constants formula.
That makes the work a methods study. Its existence conclusions are conditional on the hypotheses attached to the relevant theorems, and the paper says data sharing is not applicable because no datasets were generated or analyzed.
What the existence results establish
The existence analysis uses fixed-point arguments. In plain language, this means recasting the problem as a mapping and looking for a solution that the mapping leaves unchanged.
For the semilinear problem, the paper proves both existence and uniqueness of a mild solution under its stated hypotheses. The finding is conditional, but it gives a precise mathematical answer within the framework being analyzed.
Separate results cover more than one setting. The paper reports existence under general conditions and also under compact or immediately norm-continuous semigroup assumptions. The theorem family therefore addresses the existence question across the operator settings specified in the paper, rather than tying it to only one of them.
The framework reaches a diffusion model
The framework is then applied to a diffusion-with-memory problem. The application is treated in two phase-space cases, or two mathematical settings for organizing the system’s history.
In the first, called the Cg0 (X) phase-space case, the paper states that the diffusion-with-memory problem has a mild solution w ∈ C([0, a], L2 ([0, π])). For a general reader, the notation mainly says that the solution belongs to a specified function space in the case being studied.
In the second, the C0 × L2 (ρ, X) phase-space case, and with the stated continuity condition on φ, the problem again has a mild solution w ∈ C([0, a], L2 ([0, π])). Here too, the result is an existence statement inside the assumptions of that phase-space construction.
Taken together, the two application cases show how the general theory is carried into a concrete diffusion-with-memory equation while keeping the conclusion conditional on the chosen mathematical setting.
From solution histories to a formula
For the homogeneous problem—the version represented without the inhomogeneous forcing—the solution-history maps form an evolution family on Bτ. In ordinary language, an evolution family is the paper’s operator description of how an admissible history is carried from one time to another.
That construction uses a Dyson–Phillips type operator series defining UΛ(t, s). The series converges uniformly on Δ in L(Bτ), which supplies the convergence property stated for the operator construction.
A further theorem identifies the homogeneous solution’s state history with UΛ(t, s)φ. This ties the abstract operator family to the history generated by the homogeneous equation in the paper’s framework.
For the inhomogeneous problem, the paper identifies its representation as a variation-of-constants formula. A companion theorem concludes that the function built from that representation is a mild solution of the problem.
A conditional mathematical result
The document is an arXiv version 1 preprint dated 25 August 2026. It analyzes abstract equations rather than an empirical dataset, and its data statement says that data sharing is not applicable because no datasets were generated or analyzed.
That scope matters: the semilinear theorem, the alternative existence results, the two diffusion cases and the operator representations are all conditional on their stated hypotheses. The supplied evidence supports mathematical conclusions within that framework, not measurements from an observed system.
The paper’s results form a connected chain. Fixed-point arguments address existence; the homogeneous maps form an evolution family; the Dyson–Phillips series converges in the relevant operator space; and the inhomogeneous representation produces a mild solution.
The paper reports partial support from ANID Doctorado Nacional grant No. 21240764, ANID-FONDECYT Iniciación grant No. 11260638 and a DICYT-USACH grant. The corresponding author states that there is no conflict of interest on behalf of all authors.
Paper data and sources
Original title: Evolution families and variation of constants formula for abstract functional differential equations with time-dependent infinite delay
Authors: Claudio Carrasco, Claudio A. Gallegos, Hernán R. Henríquez, Matthieu F. Pinaud
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text