IN THIS WORK, WITHOUT ASSUMPTIONS ON THE CONTINUITY AND THE SUBADDITIVITY OF  H, WE GIVE A SIMPLE PROOF AND A CHARACTERIZATION OF THE EXISTENCE OF CARISTI-KIRK’S PROBLEM. ALSO A SUPPLEMENTARY EXAMPLE IS GIVEN.

IN THIS PAPER WE PROVE A FIXED POINT THEOREM IN COMPACT CONE METRIC SPACES BY USING IMPLICIT RELATION TECHNIQUE.

IN THE PRESENT WORK, WE EXTEND CARISTI'S FIXED POINT THEOREM, EKELAND'S VARIATIONAL PRINCIPLE AND TAKAHASHI'S MAXI-MIZATION THEOREM TO FUZZY METRIC SPACES IN THE SENSE OF GEORGE AND VEERAMANI [A. GEORGE, P VEERAMANI. ON SOME RESULTS IN FUZZY METRIC SPACES, FUZZY SETS AND SYSTEMS. 64 (1994) 395-399]. FURTHER, A DIRECT SIMPLE PROOF OF THE EQUIVALENCES BETWEEN THESE THEOREMS IS PROVIDED. ALSO SOME ILLUSTRATIVE EXAMPLES ARE GIVEN.