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Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems. (English) Zbl 0695.47044

Summary: This deals with the solvability of the nonlinear operator equations in normed spaces \({\mathcal L}x=EGx+f\), where \({\mathcal L}\) is a linear map with possible nontrivial kernel. Applications are given to the existence of periodic solutions for the third-order scalar differential equation \(x'''+ax''+bx'+cx+g(t,x)=p(t)\) under various conditions on the interaction of g(t,x)/x with spectral configurations of a, b, and c.

MSC:

47E05 General theory of ordinary differential operators
47F05 General theory of partial differential operators
34C25 Periodic solutions to ordinary differential equations
47A53 (Semi-) Fredholm operators; index theories
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