Study On Lyapunov-type Inequalities For Fractional Difference Equations And Oscillation Of Solutions

In this paper,we study Lyapunov-type inequalities for certain fractional order d-ifference equations and oscillation criteria of a class of difference equation with forced term.The known results of Lyapunov inequalities for fractional differential equations and oscillation criteria are improved.This paper is divided into four chapters.In part one,we summarize the research background of Lyapunov-type inequalities and oscillation of fractional order equations as well as the related definitions used in this paper.In part two,we establish Lyapunov-type inequalities for the following differenceequations:where x(n)(?)0,n,?Z[z,b],m ? N,p>1,r(n)is a real-valued function defined on Z,and the equation where x(n)(?)0,n ? Z[a,c],a<6<c,m ? N,p>1,r(n)is a real-valued function defined on Z.Based on some properties of difference operator,we obtain Greenes function.Some new Lyapunov-type inequalities are obtained by using some important inequalities.In part three,we establish Lyapunov-type inequalities for the following fractional difference equation with p-Laplacian:where t ?[0,b]N0,2<?,??3,b?N,p>1,f:[? + ?-6,? + ?+ b-1]N?+?-6×R?R is a continuous function,?? and ?? denote the fractional differences of order ? and ?respectively,?p is the p-Laplacian operator.The Green's function and the properties are obtained by using the definitions of fractional sum,fractional difference and factorial function.Then the fractional difference equation is transformed into equivalent sum form.Based on some related inequalities,we obtain the results.The obtained results are used to the eigenvalues of corresponding equations and establish the existence of solutions for certain fractional boundary value problems.In part four,we consider the oscillation of following fractional difference equation with forced term:where t ? Nt0,0<?<1,? is a quotient of two odd positive integers.Using Riccati transformation,the algebraic inequalities and the properties of fractional difference,some new results are obtained,and some known results are extended.