  The formation of unstaggered localized modes in dynamical lattices can be
supported by the interplay of discreteness and nonlinearity with a finite
relaxation time. In rapidly responding nonlinear media, on-site discrete
solitons are stable, and their broad inter-site counterparts are marginally
stable, featuring a virtually vanishing real instability eigenvalue. The
solitons become unstable in the case of the slowly relaxing nonlinearity. The
character of the instability alters with the increase of the delay time, which
leads to a change in the dynamics of unstable discrete solitons. They form
robust localized breathers in rapidly relaxing media, and decay into
oscillatory diffractive pattern in the lattices with a slow nonlinear response.
Marginally stable solitons can freely move across the lattice.
