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**Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.**
*(English)*
Zbl 1140.47045

Summary: 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### Citations:

Zbl 1095.47013
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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:

[1] | Ran, A. C. M., Reurings, M. C. B.: A .xed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc., 132, 1435–1443 (2004) · Zbl 1060.47056 |

[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 |

[3] | Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math., 5, 285–309 (1955) · Zbl 0064.26004 |

[4] | Cousot, P., Cousot, R.: Constructive versions of Tarski’s fixed point theorems. Pacific J. Math., 82, 43–57 (1979) · Zbl 0413.06004 |

[5] | Zeidler, E.: Nonlinear functional analysis and its applications, Vol I: Fixed-Point Theorems, Springer-Verlag, New York, 1986 · Zbl 0583.47050 |

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[7] | Heikkilä, S., Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations, Marcel Dekker, Inc., New York, 1994 · Zbl 0804.34001 |

[8] | Ladde, G. S., Lakshmikantham, V., Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, Pitman, Boston, 1985 · Zbl 0658.35003 |

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