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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 [J. J. Nieto and 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 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 06B30 Topological lattices
##### Keywords:
fixed point; poset; L-spaces
Zbl 1095.47013
Full Text:
##### References:
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