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Structure to the solution set of a nonlinear equation on Banach manifolds. (Chinese. English summary) Zbl 1089.47048

Let \(f\) be a \(C^1\) mapping between two \(C^k\) (\(k\geq 1\)) Banach manifolds \(M\) and \(N\). The authors define a triple index \((M(x), M_c(x), M_r(x))\) on \(M\) by \(f\), from which the locally fine points of \(f\) are characterized. Moreover, when \(y\) is a generalized regular value of \(f\), the authors obtain that the connected component containing \(x_0\) of the preimage \(f^{-1}(y)\) is a \(C^1\) Banach submanifold of \(M\) of dimension \(M(x_0)\). In particular, a characterization of isolated solutions of a nonlinear equation \(f(x)=y\) is given.

MSC:

47J05 Equations involving nonlinear operators (general)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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