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.