  We study modulational instability of matter-waves in Bose-Einstein
condensates (BEC) under strong temporal nonlinearity-management. Both BEC in an
optical lattice and homogeneous BEC are considered in the framework of the
Gross-Pitaevskii equation, averaged over rapid time modulations. For a BEC in
an optical lattice, it is shown that the loop formed on a dispersion curve
undergoes transformation due to the nonlinearity-management. A critical
strength for the nonlinearity-management strength is obtained that changes the
character of instability of an attractive condensate. MI is shown to occur
below(above) the threshold for the positive (negative) effective mass. The
enhancement of number of atoms in the nonlinearity-managed gap soliton is
revealed.
