The investigation of free vibrations for an asymmetric beam equation using category theory (Q2488775)

This paper concerns multiple periodic solutions to the following problem: \[ u_{tt}+u_{xxxx}+bu^+ =f(x,t,u),\quad (x,t) \in \left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)\times \mathbb R, \] \[ u(\pm\tfrac{\pi}{2},t)=u_{xx}(\pm\tfrac{\pi}{2},t)=0, \eqno(1) \] \(u\) is \(\pi\)-periodic in \(t\) and even in \(x,t;\, u^+=\max(u,0)\), \[ f(x,t,s)=| s|^{p-2} s,\quad s\geq 0, \] \[ f(x,t,s)=| s|^{q-2} s,\quad s<0, \] where \(p,q>2\), \(p\neq q\). First the associated eigenvalue problem is considered: \[ u_{tt}+u_{xxxx}=\lambda,\quad (x,t)\in (-\tfrac{\pi}{2},\tfrac{\pi}{2})\times \mathbb R, \] \[ u(\pm\tfrac{\pi}{2},t)=u_{xx}(\pm\tfrac{\pi}{2},t)=0, \eqno(3) \] \[ u(x,t)=u(-x,t)=u(x,-t)=u(x,t+\pi). \] The authors study the correspondence between negative eigenvalues of (3), \(b\) and a number of nontrivial solutions of (1).