Аннотация
We consider problem with conditions on the characteristics for second order system of hyperbolic equation with piecewise-constant argument of generalized type. By a new unknown functions this problem is transformed to a family of problems for system of differential equations with piecewise-constant argument of generalized type and unknown functions. Further, we investigate the family of differential equations with piecewise-constant argument of generalized type in the rectangular domain. This domain is divided into $$N$$ parts, the values of a solution at the interior lines of the subdomains are considered as additional functional parameters, and the family of differential equations with piecewise-constant argument of generalized type is reduced to a family of Cauchy problems for system of differential equations with parameters on the subdomains. Using the solutions to these problems, new general solutions to the family of differential equations with piecewise-constant argument of generalized type are introduced and their properties are established. Based on the general solution, boundary condition, and continuity conditions of a solution at the interior lines of the partition, the system of linear functional equations with respect to parameters is composed. Its coefficients and right hand sides are found by solving the family of Cauchy problems for linear differential equations on the subdomains. It is shown that the solvability of family boundary value problems is equivalent to the solvability of systems composed. Methods for solving problem with conditions on the characteristics for second order system of hyperbolic equation with piecewise-constant argument of generalized type are proposed, which are based on the solving of family boundary value problems.