Extensions of a fixed point theorem of Meir and Keeler
Meir and Keeler proved that an (e, 6)-contraction g of a complete metric space X has a unique fixed point r/ in X and {g"x}2"= 1 converges to t/ for all x ~ X [9]. The class of (e, 6)-contractions clearly contains the classes of (Banach) contractions and nonlinear contractions investigated by Browder [3] and by Boyd and Wong [2]. A fixed point of a selfmap g of X can be considered as a common fixed point of g and Ix, the identity map of X. In certain cases, we can replace 1 x by a continuous selfmap f of X and consider common fixed point of f and g. Jungck [8] adopted this idea and obtained a useful generalization of the Banach contraction principle to commuting selfmaps. More recently, Park [10] extended these facts and obtained a number of results on commuting selfmaps. L e t f b e a continuous selfmap of a metric space (X, d) and C I denote the class of selfmaps g of X such that f g = g f and g X c f X . A selfmap g of X is called an (~, 6)-f-contraction if for any ~ >0 , there exists 6 = 6 (~)>0 such that for all x, y~J(,