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Các biểu diễn của hàm Green trong bài toán các nghiệm giới hạn cho phương trình vi phân-đại số
Tóm tắt
Phương trình
$$\bigl (Fu'\bigr )(t)=\bigl (Gu\bigr )(t)+f(t)$$
,
$$t\in {\mathbb {R}}$$
, trong đó F và G là các toán tử tuyến tính bị giới hạn, được xem xét. Giả định rằng vô cực là một cực của phép giải quyết của cây bút chì
$$\lambda \mapsto \lambda F-G$$
và phổ của cây bút chì không giao nhau với trục số ảo. Dưới các giả định này, với mỗi hạng tử tự do f bị giới hạn trên
$${\mathbb {R}}$$
(theo nghĩa phân phối) sẽ có một nghiệm u duy nhất và
$$u(t)=\int _{-\infty }^{\infty }{\mathcal {G}}(s)f(t-s)\,ds$$
. Hạt nhân
$${\mathcal {G}}$$
được gọi là hàm Green. Trong bài báo này, các biểu diễn của hàm Green dựa trên phép tính hàm trong đại số Banach được xây dựng.
Từ khóa
#hàm Green #toán tử tuyến tính #phương trình vi phân #phép giải quyết #đại số BanachTài liệu tham khảo
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