I am trying to build an application which uses Differential BPSK /Differential QPSK modulated on to a chirp signal. (S2C method). The modulation is multiplication of message signal (Rectangular Pulse Shape(+1/-1) with a chirp signal.

1) Is RRC pulse shaping mandatory at the Txr, in order for the Fractionally spaced (T/2) equalizer to work properly or will it still work with no pulse shaping. (Rectangular pulse shape)

2) I didnt see much literature on equalizing differential modulation. Some have attempted minimizing the MSE post DQPSK decoding. Since the cost function is not quadratic, It doesnt guarantee a global minima and I think it might settle for a local minima .  So Whether constant modulus algorithm is the right candidate for equalizing DQPSK.

Your valuable insights would be highly appreciated

Are you phase-locking the receive signal and using differential decoding or are you using differential demodulation as a non-coherent signal?

If you are using coherent (phase-locked) demodulation then the usual equalization techniques should work just fine.  If you are using differential demodulation it is more difficult but still possible to do.

I'm not sure how you're using the chirps or the fractional spacing, so I can't comment on that part of it.