A common fixed point theorem for nested spaces

Raymond Earl Smithson

Vol. 82 (1979), No. 2, 533–537

DOI: 10.2140/pjm.1979.82.533

Abstract

Let X be an arcwise connected Hausdorff space in which the union of any nest of arcs is contained in an arc. Let f,g : X → X be commuting functions (not necessarily continuous), which satisfy (1) f(A) and g(A) are arcwise connected for each arc A ⊂ X, and (2) f⁻¹(x) and g⁻¹(x) are arcwise connected for each x ∈ X. The principal result of this paper is:

Theorem. The functions f and g have a common fixed point.

Mathematical Subject Classification 2000

Primary: 54H25

Milestones

Received: 14 July 1978

Revised: 15 February 1979

Published: 1 June 1979