关于Razumikhin型定理及其应用与相关结果

COMMENTS ON “AN IMPROVED RAZUMIKHIN TYPE THEOREM AND ITS APPLICATION” AND A RELATED RESULT

本文指出Ｘｕ等１９９４年“改进的Ｒａｚｕｍｉｋｈｉｎ定理”的证明是不成功的．这是由于Ｘｕ等（１９９４）定理１的条件不能保证经典的Ｒａｚｕｍｉｋｈｉｎ型定理条件ｉ）成立．于是Ｘｕ等１９９５年基于其“改进Ｒａｚｕｍｉｋｈｉｎ定理的结果仍然没得到证明．利用徐等１９９１年用李雅普诺夫定理与代数方法导出的结果，证明了Ｘｕ等１９９５年的结果对于常时滞是真的，因此Ｘｕ等（１９９４．１９９５）的结果是否成立。

It is noticed that the proof of the “improved Razumikhin Theorem” proposed by Xu et al.(1994) is unsuccessful, because the condition ii) of Razumikhin type theorem can not be ensured by the conditions of Theorem 1 in Xu et al.(1994). Then,the result in Xu (1995) based on the “improved Razumikhin type theorem” is also not proved. When the delays are constant, we show that the result in Xu (1995) is true by employing the results obtained by Lyapunov theorem and algebraic method in Xu et al.(1991). Therefore,it becomes two open problems whether the results obtained by Xu et al.(1994,1995) are true.