Problem with periodicity conditions for a degenerating equation of mixed type (Q981744)

The equation \[ K(t)u_{xx}+ u_{tt}- b^2 K(t)u= 0, \] where \(K(t)=(\text{sgn\,}t)|t|^m\), \(m> 0\), and \(b> 0\), is considered in the rectangular domain \[ D= \{(x,t)\mid0< x< 1,\,-\alpha< t<\beta\}, \] \(\alpha> 0\), and \(\beta> 0\) are given real numbers. For this equation the boundary value problem \[ u(0,t)= u(1,t),\;u_x(0,t)= u_x(1,t),\;-\alpha\leq t\leq \beta,\;u(x,\beta)= \varphi(x),\;u(x,-\alpha)= \psi(x),\;0\leq x\leq 1 \] is defined and the necessary and sufficient conditions for the unique solvability of the problem are obtained using the spectral method.