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A priori analysis of initial data for the Riccati equation and asymptotic properties of its solutions
Author(s) -
Chernyavskaya N. A.,
Schiff Jeremy,
Shuster L. A.
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp050
Subject(s) - mathematics , infinity , initial value problem , cauchy problem , a priori and a posteriori , interval (graph theory) , combinatorics , riccati equation , mathematical analysis , differential equation , philosophy , epistemology
We obtain two main results for the Cauchy problem y ′ ( x ) + 1 r ( x )y 2 = q ( x ) , y ( x ) | x = x 0= y 0 ,where x 0 , y 0 ∈ ℝ, r > 0, q ⩾ 0, 1/ r ∈L 1 loc( R ) , q ∈ L 1 loc( R )and∫ − ∞ x1 r ( t )∫ t x q ( ξ ) d ξ d t = ∫ x ∞1 r ( t )∫ x t q ( ξ ) d ξ d t = ∞ ∀ x ∈ R . (1) For given initial data x 0 , y 0 and functions r and q , we give a condition that can be used to determine whether the solution of the problem can be continued to the whole of ℝ. (2) When the solution is defined on an infinite interval, we study its asymptotic properties as the argument tends to infinity.