积分不等式解的渐近性及在稳定性中的应用

It is well known that most of differential equations do not have general solutions, and is also an original motivation of the development of dynamical systems and qualitative theory. Because of insolvability, estimation of solutions becomes very important. As a useful tool estimating solutions, integral inequalities play an important role in the study of existence, uniqueness, boundedness and stability of solutions and some significant problems, e.g., exponential dichotomy, invariant manifolds and attractors. Various types of integral inequalities were studied, and applied to the qualitative properties of solutions, especially stability. But the properties of estimates for some complex inequalities are still not clear, e.g., multiterm sum and impulsive integral inequalities, whose estimates are defined recursively by difference equations. Moreover, the conditions of stability given by these estimates are also not sharp. According to the classic result, if the estimates of integral inequalities can be dominated by the contractive linear part, e.g., bounded estimates, the solutions of equations are all stable. Until now, almost all the works depend on the boundedness, but the estimates still can be dominated by the linear part, even if they are unbounded. This program will consider the complex inequalities above, and investigate some controllable unboundedness, e.g., boundedness in α-weight and temperedness, by discussing the difference equations generated by the estimates. Depending on these properties, our works will obtain better conditions of stability.

众所周知，绝大部分微分方程是无法求出通解的，这也是动力系统和定性理论发展的原始驱动力。既然无法求解，对解作估计就变得非常重要。作为估计解的重要工具，积分不等式在研究解的存在性、唯一性、有界性、稳定性以及指数二分、不变流形与吸引子等重要问题中扮演着重要角色。前人已经对各种积分不等式给出了估计，并且运用它们研究了解的定性性质，尤其是稳定性。但是对于一些复杂的不等式，例如多项求和与脉冲积分不等式等估计式由差分方程递归地给出的情况，其性质仍然不清楚。并且目前由这些估计式给出的解的稳定性条件也远未达到最优。按照经典结果，积分不等式的估计式如果能被压缩的线性部分控制住，例如估计式有界，则方程的解是稳定的。目前几乎所有结果都依赖于有界性，但其实估计式无界，线性部分仍有可能控制住它。本项目考虑以上复杂不等式，通过研究估计式所属的差分方程来给出其α加权有界性和缓增性等可控无界性质，从而得到更好的稳定性条件。

众所周知，绝大部分微分方程是得不到通解的，这是动力系统和定性理论发展的原始驱动力。既然无法求解，对解作估计就变得非常重要。作为估计微分方程解的重要工具，积分不等式在研究解的存在性、唯一性、有界性、稳定性、不变流形和不变叶层中扮演着重要角色。本项目研究几类复杂的积分不等式，利用它们去估计原方程的解，得到解的稳定性。考虑带求和项的脉冲积分不等式，它可以用来估计多时滞和脉冲微分方程的解。我们证明该积分不等式的解由一个非自治差分方程组给出，并可以通过研究函数空间上非自治迭代的极限行为去得到解的单调性、有界性和加权有界性，从而得到原时滞和脉冲微分方程解的稳定性。另外，为了研究状态依赖时滞、混合型和迭代泛函微分方程解的定性性质，我们发展几类带未知函数自复合项的积分不等式。结合Kransnoselski不动点定理，给出混合型和迭代泛函微分方程正解和单调解的存在性、唯一性、连续依赖性和稳定性等结果。