Stability in mean for uncertain delay differential equations based on new Lipschitz conditions

Uncertain delay differential equations (UDDEs) involving a current state and a certain past state have been proposed to model an uncertain system with a delay time, such as uncertain delay logistic model. Stability of the UDDEs is a vital problem for its applications. Based on the strong Lipschitz condition, the stability in mean for UDDEs have been investigated. Actually, the strong Lipschitz condition is assumed that it only relates to the current state, it is difficult to be employed to determine the stability in mean for the UDDEs. In this paper, we propose two kinds of new Lipschitz conditions containing the current state and the past state, which are more weaker than the strong Lipschitz condition. Meanwhile, two sufficient theorems based on these new Lipschitz conditions as the tools to judge the stability in mean for the UDDEs are verified. For a special class of the UDDEs, which are proved to be stable in mean without any limited condition. Besides, some examples are discussed in this paper.