Lyapunov inequalities for a class of nonlinear dynamic systems on time scales

The purpose of this work is to obtain several Lyapunov inequalities for the nonlinear dynamic systems $$\left \{ \textstyle\begin{array}{l} x^{\Delta}(t)= -A(t)x(\sigma(t))-B(t)y(t)|\sqrt{B(t)}y(t)|^{p-2}, \\ y^{\Delta}(t)= C(t)x(\sigma(t))|x(\sigma(t))|^{q-2}+A^{T}(t)y(t), \end{array}\displaystyle \right . $$ on a given time scale interval \([a,b]_{\mathbb{T}}\) (\(a,b\in{\mathbb{T}}\) with \(\sigma(a)< b\)), where \(p,q\in (1,+\infty)\) satisfy \(1/p+1/q=1\), \(A(t)\) is a real \(n\times n\) matrix-valued function on \([a,b]_{\mathbb{T}}\) such that \(I+\mu(t)A(t)\) is invertible, \(B(t)\) and \(C(t)\) are two real \(n\times n\) symmetric matrix-valued functions on \([a,b]_{ \mathbb{T}}\), \(B(t)\) is positive definite, and \(x(t)\), \(y(t)\) are two real n-dimensional vector-valued functions on \([a,b]_{\mathbb{T}}\).