Non-oscillation theorems

Oscillation and Asymptotic Behavior of Systems of Ordinary Linear Differential Equations

“…Taking V0 = X0, the resulting reduced system (2.7) is also Y= V^W (2.5a) (2.5b) (2.6a) (2.6b) (2-7) (2. 8) Hamiltonian. This special case of the following more general transformation shows that (2.1) and its adjoint can always be transformed to reduced systems such that boundary value problems are transformed to equivalent boundary value problems.…”

“…A detailed definition of the systems involved is given in §2, and for background and motivation the reader may refer, for example, to [7], [11], or [15]. In §3 extensions are obtained of certain theorems of Hille [8], Shreve [14], and Hartman [7] on the asymptotic behavior of solutions, and partial converses of these are obtained in §4 for the selfadjoint case. Certain nonoscillation results of Reid ([9], [10]) are extended to an arbitrary non-self adjoint system in §2.…”

General solutions of second-order linear difference equations of Euler type

“…Moreover, notice that we have adopted the notation k i=j a i = 0 and k i=j a i = 1 if j > k. Then it is easy to check that equation (1.2) with m = 1 becomes equation (1.1). Furthermore, c Wydawnictwa AGH, Krakow 2017 it is known that equation (1.2) with m = 2 is called the Riemann-Weber version of the Euler type differential equation (see [8]). …”

“…Thus, we can get general solutions of equation (1.2) (for example, see [6,8,13,14,16]). As for linear difference equations which are related to equation (1.1), we can consider various types.…”

An asymptotic expansion for a nonhomogeneous linear system

“…This does not happen for certain other sets of hypotheses; see e.g. Hille [4] for a linear second order scalar equation or [l], [5] for nonlinear extensions where additional (stronger) hypotheses are required to find higher order asymptotic expansions.…”