Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)2899610.22363/2413-3639-2021-67-3-483-506ArticlesСтатьиResearch ArticleDelay Differential Equations with Differentiable Solution Operators on Open Domains in C((-∞, 0], Rn) and Processes for Volterra Integro-Differential EquationsДифференциальные уравнения с запаздыванием с дифференцируемыми операторами решений на открытых областях в C((-∞, 0], Rn) и процессы для интегродифференциальных уравнений ВольтерраWaltherHans-OttoВальтерХ.-О.Hans-Otto.Walther@math.uni-giessen.deUniversitat Gießen15122021673Dedicated to 70th anniversary of the President of the RUDN University V. M. FilippovПосвящается 70-летию президента РУДН В. М. Филиппова48350623102021Copyright ©; 2021, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2021, Современная математика. Фундаментальные направления2021Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.enhttps://journals.rudn.ru/CMFD/article/view/28996

For autonomous delay differential equations x'(t)=f(xt){x'(t)=f(x_t)} we construct a continuous semiflow of continuously differentiable solution operators x0→xt{x_0 \to x_t}, t≤0{t \le 0}, on open subsets of the Fre´chet space C((-∞,0],Rn){C((-\infty, 0], R^n)}. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application, we obtain processes which incorporate all solutions of Volterra integro-differential equations x'(t)=∫0tk(t,s)h(x(s))ds{x'(t)={\int_0}^t k(t,s) h(x(s)) ds}.

Для автономных дифференциальных уравнений с запаздыванием x'(t)=f(xt){x'(t)=f(x_t)} мы строим непрерывный полупоток непрерывно дифференцируемых операторов решений x0→xt{x_0 \to x_t}, t≤0{t \le 0} на открытых множествах пространства Фреше C((-∞,0],Rn){C((-\infty, 0], R^n)}. Для неавтономных уравнений это дает непрерывный процесс дифференцируемых операторов решения. В качестве приложения мы получаем процессы, которые включают все решения интегродифференциальных уравнений Вольтерра x'(t)=∫0tk(t,s)h(x(s))ds{x'(t)={\int_0}^t k(t,s) h(x(s)) ds}.

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