  We introduce a generalized version of the Ablowitz-Ladik model with a
power-law nonlinearity, as a discretization of the continuum nonlinear
Schr\"{o}dinger equation with the same type of the nonlinearity. The model
opens a way to study the interplay of discreteness and nonlinearity features.
We identify stationary discrete-soliton states for different values of
nonlinearity power $\sigma $, and address changes of their stability as
frequency $\omega $ of the standing wave varies for given $\sigma $. Along with
numerical methods, a variational approximation is used to predict the form of
the discrete solitons, their stability changes, and bistability features by
means of the Vakhitov-Kolokolov criterion (developed from the first
principles). Development of instabilities and the resulting asymptotic dynamics
are explored by means of direct simulations.
