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

Jie Shi; Jie Shi

doi:10.3934/math.2026926

AIMS Mathematics

2026, Volume 11, Issue 7: 22983-22998. doi: 10.3934/math.2026926

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Research article

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

Jie Shi ^,

Department of Mathematics and Statistics, Hubei Engineering University, 272 Traffic Avenue, Xiaogan, 432000, Hubei, China

Received: 06 May 2026 Revised: 10 July 2026 Accepted: 16 July 2026 Published: 29 July 2026
MSC : 47H10, 47H09

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.
- fixed point,
- relaxed asymptotic contractions,
- lower quasi-metric,
- upper quasi-metric,
- local uniform convergence,
- Boyd-Wong condition
Citation: Jie Shi. Fixed point theorems for relaxed asymptotic contractions via two quasi-metrics[J]. AIMS Mathematics, 2026, 11(7): 22983-22998. doi: 10.3934/math.2026926

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Abstract

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.

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