Research On Some New Fractional Integral Inequalities

In recent decades,with the continuous study of differential equation-s,a series of important achievements have been made.For example,the existence,uniqueness,asymptotic,boundedness and oscillation for solutions of differential equations,integral equations and difference e-quations.However,it is difficult to find the exact solution for most differential equations.Integral inequality can be used to estimate the solution of the equation,so integral inequality is an important tool to study the qualitative properties of differential equations and integral e-quations.With the development of fractional calculus theory and its wide application,the problems related to the solution of fractional dif-ferential(difference)equation have been further studied.Based on the ideas and methods of existed literatures and theories,we mainly studies some new Gronwall-Bellman type inequalities and Hermite-Hadamard type inequalities for fractional integral.According to the content,this paper is divided into the following four chapters.Chapter 1 Preference,we introduce the main background of the research for this paper and give some definitions of integrals used.Chapter 2 Based on some integral inequalities given in[31],we s-tudy the following nonlinear Gronwall-Bellman integral inequality and the results obtained are used to study the boundedness of some solutions of differential equations.Chapter 3 Based on the literature[27],the following inequalities for the generalized proportional Hadamard fractional integral are estab-lished and where(?)is the generalized proportional Hadamard fractional integral.Chapter 4 Based on the definition of generalized proportional frac-tional integral established in literature[16],we study the following Hermite-Hadamard type inequality and Hermite-Hadamard-Fejer type inequality and where(?)is the generalized proportional fractional integral.