Several Types Of Weakly Singular Integral Inequalities And Fractional Hermite-Hadamard Inequalities

Inequality is an important tool to research the existence,uniqueness,boundedness and vi-bration of solutions of differential equations.With the need of practical application and the development of calculus theory,inequality has been generalized in many ways.Gronwall-bellman inequality,Voltera-Fredholm inequality,Hermite-Hadamard inequality are widely used in mathematical model,physics,chemistry,biology,economics and many other fields.The fractional calculus equation is one of the important tools of mathematical modeling in complex mechanics and natural phenomena.In recent years,fractional differential equations have been applied to many theoretical and practical problems,many valuable results have been obtained in the qualitative properties of solutions.Therefore,the research of fractional inequality has developed.There are still many problems to be solved in the research of fractional integral inequality.In this paper,we introduce and research three kinds of fractional integral inequalities in four chapters.Chapter 1 Preference,We introduce the research status of weakly singular integral inequality and Hermite-Hadamard type inequalities for fractional integral.Chapter 2 We research the weakly singular Volterra integral inequality with maxima in two variables:(?) where (?),(?)Chapter 3 We research some Hermite-Hadamard type inequalities for fractional inte-grals via convex functions and we obtain some estimates for(?),(?)Chapter 4 We research some Hermite-Hadamard type inequalities for?-Riemann-Liouville fractional integrals via convex functions and we obtain some estimates for(?)and(?).