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Peer-reviewedCC BY 4.0Axioms

Relation-Theoretic Φ-Contractions in Multiplicative Metric Spaces for Nonlinear Volterra Integral Equations

  • Savita Rathee1
  • Priyanka Lathwal1
  • Muhammad Waseem Asghar2,3
  1. 1Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, Haryana, India
  2. 2Department of Mechanical Engineering Science, University of Johannesburg, Johannesburg 2092, South Africa
  3. 3International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore 54000, Pakistan
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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.

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Cite

BibTeXRIS
BibTeX
@article{rathee2026relationtheoretic,
  title = {{Relation-Theoretic Φ-Contractions in Multiplicative Metric Spaces for Nonlinear Volterra Integral Equations}},
  author = {Savita Rathee and Priyanka Lathwal and Muhammad Waseem Asghar},
  journal = {Axioms},
  year = {2026},
  month = nov,
  doi = {10.3390/axioms15110811},
  url = {https://www.mdpi.com/2075-1680/15/11/811},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Relation-Theoretic Φ-Contractions in Multiplicative Metric Spaces for Nonlinear Volterra Integral Equations
AU  - Savita Rathee
AU  - Priyanka Lathwal
AU  - Muhammad Waseem Asghar
PY  - 2026
DA  - 2026/11/01
JO  - Axioms
DO  - 10.3390/axioms15110811
UR  - https://www.mdpi.com/2075-1680/15/11/811
AB  - 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.
SN  - 2075-1680
ER  -