Studies On Several Types Of Integral Inequalities And Discrete Inequalities And Their Applications

It is well known that the integral and finite difference inequalities play a fundamental role in the development of the theory of differential and finite difference equations. During the past few years, many new inequalities have been discovered, which are motivated by certain applications. In this paper, some new nonlinear inequalities in n independent variables are established, which generalize several types of integral inequalities and discrete inequalities. Using our results, the boundedness of some partial differential equations and some partial difference equations are studied. This thesis is divided into two Chapters.In Chapter one, some generalizations of several integral inequalities are obtained.The inequality of Bellman-Bihari type is the nonlinear generalization of the noted linear Gronwall-Bellman inequality. In the study of the qualitative theory of differential equations, especially in the study of the stability of differential equations, the estimate of solutions and the boundedness of solutions, Gronwall-Bellman-Bihari inequality can be used as handy tools. So far, there have been many types of inequalities of Gronwall-Bellman-Bihari type, including nonlinear inequalities, the inequalities of Bihari type, the inequalities on distribution, the inequalities in several variables, discrete inequalities, matrix inequalities and so on. Some new results are established in Section one, which generalize the important inequalities of Bellman-Bihari type. We mainly study the following inequalities in n independent variablesandIn Section two, we discussed nonlinear delay integral inequalities of Ou-Iang type on R which generalized integral inequality of Ou-Iang type in n independent variables that obtained by Guo Jifeng.In Section three, nonlinear integral inequality of Mate-Navi type in n independent variables are concluded. Considering the inequalityThe inequalities given here can be used as tools in the qualitative theory of certain differential equations.The purpose of generalizing and improving inequalities is to study related qualities of differential and integral equations. Using our results, the related qualities of some partial differential equations are discussed, and we proved the importance of inequalities in course of studying equations. We mainly consider the boundedness of partial differential equationsandand a Volterra integral equationThe discrete inequalities of " G-B " type is very important in the study of the stability and asymptotic behavior of difference equations. In the past few years, people are very interested in the study and applications of discrete inequalities. In Chapter two, some important discrete inequalities are obtained, which are the discrete analogues of some results we concluded in Chapter one.In Section one, we study the generalizations of discrete inequalities of Bellman-Bihari type. We discussandand some others.In Section two, we discussed nonlinear delay discrete inequalities of Ou-Iang type in n independent variablesand some special conclusions are acquired.Using the results, we study the related qualities of nonlinear partial difference equationsand...