Chrysler Repair: concord wont crank over/ then dies quickly, starter relay, subt

Question
I have a 2001 concord, we drove it into work, parked it, then 2 hours later my wife went out to go into town, it would not turn over, plenty of battery, all lights work, let it sit over the weekend, then went out on Monday am, it started right up as usual, then drove around back, (not even a block) to air up tire, it died, would not turn over again, called the mechanic, he came down to pull it to there shop, it started up, then within 10 seconds died, still plenty of battery,
they have had it for 6 weeks now, no results, wanting to replace computer and all kinds of stuff, they said after replacing computer it wouldn't start but number 4 cylinder not firing, that would have nothing to do with it not cranking over, any ideas please before they keep just putting on parts? they had the starter off and it is fine

Answer
Hi Dean,
The starter motor refusing to turnover is not that complicated a problem, particularly if the motor itself will turnover when removed. The circuits to the motor are basically the power from the battery going through the solenoid on the motor and the power from the battery to the starter relay to activate the solenoid, and finally the power to activate the relay which depends upon the transmission being in park or neutral.
The one unusual thing is the fact that when the engine has started you say that it dies quickly for no apparent reason. This, coupled with the starter issue, makes me wonder if there is a problem with the ignition switch (although there is a separate section for the starter from the other sections needed to run the engine). There is however one fuse in the engine compartment box, fuse G (40 amp), which could be cracked and opening up spontaneously/unpredictably and then closing which could explain the combo starter/quick die behavior. So I recommend that they take a look at that fuse because it power both the starter and the fuel pump and several sections of the ignition switch proper. The fuse could have a subtle crack in its wire and not look to be 'blown'.
Other than that, I would have to believe that if the shop has the wiring diagram it can figure out why the starter won't work reliably.
Please read the PS below.
Thanks,
Roland
Roland