SOLVABILITY AND CONSTRUCTIVE SOLUTION OF A THREE-POINT BOUNDARY VALUE PROBLEM FOR A HYPERBOLIC EQUATION WITH PIECEWISE CONSTANT ARGUMENT AND A DISTRIBUTED PARAMETER

Abstract

The present study addresses a three-point boundary value problem for a linear hyperbolic system involving a piecewise constant argument and an unknown distributed parameter. The problem is supplemented by initial and boundary conditions and a three-point condition involving partial derivatives of the unknown function and the distributed parameter. With the aid of suitably chosen auxiliary functions, the initial problem is transformed into an equivalent family of three point boundary value problems for functional differential systems involving functional parameters. Subsequently, Dzhumabaev’s parametrization method is employed to reduce the resulting problems to systems of linear algebraic equations for the introduced parameters. A readily verifiable sufficient condition ensuring the unique solvability of the auxiliary problem is established through the nonsingularity of the associated block matrix, and a corresponding a priori bound for the solution is derived. A constructive algorithm is proposed that involves the sequential solution of Cauchy problems and systems of linear algebraic equations. The successive approximation technique is employed to derive sufficient conditions guaranteeing the existence and uniqueness of the solution to the original problem. These findings establish a theoretical framework for the numerical analysis of boundary value problems for hyperbolic equations involving piecewise constant arguments and unknown distributed parameters.