若干混合导数型时标动态方程的理论、方法及应用

The theory of time scales is a new research area in mathematics which unifies the continuous and discrete analyses. The study of dynamic equations on time scales combines the studies of differential equations, difference equations, and differential-difference equations reflecting the coexistence of continuous and discrete phenomena into one unified approach and hence reveals the deep nature of and relations among these equations, as a result, promotes such studies to a higher level. The theory of time scales have strong roots in other fields. For instance, various problems in mathematical biology, engineering, and physics can be better characterized by dynamic equations on time scales than by the classical differential and difference equations. Due to the scope of related problems, the development of the theory of time scales is of not only theoretical importance, but also significant values and trenmendously broad potentials in applications. .Since this theory is newly developed from 1988, there are many major tasks to be done in both the theoretical and applied branches. In particular, the research on dynamic equations on time scales involving non-delta derivatives or mixed derivatives, which arise from various real world models, is only in its initial stage. As continuation of our previous work, in this project, we will further study dynamic equations on time scales with emphasis on those of mixed derivatives. Some main topics to be worked with are as follows: further development of the qualitative analysis of dynamic equations on time scales of mixed derivatives, periodic and almost-periodic problems on time scales, relations between certain dynamic equations on time scales and their corresponding impulsive differential equations, and their applications..We believe that the research in this project will be of substantial significance and originality.

时标理论属国际前沿研究的一个新领域，它整合了连续与离散分析，它的研究不仅把微分方程和差分方程理论有机结合，而且也包含了兼有连续与离散现象共存的微分差分方程，所得结果比一般的微分方程理论和差分方程理论更为广泛。同时，时标理论在应用上有着巨大的潜力，生态、工程技术、物理等领域的许多现象应用时标动态方程来描述，更能揭示其本质属性，因此，时标理论的研究，具有重要的理论价值与广泛的应用前景。.这一理论从1988年提出至今发展时间不长，存在着许多有重要意义的、亟待解决的理论和实际问题，特别是时标上包含非Delta导数和混合导数的动态方程，迄今研究极少。本项目重点开展混合导数型时标动态方程的研究，包括：进一步研究和完善若干混合导数型时标动态方程的基本理论；建立若干时标动态方程周期和概周期问题的新结果；深化时标理论在脉冲微分方程定解问题等方面的应用。项目的研究对象和数学手段都具有明显的开拓性和创新性。

时标理论属国际前沿研究的一个新领域，它整合了连续与离散分析，它的研究具有重要的理论价值与广泛的应用前景。四年来，我们课题组按研究计划开展工作，在一类具有静止阶段的时滞非局部扩散模型的提出及其动力学行为、带有混合单调性的一类n维时滞反应扩散系统行波解的存在性、具任意阶自催化双细胞耦合等温化学系统的稳态模式、时标上关于nabla导数的两类神经网络系统的概周期解的存在唯一性和全局指数稳定性、具交错扩散Holling-II型捕食-食饵系统、带时滞的时标动力系统的稳定性和周期性、一类带脉冲的反应扩散系统的渐近播速和行波解的存在性、具有混合导数的时标动态方程非振动解的存在性、三类流体动力学方程的Cauchy问题、一类Riemann–Liouville型分数阶微分包含的逼近控制问题以及一类带扩散的Brusselator型系统的动力学行为等方面开展了一系列研究工作。