### $(m,q)$-Isometric tuples of commutative mappings on a metric space

#### Abstract

In this paper, we introduce new concepts of $(m,q)$-isometries and $(m,\infty)$-isometries tuples of commutative mappings on metrics spaces. We discuss the most interesting results concerning this class of mappings obtained form the idea of generalizing the $(m,q)$-isometries and $(m,\infty)$-isometries for single mappings. In particular, we prove that if ${\bf\large T}=(T_1,\cdots,T_n)$ is an $(m,q)$-isometric commutative and power bounded tuple, then ${\bf T}$ is a $(1,q)$-isometric tuple. Moreover, we show that

if ${\bf\large T}=(T_1,\cdots,T_d)$ is an $(m,\infty)$-isometric commutative tuple of mappings on a metric space $(E, d)$, then there exists a metric $d_\infty$ on $E$ such that ${\bf\large T}$ is a $(1,\infty)$-isometric tuple on $(E, d_\infty).$

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