What difficulties did Muslims face after Muhammad's death?

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.

They had trouble choosing a successor, or Caliph.