Classical Solution of the First Mixed Problemfor the Inhomogeneous Liouville Equation in a Half-Strip

The existence and uniqueness of a classical solution of a mixed problem in a half-strip for the nonlinear Liouville equation is studied. The solution is presented in implicit form as a solution of an integral equation whose solvability is proved with the use of the Leray–Schauder theorem, and the corresponding a priori estimate is obtained by energy methods.