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Upper and lower stability and index theory for positive mappings and applications. (English) Zbl 0755.47038

The author considers a Banach space \(E\) with a cone \(K\) such that \(K-K\) is dense in \(E\) and a continuously differentiable increasing order- compact mapping \(T: E\to E\). A fixed point \(z\) of \(T\) is said to be stable from above if for each \(\varepsilon>0\) there is a \(\delta>0\) such that \(\| T^ n x-z\|<\varepsilon\) whenever \(x\geq z\) and \(\| x- z\|\leq\delta\). A fixed point \(z\) of \(T\) is said to be isolated from above if there is a \(\delta>0\) such that \(Tx\neq z\) if \(x>z\) and \(\| x- z\|\leq\delta\). Under some technical assumptions on \(T\), the author obtains the following characterization of stability: Assume that \(a<b\), \(Ta=a\), and \(a\) is isolated from above as a fixed point in \([a,b]\) and that \(Tb\leq b\). Then \(a\) is stable from above in \([a,b]\) if and only if the fixed point index \(\text{ind}([a,b],T,a)=1\). Moreover, this condition in turn is equivalent to the existence of strict supersolutions arbitrarily close to \(a\). The author gives some applications to discrete dynamical systems and cones with empty interior, and he indicates an application to a boundary value problem in the case of an irregular boundary.
Reviewer: C.Fenske (Gießen)

MSC:

47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
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References:

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