Mathematica Eterna
Open Access

Abstract

A Unique Common Fixed Point of a pair of Self-maps on a 2-metric space

T. Phaneendra and K. Kumara Swamy

Let (M, ) be a complete metric space and f a self-map on M such that (fx, fy)  (fx, fy) for all x, y  X, where 0   <1/2. Kannan proved that f has a unique fixed point p and for each x  M the iterates f, f 2 , … will converge to p. In this paper, we extend this result to a pair of self-maps on a complete 2-metric space. Our technique is remarkable to use only elementary properties of greatest lower bound, and repeatedly employing the symmetry and the tetrahedron inequality of the 2-metric instead of routine iteration procedure. This idea was initiated for only metric spaces