The question asks about the problems while he is caliph not before being elected as caliph. The first main problem was the opposition of some people to pay the ZAKAT as they were doing in time of the prophet. He waged war against them to oblige them to do as they were doing in time of prophet.
Answer
while he was Caliph Ali (sa) and more then 100 of Sahaba did not accept him as Caliph because they said Allah and prophet have selected Ali (sa) as Caliph at Ghadir. Ali (sa) neve willingly given Bayat to Abubakr. and also many Sahaba.
The first Caliph was Abu Bakr Assidik.
Abu Bakr ruled as a Caliph for 27 months.
Perhaps you are referring to abu Bakr? Abu Bakr was one of Muhammad's fathers-in-laws and became the first Caliph.
Abu bakr
Abu Bakr
Only Sunnis agreed on Abu Bakr as first Caliph after prophet Muhammad death. The Shiites believed that the Caliph should from the prophet family and accordingly they supported Ali to be the first caliph and not Abu Bakr. However, the Shiites accepted the majority selection of Abu Bakr and supported him and Ali (God pleased with him) was a closeadviser to Abu Bakr. See for more information the related question below
Abu Bakr, the first caliph of Islam after the death of Prophet Muhammad, held the title of "Caliph" (Khalifah in Arabic), which means "successor" or "representative." He is often referred to as "Caliph Abu Bakr" or simply "Abu Bakr as-Siddiq," with "as-Siddiq" meaning "the truthful." His leadership marked the beginning of the Rashidun Caliphate, which was characterized by a focus on unifying the Muslim community and expanding Islamic territories.
Abu Bakr. Then Omar. Then Othman. Then Ali.
was the father of muhammad and he was the first caliph after his death
The first Caliph Abu Bakr Assedik
The first Rightly Guided Caliph Hazrat Abu Bakr Siddique RAU.
Abu Bakr took the office of caliph in 632 A.D. He assumed it until his death in 634 A.D, meaning that his caliphate lasted for around 2 years.