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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
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##### References:
 [1] André C. M. Ran and Martine C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435 – 1443. · Zbl 1060.47056 [2] Juan J. Nieto and Rosana Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223 – 239 (2006). · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 · doi.org [3] J.J. Nieto, R. Rodríguez-López, Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations, Acta Math. Sinica (English Series), to appear. · Zbl 1140.47045 [4] Adrian Petruşel and Ioan A. Rus, Fixed point theorems in ordered \?-spaces, Proc. Amer. Math. Soc. 134 (2006), no. 2, 411 – 418. · Zbl 1086.47026 [5] M. Fréchet, Les espaces abstraits, Gauthiers-Villars, Paris, 1928. · JFM 54.0614.02 [6] Phil Diamond and Peter Kloeden, Metric spaces of fuzzy sets, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. Theory and applications. · Zbl 0873.54019 [7] Seppo Heikkilä and V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 181, Marcel Dekker, Inc., New York, 1994. · Zbl 0804.34001
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