Schmitt Trigger Comparator 0 - 5V
The LM358 op-amp will not swing it's output to the positive supply rail. The top "transistor" in the package is configured as a Darlington-pair and this is your problem: - 



Even if the base of Q5 could reach +5V, its emitter would be 0.7V below 5V and therefore Q6's base would be also 0.7 volts below 5V and, in turn, Q6's emitter would be 1.4 volts below 5V = 3.6 volts.

Next, you say that the schmitt trigger works off an input hysteresis from 2.2 to 2.8 V. I have to disagree.

If the comparator were able to reach 5 V on the output (as you appear to have assumed originally) then the upper threshold would be: -

5V x \$\dfrac{R3}{R3+(R2||R4)}\$ = 2.705 volts

The lower threshold would be 5V x \$\dfrac{R3||R4}{R2+(R3||R4)}\$ = 2.295 volts.

Clearly, at a lower output voltage this shifts the range down but you were never ever going to see a top threshold of 2.8 volts. How did you calculate it?