Ouyang Type Integral Inequality

With the development of society and the progress of science,the study and applicationof diferential equation have reached into many areas of natural and social science,wherethe development on qualitative theory and stability theory of diferential equation furtherextends its feld of application.In qualitative theory and stability theory of diferentialequation, especially in studying the stability and boundedness of solution of diferentialequation, various inequalities play an important role. As a result, the research of the theoryand application of the inequality has been become one of important research directions inmodern mathematics.In recent years, Ou-Iang type inequalities are efective tools in researching ordinarydiferential equation, partial diferential equation and qualitative theory and stability the-ory of diference equation,so many scholars such as Enhao yang, Pachpatte, Fanwei Mengetc. have generalized and improved Ou-Iang type inequalities from continuous inequali-ties to discrete inequalities, linear inequalities to nonlinear inequalities, inequalities withone variable to inequalities in n independent variables, inequalities with non-retardationto inequalities with retardation and so on. Strengthening the study of such inequali-ties, especially searching for new ways to estimate and prove such inequalities is of greatimportance.In this paper,some Ou-Iang type integral inequalities are generalized and improved onthe basis of some existed results. Firstly,we add non-constant coefcients to establish newinequalities on the basis of some nonlinear integral inequalities in n independent variableswith retardation;Secondly, we further make some generalizations from certain parts ton parts on the basis of the above inequalities; Thirdly, we make some generalizationsfrom certain parts to n parts to establish new integral inequalities on the basis of someOu-Iang type discrete inequalities in n independent variables;Finally, by using a auxiliaryfunction, we generalize some known inequalities and established new integral inequalitiesin n independent variables with delay.