The Solutions And Their Properties Of Degenerate Differential Equations With Delay

Fractional differential system with delay plays an increasingly important role in the research of functional differential system. Many researchers have pay more attention to the degenerate fractional differential system with delay, because it is difficult to consider time delay and degenerate simultaneously in the progress of the research of fractional differential systems, which have extensive applications. This dissertation considers three important factors, namely fractional, time delay and degenerate, which can describe actual systems accurately. The results will enrich and develop the theory of functional differential system and can radiate actual systems efficiently.In this dissertation, the existence and expressions of solution for degenerate fractional differential equation with delay are discussed as well as stability, finite time stability and stability of all delay. The existence of periodic solution of de-generate differential equation with discreted delay and distributed delay is also studied. Our main results are as follows:1. The constant variation formulaes for the general degenerate fractional dif-ferential equations with delay are obtained. Firstly, the normalization of the general degenerate fractional differential equations with delay is considered. We get the constant variation formulae for the general degenerate fractional differential equa-tions with delay under the condition of the normalization. Next, basing on the solvable matrix pair method, we establish the expressions of the general solution for the general degenerate fractional differential equations with Caputo derivative. By combining step and step method, we prove the existence and uniqueness of the solution for the general degenerate fractional differential equations with delay with Caputo derivative. In addition, we obtain the constant variation formulae for the general degenerate fractional differential equation with delay by applying Laplace transformation.2. Some results of stability for degenerate fractional differential equations with delay are obtained. Firstly, basing on the Drazin inverse matrix method, the constant variation formulae for degenerate fractional homogeneous differential equation is obtained. Next, we give stability conditions of fractional homogeneous differential equations by using the two-parameter Mittag-Leffler function and the eigenvalues. In addition, the conditions of the stability of all delay for degenerate fractional differential equations with delay are established. Finally, The exponential estimation of the degenerate fractional differential system with delay with Caputo derivative and sufficient conditions for the finite time stability of the system are derived.3. The existence of periodic solutions of degenerate differential equations with delay is studied. By combining the theory of exponential dichotomies of linear system and the theorem of Krasnoselskii, some sufficient conditions that guarantee the existence and uniqueness of periodic solutions of the systems are given. Next, by combining the theory of characteristic equation and the theory of Fourier series, necessary and sufficient conditions for the existence of periodic solution of degen-erate differential equation with distributed delay are obtained. Especially, we give the criteria of the existence of periodic solution for two-dimensional degenerate dif-ferential equation with distributed delay, at the same time, we provide a method of getting periodic solution for degenerate differential equation with discreted delay and distributed delay.