A B737-500 FLC HAD THE AUTOPLT ENGAGED WITH A TRAILING EDGE FLAP PROB. USE OF THE AUTOPLT DURING A GAR RESULTED IN A STICK SHAKER WARNING.

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MEMPHIS WAS LNDG ON RWY 36R. APCH CTL HAD US ON A VERY HIGH R DOWNWIND. THE FO AS THE PF CALLED FOR FLAPS 1 DEG. APCH SAID SLOW TO 180 KTS; TURN R HDG 270 DEGS; PUTTING US A BASE LEG; STILL HIGH. FO SAID FLAPS 5 DEGS; AND I SELECTED IT. APCH SAID TURN R HDG 330 DEGS; CLRED FOR THE APCH; SLOW TO 170 DEGS AND HOLD THAT TO THE MARKER. FO SAID FLAPS 15 DEGS; GEAR DOWN; LNDG CHKLIST. HERE'S WHERE THE PROB BEGAN. IT WAS AT THIS POINT; SLOWING THROUGH MAYBE 180 KTS THAT I NOTICED THAT THE TRAILING EDGE FLAPS WERE STUCK IN AN ASYMMETRICALLY SPLIT POS BTWN 0 AND 1 DEG. THE REASON I'M WRITING THIS RPT IS BECAUSE AT THIS POINT; WE WERE ABOUT 15 OR 20 KTS SLOWER THAN THE APPROVED MINIMUM MANEUVERING SPD FOR THAT FLAP SETTING. WE WERE BOTH GUILTY OF NOT NOTICING THAT AFTER CALLING FOR FLAP EXTENSION; THE FLAPS HAD NOT EXTENDED PROPERLY. I CAUGHT THE SLIP AND IMMEDIATELY TOLD THE FO TO ACCELERATE BACK UP TO THE CLEAN MANEUVERING SPD. THEN I TOLD APCH THAT WE HAD A LITTLE FLAP PROB AND WOULD NEED TO ORBIT SOMEWHERE TO RUN A LENGTHY CHKLIST AND TROUBLESHOOT THE PROB. APCH THEN TOLD US TO TURN R HDG 090 DEGS AND MAINTAIN OUR ALT (WHICH I THINK WAS ABOUT 3000 FT MSL AT THAT POINT). THE FO WAS STILL FLYING; THE AUTOPLT WAS ENGAGED; AND WHEN HE TWIRLED THE AUTOTHROTTLE BUG UP TO 220 KTS; IT WAS SLUGGISH TO ADVANCE THE THROTTLES (THEY NEVER ADVANCE RIGHT AWAY). WHEN THE FO TWIRLED THE HDG BUG OVER TO 090 DEGS; THE BANK LIMITER WAS IN ITS NORMAL POS AT 25 DEGS; SO THAT'S THE ANGLE TO WHICH THE AIRPLANE BANKED. I NOTICED THAT THE AUTOTHROTTLES WERE STILL NOT ADVANCING; SO I SHOVED THE PWR UP; TOLD THE FO THAT'S WHAT I WAS DOING; SO HE WOULDN'T THINK THAT I WAS SIMPLY TAKING THE AIRPLANE AWAY FROM HIM; AND I SAID 'SPD' IN A SLIGHTLY WARNING TONE OF VOICE. THE AIRPLANE STARTED TO ACCELERATE. BUT BECAUSE OF THE BANK AND THE ASSOCIATED INCREASE OF THE WING'S ANGLE OF ATTACK AND G FORCE; THIS COMBINATION PUT US AT THE STICK SHAKER ALPHA; AND WE GOT A LITTLE BURST OF IT FOR MAYBE 1 SECOND. WE HAD ALREADY BEEN CORRECTING BEFORE THE STICK SHAKER; BUT TO MOVE THINGS ALONG A LITTLE FASTER; I TOOK THE AIRPLANE JUST LONG ENOUGH TO SIMULTANEOUSLY PUSH THE NOSE DOWN (SO WE WERE STILL ON AUTOPLT; BUT THIS TOOK US OUT OF HDG AND ALT HOLD MODES AND INTO THE CTL WHEEL STEERING MODE); PUSH TO PWR UP EVEN MORE; AND ROLL LEVEL. WITH THE ACFT BACK AT THE CLEAN MINIMUM MANEUVERING SPD; WE RAN THE 'TRAILING EDGE FLAPS MALFUNCTION' CHKLIST. WITH ALTERNATE FLAPS SWITCH ARMED; I BRIEFLY HELD DOWN THE ALTERNATE FLAP SWITCH TO ACHIEVE LED FULL EXTENSION; AND THE TRAILING EDGE ASYMMETRY DISAPPEARED. WITH THE AIRPLANE PROPERLY CONFIGURED FOR A FLAPS 15 DEG LNDG; I TOOK CTL OF THE AIRPLANE AND WE LANDED WITHOUT FURTHER INCIDENT. BEFORE LNDG; I INFORMED AND BRIEFED ALL THE APPROPRIATE PARTIES; INCLUDING THE FLT ATTENDANTS; THE PAX; THE COMPANY; MAINT; CALLED FOR THE EMER VEHICLES AS A PRECAUTION.