非线性分数阶耦合泛函微分方程组边值问题的可解性

Solubility of Boundary Value Problems for Nonlinear Fractional Coupled Functional Differential Systems

由于运动速度是有限的,因此在信号传输等过程中时滞现象往往是不可避免的。分数阶泛函微分方程是研究时滞系统运动规律的重要模型,当系统中具有两个或多个状态变量且这些状态变量相互作用时,常常运用耦合微分方程组来刻画。对一类具有Riemann-Liouville分数阶导数的非线性时滞耦合泛函微分方程组边值问题正解的存在唯一性进行了研究。首先,根据方程与边界条件的特点,建立了比较定理,构造了上解与下解的单调序列,并确定了上下解的关系。运用上下解的方法建立并证明了边值问题正解的存在性定理,同时得到了正解的取值范围。然后,利用迭代技术建立并证明了边值问题正解的存在唯一性定理。最后,给出了具体例子用于说明所得主要结论的适应性与广泛性。

Since the speed of all motions are limited, the time-delay phenomena are often inevitable in the signal transmission or other process. The fractional functional differential equations are important models to study the movement of time-delay systems. When there are two or more interact state variables in the system, they could always be characterized by coupled differential equations. The existence and uniqueness of positive solutions for boundary value problems of a class of nonlinear delay coupled functional differential systems with Riemann-Liouville fractional derivatives are studied. Firstly, according to the characteristics of equations and boundary conditions, a comparison theorem for the system is constructed, the monotonic sequence of upper and lower solutions is obtained, and the relationship between the upper and lower solutions is determined. Secondly, the existence theorems for positive solution of boundary value problem are established and proved by using the method of upper and lower solutions, and the value range of positive solutions is obtained. And thirdly, the existence and uniqueness theorem for positive solution of the boundary value problem is established and proved by iterative technique. Finally, a specific example is given out to illustrate the adaptability and universality of the main results.