{
  "abstract": "In this research paper, we develop a relation-theoretic Φ-contraction framework in multiplicative metric spaces. We introduce the concept of a relation-theoretic Φ-contraction, in which the contractive condition is imposed only on pairs satisfying a prescribed binary relation. The existence and uniqueness of fixed point results are established under suitable relation-dependent completeness, continuity, and closedness assumptions. Two finite examples are given. The first shows that the logarithmic quantity associated with the chosen Φ-function need not satisfy the triangle inequality; in the second, the map is neither a global Φ-contraction nor a contraction of the logarithmic metric, so that the relation is essential. The theorem is then applied to a nonlinear Volterra-type integral equation whose kernel is only locally contractive, the region of contractivity being encoded by the binary relation.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, Haryana, India",
    "Department of Mechanical Engineering Science, University of Johannesburg, Johannesburg 2092, South Africa",
    "International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore 54000, Pakistan"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Savita Rathee"
    },
    {
      "affs": [
        0
      ],
      "name": "Priyanka Lathwal"
    },
    {
      "affs": [
        1,
        2
      ],
      "name": "Muhammad Waseem Asghar",
      "orcid": "0000-0001-7538-2128"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1v6psdz1gx3d3.bib",
  "cited_by": 0,
  "doi": "10.3390/axioms15110811",
  "fulltext_url": null,
  "id": "1v6psdz1gx3d3",
  "issns": [
    "2075-1680"
  ],
  "landing_url": "https://www.mdpi.com/2075-1680/15/11/811",
  "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": "2026-11-01",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/1v6psdz1gx3d3.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:20:59Z",
      "native_id": "10.3390/axioms15110811",
      "source": "crossref",
      "url": "https://www.mdpi.com/2075-1680/15/11/811"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "journal-openalex:F26",
      "subfield": "general"
    }
  ],
  "title": "Relation-Theoretic Φ-Contractions in Multiplicative Metric Spaces for Nonlinear Volterra Integral Equations",
  "title_text": "Relation-Theoretic Φ-Contractions in Multiplicative Metric Spaces for Nonlinear Volterra Integral Equations",
  "type": "peer-reviewed",
  "updated_at": "2026-11-06T00:21:06Z",
  "url": "https://research.draconitic.com/paper/1v6psdz1gx3d3",
  "venue": "Axioms",
  "versions": []
}