## Periodic solutions of a piecewise linear beam equation damping and nonconstant load.(English)Zbl 1015.35061

By using the Lyapunov-Schmidt reduction method, existence and multiplicity of periodic solutions is shown for a piecewise linear beam equation of the form $$u_{tt}+u_{xxxx}+\delta u_t+bu^+-au^-=h(x,t)$$ on the domain $$[-\pi/2,\pi/2]\times \mathbb R$$ with the boundary value conditions $$u(\pm \pi/2,t)=u_{xx}(\pm \pi/2,t)=0$$. Moreover $$u$$ is $$\pi$$-periodic in $$t$$ and even in $$x$$. The source term $$h(x,t)$$ has the form $$h(x,t)=\alpha \cos x+\beta \cos 2t\cos x+\gamma \sin 2t\cos x$$. It is also established the relationship between the constants $$a,b,\delta$$ and the function $$h(x,t)$$, so that there exists a sign-changing periodic solution of the beam equation while $$h(x,t)$$ is of single-sign.

### MSC:

 35L35 Initial-boundary value problems for higher-order hyperbolic equations 35B34 Resonance in context of PDEs 35B10 Periodic solutions to PDEs 35L75 Higher-order nonlinear hyperbolic equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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