Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.

*(English)*Zbl 1140.47045Summary: We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case [Order 22, No. 3, 223–239 (2005; Zbl 1095.47013)], we consider in this paper nonincreasing mappings as well as non-monotone mappings. We also present some applications to first-order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.

##### MSC:

47H10 | Fixed-point theorems |

47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |

47N20 | Applications of operator theory to differential and integral equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

##### Keywords:

fixed point; partially ordered set; first-order differential equation; lower and upper solutions
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\textit{J. J. Nieto} and \textit{R. Rodríguez-López}, Acta Math. Sin., Engl. Ser. 23, No. 12, 2205--2212 (2007; Zbl 1140.47045)

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##### References:

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[2] | Nieto, J. J., Rodríguez-López, R.: Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22, 223–239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 |

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