Báo cáo toán học: Applying Fixed Point Theory to the Initial Value Problem for the Functional Differential Equations with Finite Delay
Số trang: 18
Loại file: pdf
Dung lượng: 167.01 KB
Lượt xem: 4
Lượt tải: 0
Xem trước 2 trang đầu tiên của tài liệu này:
Thông tin tài liệu:
Dịch văn bản hoặc trang webNhập văn bản hoặc địa chỉ trang web hoặc dịch tài liệu.HủyBản dịch từ Tiếng Anh sang Tiếng Việtgiấy của ông là dành cho các nghiên cứu về sự tồn tại và tính độc đáo củacác giải pháp mạnh mẽ cho các phương trình vi phân chức năng với sự chậm trễ hữu hạn. Chúng tôi cũng nghiên cứu sự ổn định tiệm cận của các giải pháp và sự tồn tại của các giải pháp định kỳ. Hơn nữa, theo một số giả định phù hợp về chức năng nhất định, chúng tôi chứng...
Nội dung trích xuất từ tài liệu:
Báo cáo toán học: "Applying Fixed Point Theory to the Initial Value Problem for the Functional Differential Equations with Finite Delay " Vietnam Journal of Mathematics 35:1 (2007) 43–60 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 9$ 67 Applying Fixed Point Theory to the Initial Value Problem for the Functional Differential Equations with Finite Delay Le Thi Phuong Ngoc Educational College of Nha Trang, 1 Nguyen Chanh Str., Nha Trang City, Vietnam Received March 1, 2006 Revised November 16, 2006Abstract. This paper is devoted to the study of the existence and uniqueness ofstrong solutions for the functional differential equations with finite delay. We alsostudy the asymptotic stability of solutions and the existence of periodic solutions.Furthermore, under some suitable assumptions on the given functions, we prove thatthe solution set of the problem is nonempty, compact and connected. Our approach isbased on the fixed point theory and the topological degree theory of compact vectorfields.2000 Mathematics Subject Classification: 34G20.Keywords: The fixed point theory, the Schauder’s fixed point theorem, contractionmapping.1. IntroductionIn this paper, we consider the initial value problem for the following functionaldifferential equations with finite delay x (t) + A(t)x(t) = g(t, x(t), xt), t ≥ 0, (1.1) x0 = ϕ, (1.2)in which A(t) = diag[a1(t), a2(t), ..., an(t)], ai ∈ BC [0, ∞) for all i = 1, ..., n,44 Le Thi Phuong Ngocwhere BC [0, ∞) denotes the Banach space of bounded continuous fuctions x :[0, ∞) → Rn . The equation of the form (1.1) with finite or infinite delay has been studiedby many authors using various techniques. There are many important resultsabout the existence and uniqueness of solutions, the existence periodic solu-tions and the asymptotic behavior of the solutions; for example, we refer to the[1, 3, 5, 7, 9, 10, 11] and references therein. In [1, 3], the authors used the notion of fundamental solutions to study thestability of the semi-linear retarded equation x (t) = A(t)x(t) + B (t)xt + F (t, xt) , t ≥ s ≥ 0, xs = ϕ ∈ C ([−r, 0], E ) or xs = ϕ ∈ C0((−∞, 0], E ),where (A(t), D(A(t)))t≥0 generates the strongly continuous evolution family(V (t, s))t≥s≥0 on a Banach space E , and (B (t))t≥0 is a family of bounded linearoperators from C ([−r, 0], E ) or C0((−∞, 0], E ) into E. In [9, 10], the authors studied the relationship between the bounded solutionsand the periodic solutions of finite (or infinite) delay evolution equation in ageneral Banach space as follows u (t) + A(t)u(t) = f (t, u(t), ut) , t > 0, u(s) = ϕ(s), s ∈ [−r, 0] (or s ≤ 0),where A(t) is a unbounded operator, f is a continuous function and A(t), f (t, x, y)are T -periodic in t such that there exists a unique evolution system U (t, s),0 ≤ s ≤ t ≤ T, for the equation as above. In [5], by using a Massera type criterion, the author proved the existence ofa periodic solution for the partial neutral functional differential equation d (x(t) + G(t, xt)) = Ax(t) + F (t, xt), t > 0, dt x0 = ϕ ∈ D,where A is the infinitesimal generator of a compact analytic semigroup of linearoperators, (T (t))t≥0, on a Banach space E. The history xt, xt(θ) = x(t + θ),belongs to an appropriate phase D and G, F : R × D → E are continuousfunctions. In [7], the existence of positive periodic solutions of the system of functionaldifferential equations x (t) = A(t)x(t) + f (t, xt ), t ≥ s ≥ 0,was established, in which A(t) = diag[a ...
