  We study the existence of solutions arising from the modelling of elastic
materials using generalized theories of continua. In view of some evidence from
physics of meta-materials we focus our effort on two recent nonstandard relaxed
micromorphic models including novel micro-inertia terms. These novel
micro-inertia terms are needed to better capture the band-gap response. The
existence proof is based on the Banach fixed point theorem.
