Fixed-Point Results in Metric Spaces for Large Triangle–Perimeter Contractions

We introduce a new class of mappings, referred to as large triangle–perimeter contractions, which simultaneously extend Petrov’s triangle–perimeter contractions and Burton’s large contraction principle. The proposed approach combines a strict local reduction of triangle perimeters with a nonuniform contractive mechanism that becomes effective whenever the underlying triangle is sufficiently nondegenerate. Within this two-scale setting, we establish a fixed-point theorem showing that every such mapping defined on a complete metric space admits a unique fixed point, provided that one orbit is bounded. The proof follows the spirit of Burton’s decay technique, adapted here to control the behavior of triangle perimeters rather than pairwise distances. Several illustrative examples, including both continuous and discrete cases, demonstrate that this class strictly contains mappings that fail to satisfy Petrov’s uniform perimeter contraction condition.