Existence Of Solutions For Boundary Value Problems Of Fractional Dynamic Equations On Time Scales

Fractional calculus is a discipline for arbitrary order derivative and integral, it is an extension and expansion of integral-order calculus. However, it has a big difference with integer-order calculus. The study on the theory for the boundary value problem of fractional differential equation has been widely used in physics, biology, communications engineering and other fields. Fractional difference equation appears in rheology, self-similarity in rheological dynamics and porous structure, popular, power network, viscoelastic, chemical physics and many other branches of science addition to the field of mathematics. There are similarities and differences between fractional differential equation and integer order differential equation, and there are also similarities and differences between fractional differential equation and fractional difference equation. Many experts and scholars started to combine the theory of time scales calculus with fractional calculus. They paid close attention to the research of fractional dynamic equations and their applications on time scales. The research can study continuous and discrete systems in a unified framework. It can provide the theoretical basis for the development of differential equations, and it has important significance and value.This paper is devoted to the study of the existence of solutions for the boundary value problem of fractional dynamic equation, which includes boundary value problems with parameters, hybird boundary value problems, boundary value problems with ?-Laplace operator, boundary value problems with integral conditons and boundary value problems on time scales and other circumstances. We study the existence, multiplicity, uniqueness of solution or positive solution and obtain some new results.In chapter one, we introduce research background, history of development and present situation of fractional calculus, present situation and significance of research of the boundary value problems of fractional dynamic equation. We list some basic definition of fractional calculus theory, some related lemmas and the main tool used in this paper. Also we give the main content of this paper.In chapter two, we investigate the existence of solutions for boundary value problems offractional dynamic equation with parameters. By using Guo-Krasnosel'skii fixed point theorem, some sufficient conditions of the existence of solutions are obtained.In chapter three, we consider the existence of solutions for boundary value problems of fractional hybrid differential equation. By using fixed point theorem, some existence results of solutions are obtained.In chapter four, we discuss the existence of positive solutions for boundary value problems of fractional differential equations with ?-Laplace operator. By means of Schauder fixed point theorem, some sufficient conditions of existence of positive solutions are obtained.In chapter five, we study the existence of solutions for boundary value problem of fractional differential equations with integral conditions. By using Banach contraction principle, Schaefer fixed point theorem and Schauder fixed point theorem, some sufficient conditions of the existence and uniqueness of solutions are obtained.In chapter six, we study the existence of solutions for boundary value problem of fractional dynamic equations on time scales. By using Banach contraction principle and Schauder fixed point theorem, some existence results of solutions are obtained.In chapter seven, we summarize the main results and the innovations in this paper.Finally, we prospect some future research work based on this paper.