Inverse Problem for Equations of Mixed Type with Lavrentev–Bitsadze Operator

For the equation of mixed elliptic-hyperbolic type
$$ u_{xx}+(\operatorname{sgn}y)u_{yy}-b^2u=f(x) $$
in a rectangular domain $D=\{(x,y)\mid 0<x<1,\,-\alpha<y<\beta\}$, where $\alpha$, $\beta$, and $b$ are given positive numbers, we study the problem with boundary conditions
\begin{gather*} u(0,y)=u(1,y)=0,\qquad-\alpha\le y\le \beta, \\ u(x,\beta)=\varphi(x),\quad u(x,-\alpha)=\psi(x),\quad u_y(x,-\alpha)=g(x),\qquad 0\le x\le 1. \end{gather*}
We establish a criterion for the uniqueness of the solution, which is constructed as the sum of the series in eigenfunctions of the corresponding eigenvalue problem and prove the stability of the solution.