Is there a version of the implicit function theorem in some space of functions from $[0,1]$ to a Hilbert space $H$ that contains as a special case the unique solvability of the initial-value problem $\dot u(t) = iAu(t)$, $u(0)=u_0\in H$ when $A$ is a selfadjoint operator on a dense subspace of $H$?
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$\begingroup$ Are you asking out of sheer curiosity, or do you have a specific problem in mind? Isn't Stone's theorem enough? $\endgroup$– Pedro Lauridsen RibeiroCommented Jun 29, 2016 at 15:35
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1$\begingroup$ @PedroLauridsenRibeiro: I want to look at some nonlinear evolution equations (modeling some application where it is not yet clear how the equations will actually look like) whose linear constant coefficient approximation is of the kind described. So I wonder whether the techniques used in the linear case have already been extended in some way to a nonlinear setting. $\endgroup$– Arnold NeumaierCommented Jun 29, 2016 at 16:14
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$\begingroup$ Hmm... This is usually done using semigroup methods, which were introduced into the study of nonlinear evolution equations mainly by Kato. Good references for this are Kato's papers and the books of Tanabe and Pazy. $\endgroup$– Pedro Lauridsen RibeiroCommented Jun 29, 2016 at 16:18
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