Journal Nonlinear Analysis and Application 2015 No.2 (2015) 105-110 Available online at www.ispacs.com/jnaa Volume 2015, Issue 2, Year 2015 Article ID jnaa-00283, 6 Pages doi:10.5899/2015/jnaa-00283 Research Article A new type of contraction in a complete G-metric space Nidhi Malhotra 1∗ , Bindu Bansal 1 (1) Department of Mathematics, Hindu College, University of Delhi, Delhi, India Copyright 2015 c ⃝ Nidhi Malhotra and Bindu Bansal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we extend and generalize the concept of F -contraction to F -weak contraction and prove a fixed point theorem for F -weak contraction in a complete G-metric space. The article includes a nontrivial example which verify the effectiveness and applicability of our main result. Keywords: fixed point, F -contraction, F -weak contraction, complete G-metric spaces. 1 Introduction The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions (see [1], [2], [3], [4], [7], [9], [10], [11] and [12]) and the reference therein. In [5], Dhage introduced D-metric space as a generalization of metric space and proved many results for this metric. But in 2005, Mustafa and Sims [8] proved that these results are not true in topological structure and hence they introduced G-metric space as a generalized form of metric space. Since then, many fixed point results have been developed by different authors in G-metric spaces. In 2012 ,Wardowski [13] introduced a new concept of F -contraction and proved a fixed point theorem for such a map on a complete metric space which generalizes Banach contraction principle in a different direction. Recently in 2014, Wardowski and Van Dung [14] defined the notion of F -weak contraction in metric spaces and generalized the theorem of Wardowski [13]. Also, Gupta [6] in 2014, introduced the notion of F -contraction in G-metric space and proved a fixed point theorem concerning F -contraction. In this paper, we introduce the notion of F -weak contraction in a complete G-metric space which is a generalization of the concept of F -weak contraction due to Wardowski and Van Dung [14]. We also extend and generalize the fixed point theorem due to Gupta. 2 Preliminaries and notations Definition 2.1. [8]Let X be a nonempty set, G : X × X × X → ℜ + be a function satisfying the following properties: (G1) G(x, y , z)= 0 if x = y = z, (G2) 0 < G(x, x, y) for all x, y ∈ X with x = y, (G3) G(x, x, y) ≤ G(x, y , z) for all x, y , z ∈ X with x ̸= y , (G4) G(x, y , z)= G(x, z, y)= G(y , z, x)= ....(symmetry in all three variables), (G5) G(x, y , z) ≤ G(x, a, a)+ G(a, y , z) for all x, y , z, a ∈ X (rectangle inequality). Then the function G is called a generalized metric or a G-metric on X , and the pair (X , G) is called a G-metric space. ∗ Corresponding Author. Email address: nidmal25@gmail.com 105