The engine warning light on my 2008 malibu is on. I also have had problems with the acceleration of the vehicle. My local mechanic ran an emissions check and showed a defective catalitic convertor. I have taken the car on three seperate occasions to the local chevy dealer who has replaced several sensors which corrects the problem for a day or two and then the problems starts up again. The dealer continues to state that it is not the catalytic convertor. should I take my car to another dealership for a second opinion?

Why take it to the dealer , find a local independent repair shop . Replacing a sensor and the problem comes back in a couple days means it wasn't the sensor in the first place . Knowing how the computer systems work on these vehicles is very important . Some problems after the check engine light is turned off , by clearing the codes with a scan tool , won't set for a couples day's because it is a two trip code . The vehicle has to sit over nignt , started an ran till fully warmed , then let sitting over night again. The computer has temp sensor inputs an knows this .
What codes did your mechanic find ? P0420 - P0430 ? These are codes for Catalytic converter P0420 Catalyst System Efficiency Below Threshold

Read more at: https://www.obd-codes.com/p0420
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had problems with the acceleration , your vehicle has electronic throttle control controlled by the PCM/ECM - engine computer ! There are position sensors in the gas pedal assembly an the throttle body . After time these sensors wear causing acceleration problems . All vehicles starting back in 2004 have this .
Electronic Throttle Control Ford , GM , Chrysler ,toyota , BMW etc... They all wear .

That car sounds like it has too many problems.
If it were me, I would not waste anymore money and time
on it and find another vehicle if possible.