Almost diagonal systems in asymptotic integration

Source abstract

Consider the ordinary linear matrix differential system ψ( x ) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C 1 ([a,∞)) for some integer l . The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [ 2 ], Levinson [ 24 ], Hartman and Wintner [ 20 ], Devinatz [ 9 ], Fedoryuk [ 11 ], Harris and Lutz [ 16,17,18 ] and Cassell [ 30 ]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [ 36 ], Eastham and Grundniewicz [ 10 ] and [ 8,9 ]. Harris and Lutz [ 16,17,18 ] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “( 1 + Q )” linear transformations.