On the Solvability of a Nonlinear Mixed Problem of Bitsadze–Samarskii Type

This paper investigates a nonlinear Bitsadze–Samarskii type boundary value problem for an elliptic-hyperbolic operator. First, we establish necessary and sufficient conditions for the unique solvability of the corresponding linear mixed-type problem using the abstract theory of correct restrictions and extensions of operators. Explicit solution formulas are obtained via Green’s function for the elliptic part and integral representations for the hyperbolic part. These results are then extended to a nonlinear mixed operator of power-law type by applying a bijective transformation that reduces the nonlinear problem to the linear case. The key condition for solvability is the continuity and invertibility of a functional coefficient in the boundary condition. Our work provides a systematic framework for handling non-standard boundary conditions in mixed-type problems and highlights the role of operator-theoretic methods in nonlinear settings.