  We analyze the existence and stability of two kinds of self-trapped spatially
localized gap modes, gap solitons and truncated nonlinear Bloch waves, in
one-and two-dimensional optical or matter-wave media with self-focusing
nonlinearity, supported by a combination of linear and nonlinear periodic
lattice potentials. The former is found to be stable once placed inside a
single well of the nonlinear lattice, it is unstable otherwise. Contrary to the
case with constant self-focusing nonlinearity, where the latter solution is
always unstable, here, we demonstrate that it nevertheless can be stabilized by
the nonlinear lattice since the model under consideration combines the unique
properties of both the linear and nonlinear lattices. The practical
possibilities for experimental realization of the predicted solutions are also
discussed.
