Well-posedness criteria for one family of boundary value problems

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained

Bounder solution on a strip to a system of nonlinear hyperbolic equations with mixed derivatives

The system of nonlinear hyperbolic equations with mixed derivatives is considered on the strip. Time variable of the unknown function changes on the whole axis, and the spatial variable belongs to a finite interval. A function, the partial derivative with respect to the spatial variable, is denoted as unknown function, and problem of finding a bounded on the strip solution to the origin system is reduced to the problem of finding a bounded on the strip solution to a system of integro - partial differential equations. The whole axes is divided into parts, and additional functional parameters are introduced as the values of unknown function on the initial lines of sub - domains. For the fixed values of functional parameters, the new unknown functions in the sub - domains are defined as the solutions to the Cauchy problems for integro - partial differential equations of the first order. Using the continuity conditions of the solution on the partition lines, the two - sided infinite system of nonlinear Volterra integral equations of the second kind with respect to introduced functional parameters is obtained. Algorithms for finding solutions of problem with functional parameters are proposed. Conditions for the convergence of algorithms, and existence of bounded on the strip solution of the system of nonlinear hyperbolic equations with mixed derivatives are obtained