The problem with periodicity conditions for the equations of mixed type with characteristic degeneracy

For mixed type equation
$$ Lu\equiv u_{xx}+sgny\cdot |y|^m u_{yy}=0,\: 0<m<1\nonumber $$
\noindent in a rectangular domain $\{(x,y)|\quad 0<x<1,-\alpha<y<\beta\}$, where $m,\alpha,\beta$ – defined positive numbers, theorems of existence and uniqueness of the problem solvability with boundary solutions $u(0,y)=u(1,y)$, $u_x(0,y)=u_x(1,y)$, $-\alpha\leq y\leq \beta$; $u(x,\beta)=f(x)$, $u(x,-\alpha)=g(x),$ $0\le x\le 1$ are proved by the method of spectral analysis.