Banach空间中非线性复合分数阶微分方程的若干研究

The fractional derivatives provide an excellent tool for the　description of memory and hereditary properties of various materials and processes. With this advantage, the fractional-order models are more realistic than the classical integer-order models to describe practical problems. Very recently, many　researchers focus on the theory of fractional differential equations. Noting that the difference between the fractional derivatives and integer-order ones, we will adopt new methods and techniques to study the fractional differential equations. In this project, by using the regularized resolvent families, we will study the nonlinear composite fractional differential equations which are the generalizations of the classical fractional differential equations, and obtain the results about solvability of the nonlinear composite fractional differential equations with delay or impulsive effects, Cauchy problem for nonlinear composite fractional differential equations of Sobolev type and existence of almost periodic mild solutions to nonlinear composite fractional differential equations with singular operator. Based on above studies, we will try to study more complicated nonlinear composite fractional differential equations. This project will enrich and develop the theory of fractional differential equations, which can be applied to solve more practical problems.

分数阶导数在描绘不同物质和进程的记忆和遗传方面有着极大的优势，故分数阶模型能更真实地反映那些经典整数阶模型所无法刻画的实际问题。目前，非线性分数阶微分方程的理论研究已成为国际上热点关注的问题之一。由于分数阶导数与整数阶导数的本质差异，我们必须采用不同于整数阶微分方程的新的研究方法和技术。本项目拟利用正则预解算子族的相关理论研究非线性复合分数阶微分方程的相关问题。该类方程是经典分数阶微分方程的推广，具有更一般的形式并能刻画更多实际问题。本项目拟研究两类非线性复合分数阶微分方程在延迟条件或脉冲条件影响下的适定性；Sobolev型非线性复合分数阶微分方程的Cauchy问题；含奇异算子的非线性复合分数阶微分方程的概周期解的存在性等问题。在此基础上，我们尝试进一步研究其它更复杂的非线性复合分数阶微分方程。本项目将丰富和完善分数阶微分方程理论，使其解决更多的实际问题。

分数阶导数在描绘不同物质和进程的记忆和遗传方面有着极大的优势，故分数阶模型能更真实地反映那些经典整数阶模型所无法刻画的实际问题。目前，非线性分数阶微分方程的理论研究已成为国际上热点关注的问题之一。由于分数阶导数与整数阶导数的本质差异，我们必须采用不同于整数阶微分方程的新的研究方法和技术。本项目按计划研究了两类非线性复合分数阶微分方程在脉冲条件影响下的适定性；Sobolev型非线性复合分数阶微分方程的Cauchy问题；含奇异算子的非线性复合分数阶微分方程的概周期解的存在性等问题。在此基础上，我们进一步研究了更多类型的非线性分数阶微分方程的S-渐近概周期解的存在性、全局解的吸引性、概自守及概周期性等；以及若干类型的非线性分数阶微分方程边值问题的适定性。上述共涉及17类不同的问题，我们分别利用正则预解算子族、解析半群、几乎扇形算子生成的奇异半群、二阶余弦正则算子函数、广义Mittag-Leffler函数、Wright函数等研究工具研究上述问题，产生了一些新技巧和新技术，此类技术还可进一步应用到解决更复杂类型的分数型发展方程中。