Fixed point theory on Banach spheres

Abstract

In this paper, we consider and study the concept of the Banach sphere. The usual spherical distance of the unit sphere of a Hilbert space is defined by its inner product and the inverse of the cosine function. Therefore, we cannot apply this notion to Banach spheres in general. We first introduce a two-variable function like a spherical distance and a notion of convex combination on a Banach sphere. After that, we define a projection onto a subset of a Banach sphere, and prove a fixed point theorem and fixed point approximation for a mapping having a kind of nonspreadingness. Our work is a challenge to construct optimisation theory on spherical planes without geodesics.