Abstract
This article investigates a boundary value problem for an impulsive hyperbolic equation with a piecewise constant argument. Impulsive hyperbolic equations with a piecewise constant argument arise as mathematical models of physical processes in neural networks, continuous dynamical systems, hybrid systems, and other fields. Issues related to the existence of boundary value problems and the construction of their solutions for such equations remain among the most relevant and challenging problems at present. To obtain solvability conditions for the considered problem, the Dzhumabaev parametrization method is employed, and an iterative algorithm for constructing an approximate solution is developed. For each step of the iterative process, integral formulas are derived and expressed in terms of the matrix Q(x), which describes the relationship between the functional parameters. If this matrix is invertible, the existence and uniqueness of the solution are proved both for the parametric and the original problems. The proposed method is not limited to proving theoretical solvability; it also provides a concrete constructive procedure for obtaining the solution. This is of significant importance for subsequent numerical implementations and for the analysis of solution stability. Moreover, the proposed approach can be applied to other types of problems with a piecewise constant argument, including systems with impulsive conditions, neural networks with memory effects, and nonlinear hybrid models.