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In this booked point theorems are proved for independent types of contraction mappings; and, also give a generalization of these cases by Cirics contraction mapping is given. All these results are proved in complete, orbitally complete or chainable orbitally complete G-metric space in two states goloblly or locally. Secondly, the concepts of compatible, semi-compatible or weakly commuting are applied to prove common fixed point theorems for two mappings. Also other results are formulated without any commuting condition to get common fixed points for two or more than two mappings. Finally, the new concept of G-distance are presented in this book and used to introduce some important results in G-metric space, such as fixed point theorems, non-convex minimization theorem, -variational principle, generalization of Carsitis theorem and infinite fixed points Carsitis theorem.