Existence And Controllability Of Solutions For Several Types Of Fractional Differential Equations

Fractional calculus refers to the differentiation and integration of orders of any real or complex number.Classical integral order calculus is just a special case of its order taking integers.Moreover,when establishing mathematical models to describe complex phenomena or systems,fractional calculus can use fewer parameters,but achieve better characterization.Therefore,the research on fractional differential systems has more theoretical significance and practical application value.This article first introduces the origin and development of fractional calculus,basic concepts and related theorems.It aims to give readers a preliminary understanding of the theory of fractional calculus and lay the theoretical foundation for subsequent work.Then,using these basic theories and some methods and techniques,the existence of solutions of fractional order nonlinear differential equations and the controllability of fractional order differential dynamical systems under several types of boundary conditions are studied.Specifically,using the topological degree theory and the Leray-Schauder fixed point theorem to verify that the fractional differential system with one-sided Lipschitz condition has a unique solution of rotational period boundary value,and basing on this research conclusion,two application examples of neural network model and differential control system with memory function are listed.Using the Leray-Schauder fixed point theorem and Krasnoselskii fixed point theorem,the existence and uniqueness of solutions of fractional nonlinear differential equations with growth conditions under nonlocal conditions are verified from two perspectives.The sufficient conditions for the precise controllability of a class of fractional differential dynamical systems with time-varying delay are studied by Laplace transform,fixed point theory,transfer function theory,and semigroup theory.