We prove some fixed point results for mapping satisfying sufficient conditions on complete Gmetric space, also we showed that if the G-metric space X, G is symmetric, then the existence and uniqueness of these fixed point results follow from well-known theorems in usual metric space X, d G , where X, d G is the usual metric space which defined from the G-metric space X, G .

Solving systems of nonlinear equations is a relatively complicated problem for which a number of different approaches have been proposed. In this paper, we employ the Homotopy Analysis Method (HAM) to derive a family of iterative methods for solving systems of nonlinear algebraic equations. Our approach yields second and third order iterative methods which are more efficient than their classical counterparts such as Newton's, Chebychev's and Halley's methods.

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