Is there any kind of Warranty when we purchase used car
That's a question for the seller. All vehicles have a manufacturer warranty period, typically 3 years or 36,000 on basically everything but wearable items, 5 years or 60,000 miles on power-train components. Some makes up to 100,000; regardless of how many owners. If they vehicle is beyond this, it is likely as-is/as seen unless otherwise stated in writing. There is no return period, or much you can do unless you have a contract signed.
Emission control items, including the PCM and catalytic converter are warranted for either 5 years/50,000 miles or 8 years/80,000 miles from the original new purchase date depending on where you are. HTH. -Jim