The Study On Fractional Difference Operators And Discrete Hermite-Hadamard Inequalities

The theory of discrete fractional calculus is a generalization of fractional calculus and unifies discrete calculus(time scale calculus)and fractional calculus.With the rapid development of the discrete fractional calculus,fractional sums and differences for different types and their uses have been introduced,and correspond to many models in practical applications.As a significant result of the theory of convex functions,the Hermite-Hadamard inequalities have been widely concerned.Furthermore,there is much interest in studying fractional Hermite-Hadamard inequalities involving various fractional integrals even fractional sums.In this dissertation,we consider h-Riemann-Liouville fractional sums and differences in the discrete fractional calculus.Under this framework,we introduce a new definition of fractional sums.Besides,resorting to the substitution rules and variable transformations,we prove some generalized Hermite-Hadamard inequalities on time scales.The principal results are as follows:In Chapter 1,we introduce the research background and basic knowledge of this dissertation.In Chapter 2,we consider the right-and left-sided Hilfer type generalized proportional nabla fractional difference operators,(?)where a▽h-β(n-α),ρ(·)and h▽b-β(n-α),ρ(·)are generalized proportional fractional sum operators,respectively.Meanwhile,we derive a few important properties and the discrete Laplace transform for new operators.We also discuss the generalized solution of a fractional difference equation with initial conditions,which is one of the most important tools for studying the qualitative properties of exact solutions.In Chapter 3,we study the generalized Hermite-Hadamard inequalities on the time scale Nc,h.First,considering convex functions on Nc,h,we investigate the integer HermiteHadamard inequality and the fractional Hermite-Hadamard inequality involving h-RiemannLiouville fractional sums.Next,we study integer and fractional inequalities of HermiteHadamard type for s-convex mapping in the second senses f which is defined in a time scaleIn Chapter 4,we establish generalized Hermite-Hadamard inequalities relating to the midpoint(a+b)/2 for discrete convex functions on Z and Nc,h.Besides,we give two examples to illustrate the integer result.In Chapter 5,we provide a summary of the article and further prospects.