DEMONSTRATIO MATHEMATICA Vol. XLV No 4 2012 Saurabh Manro, Sanjay Kumar, S. S. Bhatia WEAKLY COMPATIBLE MAPS OF TYPE (A) IN G-METRIC SPACES Abstract. In this paper, we introduce the concept of compatible maps and weakly compatible maps of type (A) in G-metric spaces. 1. Introduction In 1922, Banach proved fixed-point theorem (“Let (X, d) be a complete metric space. If T satisfies d(Tx,Ty) ≤ kd(x, y) for each x, y in X where 0 <k< 1, then T has a unique fixed point in X .”), which ensures under appropriate conditions, the existence and uniqueness of a fixed-point. This theorem had many applications, but suffers from one drawback-the definition requires that T be continuous throughout X . Then there followed a flood of papers involving contractive definition that do not require the continuity of T . This result was further generalized and extended in various ways by many authors ([1], [4], [5]). This theorem provides a technique for solving a variety of applied problems in mathematical sciences and engineering. In 1963, Gahler [3] introduced the concept of 2-metric spaces and claimed that a 2-metric is a generalization of the usual notion of a metric, but some authors proved that there is no relation between these two functions. It is clear that in 2-metric d(x, y, z ) is to be taken as the area of the triangle with vertices at x, y and z in R 2 . However, Hsiao [2] showed that for every contractive definition with x n = T n x 0 , are trivial in the sense that the iterations of mapping are collinear. In 1992, Dhage [1] introduced the concept of D-metric space. The sit- uation for a D-metric space is quite different from 2-metric spaces. Geo- metrically, a D-metric D(x, y, z ) represent the perimeter of the triangle with vertices x, y and z in R 2 . Recently, Mustafa and Sims [6] showed that most Key words and phrases : G-metric space, compatible maps of type (A), weakly com- patible maps of type (A). 1991 Mathematics Subject Classification : 47H10, 54H25.