Convergence Analysis Of The Solutions Of Some Kinds Of Nonlinear Differential Equations

This paper focuses on the convergence of solutions of several kinds problems of nonlinear differential equations.It is very difficult to find the exact solutions of nonlinear differential equations due to the complexity of the real world.Therefore studying the approximate solutions of nonlinear differential equations is very important in both theoretical and practical sence.The quasilinearization method is one of the effective methods to obtain approximate solutions of nonlinear differential equations.This paper discusses three kinds of nonlinear differential equations with the sum of two terms as the right hand function,including fractional differential equations,set-valued differential equations and set-valued differential equations on time scales.The main content is as follows.Firstly,we study the convergence for the solutions of Caputo fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions of order8)(8)? 1).Comparison principle corresponding the equations is constructed to obtain the sequences of approximate solutions.Then we establish the convergence of order 6)(6)? 2).Secondly,the convergence of the solutions of periodic boundary value problems for setvalued differential equations is discussed.We define the partial derivatives of set-valued functions in the sense of Hukuhara derivative,give the comparison principle of set-valued differential systems,employ the generalized quasilinearization method combined with the theory of upper and lower solutions to construct the sequences of the approximate solutions and obtain the uniform and quadratic convergence of the above sequences.Finally,we introduce the convergence of the solutions of set-valued differential equations on time scales.The comparison principle of set-valued differential systems on time scales is given.By introducing the concept of partial derivatives of set-valued functions on time scales in the sense of Hukuhara derivative,and using the generalized quasilinearization method,the results of the quadratic convergence of the approximate solutions are obtained.