R. B. Fraser and S. Nadler have
recently proved the following theorem: If X is a locally compact metric space, if
fn→ f0 pointwise, where each fn,n = 0,1,2,⋯ is a contractive map with fixed point
an, then fn→ f0 uniformly on compacta, and an→ a0. Their method of
proof actually showed more. In fact it implied that if a0 was a fixed point of
f0, and if U is a compact neighborhood of a0, then there exists a natural
number N(U) such that if n ≧ N(U) then fn had a fixed point an∈ U, and
an→ a0. In 196S, W. J. Kammerer and R. H. Kasriel proved a theorem giving
conditions for existence and uniqueness of fixed points of a general type
contractive map on a uniform space. Edelstein in 1965, was able to considerably
strengthen their results and achieved a significant extension of the Banach
fixed point theorem. In this paper we show that the theorem of Fraser and
Nadler may be extended with minor alteration to include locally compact
uniform spaces. It was evident in the context of uniform spaces that the
covergent sequences of their theorem should be replaced by covergent nets. Our
method of proof is similar to their proof and used Edelstein’s fixed point
theorem.
