Fixed Point of Pseudo Contractive Mapping in a Banach Space

Abstract

Let X be a Banach space, B a closed ball centred at origin in X, $f : B\to X$ a pseudo contractive mapping i.e. $(\alpha-1)\| x-y\| \le\|(\alpha I-f)(x) - (\alpha I-f)(y)\|$ for all x and y in B and $\alpha> 1$. Here we shown that Mapping f satisfies the property that $f(x) = -f(-x)\ \ \forall \ \ x$ in $\partial B$ 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.