ON THE SOLUTION OF A BOUNDARY VALUE PROBLEM FOR AN IMPULSIVE HYPERBOLIC EQUATION WITH A PIECEWISE-CONSTANT ARGUMENT

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.

1. Wiener, J. Generalized Solutions of Functional Differential Equations (Singapore: World Scientific, 1993).

2. Samoilenko, A.M., and Perestyuk, N.A. Impulsive Differential Equations (Singapore: World Scientific, 1995).

3. Imanchiyev, A.E., Assanova, A.T., and Molybaikyzy, A. Properties of a nonlocal problem for hyperbolic equations with impulse discrete memory. Lobachevskii Journal of Mathematics, 44 (10), 4299–4309 (2023). https://doi.org/10.1134/S1995080223100177

4. Dzhumabayev, D.S. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. U.S.S.R. Computational Mathematics and Mathematical Physics, 29 (1), 34–46 (1989).

5. Assanova, A.T. Hyperbolic equation with piecewise-constant argument of generalized type and solving boundary value problems for it. Lobachevskii Journal of Mathematics, 42, 3584–3593 (2021).

6. Assanova, A.T., and Molybaikyzy, A. Solution to the periodic problem for the impulsive hyperbolic equation with discrete memory. Kazakh Mathematical Journal, 25 (1), 16–27 (2025).

7. Akhmet, M.U. Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlinear Analysis, 66 (3), 367–383 (2007). https://doi.org/10.1016/j.na.2005.11.032

8. Akhmet, M.U. Almost periodic solutions of differential equations with piecewise constant argument of generalized type. Journal of Mathematical Analysis and Applications, 336 (4), 646–663 (2007). https://doi.org/10.1016/j.jmaa.2007.03.010

9. Nieto, J.J., and Rodriguez-Lopez, R. Green’s function for second order periodic BVPs with piecewise constant argument. Journal of Mathematical Analysis and Applications, 304 (1), 33–57 (2005). https://doi.org/10.1016/j.jmaa.2004.09.023

10. Aviltay, N., Akhmet, T., and Zhamanshin, A. Asymptotic solutions of differential equations with singular impulses. Carpathian Journal of Mathematics, 40 (3), 581–598 (2024).