Dynamic Properties For Some Differential (Integral) Equations

With the increasing development of science and technology, the differential equa-tions arise in numbers of important applications, including Physics, Astronomy, Biolo-gies, Medicines, Population dynamics, Economies, Engineering and theory of controls. The differential equations are powerful tools to investigate the law of nature. Since it is difficult to find their general solutions, there has been an increasing interest in the study of the properties of solutions of differential equations in theory. At the 1830s, C. F. Sturm applied the theory of the distribution of zeros of Polynomial to differential equations and gained the famous Sturm's Theorems—the Comparison Theorem and the Separation Theorem. In the late 19th century, the France Mathematician and the Russian Mathematician, J. H. Poincare and A. M. Lyapunov apply the qualitative theory to the research of the basic problem of celestial mechanics. In this theory, the properties to the solutions of the equations are studied no longer by solving the equa-tions, but by investigating the composition and the coefficients of the equations. From then on a series of qualitative theories of nonlinear differential equations have been formed. Poincare's《On the integral curves defined by differential equations》and Lyapunov's《Stability theory of motions》are the classical works in qualitative theo-ries. Since then a new age of studying the dynamic properties to the solutions of the differential equations in terms of theory has been coming.The thesis is divided into six chapters, main contents are as follows.In ChapterⅠ, we give a survey to the development and current state of the quali-tative theory (for example, oscillatory theory), stable theory of differential equations.In ChapterⅡ, using the generalized Riccati technique, the averaging integral tech-nique and differential inequalities, for linear Hamiltonian systems and some even order nonlinear partial functional differential equations of neutral type with continuous dis-tributed delay, new oscillation criteria are obtained without some assumptions which have been required for related results given before. The results are independent and have improved some previous results to a great extent. Some examples are included to show the versatility of our results.It is well known that Gronwall type inequalities play a dominant role in the study of quantitative properties of solutions of differential and integral equations, and often they are used to study the boundedness and the asymptotic behavior of the solutions of integral equations. Lipovan [62] study the global existence of the solutions of differential systems using the integral inequalities.In ChapterⅢ, we generalize some integral inequalities of Gronwall-Bellman-Ou-Iangto type. We also show the usefulness of our results in investigating the global existence of the solutions of differential systems. Our results generalize Lipovan's results in [62].Chapter IV focuses on the study of some inequalities of double integral and quadratic integral and their applications. We show the usefulness of our results by establishing some theorems about the stability on the solutions of differential equa-tions (systems) and integro-differential equations (systems) with time delay.Akinyele [78] introduced the notion ofΨ-tability of degree k with respect to a functionΨ∈C(R+,R+); Morchalo [89] introduced the notions ofΨ-stability,Ψ-uniform stability, andΨ-asymptotic stability of trivial solution of the nonlinear system x'=f(t,x).In Chapter V, in§5.1, for the nonlinear differential systems x'= f(t,x), x' f(t, x)+g(t, x) and nonlinear Volterra integro-differential system x'=f(t, x)+ integral from n=0 to t F(t, s,x(s))ds, we give some sufficient conditions forΨ-(uniform) stability of their trivial solutions. The innovation and difficulty in the dissertation is the fundamental matrix solution of the linear differential system x'=A(t)x is in the type of Y(t), whereas the fundamental matrix solution of the nonlinear differential system x'= f{t,x) is in the type ofΦ(t,to,xo). In§5.2, we study theΨ-(uniform) stability of some nonlinear Volterra integro-differential systems with time delay using some integral inequalities.In ChapterⅥ, using the integral inequalities we generalized in ChapterⅢand IV, we establish bounds and the asymptotic behaviors on the solutions of some integral equations.Finally, we give the prospect on the study of the dynamic properties of the solu-tions of dynamic equations on time scales, fractional differential equations and differ-ence equations, as well as ordinary (partial) differential equations and integral equa-tions.