IN THIS PAPER, WE WILL FIRST INTRODUCE THE APPROXIMATE fixed point PROPERTY FOR A CYCLIC MAP T ON G-METRIC SPACE. ALSO, WE PROVE GENERAL LEMMA WHICH REGARDING APPROXIMATE fixed point OF CYCLIC MAPS ON G-METRIC SPACE. USING THESE RESULTS WE PROVE DIAMETER APPROXIMATE fixed point THEOREMS FOR VARIOUS TYPES OF THE WELL-KNOWN GENERALIZED CONTRACTIONS ON G-METRIC SPACE.

LET X BE A COMPLEX BANACH SPACE WITH DIM X³3 AND B (X) THE ALGEBRA OF ALL BOUNDED LINEAR OPERATORS ON X. SUPPOSE F: B (X) ®B (X) IS A SURJECTIVE MAP SATISFYING THE FOLLOWING PROPERTY: (FORMULA) THEN THE FORM OF F IS CHARACTERIZED, WHERE FIX (T) IS THE SET OF ALL fixed pointS OF AN OPERATOR T.

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.

Huang and Zhang generalized the concept of a metric space by considering vector-valued metrics with values in an ordered real Banach space. They obtained some fixed point theorems for mappings satisfying different contractive conditions. In this paper we prove some fixed point theorems in cone metric spaces, including results which generalize those from Huang and Zhang’s work.

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