Generalization Of Nonlinear Integral Inequality With Delay

Diferential equation and calculus are the product of the same contemporary. Sincethe appearance of diferential equation, numerous scholars like Newton and Euler havetaken a great interest in it and conducted unceasing research. With the development ofdiferential equation, the study on it becomes more and more closely related to humansociety. Nowadays, the study and application of diferential equation have reached intomany felds of natural and social science, where the development on stability theory andqualitative theory of diferential equation further extends its range of application. Vari-ous inequalities play an important role in the research of stability theory and qualitativetheory of diferential equation, especially in the study of diferential equation’s stability,estimation and boundedness of solution. In consideration of the signifcant role in promot-ing research and application of diferential equation, the study on theory and applicationof inequality has become one of the most important research directions in modern math-ematics.With the appearance of Ou-Iang type inequalities in1957, a lot of scholars such asYang enhao, Meng fanwei, etc have generalized and improved it from linear inequalities tononlinear inequalities, from inequalities with one variable to inequalities in n independentvariables, from continuous inequalities to discrete inequalities, from inequalities with non-retardation to inequalities with retardation and so on, forming a grand and large family ofOu-Iang type inequalities. In recent years, Ou-Iang type inequalities have become powerfuland efective tools in researching diferential equation. Nowadays people have realized thata feasible approach to develop diferential equation is to strengthen the study on Ou-Iangtype inequalities. As a result, more and more people begin paying close attention to searchfor new ways and approaches to construct and extend new Ou-Iang type inequalities, andto prove such inequalities.The progress of science lies in the continuous exploration and research。In this pa-per,some Ou-Iang type inequalities will be generalized and improved on the basis of someexisted research results. This thesis contains three chapters.Chapter One, introduces the appearance of Ou-Iang type inequalities, and the variousforms of promotion and improvement of Ou-Iang type inequalities by many scholars.Chapter Two, based on some references，integrates and promotes the inequalities In the process of proof, the method used in this paper is better than the method used inliterature [6] and literature [9] in terms of innovation. The results we get are generalizationand improvement of relevant conclusions of Literature [6] and [9].In Chapter Three,on the basis of some existed references，by integrating and pro-moting the inequalities (1.1.1)the more general results by showing that: By using the methods in the literature [4],[5]and [11], and the method of auxiliary function,we get the conclusions on the basis of existed nonlinear integral inequality with retardation. The results are promotion of relevant conclusions of Literature [4],[5],[11].