A new class of hybrid contractions with higher-order iterative Kirk's method for reckoning fixed points

The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched $F$-contraction and enriched $F^{\mathrm{\prime }}$-contraction associated with $u$-fold averaged mapping, which are involved with Kirk's iterative technique with order $u$. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, $u$-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the $u$-fold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the $u$-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a $u$-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the $u$-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature.