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

Abstract

In this paper, the non-eigenvalue forms of Liouville’s formulas for delta, nabla and α -diamond matrix dynamic equations on time scales are given and proved. Meanwhile, a diamond matrix exponential function (or α -matrix exponential function) is introduced and some classes of homogenous linear diamond- α dynamic equations which possess the α -matrix exponential solutions is studied. The difference and relation of non-eigenvalue forms of Liouville’s formulas among these representative types of dynamic equations is investigated. Moreover, we establish some sufficient conditions to guarantee transformational relation of Liouville’s formulas and exponential solutions among these types of matrix dynamic equations. In addition, we provide several examples on various time scales to illustrate the effectiveness of our result.

Keywords

• liouville’s formula
• matrix dynamic equations
• time scales