Existence Of Solutions For A Class Of Nonlinear Differential Equations

Based on the study of [30]-[32], in this paper, we will study the existence of solutions for a class of nonlinear fractional differential equations.For the linear ordinary differential equations of integral order, we have learned very much and have been widely used in real life. With the development and deeply research of the subject, it is generalized and reformed on the linear ordinary differential equations, in the recent years, many researchers began to make deeply and extensively research on nonlinear fractional differential equations, and many classical conclusions have been obtained, for example the literature of [5], by using the super-linear and sub-linear conditions to prove the existence of solution; the literature of [6]-[9], by using the fixed point theorem on partially ordered sets to obtained the existence of solution; the literature of [10]-[13], applied the iterative method to prove the existence of solution, in addition, literature [17], [22], [24]-[29], applied the other different methods to solve the existence of solutions for nonlinear fractional differential equations.In this paper, inspired by literature of [1]-[4], [14]-[16], [18]-[23], we researched exis-tence and uniqueness of solution for the high-order Caputo type of fractional differential equation with integral boundary value condition and existence and uniqueness of solu-tions for Riemann-Liouville type of fractional differential equations with the nonlocal boundary value problem.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 high-order Caputo type of fractional differential equation with integral boundary value condition: existence and uniqueness of solution, which n-1≤α<n, n is natural numbers, i0∈[1,n-2] is fixed integer and α>2, α-i0>1, cD0+α as the standard of Caputo type differential,f∈C([0,1]×R+, R+) is the nonlinear terms, g(t)∈C[0,1], it allows the function h(t) on t=0,t=1 be singular. In this paper, by using the properties of Green function and the theory on partially ordered Banach space to obtain the existence and uniqueness of positive solution, through the establishment of highly function to obtain the conclusion that the positive solution is strictly increasing. Furthermore, in this paper, we also attempts to use fixed point theorem for weakly contractive mappings in partially ordered sets to prove existence of the positive solution for the researched equation.In Chapter 3, we consider for Riemann-Liouville type of fractional differential e-quations with the nonlocal boundary value problem: existence and uniqueness of solution, which 2<α<3, A0, B0∈R+ and B0 Ao, A0+B0<Г(α-1),f:[0,1]×R3â†'R+ is continuous function, D0+α as the standard of Riemann-Liouville type differential. In this paper, by applying the iterative technique to obtain the conclusions.