Existence of dark solitons in discrete nonlinear Schrödinger equations with non-weak couplings

The anti-integrable limit for the discrete nonlinear Schrödinger equation is defined as the limit of vanishing couplings. There are an infinite number of trivial discrete dark solitons in this limit. The existence of discrete dark solitons continued from them has been proved only for sufficiently weak couplings. In the present paper, we focus on the case of non-weak couplings and prove the existence of discrete dark solitons over an explicitly given range of the coupling constant.