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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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