Small time-periodic solutions to a nonlinear equation of a vibrating string. (English) Zbl 0653.35063

Making use of the accelerated convergence method the author proves existence of at least one \(\omega\)-periodic \((u_ 1,u_ 2)\) to the system \[ u_{1tt}+\sigma_ 1(u_{1x},u_{2x},u_{1xx},u_{1t})=f_ 1,\quad -u_{2xx}+\sigma_ 2(u_{1x},u_{1xx})=f_ 2 \] with Dirichlet boundary conditions where \(f_ j\) are \(\omega\)-periodic in t, small and sufficiently smooth. In a special case, the system above reduces to an integrodifferential equation describing vibrations of a damped extensive string.
Reviewer: O.Vejvoda


35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
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