Picard problem on the plane for a quasilinear hyperbolic equation of the second order

Classical solutions of problems for a quasilinear hyperbolic equation of the second order in the case of two independent variables with given conditions for the desired function in combination both on characteristic lines and on non-characteristic lines are obtained in the paper. The problems are reduced to a system of equations with a completely continuous operator. Solutions are constructed using the method of successive approximations. In addition, for each problem considered, the uniqueness of the resulting classical solution is shown. Necessary and sufficient matching conditions of given functions are proved in the case of each of the problems considered in the paper, under which classical solutions exist in the presence of a certain smoothness of the given functions.