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Hard implicit function theorem and small periodic solutions to partial differential equations. (English) Zbl 0567.35007
Existence of small solutions to an abstract equation \(F(u)=h\) in a system of Banach spaces is studied. The abstract problem represents in applications in PDE’s a situation when some a priori estimates with a ”lost of derivative” hold and therefore the classical Newton’s method does not converge. The abstract result is proved using the Nash’s iteration procedure. It is done in an elementary framework and under minimal smoothness assumptions about the data. The abstract theorem is applied to the proof of existence of classical priodic solutions of equations of the type \(\Phi (u,u_ t,u_ x,u_{tt},u_{xx},u_{tx})=h(t,x)\) on (0,1) with trivial Dirichlet boundary conditions.
Reviewer: M.Kučera

35B10 Periodic solutions to PDEs
35G30 Boundary value problems for nonlinear higher-order PDEs
47J25 Iterative procedures involving nonlinear operators
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