FA20 FLT CREW EXPERIENCES LNDG GEAR RETRACTION ANOMALY AND A TCAS RESOLUTION ADVISORY DURING INITIAL CLB.

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2 PROBS ACTUALLY. FIRST OF ALL; AFTER TKOF I BROUGHT THE GEAR HANDLE UP; GOT A FLASHING RED LIGHT IN THE GEAR HANDLE ALONG WITH A RED LIGHT ON THE L MAIN LNDG GEAR INDICATOR. CLBING RAPIDLY WITH A PLANNED LEVELOFF AT 3000 FT MSL; TWR INSTRUCTED US TO STOP THE CLB AT 2000 FT; ADVISED TFC; AND INSTRUCTED US TO CALL DEP CTL. WE WERE ABOUT TO PASS THROUGH 2000 FT WHEN WE GOT THE CALL FROM TWR. WE SAW THE TFC AT 1-2 MI; 1 O'CLOCK POS MOVING R TO L AT APPROX 2500 FT MSL. WE GOT A TA AT ABOUT THE TIME THE TWR TOLD US TO LEVEL OF AT 2000 FT AND CALL DEP. THE TA WAS FOLLOWED SHORTLY AFTER BY AN RA. I CALLED OUT TO THE PF TO TURN R TO AVOID THE TFC. ONCE WE GOT CLR OF THE CONFLICTING TFC IT WAS TIME TO DEAL WITH THE OTHER PROB OF THE MOMENT; THE LNDG GEAR. I CONTACTED DEP AND INFORMED THEM OF OUR SITUATION. WE LEVELED OFF AT 4000 FT AND GOT VECTORS WHILE WE TROUBLESHOT THE SYS. WHILE DEALING WITH THE TA AND RA; WE MANAGED TO KEEP THE SPD BACK AT VLO AND LOWER THE GEAR HANDLE IN AN ATTEMPT TO CYCLE THE GEAR. I HAD PUT THE GEAR HANDLE DOWN POS AND WE MADE NOTE THAT THE R MAIN GEAR AND THE NOSE GEAR APPEARED TO HAVE GONE DOWN AND LOCKED WHILE THE L MAIN REMAINED NEITHER DOWN NOR UP. I WENT INTO THE CABIN TO BRIEF THE PAX ON THE SITUATION AND AS I RETURNED TO THE COCKPIT A COUPLE OF MINS LATER WE HEARD AND FELT THE L MAIN DROP DOWN. THE PAX HAD RPTED HEARING GRINDING NOISES FROM THE AREA OF THE L MAIN AFTER TKOF. NOW WE HAD THE GEAR INDICATING DOWN BUT HAD SEVERAL FAULTS DISPLAYED AND CREW ALERTING SYS MESSAGES; INCLUDING NOSEWHEEL STEERING FAILED. I GOT ON THE SATPHONE AND CALLED THE DASSAULT HELP DESK AND TALKED WITH THE ENGINEER ON DUTY ABOUT OUR SITUATION. WE WENT THROUGH ALL OF THE APPROPRIATE CHKLISTS. ENDED UP PULLING THE RED GEAR PULL HANDLE WHICH IS BASICALLY A HYD BACKUP ASSIST FOR THE GEAR. WITHOUT HAVING A MIRROR TO CHK THE GEAR BELOW; WE COULDN'T REALLY BE SURE WHAT HAD HAPPENED TO THE L MAIN OR HOW MUCH DAMAGE IF ANY IT HAD SUSTAINED; OR IF THE TIRES WERE IN TACT. APCH CTL VECTORED US FOR THE VISUAL APCH TO RWY XX. TWR CHKED THE GEAR AND ASKED OUR INTENTIONS. WE DID THE LOW APCH FOLLOWED BY R TFC VFR AND LANDED UNEVENTFULLY. WE ROLLED OUT NEAR THE END AND CAREFULLY TURNED OFF; NOSEWHEEL STEERING WORKING. TALKED WITH GND CTL; THEN SHUT DOWN THE PLANE AND DEPLANED THE PAX. THE TUG PULLED UP AND TOWED US IN AND THE VAN WHISKED AWAY THE PAX. WE WERE ALL PLEASANTLY SURPRISED TO FIND NO VISIBLE DAMAGE AND EVERYTHING INTACT. THE FIRE TRUCKS HAD BEEN STANDING BY. MAINT TECHNICIANS LOOKED AT THE ACFT TODAY AND DETERMINED THE PROB TO BE A FAULTY L MAIN LNDG GEAR DOOR ACTUATOR PROX SENSOR OR POSSIBLY THE 'TARGET MECHANISM' WITHIN THE ACTUATOR ITSELF.