(m,q)-isometric and (m,∞)-isometric tuples of commutative mappings on a metric space

In this paper, we introduce new concepts of (m,q)-isometries and (m,?)-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,?)-isometries for single mappings. In particular, we prove that if T = (T1,..., Tn) is an (m,q)-isometric commutative and power bounded tuple, then T is a (1,q)-isometric tuple. Moreover, we show that if T = (T1,...,Td) is an (m,?)- isometric commutative tuple of mappings on a metric space (E,d), then there exists a metric d? on E such that T is a (1,?)-isometric tuple on (E,d?).