Remarks on brouwer fixed point theorem for some surfaces in R³

Abstract

© 2019 by the Mathematical Association of Thailand. Let X be a surface in R3. A subset E of X is said to be convex if, for each p, q ∈ E, it contains each shortest geodesic joining p and q. A surface in R3 is said to have the fixed point property if each continuous mapping T: E → E from a compact convex subset E of X has a fixed point. In this paper, we give some examples of surfaces in R3 that do not have the fixed point property. Moreover, we show that the surface z = y2 and the upper hemisphere of the sphere of radius r centered at (0, 0, 0) with north pole and equator removed have the fixed point property.