Fractional Differential Equations With Nonlinear Terms Of Different Properties

Fractional differential equations have good memory and heredity in the process of describing practical problems,and can more accurately and objectively describe the properties of nonlinear phenomena or states.They have been widely applied in economics,biomedicine,physics,electrochemistry and other fields.In practical application,many complex problems can be described by fractional differential equation models with complex nonlinear terms.Since nonlinear operator theory is an important tool to investigate the existence and uniqueness of solutions for fractional differential equations,this paper mainly uses nonlinear operator theory and monotone iterative technique to study several kinds of fractional differential equations with nonlinear terms of different properties,not only obtaining the existence and uniqueness of the solutions for the equations,but also constructing two iterative sequences to approximate the solution,generalizing and enriching the research results of related problems.The main research content has the following three parts:The first part discusses the fractional differential equations with nonlinear terms having derivatives.Their nonlinear terms have derivatives,and boundary value conditions have the cases of derivatives of different orders.The main research idea is to first transform the differential equations into the corresponding integral equations,and then investigate the properties of Green’s function.By virtue of fixed point theorems for two classes of nonlinear sum-type operators,we theoretically prove the existence and uniqueness of the solutions.Finally,two examples are given to verify the correctness of the conclusions.The second part is in a ordered Banach space,by using the fixed point theorem for mixed monotone operators,we discuss the fixed point theorems for a class of nonlinear sum-type operators.Then they are used to study the existence and uniqueness of the solutions for fractional differential equations with sum nonlinear terms.At the same time,an example is given for verification.This study is more general and generalizes the relevant results in the existing literature.The third part is by using the fixed point theorems for a class of nonlinear sum-type operators,we establishes the sufficient conditions for the existence and uniqueness of the solutions for fractional differential equations with concave-convex nonlinear terms,and gives an example for verification.