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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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