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Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence. (English) Zbl 1127.47049
Let \(E\) 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.

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
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References:
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