c International Journal of Research (IJR) e-ISSN: 2348-6848, p- ISSN: 2348-795X Volume 3, Issue 01, January 2016 Available at http://internationaljournalofresearch.org Available online:http://internationaljournalofresearch.org/ Page | 572 Coincidence and Common Fixed Point Using Sequentially Weak Compatible Mapping Kamal Kumar Research Scholar Department of Mathematics Sunrise University Alwar, India (kamalg123@yahoo.co.in ) Rajeev Jha Department of Mathematics Teerthanker Mahaveer University, Moradabad (U.P), India (jhadrrajeev@gmail.com ) Abstract. In this paper we are going to introduce the concept of sequentially weak contraction using sequence of function which is uniformly convergent to a continuous function . The concept of sequence of function is already given by Dutta et. al.[5]. Mathematics Subject Classification: (2001) AMS 47H10, 54H25. Keywords: Banach Contraction Principle, complete metric space, Cauchy sequence, Weakly compatible maps, Weak contraction, Generalized weak contraction. 1. Introduction Stability of fixed points of contraction mappings has been studied by Bonsall et.al. [2] and Nadler et.al. [11]. These authors consider a sequence (T n ) of maps defined on a metric space ( X,d) into itself and study the convergence of the sequence of fixed points for uniform or pointwise convergence of (T n ), under contraction assumptions of the maps. Theorem.1.1. Let (X, d) be a complete metric space and let T : X → X be a self-mapping satisfying the inequality (1.1) (d(Tx, Ty)) ≤(d(x, y)) – (d(x, y)) where φ,  : [0, ∞) → [0, ∞) are both continuous and monotonic nondecreasing functions with (t) = 0 = (t) if and only if t = 0. Then T has a unique fixed point. This theorem can be restated using sequence of function as: Definition.1.2. A mapping T : X → X, where (X, d) is a metric space, is said to be sequencially weakly contraction if (1.2) d(Tx, Ty) ≤ d(x, y) - fn(d(x,y)) (f n :I (interval or subset of R) → R ) where x,y∈ X and f n (t) is a sequence of function which converges uniformly to t, and monotonic function such that f n (t) = 0 if and only if t = 0. If one takes f n (t) = kt where 0 < k < 1 and t = 1, then (1.2) reduces Banach Contraction Principle, which states that “Let (X, d) be a complete metric space. If T satisfies (1.3) d(Tx, Ty) ≤ k d(x, y)