Study On Fractional Gronwall-type Inequalities And Oscillation Of A Class Of Difference Equations

The fractional calculus is a theory that studies the mathematical properties and applications of any order differential and integral,it is extended and expanded by traditional integer calculus.Gronwall type inequality has been concerned by many scholars,it plays an important role in the qualitative and quantitative analysis of the solutions of the equations.For example,the existence of solution,asymptotic behavior,the estimation of the solution,etc.With the development of the theory of fractional order,the research and extension of fractional order Gronwall inequalities have become a hot topic for many scholars.In this paper,we mainly study fractional Gronwall type inequalities and the oscillation of the solutions of the fractional difference equations,give its application and generalize the related theories.This paper is divided into four chapters.The first chapter mainly summarizes the general situation of the research on the fractional calculus theory and the Gronwall type inequality at home and abroad and some related definitions and lemmas which will be used in this paper.The second chapter discusses the fractional Gronwall type inequalities on time scales T1,and some new results are obtained by using the correlation lemmas and the algebraic inequalities.They are applied to the fractional differential equations,and the new bounds of the solutions of the equations are obtained.The third chapter studies the discrete fractional Gronwall type inequalities,by using the related algebraic inequalities and some operational techniques as well as analysis,we obtain a new upper bound of the unknown function.The fourth chapter mainly discusses the following oscillation of solutions of the fractional difference equationand the oscillation properties of a class of equations with initial conditions and forcing terms are as followsBy using the Riccati transform and the fractional difference equations transformed into equivalent sum forms as well as a series of derivation,the sufficient conditions for the oscillation of the solutions of the above two kinds of equations are obtained.The results are applied to the specific equations,and the oscillation properties of the solutions of the equations are verified.