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Newton's method and periodic solutions of nonlinear wave equations
Author(s) -
Craig Walter,
Wayne C. Eugene
Publication year - 1993
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160461102
Subject(s) - mathematics , eigenfunction , nonlinear system , mathematical analysis , dirichlet boundary condition , boundary value problem , periodic boundary conditions , dirichlet distribution , wave equation , eigenvalues and eigenvectors , physics , quantum mechanics
We prove the existence of periodic solutions of the nonlinear wave equation\documentclass{article}\pagestyle{empty}\begin{document}$$ \mathop \partial \nolimits_t^2 u = \mathop \partial \nolimits_x^2 u - g(x,u), $$\end{document}satisfying either Dirichlet or periodic boundary conditions on the interval [ O , π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g ( x, u ) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.