96 Ramakant Bhardwaj ,Balaji R Wadkar * , Basant Kumar Singh ** International Journal of Computer & Mathematical Sciences IJCMS ISSN 2347 – 8527 Volume 4, Special Issue September 2015 Fixed Point Theorem in Generalized Banach Space Ramakant Bhardwaj ,Balaji R Wadkar * , Basant Kumar Singh ** Deputy Director (R & D) ,TIT Group of Institutes, Bhopal * Department of Mathematics “Shree Ramchandra College of Engineering”, Lonikand,(Pune), ** Principal, AISECT University, Bhopal Abstract In this paper we have obtained some fixed point theorem on generalized Banach space which is an extension of some known result & again extended to generalized 2-Banach space. Keywords: Generalized normed linear space, Generalized 2-normed linear space, Generalized Banch space. Introduction: In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892–1945), and was first stated by him in 1922. Although metric fixed point theory is vast field of the study and is capable of solving many equations. Toovercome the problem of measurable functions w. r.t.a measure andtheir convergence, czerwik [8 ] needs an extension of metric space. Using this idea , he presented a generalization of Banach fixed point theorem theorem in b –metric space. See [ 9, 10 , 11]. Many researchers studied the extension of fixed point theorem in b- metric space. (see [1-7,12-24 ] ). In this paper our aim is to show the validity of some important fixed point theorem in generalized Banach space. This paper is divided into two parts Section I: some fixed point theorems in generalized Banach space. Section II: some fixed point theorems in generalized 2-Banach space. 2 Preliminaries: We recall some definitions and properties of generalized normed linear space. Definition 2.1:If ) (   M is a linear space having ) ( s R  , let . denotes a function from linear space M into R that satisfies the following axioms: i. M x   , 0 , 0   x x iff 0  x ii. M y x   , ,   y x s y x    iii. R M x     , , x x    x is called norm of x and   . , M is called generalized normed linear space. If for s = 1, it reduces to standard normed linear space. Definition 2.2:A Banach space   . , M is a normed vector space such that M is complete under the metric induced by the . . Definition 2.3:The set of continuous function on closed interval of real line with the norm . of a function f given by ) ( sup x f f M x  is a Banach space, where sup denoted the supremum.