kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-3-221-230kaz29-2118Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО РАЗРЕШИМОСТИ НЕЛОКАЛЬНОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ С ИНВОЛЮЦИЕЙON THE SOLVABILITY OF A NONLOCAL BOUNDARY VALUE PROBLEM FOR AN INTEGRO-DIFFERENTIAL EQUATION WITH INVOLUTIONhttps://orcid.org/0000-0002-4311-5807УсмановК. И.UsmanovK. I.

к.ф.-м.н, доцент

г. Туркестан

Cand.Phys.-Math.Sc., Associate Professor

Turkestan

kairat.usmanov@ayu.edu.kzhttps://orcid.org/0000-0002-2093-1879НазароваК. Ж.NazarovaK. Zh.

к.ф.-м.н, доцент

г. Туркестан

Cand.Phys.-Math.Sc., Associate Professor

Turkestan

kulzina.nazarova@ayu.edu.kzhttps://orcid.org/0000-0001-9680-2347ТурганбаеваЖ. Н.TurganbayevaZh. N.

PhD

г. Туркестан

PhD

Turkestan 

zhannur.turganbaeva@ayu.edu.kzМеждународный казахско-турецкий университет им. А.ЯсавиКазахстанInternational Kazakh-Turkish University named after H.A. YasawiKazakhstan202527092025223221230Copyright © Усманов К.И., Назарова К.Ж., Турганбаева Ж.Н., 20252025Усманов К.И., Назарова К.Ж., Турганбаева Ж.Н.Usmanov K.I., Nazarova K.Z., Turganbayeva Z.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/2118

В данной научной работе рассматривается нелокальная краевая задача для одного класса интегро-дифференциальных уравнений, в структуре которых присутствует инволютивное преобразование. Основное внимание сосредоточено на применении метода параметризации, разработанного и предложенного профессором Д. Джумабаевым, целью которого является исследование условий существования и единственности решения для подобного рода задач, а также определение спектра собственных значений соответствующей краевой задачи. Как известно из теории, задача Коши для уравнений, содержащих инволюции, не всегда имеет единственное решение. Для преодоления данной трудности вводятся параметры  в середине рассматриваемого отрезка и осуществляется преобразование , которое обеспечивает существование единственного решения задачи Коши. Это преобразование позволяет разделить исходную нелокальную краевую задачу на две части: во-первых, на специальную задачу Коши и, во-вторых, на систему линейных алгебраических уравнений относительно введенных параметров. После подстановки решения в краевые условия строится система уравнений, разрешимость которой зависит от невырожденности соответствующей матрицы. Кроме того, рассматривается случай неоднозначности решения, при котором исследуются собственные значения и формулируются условия разрешимости исходной задачи.

This scientific paper considers a nonlocal boundary value problem for a certain class of integro-differential equations that include an involutive transformation in their structure. The main focus is on the application of the parameterization method developed and proposed by Professor D. Dzhumabayev, the aim of which is to study the conditions for the existence and uniqueness of solutions for such problems, as well as to determine the spectrum of eigenvalues of the corresponding boundary value problem. As is known from theory, the Cauchy problem for equations involving involutions does not always have a unique solution. To overcome this difficulty,  parameters are introduced at the midpoint of the considered interval, and a transformation  is performed that ensures the existence of a unique solution to the Cauchy problem. This transformation allows the original nonlocal boundary value problem to be divided into two parts: first, a special Cauchy problem, and second, a system of linear algebraic equations with respect to the introduced parameters. After substituting the solution into the boundary conditions, a system of equations is constructed, the solvability of which depends on the non-degeneracy of the corresponding matrix. In addition, the case of non-uniqueness of the solution is considered, in which the eigenvalues are studied and the paper establishes criteria ensuring the existence of solutions to the initial boundary value problem.

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The authors declare that there are no conflicts of interest present.