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

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.