Research On The Solution Of Nonlinear Differential Equations With Perturbation Terms

The paper studies the solutions of nonlinear differential equations with perturbing terms.We are very familiar with the nonlinear integer order ordinary differential equations through the continuous research of them by experts and scholars, which are widely used in physics, biology, economics and many other fields. With the development of science and constantly in-depth study, by promoting and reforming nonlinear ordinary differential equations, we have done some research on nonlinear differential equations with perturbing terms, and also have obtained breakthrough results. For example,literature [1] proves the existence of the solution of nonlinear differential equation with perturbing terms by making use of Schauder fixed point theorem. And literature [2]and literature [3] obtain the existence of the solution of equations under study by using the cone Krasnoselskii's fixed point theorem.In this paper, inspired by literature of [1] - [5], we researched nonlinear fractional boundary value problem with perturbation.According to the content, the paper is divided by three parts as follows:In Chapter 1, we mainly collect some basic definitions and facts that will be used in this article.In Chapter 2, we consider the boundary value problems of nonlinear fractional differential equations existence of solutions. Which D0+?u?t? is Riemann-Liouville fractional differential equa-tion,?? 2,1<?-? ?n - 1,n - 1 ? ? ? n,?i > 0?i = 1,2,...,m - 2?,0 <?₁ < ?₂ <... < ?_m-2 < 1,f :?0,1? × ?0,?? × ?0,????0,?? is continuous,e?t??L1?[0,1],R?may change sign.In this paper,by using the Schauder fixed point theorem to obtain the conclusions.In Chapter 3, we consider the eigenvalue problems of nonlinear fractional differen-tial equations with perturbation which ? is a positive parameters, D1qx?t? is the standard Riemann-Liouville fractional derivative, qq ? 2,n-1 ? ?n,i ? N, 0 < i < n - 2, aj > 0?j = 1, 2,..., m - 2?, 0 <b₁ < b₂ < … < b_m-2 <1, ???,f ? C?0,1? × ?0,?? ? [0,?)and t 0,1 singular, e?t? ?L1?[0,1], R? may changing-sign. By applying the cone Krasnoselskii's fixed point theorem, some new results are established and an example is given to demonstrate the application of our main results.