The Stability And Existence Of Almost Periodic Solution For Some Neutral Type Differential Equations

In 1892, Russian mathematician, mechanician and physicist, Lyapunov, proposed the rigorous mathematics definition of the stability of motion and its general research method in his doctoral dissertation "The general problem of the stability of motion", which established the foundation of the stability theory. With scientific progress and social development, Lyapunov theory is being expanded and developed constantly and is applied extensively in delay differential equations. however, Lyapunov method need construct Lyapunov v- function, which are difficult to some delay differential equations .Therefore, some scholars try to seek others ways to solve stability problems of delay differential equations to avoid constructing Lyapunov v- function. In 1960s, some delay differential inequalities were introduced by Halanay and the other scholars'. The stability problems of delay differential equations are solved by using the delay differential inequalities. The methods provided new judging criterion for the stability of delay differential equations. In recent years, delay differential inequalities have embodied particular advantage in solving the stability problems of delay differential equation, which formed a distinctive branch in the delay system theory.In the present study, some new stability results for a class of neutral differential equations with variable coefficients and multiple delays in the metric space C₁ and a class of neutral integral differential equations are obtained by using mixed type differential-difference inequalities. In addition, the existence, uniqueness and stability of almost periodic solutions of a class of neural integral differential equations areinvestigated by using the theory of exponential dichotomy and fixed point.This thesis has six sections.The prologue first introduces the current research situation of the stability of the zero solution and almost periodic solution to the neutral differential equation, and then emphasizes the significance of our work in this thesis.In the first chapter, the problems are posed.In the second chapter, the basic definitions and lemmas are presented.In the third chapter, we study the stability of a class of neutral differential equations, and generalize the stability of the differential equations in [1-4] from metric space Q to Cu and obtain the conclusions that can't be got by [2⁴].In the fourth chapter, we study the stability of a class of neutral integral differential equations, and changed the sufficient conditions for the stability of the equations in [5⁶], and extend the results in [5-6].In the fifth chapter, we prove the existence, uniqueness and the stability of almost periodic solutions to a class of neural integral differential equations, and generalize the results in [7-11 ].