The Research Of Qualitative Theory Of Some Classes Of Functional Differential Equations And Their Application

The fractional calculus is an extension and expansion of the integer order calculus on the basis of a discipline, sub-properties and applications of theory is the study of the calculus of any order operator, but also accompanied by the development of fractional calculus equations, so it has a profound theoretical significance and a wide range of practical applications. In the recent years, with the fractional calculus theory is widely used in the fields of physical, mechanical, biological, ecological, and engineering, the fractional differential equations has been the concern of many scholars and researchers. Meanwhile, the linear and nonlinear fractional differential equations boundary value problems caused widespread concern in many mathematical researchers, existence of solutions for theoretical research and application has become a hot issue.The paper is divided four parts.In the first part, we introduce the history of the development of the theory of fractional calculus, fractional differential equations, at home and abroad on the existence of solutions of boundary value problems of fractional differential equations research, the basic definition of the fractional differential equations and lemma. Riemann-Liouville fractional differential equations, the concept of the Caputo fractional differential equations, the meaning of various symbols and definitions, theorems, lemmas and propositions of this chapter need theoretical knowledge.In the second part, we using the Green function properties to study a class of nonlinear fractional differential equations boundary value problems.In the third part, we discuss the existence of positive solutions for a class of fractional nonlinear fractional differential equations boundary value problems with parameters. By the properties of the Green function and Krasnosel’skii fixed point theorem on a cone, eigenvalue intervals of nonlinear fractional differential equations boundary value problems are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. In the fourth part, we make a summary of the article, and the outlook for the fractional calculus equations.