On the asymptotic behavior of solutions of second order linear differential equations (original) (raw)

The asymptotic behaviour of solutions to linear systems of ordinary differential equations

Pacific Journal of Mathematics, 1970

This paper is concerned with the system of differential equations (1) x' = A(t)x, te[0,ώ) where A(t) is an n x n matrix of locally integrable complexvalued functions on [0, ώ) and x(t) is an ?z-dimensional column vector. The class of matrices A(t) such that (1) has a nontrivial solution x o (t) satisfying lim*-™ | x o (t) | = 0 is denoted by Ω o ; the class of matrices Ait) such that (1) has a solution Xoo(t) satisfying lim*ω I Xco(t) I-+ oo is denoted by Ωco. If P is a projection then ΩP 0 denotes the class of matrices A(t) such that (1) has a nontrivial solution x o (t) satisfying lim^ω I Pxo{t) I = 0. Sufficient conditions are given for A(t) G Ω o , A(t) e Ω^ and A(t) e ΩP 0 ; the result, obtained include as special cases theorems of Coppel, Hart man, and Milloux.

On the Complex Oscillation of Some Linear Differential Equations

Journal of Mathematical Analysis and Applications, 1997

We treat the linear differential equation (∗)f(k)+A(z)f=0, wherek≧2 is an integer andA(z) is a transcendental entire function of order σ(A). It is shown that any non-trivial solution of the equation (∗) satisfies λ(f)≧σ(A), where λ(f) is the exponent of convergence of the zero-sequence off, under the conditionKN̄(r,1/A)≦T(r,A),r∉Efor aK>2kand an exceptional setEof finite linear measure. The second order equationf″+(eP1(z)+eP2(z)+Q(z))f=0, whereP1(z),P2(z) are