Solution to the periodic problem  for the impulsive hyperbolic equation  with discrete memory

In this article, we consider the periodic problem for the impulsive hyperbolic equation with discrete memory. Impulsive hyperbolic equations with discrete memory arise as a mathematical model for describing physical processes in the neural networks, discontinuous dynamical systems, hybrid systems, and etc. Questions of the existence and construction of solutions to periodic problems for impulsive hyperbolic equations with discrete memory remain important issues in the theory of discontinuous partial differential equations. To find the solvability conditions of this problem we apply Dzhumabaev’s parametrization method. The coefficient conditions for the existence and uniqueness of the periodic problem for the impulsive hyperbolic equation with discrete memory are established. We offer an algorithm for determining the approximate solution to this problem and show its convergence to the exact solution of the periodic problem for the impulsive hyperbolic equation with discrete memory.