As a natural extension of the concept of $(m,p)$-hyperexpansive and $(m,p)$-hypercontractive of a single operator, we introduce and study the concepts of $(m,p)$-hyperexpansivity and $(m,p)$-hypercontractivity for $d$-tuple of commuting operators acting on Banach spaces. These concepts extend the definitions of $m$-isometries and $(m,p)$-isometric tuples of bounded linear operators acting on Hilbert or Banach spaces, which have been introduced and studied by many authors.

This paper introduces and explores new concepts of $(m,q)$-isometric multimappings in the context of extended metric structures. These newly defined concepts serve as extensions of the existing theory of $(m,q)$-isometric multimappings in traditional metric spaces, as well as $(m,q)$- $\mathcal{G}$-isometric single mappings in generalized metric spaces. The study aims to broaden the understanding of isometric multimappings properties and their interactions within these extended spaces.