Control Theory and Informatics www.iiste.org ISSN 2224-5774 (Paper) ISSN 2225-0492 (Online) Vol.4, No.6, 2014 33 Unique Fixed Point Theorem For Asymptotically Regular Maps In Hilbert Space 1 Pravin B. Prajapati, 2 Ramakant Bhardwaj, SabhaKant Dwivedi 3 1 S.P.B.Patel Engineering College, Linch 1 The Research Scholar of Sai nath University ,Ranchi(Jharkhand) 2 Truba Institute of Engineering & Information Technology, Bhopal (M.P) 3 Department of Mathematics, Institute for excellence in higher education Bhopal E-mail: rkbhardwaj100@gmail.com Abstract: The object of this paper is to obtain unique fixed point theorems for asymptotically regular maps and sequence in Hilbert Space. Keywords: Hilbert Space, Asymptotically Regular Map, Asymptotically Regular Sequence. INTRODUCTION Banach fixed point theorem and its applications are well known. Many authors have extended this theorem, introducing more general contractive conditions which imply the existence of a fixed point. Almost all of conditions imply the asymptotic regularity of the mappings under consideration. So the investigation of the asymptotically regular maps plays an important role in fixed point theory. Sharma and Yuel [8] and Guay and Singh [6] were among the first who used the concept of asymptotic regularity to prove fixed point theorems for wider class of mappings than a class of mappings introduced and studied by Ciric [4]. The purpose of this short paper is to study a wide class of asymptotically regular mappings which possess fixed points in Hilbert spaces. In present paper we extend the results of Gopal and Ranadive [5] in Hilbert space. 2. PRELIMINARIES Definition 2.1: A self-mapping T on a metric space   d X , is said to be asymptotically regular at a point x in X, if as n Where denotes the nth iterate of T at x. Definition 2.2: Let C be a closed subset of a Hilbert space H. A sequence in C is said to be asymptotically T – regular if as n Definition 2.3: A self- mapping T on a closed subset C of a Hilbert space H is said to be asymptotically regular at a point x in C if As n Where denotes the nth iterate of T at x.