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Zbl 1126.47045
Nieto, Juan J.; Pouso, Rodrigo L.; Rodr{\'\i}guez-López, Rosana
Fixed point theorems in ordered abstract spaces. (English)
[J] Proc. Am. Math. Soc. 135, No. 8, 2505-2517 (2007). ISSN 0002-9939; ISSN 1088-6826/e

The authors continue their discussion of the extension of the Banach fixed point theorem to partially ordered sets in [{\it J.\,J.\thinspace Nieto} and {\it R.\,Rodr{\'\i}guez--López}, Order 22, No.\,3, 223--239 (2005; Zbl 1095.47013)]. In that paper, they extended the Banach fixed point theorem to ordered metric spaces and showed that if $X$ is a completely ordered metric space and $f: X\to X$ is a monotone continuous mapping satisfying the conditions that $f$ is order-contractive and the fixed pont equation $x=f(x)$ has a lower solution or an upper solution, then $f$ has a fixed point. In the present paper, this fixed point theorem is extended to ordered $L$-spaces. An ordered $L$-space is a nonempty set with a limit operation of sequences and a partial order which is compatible with the limit operation.
[Yongxiang Li (Lanzhou)]

MSC 2000:: *47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H07 Positive operators on ordered topological linear spaces
06B30 Topological lattices

Keywords: fixed point; poset; L-spaces

Citations: Zbl 1095.47013

Cited in: Zbl 1227.54053

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