  The problem of showing the existence of localised modes in nonlinear lattices
has attracted considerable efforts from the physical but also from the
mathematical viewpoint where a rich variety of methods has been employed. In
this paper we prove that a fixed point theory approach based on the celebrated
Schauder's Fixed Point Theorem may provide a general method to establish
concisely not only the existence of localised structures but also a required
rate of spatial localisation. As a case study we consider lattices of coupled
particles with nonlinear nearest neighbour interaction and prove the existence
of exponentially spatially localised breathers exhibiting either even-parity or
odd-parity symmetry under necessary non-resonant conditions accompanied with
the proof of energy bounds of the solutions.
