Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence.

*(English)*Zbl 1127.47049Let \(E\) be a complete metric space with a partial order, \(T: C([a,b],E)\to E\) be a monotonically nondecreasing operator. A fixed point \(\phi\in C([a,b],E)\) of \(T\) means that there exists some \(c\in [a,\,b]\) such that \(T\phi= \phi(c)\). The present paper discusses the existence and uniqueness of the fixed points of \(T\) under the conditions that \(T\) is order-contractive and the fixed point equation \(\phi(c)=T\phi\) has a lower solution. The obtained fixed point theorem is applied to a periodic boundary value problem of a delay ordinary differential equation, and a unique existence result for periodic solutions is obtained.

Reviewer: Yongxiang Li (Lanzhou)

##### 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 |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |

##### Keywords:

fixed point; partially ordered set; PPF dependence; lower solution; periodic boundary value problem with delay
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\textit{Z. Drici} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 2, 641--647 (2007; Zbl 1127.47049)

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##### References:

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