The equation is under consideration, where
at and at , , are given complex numbers. Let be a simple connected domain, located in the plane of independent variables , , at sectionally bounded , , and at bounded with characteristics , of the above equation. The following notations are introduced:
, , , and are points of intersection of characteristics, which are outgoing from the point with characteristics , respectively, i.e.,
The following non-local problems are defined:
Problem : Find a regular solution of the above equation in satisfying the conditions
Problem : Find regular solution of the above equation in satisfying the condition the first conditionn in Problem and
Here , , are given real-valued functions, moreover , for all and , , are, generally speaking, complex-valued functions. Here is the square root of the complex number with .
In the present-paper the unique solvability of the both non-local problems and for the considered mixed parabolic-hyperbolic type equation with complex spectral parameter is proved. Further sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found.