  We investigate the attractor mechanism for spherically symmetric extremal
black holes in Einstein-Born-Infeld-dilaton theory of gravity in
four-dimensions, in the presence of a cosmological constant. We look for
solutions analytic near the horizon by using perturbation method. It is shown
that the values of the scalar fields at the horizon are only dependent on the
charges carried by the black hole and are irrelevant in their asymptotic
values. This analysis supports the validity of non-supersymmetric attractors in
the presence of higher derivative interactions in the gauge fields part and in
non-asymptotically flat spacetime.
