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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:XX 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:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H07Monotone and positive operators on ordered topological linear spaces
06B30Topological lattices, order topologies