DC9-30 CREW HAD TCASII RA'S FROM A MALFUNCTIONING TCASII SYS.

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WHILE CLBING THROUGH 15000 FT; GOT A TCASII RA 'REDUCE CLB;' FOLLOWED BY 'DSND; DSND;' FOLLOWED BY 'INCREASE DSCNT;' FOLLOWED BY 'MONITOR VERT SPD;' FOLLOWED BY 'CLR OF CONFLICT.' WE COMPLIED WITH ALL THE RA'S. I IMMEDIATELY TOLD ATC WHAT WAS HAPPENING; AND THEY ADVISED ME THAT THERE WAS NO TFC IN THE VICINITY. WE STARTED CLBING AGAIN; AND A COUPLE OF MINS LATER; HAD THE SAME TCASII DISPLAYS AND RA'S COMMANDS AS DESCRIBED ABOVE. WE ONCE AGAIN COMPLIED WITH RA'S AND ADVISED ATC THAT WE WERE DSNDING; BECAUSE OF THE RA'S. ATC SAID THEY HAD NO TFC IN THE VICINITY. AT THIS POINT; WE WERE NOT SURE WHAT WAS HAPPENING; BUT SUSPECTED SOMETHING WRONG WITH THE TCASII. AT NO TIME; HOWEVER; DID WE HAVE A TCASII INOP DISPLAYED; NOR DID WE HAVE A TFC SYMBOL OR TARGET ACFT DISPLAYED ON THE TCASII. THE FO AND I DISCUSSED WHAT WAS HAPPENING; AND WE AGREED THAT THERE MUST BE SOMETHING WRONG WITH OUR EQUIP. A COUPLE OF MINS LATER; THE TCASII RA WENT OFF AGAIN WITH THE SAME SCENARIO; BUT THIS TIME WE DISREGARDED; TALKED AGAIN TO ATC; AND ONCE AGAIN WAS TOLD THAT THERE WAS NO TFC IN THE AREA. WE PROCEEDED TO OUR DEST; LANDED UNEVENTFULLY AND WROTE THE TCASII UP IN THE LOGBOOK. THE FACT THAT NO TARGET ACFT WAS DISPLAYED; THEREFORE; GIVING US NO HINT OF WHERE TO LOOK FOR CONFLICTING TFC; AND THE URGENCY OF THE RA COMMANDS CREATED A SIT WHERE YOU HAD TO ACT NOW; FOLLOW THE RA COMMANDS NOW; AND LATER ANALYZE WHAT YOU HAD. NOT UNTIL WE HAD GONE THROUGH 2 FULL CYCLES OF RA'S; TAKING ACTION; AND BEING CLRED BY ATC; DID WE CONSIDER THAT WE MAY HAVE A FAULTY TCASII.