  We report that infinite and semi-infinite lattices with spatially
inhomogeneous self-defocusing (SDF)\ onsite nonlinearity, whose strength
increases rapidly enough toward the lattice periphery, support stable
unstaggered (UnST) discrete bright solitons, which do not exist in lattices
with the spatially uniform SDF nonlinearity. The UnST solitons coexist with
stable staggered (ST) localized modes, which are always possible under the
defocusing onsite nonlinearity. The results are obtained in a numerical form,
and also by means of variational approximation (VA). In the semi-infinite
(truncated) system, some solutions for the UnST surface solitons are produced
in an exact form. On the contrary to surface discrete solitons in uniform
truncated lattices, the threshold value of the norm vanishes for the UnST
solitons in the present system. Stability regions for the novel UnST solitons
are identified. The same results imply the existence of ST discrete solitons in
lattices with the spatially growing self-focusing nonlinearity, where such
solitons cannot exist either if the nonlinearity is homogeneous. In addition, a
lattice with the uniform onsite SDF nonlinearity and exponentially decaying
inter-site coupling is introduced and briefly considered too. Via a similar
mechanism, it may also support UnST discrete solitons, under the action of the
SDF nonlinearity. The results may be realized in arrayed optical waveguides and
collisionally inhomogeneous Bose-Einstein condensates trapped in deep optical
lattices. A generalization for a two-dimensional system is briefly considered
too.
