  We study the formation of localized modes around a generalized nonlinear
impurity which is located at the boundary of a semi-infinite square lattice,
and where we replace the standard discrete Laplacian by a fractional one,
characterized by a fractional exponent $0<\alpha<1$ where $\alpha=1$ marks the
standard, non-fractional case. We specialize to two impurity cases: impurity at
an "edge" and impurity at a "corner" and use the formalism of lattice Green
functions to obtain in closed form the bound state energy and its mode
amplitude. It is found that, for any fractional exponent and for impurity
strengths above a certain threshold, there is always a single bound state for
the linear impurity, while for the nonlinear (cubic) case, up to two bound
states are possible. At small fractional exponents, the energy of the impurity
mode becomes directly proportional to the impurity strength.
