Approximation of odd mappings (Q1090877)

The author proves the following theorem: Let \(f: R^ n\to R^ n\) be a continuous odd mapping. For any \(\epsilon >0\), there exists an infinitely smooth odd mapping \(f_{\epsilon}: R^ n\to R^ n\), such that the point 0 is a regular value, and for each \(x\in R^ n\) we have the estimate \(\| f(x)-f_{\epsilon}(x)\| <\epsilon\). This result is then used to obtain, in an easy way, positive solutions of two problems formulated by \textit{L. Nirenberg} [Topics in Nonlinear Functional Analysis (1974; Zbl 0286.47037)]: 1) Let X be an open bounded set in \(R^ n\), symmetric about zero and not containing zero in its closure cl X, and let \(f:\partial X\to R^ n/\{0\}\) be a continuous odd mapping. Is it true that for any number \(\epsilon >0\) there exists an odd mapping \(f_{\epsilon}: cl X\to R^ n\) of the class \(C^ 1\) into X, such that \(\| f(x)- f_{\epsilon}(x)\| <\epsilon,x\in \partial X\), and for which the point 0 is a regular value ? 2) Moreover, if the mapping f is defined (is odd and continuous) on the set cl X, then can it be approximated by the above mentioned mappings \(f_{\epsilon}\), uniformly on the whole of cl X ?. These two problems are related to the theory of topological degree. The above theorem is proved by directly constructing an approximate mapping.