Systems of Evolution Equations with GPCA and Impulsive Multipoint Constraints

A. D. Abildayeva, A. T. Assanova, Askarbek E. Imanchiyev, Sayakhat G. Karakenova

2026

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DOI 10.1134/s1995080226616395
Data confidence 70 medium
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Abstract

Abstract In this paper, we investigate a system of evolution equations with a generalized piecewise-constant argument subject to impulsive multipoint constraints. To analyze the impulsive multipoint problem, the Dzhumabaev parametrization method is employed. This approach reduces the original problem to an equivalent initial value problem for a system of evolution equations with a generalized piecewise-constant argument, impulses, and an additional parameter. The initial value problem is solved successively on each subinterval. First, the problem is solved on the initial interval. Then, using the impulse condition at the right endpoint of this interval, the initial value problem is formulated and solved on the next interval, and the procedure is continued in this manner. On the final interval, the initial value problem is again solved by incorporating the impulse condition at the left endpoint.The solutions on each interval are constructed with the aid of the fundamental solution of the evolution system and depend explicitly on the introduced parameter. To determine this parameter, an equation is derived using the obtained solutions together with the matrices defining the multipoint condition. On the basis of this construction, an algorithm is proposed for solving the initial value problem for systems of evolution equations with generalized piecewise-constant argument, impulses, and parameter. Sufficient conditions ensuring the applicability of this algorithm are established, which guarantee the existence and uniqueness of the solution to the corresponding initial value problem. Finally, exploiting the equivalence between the original impulsive multipoint problem and the associated initial value problem, conditions for the unique solvability of the systems of evolution equations with generalized piecewise-constant argument and impulsive multipoint constraints are obtained.