Vol. 35, No. 2, 1970

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On nets of contractive maps in uniform spaces

Gerald L. Itzkowitz

Vol. 35 (1970), No. 2, 417–423

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.

Mathematical Subject Classification
Primary: 54.85
Received: 21 November 1969
Revised: 12 February 1970
Published: 1 November 1970
Gerald L. Itzkowitz