Air carrier pilot reported intermittent malfunctioning of the autopilot after takeoff and a temporary loss of aircraft control during descent due to VNAV malfunction. Flight crew continued flight with autopilot disengaged.

Narrative

On Day 0 I was operating on a flight from ZZZ1 to ZZZ. I was the Relief Officer. The aircraft had a degraded center Autopilot and no other MELs. Shortly after takeoff the aircraft started to have intermittent malfunctions. The FMAs for both the pilot monitoring and pilot flying had sporadic faults; and both the left and right Autopilot would kick off at random intervals. The crew discussed these issues; consulted with the maintenance team onboard; and called Dispatch to consult with the headquarters Maintenance and Operations teams. It was determined the crew could safely continue the flight. The flight attendant team was notified of the malfunction and told that proceeding as scheduled was the current plan.During descent the pilot flying noticed the FMC was commanding (bugging) an airspeed just below the overspeed tape (348 kts). The pilot flying disconnected the Autopilot; and hand flew the aircraft to avoid an overspeed. The pilot flying directed the pilot monitoring the check the VNAV descent speeds. The box was programed correctly and showed an econ descent speed of 285 kts. The PF directed the PM to reselect econ descent speed; and the bug moved to 285. The PF engaged Autopilot and VNAV. Shortly after the speed bug jumped to 348 without command and the aircraft entered a steep nose down attitude to obtain 348 kts. The pilot PF again disconnected Autopilot and arrested the speed. While preventing an overspeed the aircraft entered a slight climb. ATC called us and enquired about our status and wanted to make sure we were 'okay.' The PM explained to ATC that we had a malfunction and had corrected the issues. The crew then talked over options on how to proceed while the PF maintained aircraft control manually. It was decided that a hard speed of 285 kts should be entered instead of econ. The VNAV system worked properly for the duration of the flight with the Autopilot disengaged.