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Approximation theorems and fixed point theorems in cones. (English) Zbl 0653.47033

A famous generalization of Schauder’s fixed point theorem due to K. Fan states that, if K is a nonempty compact convex set in a normed linear space X and f: \(K\to X\) is continuous, then \(\| x_ 0-f(x_ 0)\| =dist(f(x_ 0),K)\) for some \(x_ 0\in K\). In the present paper, parallel results are obtained for continuous condensing maps on a closed ball or annulus in cones. Some recent results of G. Li [Proc. Am. Math. Soc. 97, 277-280 (1986; Zbl 0592.47046)] are also generalized to continuous condensing maps.
Reviewer: J.Appell

MSC:

47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
41A50 Best approximation, Chebyshev systems
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0592.47046
Full Text: DOI