Qualitative Analysis Of Several Kinds Of Nonlinear Differential Equations And Difference Equations

It is well known that for most integer order differential equations, fractional order differential equations and fractional order difference equations, it is difficult to find their general solutions, sometimes it is impossible. The properties, such as existence, bound-edness, oscillation, asymptotic behavior and stability, to the solutions of the equations are studied no longer by solving the equations, but by investigating the composition of the equations. From then on a series of qualitative theories of nonlinear equations have been formed. This thesis is concerned with the oscillation of some integer order differ-ential equations, the existence of the periodic solutions for a predator-prey system with impulsive effects, and the boundedness of some classes of fractional order differential equations and fractional order difference equations. The results obtained extend and improve a number of results reported in the literature. The main results are described as follows:In Chapter ?, we briefly summarize background and development of the oscillation of some integer order differential equation, the existence of the periodic solutions for a predator-prey system with impulsive effects, and the boundedness of some classes of fractional order differential equations and fractional order difference equations. The main results in this thesis are also introduced.In Chapter ?, we study the oscillatory behavior of a second-order damped dif-ferential equations with nonlinearities given by Riemann-Stieltjes integrals and vari-able exponent by using generalized Riccati technique, inequality and integration av-erage technique. Section 2.1 is concerned with the oscillatory behavior of a one class of second-order damped differential equations with nonlinearities given by Riemann-Stieltjes integrals and variable exponent. El-Sayed type and Kamenev type oscillation criteria are presented that improve. some known results in the literature. Section 2.2 is concerned with oscillatory behavior of a certain class of second-order damped differen-tial equations with p-Laplacian and nonlinearities given by Riemann-Stieltjes integrals and variable exponent. Some new results are presented that complement and improve those related results in the literature. 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.In Chapter ?, we investigate a predator-prey system of Holling type IV function response with mutual interference and impulsive effects. By using Mawhin's continua-tion theorem of coincidence degree theory we obtain some sufficient conditions which guarantee the existence of positive periodic solutions for the model. In particular, our results indicate that under the appropriate linear periodic impulsive perturbations, the above impulsive differential system preserves the original periodicity of the nonimpul-sive differential system.In chapter ?, we discuss two classes of integral inequalities containing weakly singular kernels and use them in the research of the boundedness and other properties of the solution of some fractional differential equations. The results obtained complement and improve those results reported in the literature.In chapter ?, We investigate the boundedness of the solution of some discrete fractional equations. In section 5.2, we employ the Riemann-Liouville definition of the fractional difference to establish some Gronwall-Bellman type discrete fractional sum inequalities. In section 5.3, we employ the Riemann-Liouville definition of the fractional difference to establish some Volterra-Fredholm type discrete fractional sum inequalities. In section 5.4, we use the above inequalities to investigate the boundedness and other properties of the solution of certain classes of fractional difference equations and some Volterra-Fredholm type fractional sum-difference equations.In Chapter ?, we give the prospect on the the future research.