Fixed point theorems for relaxed asymptotic contractions via two quasi-metrics

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