MULTIPOINT BOUNDARY VALUE PROBLEMS FOR IMPULSIVE HYPERBOLIC EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

Abstract

This paper investigates a multipoint boundary value problem for an impulsive hyperbolic equation with a generalized piecewise constant argument. The proposed formulation simultaneously incorporates impulsive effects, a generalized piecewise constant argument, and multipoint boundary conditions, making the solvability analysis substantially more challenging than that of previously studied models. The relevance of the study is motivated by the need to develop adequate mathematical models for processes combining continuous dynamics with discrete information updating and instantaneous control actions. To investigate the problem, Dzhumabayev’s parameterization method is employed, reducing the original boundary value problem to an equivalent family of parameterized Cauchy problems together with a system of functional equations for the unknown functional parameters. Sufficient conditions for the existence and uniqueness of the solution are established under the invertibility of the constructed parameter matrix. Furthermore, a constructive iterative algorithm for the sequential construction of the solution is developed, providing a theoretical basis for its subsequent numerical implementation. The scientific novelty of the paper lies in extending the parameterization method to a new class of multipoint boundary value problems for impulsive hyperbolic equations with generalized piecewise constant arguments. The principal advantage of the proposed approach is its ability to investigate, within a unified mathematical framework, problems involving impulsive effects, generalized piecewise constant arguments, and multipoint boundary conditions simultaneously. The obtained results can be applied to mathematical modeling of automatic control systems, heat transfer processes, and technological systems involving discrete information updating and impulsive control actions.