What difficulties did muslims face after muhammad's death?

They had trouble choosing a successor, or Caliph. 

Everyone started to kill each other, Muslims were not gaining power. Wars broke out because after the death of Muhammad. All the muslims got into a deep grief, because of which they lost many wars. The main problem was the succession. There was a disagreement between those who thought they should run in the Prophet's immediate family and those who did not. In the end ,the latter won with a stopgap candidate, the aged Abu Bakr. There was also an apostate rebellion led by one Moseilama (real name prob.Musa). It was crushed.