Surjectivity in Fréchet Spaces
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We prove surjectivity result in Fréchet spaces of Nash–Moser type, that is, with uniform estimates over all seminorms. Our method works for functions, which are only continuous and strongly Gâteaux differentiable. We present the results in multi-valued setting exploring the relevant notions of map regularity. The key to our method is in geometrizing the tameness estimates and thus reducing the problem to a spectrum of problems on suitable Banach spaces. For solving the latter problems, we employ an abstract iteration scheme developed by the authors.
KeywordsSurjectivity Metric regularity Multi-valued map Fréchet space Nash–Moser–Ekeland theorem
Mathematics Subject Classification49J53 47H04 54H25
We express our sincere gratitude to Prof. Asen Dontchev for constantly pushing us towards this subject. We are deeply thankful to the anonymous referee for pointing some inaccuracies and incompleteness in the initial version. Special thanks are due to Prof. Radek Cibulka for reading the manuscript very carefully and pointing out that in metric space the definition of differentiability we use is more demanding than standard Gâteaux differentiability. Research is supported by the Bulgarian National Fund for Scientific Research, Contract KP-06-H22/4.