### INTERVAL OSCILLATION THEORY FOR LINEAR MATRIX HAMILTONIAN SYSTEMS

#### Abstract

By means of Riccati transformation technique and averaging method, interval

oscillation theorems are established for linear matrix self-adjoint

Hamiltonian system

\[

\left\{

\begin{array}{c}

U^{\prime }(x)=A(x)U(x)+B(x)V(x), \\[8pt]

V^{\prime }(x)=C(x)U(x)-A^{*}(x)V(x),

\end{array}

\right.

\]

that are different from most known ones in the sense that properties of $A,B$

and $C$ are only needed on a sequence of subintervals of $[x_{0},\infty )$,

rather than on the whole half-line. Examples that dwell upon the importance

of our results are also included.

oscillation theorems are established for linear matrix self-adjoint

Hamiltonian system

\[

\left\{

\begin{array}{c}

U^{\prime }(x)=A(x)U(x)+B(x)V(x), \\[8pt]

V^{\prime }(x)=C(x)U(x)-A^{*}(x)V(x),

\end{array}

\right.

\]

that are different from most known ones in the sense that properties of $A,B$

and $C$ are only needed on a sequence of subintervals of $[x_{0},\infty )$,

rather than on the whole half-line. Examples that dwell upon the importance

of our results are also included.

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