Nội dung trích xuất từ tài liệu:
Báo cáo toán học: "Applying Fixed Point Theory to the Initial Value Problem for the Functional Differential Equations with Finite Delay " Vietnam Journal of Mathematics 35:1 (2007) 43–60 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 9$ 67 Applying Fixed Point Theory to the Initial Value Problem for the Functional Differential Equations with Finite Delay Le Thi Phuong Ngoc Educational College of Nha Trang, 1 Nguyen Chanh Str., Nha Trang City, Vietnam Received March 1, 2006 Revised November 16, 2006Abstract. This paper is devoted to the study of the existence and uniqueness ofstrong solutions for the functional differential equations with finite delay. We alsostudy the asymptotic stability of solutions and the existence of periodic solutions.Furthermore, under some suitable assumptions on the given functions, we prove thatthe solution set of the problem is nonempty, compact and connected. Our approach isbased on the fixed point theory and the topological degree theory of compact vectorfields.2000 Mathematics Subject Classification: 34G20.Keywords: The fixed point theory, the Schauder’s fixed point theorem, contractionmapping.1. IntroductionIn this paper, we consider the initial value problem for the following functionaldifferential equations with finite delay x (t) + A(t)x(t) = g(t, x(t), xt), t ≥ 0, (1.1) x0 = ϕ, (1.2)in which A(t) = diag[a1(t), a2(t), ..., an(t)], ai ∈ BC [0, ∞) for all i = 1, ..., n,44 Le Thi Phuong Ngocwhere BC [0, ∞) denotes the Banach space of bounded continuous fuctions x :[0, ∞) → Rn . The equation of the form (1.1) with finite or infinite delay has been studiedby many authors using various techniques. There are many important resultsabout the existence and uniqueness of solutions, the existence periodic solu-tions and the asymptotic behavior of the solutions; for example, we refer to the[1, 3, 5, 7, 9, 10, 11] and references therein. In [1, 3], the authors used the notion of fundamental solutions to study thestability of the semi-linear retarded equation x (t) = A(t)x(t) + B (t)xt + F (t, xt) , t ≥ s ≥ 0, xs = ϕ ∈ C ([−r, 0], E ) or xs = ϕ ∈ C0((−∞, 0], E ),where (A(t), D(A(t)))t≥0 generates the strongly continuous evolution family(V (t, s))t≥s≥0 on a Banach space E , and (B (t))t≥0 is a family of bounded linearoperators from C ([−r, 0], E ) or C0((−∞, 0], E ) into E. In [9, 10], the authors studied the relationship between the bounded solutionsand the periodic solutions of finite (or infinite) delay evolution equation in ageneral Banach space as follows u (t) + A(t)u(t) = f (t, u(t), ut) , t > 0, u(s) = ϕ(s), s ∈ [−r, 0] (or s ≤ 0),where A(t) is a unbounded operator, f is a continuous function and A(t), f (t, x, y)are T -periodic in t such that there exists a unique evolution system U (t, s),0 ≤ s ≤ t ≤ T, for the equation as above. In [5], by using a Massera type criterion, the author proved the existence ofa periodic solution for the partial neutral functional differential equation d (x(t) + G(t, xt)) = Ax(t) + F (t, xt), t > 0, dt x0 = ϕ ∈ D,where A is the infinitesimal generator of a compact analytic semigroup of linearoperators, (T (t))t≥0, on a Banach space E. The history xt, xt(θ) = x(t + θ),belongs to an appropriate phase D and G, F : R × D → E are continuousfunctions. In [7], the existence of positive periodic solutions of the system of functionaldifferential equations x (t) = A(t)x(t) + f (t, xt ), t ≥ s ≥ 0,was established, in which A(t) = diag[a ...
Tìm kiếm theo từ khóa liên quan:
báo cáo của tạp chí Vietnam Journal of Mathematics tài liệu báo cáo nghiên cứu khoa học cách trình bày báo cáo kiến thức toán học báo cáo toán họcGợi ý tài liệu liên quan:
-
HƯỚNG DẪN THỰC TẬP VÀ VIẾT BÁO CÁO THỰC TẬP TỐT NGHIỆP
18 trang 355 0 0 -
Hướng dẫn thực tập tốt nghiệp dành cho sinh viên đại học Ngành quản trị kinh doanh
20 trang 231 0 0 -
Đồ án: Nhà máy thủy điện Vĩnh Sơn - Bình Định
54 trang 220 0 0 -
23 trang 205 0 0
-
40 trang 200 0 0
-
Báo cáo môn học vi xử lý: Khai thác phần mềm Proteus trong mô phỏng điều khiển
33 trang 180 0 0 -
BÁO CÁO IPM: MÔ HÌNH '1 PHẢI 5 GIẢM' - HIỆN TRẠNG VÀ KHUYNH HƯỚNG PHÁT TRIỂN
33 trang 175 0 0 -
8 trang 172 0 0
-
Tiểu luận Nội dung và bản ý nghĩa di chúc của Chủ tịch Hồ Chí Minh
22 trang 166 0 0 -
8 trang 158 0 0