Int. J. Math. And Appl., 8(2)(2020), 1–5 ISSN: 2347-1557 Available Online: http://ijmaa.in/ A p p l i c a t i o n s • I S S N : 2 3 4 7 - 1 5 5 7 • I n t e r n a t i o n a l J o u r n a l o f M a t h e m a t i c s A n d i t s International Journal of Mathematics And its Applications Fixed Point of Pseudo Contractive Mapping in a Banach Space Chetan Kumar Sahu 1, * , S. Biswas 1 and Subhash Chandra Shrivastava 2 1 Department of Mathematics, Kalinga University, Raipur, Chhattisgarh, India. 2 Department of Mathematics, Rungta College of Engineering & Technology, Bhilai, Chhattisgarh, India. Abstract: Let X be a Banach space, B a closed ball centred at origin in X, f : B → X a pseudo contractive mapping i.e. (α − 1)‖x − y‖≤‖(αI − f )(x) − (αI − f )(y)‖ for all x and y in B and α> 1. Here we shown that Mapping f satisfies the property that f (x)= −f (−x) ∀ x in ∂B called antipodal boundary condition assures existence of fixed point of f in B provided that ball B has a fixed point property with respect to non expansive self mapping. Also included some fixed point theorems which involve the Leray-Schauder condition. Keywords: Fixed point, Banach space, Non expansive mapping, Pseudo Contractive Mapping, Cauchy Sequence, Lipschitzian Map- ping. c  JS Publication. 1. Introduction Let X be a real Banach space and D be a subset of X. An operator f : D → X is said to be k-pseudo contractive (k> 0) if for each x and y in D and α>k (α − k)‖x − y‖≤‖(αI − f )(x) −‖(αI − f )(y)‖ for k ≤ 1 such operator is called strongly pseudo contractive. In addition to generalizing the non-expansive mappings. The pseudo- contractive mappings are characterized by the important fact that a mapping f : D → X is pseudo-contractive if and only if the mapping T = I − f is accretive on D. It is well known that if D is a bounded open convex subset of a uniformly convex Banach space X and if f is a non-expansive mapping defined from the closure D of D into X, then the Leray-Schauder boundary condition which asserts that for z ∈ D,(L − S)T (x) − z = k(x − z) for all x ∈ ∂D, k> 1 is sufficient of guarantee existence of a fixed point for T. Our main objective here is to study the question mentioned above under two different boundary conditions apparently stronger than (L − S). 2. Preliminaries This section is devoted to some basic definitions, prepositions and lemmas which are needed for the further study of this Article. ∗ E-mail: revathikrishna79@gmail.com (Research Scholar)