$F$-Contraction with an Auxiliary Function and Its Application to Terrain-Following Airplane Navigation

TL;DR

This paper introduces new contraction concepts in super metric spaces, generalizes existing fixed point theories, and applies these results to terrain-following airplane navigation.

Contribution

The paper defines $S^F$-contraction and Bianchini $S^F$-contraction, extending fixed point theory in super metric spaces with practical application to airplane terrain-following.

Findings

Established existence and uniqueness of fixed points for new contractions.

Provided nontrivial examples demonstrating the generalizations.

Applied fixed point results to terrain-following airplane navigation model.

Abstract

This paper aims to integrate the concepts of $F$-contraction and $S^B$-contraction within the context of super metric spaces. Specifically, we introduce the concepts of $S^F$-contraction and Bianchini $S^F$-contraction. We demonstrate that these new concepts are genuine generalizations of $S^B$- and $S^K$-contractions by providing nontrivial examples. Furthermore, we establish the existence and uniqueness of fixed points for mappings that satisfy these contractions. Lastly, we apply our findings to a model describing an airplane capable of automatically following a terrain.