Common fixed-point theorems for commuting maps on a metric space

2. In this section, we prove a fixed point theorem which is a generalization of Theorem 1.1. Let X be a complete metric space. Let f be a continuous self-map on X and let g be any self-map on X such that (1.1) is satisfied. We define a sequence of points {xn} as follows. For xo (E X) arbitrary, let x1 (EX), guaranteed by (1. 1), be such that g(xO) = f(x1). Having defined xn ( E X), let xn I 1 ( E X) be such that g(xn) = f(Xn + Letting g(xn) (= f(xn I 1)) = Yn (n = 0, 1, 2,... ) we denote by O(Yk; n) the set of points yk,yk+ I *.* Yk+In). Let us assume that f and g satisfy the following condition: There exists a constant a E (0, 1) such that for every x, y in X,