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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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