Studies On Some Inequalities And The Existence Results Of Fractional Dynamic Equations On Time Scales

In recent years, with the progress of the society and the development of science and technology, a lot of mathematical models which are described by dynamic equations on time scales have been proposed in the field of natural science. Therefore, the research for dynamics theory has become an important subject. At the same time, the fractional order di?erential equations have been proved that it is useful in the fields of applied science and engineering, such as chemistry, mechanics, physics and so on. Fractional equations on time scales have become an important branch of di?erential equations.With the development of dynamic equations on time scales, it is widely recognized that integral inequalities are very important, especially in the study of the existence and asymptotic behavior of solution, integral inequalities provide a new powerful tool for the research on time scales. The existence of solution for impulsive fractional order dynamic equations is an important content of qualitative of fractional di?erential equations on time scales. Because impulsive equations can exhibit several real-world phenomena, there has been an increasing interest in the study of them.In this paper, we study some integral inequalities and the existence of solution for the dynamic equations on time scales. The paper is divided into five chapters.In Chapter 1, we give a survey of the development and current state about dynamic equations on time scales and the fractional order di?erential equations.In Chapter 2, we consider Opial-type inequalities involving higher order delta derivatives on time scales. Using Taylor’s theorem, H¨older’s inequality, and Schwarz’s inequality, Opial-type inequalities are generalized to time scales.In Chapter 3, we investigate Lyapunov-type inequalities for nonlinear Hamiltonian systems on time scales. For the system, using the definition of generalized zero, the calculus theory on time scales, H¨older’s inequality, and the triangle inequality, we get Lyapunov-type inequalities.In Chapter 4, we discuss the existence of solution for impulsive fractional dynamic equations on time scales. By considering the integral equation which is equivalent to the equation, constructing an operator and using the fixed point theory, we can obtain the su?cient conditions for the existence of solution. By the Gronwall-type inequality, we get some su?cient conditions for the existence and the uniqueness of solution. At last,an example is given to illustrate the applicability of the conclusion.In Chapter 5, we discuss the existence of solution for nonlinear impulsive fractional order partial di?erential equations with delay on time scales. For a partial di?erential equation containing two variables, using the negativity of characteristic function for the equivalent integral equation, constructing functions and transforming the equivalent binary integral equation into an integral equation with one variable, and combining Schauder’s fixed point theorem, we can get the su?cient conditions for the existence of solution.