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Fixed point theorems in ordered abstract spaces. (English) Zbl 1126.47045
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. Rodríguez--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.

##### MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H07 Monotone and positive operators on ordered topological linear spaces 06B30 Topological lattices, order topologies
##### Keywords:
fixed point; poset; L-spaces
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