Establishment of Fixed-Point Results via a Triplet of Auxiliary Functions in Relational Partial Metric Spaces

This paper establishes fixed-point results for self-mappings on partial metric spaces endowed with a binary relation. The study is motivated by relation-theoretic fixed-point theory and by recent contractive conditions involving a triplet of auxiliary functions. We introduce a relational (ψ, φ, θ)-contractive condition into the framework of partial metric spaces and use suitable notions of R-completeness, R-continuity, and R-self-closedness. Under the assumptions that the relation is T-closed and locally T-transitive, we prove an existence theorem for fixed points. A uniqueness result is also obtained by imposing a finite relational path condition between fixed points. The results extend and unify several known fixed-point theorems, including relation-theoretic contractions, auxiliary-function contractions, and partial-metric fixed-point results. Corollaries are presented to show how earlier metric and partial-metric results can be recovered as special cases. Finally, an application to a Fredholm integral equation of the second kind is discussed, showing how the proposed framework can be used to establish existence and uniqueness of solutions for nonlinear integral equations. The work contributes to the further development of relation-theoretic fixed-point theory in generalized metric structures and provides a basis for future studies involving coincidence results, matrix equations, and other generalized metric-type spaces. These directions naturally follow from the theoretical framework developed in this study.