General aviation pilot reported an NMAC in the traffic pattern at a non-towered airport. The other aircraft descended on top of the C172 requiring evasive action. The aircraft also caused an NMAC with another aircraft.

Narrative

Offending aircraft - Aircraft YI had taken off from MJX closed traffic and was turning from left crosswind to left downwind when Aircraft Y was entering the 45 degree left downwind. They made a call saying they would be behind us but were overtaking us from behind. I called them to ask if they had the downwind traffic in sight and they said they did and would be behind us. At this point I started descending aggressively as they were 300ft-400ft above us (from ADSB-in reading) and overtaking quickly. I queried them again and they said that I must be mistaken. They were still catching us so I leveled off around 500-600ft MSL and increased speed and turned right to escape the pattern area. I turned around to re-enter the downwind and noticed Aircraft Y turning left base very close to traffic on final. The traffic on final asked Aircraft Y if they had the traffic in sight and Aircraft Y said they did and would be behind them. The traffic on final decided to go around for separation; as they were very close laterally and vertically. Once Aircraft Y landed; I called them to inform them they had two traffic issues but they did not respond. They continued in the traffic pattern and were making a few nonstandard radio calls of their call sign. I left after two more laps in the pattern.