On exponential dichotomy for abstract differential equations with delayed argument

• Дата: 2023
• Автори: Chaikovs'kyi, A., Lagoda, O., Чайковський, Андрій, Лагода, Оксана
• Формат: Стаття
• Мова: Українська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2023
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/7576
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.9 We consider linear differential equations of the first order with delayed arguments in a Banach space. We establish conditions for the operator coefficients necessary for the existence of  exponential dichotomy on the real axis. It is proved that the analyzed differential equation is equivalent to a difference equation in a certain space. It is shown that, under the conditions of existence and uniqueness of a solution  bounded on the entire real axis, the condition of exponential dichotomy is also satisfied for any bounded known function. The explicit formula for projectors, which form the dichotomy, is found for the case of a single delay.