Certain Fixed-Point Results for (<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="fraktur">e</mi></mrow></semantics></math></inline-formula>,<i>ψ</i>,Φ)-Enriched Weak Contractions via Theoretic Order with Applications

Abstract

This paper aims to establish several fixed-point theorems within the framework of Banach spaces endowed with a binary relation. By utilizing enriched contraction principles involving two classes of altering-distance functions, the study encompasses various types of contractive mappings, including theoretic-order contractions, Picard–Banach contractions, weak contractions, and non-expansive contractions. A suitable Krasnoselskij iterative scheme is employed to derive the results. Many well-known fixed-point theorems (FPTs) can be obtained as special cases of these findings by assigning specific control functions in the main definitions or selecting an appropriate binary relation. To validate the theoretical results, numerous illustrative examples are provided. Furthermore, the paper demonstrates the applicability of the findings through applications to ordinary differential equations.

Keywords

• theoretic-order contraction
• binary relations
• enriched contraction
• weak contractions
• altering-distance functions
• 1st- and 2nd-order ordinary differential equations