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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
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