To the problem for the impulsive system of differential equations with generalized piecewise-constant argument
DOI:
https://doi.org/10.70474/88j62974Keywords:
impulsive system of differential equations, boundary value problem, generalized piecewise-constant argument, parameter, initial value problem, solvability conditionsAbstract
This paper investigates a boundary value problem for a system of impulsive differential equations involving a generalized piecewise-constant argument. To address the problem, we employ the Dzhumabaev parametrization method, reducing the boundary value problem to an equivalent initial value problem for an impulsive system with a parameter. The solution is constructed interval by interval, using impulse conditions at the interval endpoints to iteratively extend the solution. Fundamental solutions of the differential system, depending on the parameter, are utilized to obtain explicit solutions on each interval. An equation involving the parameter is derived from the boundary conditions and the constructed solutions. Based on this framework, we propose an algorithm for solving the initial value problem, detailing the conditions under which it guarantees a unique solution. The equivalence between the original boundary value problem and the parameter-dependent initial value problem is established, and conditions for the unique solvability of the boundary value problem are formulated.
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