Fixed Point Theorems for Relaxed Asymptotic Contractions via Two Quasi-Metrics

We introduce a new class of asymptotic contractions that employs two quasi-metrics defined directly in terms of the underlying mapping. The contraction condition compares these two quantities via a sequence of bounding functions that converge locally uniformly to a Boyd-Wong function. This framework relaxes the hypotheses of Kirk's asymptotic fixed point theorem and strictly contains it as a special case. Assuming only the continuity of the map and the boundedness of some orbit in a complete metric space, we prove both the existence and uniqueness of a fixed point, along with the convergence of all iterates to that point.