## Existence of a countable set of periodic solutions of the problem of forced oscillations for a weakly nonlinear wave equation.(English. Russian original)Zbl 0683.35054

Math. USSR, Sb. 64, No. 2, 543-556 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 4(8), 546-560 (1988).
Consider the boundary value problems of the semilinear wave equation $u_{tt}-u_{xx}=| u|^{p-2}+h(t,x),\quad 0<x<\pi$
$u(t,0)=u(t,\pi)=0,\quad u(t+2\pi,x)=u(t,x)$ and $u_{tt}-u_{xx}=- | u|^{p-2}-h(t,x)$ where h(t,x) is a $$2\pi$$-periodic forced oscillation term. The solvability of the problem in the space of $$2\pi$$- periodic functions is proved.
Reviewer: J.Tian

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35B10 Periodic solutions to PDEs
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