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Fourier approximations of periodic solutions of nonlinear differential equations. (Russian) Zbl 0617.34032
The equation (1) $$\dot x=f(t,x)$$, where $$f: R\times R^ n\to R$$ is continuous and satisfies the Lipschitz condition with respect to x, is investigated. The solution of equation (1) is constructed in the form $$x(t)=c+T_ m(t)+r_ m(t),$$ where $$c=const$$, $$T_ m(t)=\sum^{m}_{k=1}a_ k\cos kt+b_ k\sin kt$$ $$(m<\infty)$$ and $$r_ m(t)$$ satisfies the conditions $$\int^{2\pi}_{0}r_ m(\tau)d\tau =0,\int^{2\pi}_{0}r_ m(\tau)\cos k\tau d\tau =0,\int^{2\pi}_{0}r_ m(\tau)\sin k\tau d\tau =0$$ $$(k=1,2,...,m)$$. Conditions for the existence of periodic solutions are given.
Reviewer: V.Kozobrod

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations 42A10 Trigonometric approximation
##### Keywords:
first order differential equation; Lipschitz condition