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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
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