Voltage Supervisor IC for PIC
It sounds like your circuit is turning on and loading the source so that it drops below the 2.88V (nominal) threshold, triggering the reset circuit and causing it to restart many times. 

One approach which may work if you're not turning on really heavy loads is to use a supply voltage supervisory circuit with some hysteresis. For example, the TPS3808 has 6mV of hysteresis built-in and it can be increased. There are many such chips available. 

If you are turning on really heavy loads you might have to measure the supply voltage with the ADC before making a decision as to whether to power them.