In this paper we prove some fixed point results for mapping satisfying sufficient contractive conditions on a complete G-metric space, also we showed that if the G-metric space <img border=0 width=46 height=25 src="http://fbpe/img/cubo/v12n1/img39.jpg">is symmetric, then the existence and uniqueness of these fixed point results follows from Reich theorems in usual metric space <img border=0 width=53 height=25 src="http://fbpe/img/cubo/v12n1/img38.jpg">, where <img border=0 width=53 height=25 src="http://fbpe/img/cubo/v12n1/img38.jpg">the metric induced by the G-metric space <img border=0 width=46 height=25 src="http://fbpe/img/cubo/v12n1/img39.jpg">.