Perimetric Expansive Maps in Perturbed Metric Spaces

This paper develops a fixed point framework for mappings that expand triangular perimeters within the setting of perturbed metric spaces. By decomposing the distance structure into an exact metric and an adjustable perturbation, we show that surjectivity transforms forward perimetric expansion into backward contractive behavior, forcing geometric convergence of inverse orbits. This mechanism yields a powerful expansive analogue of classical triangular contraction principles and ensures the existence and uniqueness of fixed points under mild regularity and the absence of 2–cycles. Detailed examples reveal both borderline nonexpansive behavior and genuinely strict expansive regimes arising from hierarchical folding dynamics. Three applications demonstrate the breadth of the theory: (i)a nonlinear integral operator whose structural expansiveness ensures the existence of a unique equilibrium solution to a Volterra-type equation. (ii) a hierarchical deduplication model in which strict perimetric expansion enforces a unique canonical representative under repeated merging: and (iii) a cryptographic state–evolution scheme whose perimetric geometry captures collision resistance and guarantees a unique master seed. These results highlight the versatility of perimetric expansion in nonlinear analysis, structured data aggregation, secure computation, and integral-equation models.