时间测度链上二阶动力方程的振动和非振动准则

 时间测度链上的分析理论不仅有效地统一了连续分析和离散分析理论，而且在理论和实际中具有非常广泛的应用。随着时间测度链的不同，动力方程被推广到微分方程和差分方程。而时间测度链上中立型时滞动力方程的振动性与非振动性理论作为中立型动力方程定性理论中的重要内容，更是引起了学术界广泛兴趣和高度关注。本文研究了时间测度链上的一类二阶非线性中立型时滞动力方程的振动和非振动性质。首先，利用Banach空间的不动点定理和分析技巧，得到该类方程存在有界的最终正解的判别准则；其次，通过引入广义Riccati变换，借助时间测度链理论，得到该类方程振动的几个充分条件。所得结果有助于统一微分方程和差分方程的有关结论。

Oscillation and Non-oscillation Criteria for Second-order Dynamic Equation on Time Scales

In recent years, the dynamic equation theory not only finds important applications in such fields as physics, space satellite, etc, but also becomes an indispensable mathematical tool in such domains of natural and social sciences as economics, biology, control theory, etc. Moreover, oscillation and non-oscillation criterion is the key concern of qualitative study of the neutral dynamic equations, as have attracted much attention. In this paper, the oscillation for a class of second order nonlinear neutral delay dynamic equation on time scales is discussed. Using the fixed point theorem in Banach space, a new non-oscillation criterion for the equation is obtained by the generalized Riccati transformation, with the time scale theory and some necessary analytic techniques. In addition, some sufficient conditions for oscillation of the equation are proposed. These criteria can improve the restrictive conditions for the equation, and unify results about oscillation for delay differential equation and delay difference equation. Some results in the literature are improved and extended.