Local Contractions in Metric Spaces1

It is shown that a theorem of E. Rakotch for locally contractive mappings can be deduced from Banach's contraction mapping theorem, and a counterexample to an assertion of R. D. Holmes concerning local radial contractions is given. Let (X, d) be a metric space, k G (0, 1), and g a mapping of X into X. If for each x G X there exists a neighborhood A^(x) of x such that for each u, v G N(x), d(g(u), g(v)) < kd(u, v), then g is said to be locally contractive (with constant k); if it is assumed only that d(g(u), g(x)) < kd(u, x) for all u G A^x), then g is said to be a local radial contraction. These are two of the standard localizations of the contraction mapping condition (cf. Rakotch [3] and Holmes [2], respectively). It is proved in [3] that if A1 is a complete metric space and g: X —> X locally contractive, and if in addition it is the case that for some x0 G X the points x0 and g(x0) are joined by a rectifiable path (i.e., a path of finite length) then g has a unique fixed point in X. An example, also given in [3], shows that this pathwise connectedness assumption cannot be dropped. In a related subsequent development, R. D. Holmes has proved [2, Theorem 1] that if (A', d) is connected and locally connected, and if g: X -» X is a homeomorphism of X onto X which is also a local radial contraction, then there exists a metric S on A' which is topologically equivalent to d such that g is a contraction globally on (X, 8). In addition, it is asserted in [2, Corollary, p. 87] that completeness of (X, 8) follows from completeness of (X, d), and hence that such a mapping g will always have a fixed point via application of the Banach contraction mapping theorem if (X, d) is complete. We give an example below, however, of a complete metric space X and a mapping g such that g and X satisfy all the assumptions of Holmes' Theorem 1, yet g has no fixed point. This shows the assertion to be false. On the other hand, we show that the Rakotch result is a direct consequence of Banach's theorem, even for the more general local radial contractions. Theorem 1. Let (X, d) be a complete metric space and g: (X, d) —> (X, d) a local radial contraction. Suppose for some x0 G X the points x0 and g(x0) are joined by a path of finite length. Then g has a unique fixed point in X. Received by the editors June 17, 1977. AMS {MOS) subject classifications (1970). Primary 54H25.