We prove the unique solvability of a mixed boundary value problem for the p -Laplace operator by means of variational methods. Using the obtained results, we construct an iterative procedure for solving the Cauchy problem for the p -Laplace operator.

We study a special class of lower trigonometric matrix value boundary problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial hyperbola is given in bilinear form and it shown that only one. Secondly, problem triangular presented transformed into four related problems. Finally, Liouville theorem Painlevé polynomials are used to give solutions.