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

Irom Shashikanta Singh, Yumnam Mahendra Singh

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.

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

Abstract

This paper aims to integrate the concepts of -contraction and -contraction within the context of super metric spaces. Specifically, we introduce the concepts of -contraction and Bianchini -contraction. We demonstrate that these new concepts are genuine generalizations of - and -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.
Paper Structure (7 sections, 20 theorems, 136 equations, 2 figures)

This paper contains 7 sections, 20 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. SF-contraction: A generalization of SB-Contraction
  3. Corollaries
  4. Bianchini SF-Contraction: A generalization of SK-Contraction
  5. Corollaries
  6. Application
  7. Conclusion

Key Result

Theorem 1

reich Let $(W, \eth)$ be a complete metric space, and $\Upsilon$ be a mapping from $W$ into itself, satisfying the following condition, where $a$, $b$, and $c$ are non-negative numbers satisfying $a + b + c < 1$. Then, $\Upsilon$ has a unique fixed point.

Figures (2)

  • Figure 1: Hierarchy of Kannan contraction and its generalizations
  • Figure 2: Block diagram of the automatic flight-path generator

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 3
  • Definition 4
  • Theorem 4
  • Definition 5
  • Theorem 5
  • ...and 42 more