{
  "abstract": "We study the variable-order fractional abstract Cauchy problem SD0α(t)u(t)=Au(t)+f(t), u(0)=u0, in a Banach space X, where A is sectorial and α(t):[0,∞)→(0,1) is a Scarpi transition function. Using the Sonine-pair structure of the Scarpi kernels, we convert the problem into a Volterra integral equation and, adapting the resolvent-family framework of the constant-order theory to this variable-order setting, introduce a variable-order resolvent family through Laplace transform techniques. In contrast to the constant-order case, the variable-order resolvent family is strongly continuous and exponentially bounded, but no analyticity in t is claimed or used. For exponentially bounded transitions α(t)=α2+(α1−α2)e−ct, the factorization sq(s)=sα1·s(α2−α1)c/(s+c) links the variable-order resolvent to the constant-order α1-resolvent. We prove the existence of this resolvent family, establish the existence of mild solutions by direct construction via the variable-order resolvent family and Laplace transform methods, prove the uniqueness in the class of strong solutions of the Volterra integral equation via a duality argument, and derive continuous dependence on the data. To the best of our knowledge, this provides the first complete well-posedness theory for inhomogeneous Scarpi-type variable-order abstract Cauchy problems in general Banach spaces. Two examples (A=−λI and A=Δ) illustrate the theory; the latter yields well-posed variable-order subdiffusion equations, of which a numerical illustration is provided at the end of the paper.",
  "abstract_shown": true,
  "affiliations": [
    "School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China",
    "Key Laboratory of Data Science and Intelligence Education, Ministry of Education, Hainan Normal University, Haikou 571158, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "affs": [
        0,
        1
      ],
      "name": "Shiyou Lin",
      "orcid": "0000-0002-8390-398X"
    },
    {
      "affs": [
        0
      ],
      "name": "Junying Chen",
      "orcid": "0009-0000-6176-4555"
    },
    {
      "affs": [
        0
      ],
      "name": "Aijie Yang",
      "orcid": "0009-0009-4436-6219"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3tn5q1dzzcyj8.bib",
  "cited_by": 0,
  "doi": "10.3390/fractalfract10110759",
  "fulltext_url": null,
  "id": "3tn5q1dzzcyj8",
  "issns": [
    "2504-3110"
  ],
  "landing_url": "https://www.mdpi.com/2504-3110/10/11/759",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/3tn5q1dzzcyj8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:32Z",
      "native_id": "10.3390/fractalfract10110759",
      "source": "crossref",
      "url": "https://www.mdpi.com/2504-3110/10/11/759"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2504-3110",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Well-Posedness of Variable-Order Fractional Abstract Cauchy Problems Under the Scarpi Framework",
  "title_text": "Well-Posedness of Variable-Order Fractional Abstract Cauchy Problems Under the Scarpi Framework",
  "type": "peer-reviewed",
  "updated_at": "2026-10-31T00:20:47Z",
  "url": "https://research.draconitic.com/paper/3tn5q1dzzcyj8",
  "venue": "Fractal and Fractional",
  "versions": []
}