A remark on Liouville's formula on small time scales

Abstract

We present a new proof of the Liouville formula for a d-dimensional linear dynamic system x ∆ = A(t)x on a time scale T, where T is in a sense small. Our proof demonstrates that Liouville's formula on small time scales is a direct consequence of its well-known counterpart for ordinary differential equations.  2004 Elsevier Inc. All rights reserved.

The Non-Eigenvalue Form of Liouville’s Formula and α-Matrix Exponential Solutions for Combined Matrix Dynamic Equations on Time Scales

“…Please note that Liouville's formula given by the form (1) is very convenient to use due to the main reason that trA + µ det A is independent of the eigenpolynomial and eigenvalue of A. The Liouville's formula for n × n-matrix dynamic equations on time scales were studied in [11,12]. In [11,12], the authors provided the nice form of Liouville's formula by considering the eigenpolynomial and eigenvalue of the coefficient matrix of the dynamic equations.…”

“…The Liouville's formula for n × n-matrix dynamic equations on time scales were studied in [11,12]. In [11,12], the authors provided the nice form of Liouville's formula by considering the eigenpolynomial and eigenvalue of the coefficient matrix of the dynamic equations. However, if A is a n × n-matrix-valued function for n sufficiently large, the calculation of eigenpolynomial and eigenvalue of A becomes a complicated task and cannot be always achieved, so it will be a better way to provide a matrix form of Liouville's formula for the case of A : T → R n×n to avoid the calculation of eigenpolynomial and eigenvalue of A, similar to (1).…”

Interpolation in dynamic equations on time scales

“…The case when R\T consists of "small" intervals-the case of "small" time-gaps-was considered and, by using local constructions, satisfactorily solved [2,7]. ("Small" time-gaps are helpful in deriving the analogue of Liouville's formula on time scales [3].) Our present paper is devoted to the case when also large time-gaps are allowed.…”