| In this thesis, we use Lyapunov stability theory and fixed point theory to study the stability and boundedness of solutions for several kinds of nonlinear dif-ferential equations with delays. The thesis is divided into four sections according to the contents.Chapter 1 briefly introduces the researched background of problems and the main work of the paper.In Chapter 2, we establish some sufficient conditions which guarantee asymp-totic stability of the null solution and boundedness of all the solutions of the following third order nonlinear differential equation with variable delay r(t): x'"(t)+a(t)x"(t)+b(t)g1(x'(t-r(t)))+g2(x'(t))+h(x(t-r(t))) =p(t, x(t), x'(t),x(t-r(t)), x'(t-r(t)), x"(t)). By defining an appropriate Lyapunov functional, we prove two new theorems on the stability and boundedness of the solutions of the above equation. Our results extend the previously known results.In Chapter 3, sufficient conditions of the boundedness and ultimate bounded-ness for solutions to the following third order nonlinear delay differential equation x'"(t)-f(x"(t-r(t)))+g(x'(t-r(t)))+h(x(t-r(t)))=p(t, x(t), x'(t), x"(t)), are given by the Lyapunov's second method. The appropriate Lyapunov function is given explicitly. Our results extend some well known results on the third order differential equations in the literature.In Chapter 4, we consider the stability of solutions of the first order nonlinear neutral differential equations with variable delays by using fixed point theory. An asymptotic stability theorem with a necessary and sufficient condition is proved, which generalizes and improves some results on the first order differential equations. An example is also given to illustrate our results. |
