An analogue of the Bitsadze-Samarskii problem for a class of hyperbolic equations (Q1594389)

There is a presentation of some results regarding the Bitzadze-Samarskii problem for the hyperbolic equation \[ -(-y)^mu_{xx}+ u_{yy}+{\alpha_0 \over(-y)^{1-m/2}} u_x+{\beta_0\over y}u_y=0 \] in a domain \(D\) of the half plane \(\text{Im} z<0\) bounded by the characteristics of the above equation and by \(y=0\). In the plane of the parameters \(\alpha_0,\beta_0\) one defines a square, bounded by straight lines with coefficients depending upon \(m\). For different locations of a point \(P(\alpha_0,\beta_0)\) within this square one states and examines various boundary value problems for the above equation. The stated problems can be reduced to a Volterra integral equation of second kind or to the solution of an integro-functional equation for which a sufficient condition of solvability is indicated.