Global correctness theorem to the first mixed problem for the general telegraph equation with variable coefficients on a segment

The global theorem to Hadamard correctness to the first mixed problem for inhomogeneous general telegraph equation with all variable coefficients in a half-strip of the plane is proved by a novel method of auxiliary mixed problems. Without explicit continuations of the mixed problem data outside set of mixed task assignments the recurrent Riemann-type formulas of a unique and stable classical solution for the first mixed problem on a segment are derived. This half-strip of the plane is divided by the curvilinear characteristics of a telegraph equation into rectangles of the same height, and each rectangle into three triangles. The correctness criterion consists of smoothness requirements and matching conditions on the right-hand side of the equation, initial and boundary conditions of the mixed problem. The smoothness requirements are necessary and sufficient for twice continuous differentiability of the solution in these triangles. The matching conditions together with these smoothness requirements are necessary and sufficient for twice continuous differentiability of solution on the implicit characteristics in these rectangles.