{
  "abstract": "For $p:[0,1]\\to(0,1)$ a continuous function, we consider the Luxemburg functional \\[ Λ_{p(\\cdot)}(u):=\\inf\\left\\{δ>0:\\int_0^1\\left|\\frac{u(t)}δ\\right|^{p(t)}\\,dt\\leq 1\\right\\}. \\] We develop a suitable functional analytic framework for $Λ_{p(\\cdot)}$ so that it can be incorporated into a positive order cone, which can then be used in the analysis of nonlocal differential equations having the general form \\[ -A\\left(\\int_0^1 b(1-s)(u(s))^{p(s)}\\,ds\\right)u''(t) =λf(t,u(t)),\\qquad t\\in(0,1). \\] By means of a specific numerical example, we demonstrate that, as opposed to a methodology relying only on the supremum norm, our methodology via the Luxemburg functional yields an appreciably sharper localisation of any solutions of the nonlocal equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35878",
  "author_total": 2,
  "authors": [
    {
      "name": "Christopher S. Goodrich"
    },
    {
      "name": "Gabriel Nakhl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2bdyxanvfz79z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35878",
  "fulltext_url": null,
  "id": "2bdyxanvfz79z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35878",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Differential Equations Analysis"
    ]
  ],
  "openalex": "W7224125058",
  "pdf_url": "https://arxiv.org/pdf/2610.35878",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2bdyxanvfz79z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35878",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35878"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Luxemburg Functional in the Concave Regime with an Application to Nonlocal Differential Equations",
  "title_text": "On the Luxemburg Functional in the Concave Regime with an Application to Nonlocal Differential Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/2bdyxanvfz79z",
  "venue": "arXiv",
  "versions": []
}