On bounded solutions of a class of nonlinear integral equations on the plane and the Urysohn equation in a quadrant of the plane

• Дата: 2021
• Автори: Khachatryan , Kh. A., Petrosyan , H. S., Хачатрян, Х. А., Петросян, А. С., Петросян , А. С.
• Формат: Стаття
• Мова: Українська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2021
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/6541
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.968.4 We study a class of two-dimensional integral equations in the plane with monotonic nonlinearity. These equations have a lot of applications in many fields of natural science. For example, such equations arise in the dynamic theory of $p$-adic open-closed strings, in the mathematical theory of spatio-temporal spread of epidemics, in the kinetic theory of gases (the Boltzmann kinetic equation in the framework of various models), in the theory of radiative transfer. We prove a constructive existence theorem for bounded nontrivial solutions and for solutions with alternating sign. It is shown that obtained results have applications in the theory of $p$-adic open-closed strings and in mathematical biology. The methods used in the proof of the theorem make it possible to investigate a class of two-dimensional integral equations of the Urysohn type in a quadrant of the plane. At the end of the paper, we provide specific examples of applications of these equations to illustrate the obtained results.