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Covering dimension and nonlinear equations. (English) Zbl 0940.47049
The author proves the following result:
Theorem. Let \(X\), \(Y\) be Banach spaces, \(\Phi\) be a continuous, linear, surjective operator from \(X\) into \(Y\) and \(\Psi\) be a completely continuous operator with bounded range from \(X\) into \(Y\). Then \[ \dim(\{x\in X: \Phi(x)= \Psi(x)\})\geq \dim(\Phi^{-1}(0)). \] This theorem improves the corresponding result of the author [C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 1, 65-70 (1997; Zbl 0884.47043)].
MSC:
47J05 Equations involving nonlinear operators (general)
47H10 Fixed-point theorems
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