Research On The Solutions Of Several Kinds Of Fractional Differential Equations

Nonlinear functional analysis is a research subject in applied mathematics that has both profound theory and widespread application. It takes the nonlinear problems ap-pearing in mathematics and natural sciences as background to establish some general theories and methods to handle nonlinear problems.The thesis is divided into three chapters. The chapter 1, we study the following a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations where cD0+?, cD0+?,D?v(t), D?u(t) stand for Caputo fractional derivative, 1 < ?, ?<2, 0 < ?, ? ? 1, ?, ??C([0,1] × R × R × R, R), h, g, k, f ? C(R,R),are Riemann-Stieltjes integrals, A, B, C, D are functions of bounded variation. As given above, u(tj+),u'(tj+) are right and u(tj-), u'(tj-) are left limits, respectively, t =tj (j = 1,2,...,m).v(ti+),v'(ti+) are right and v(ti-), v'(ti-) are left limits, respectively,t = ti (i = 1,2,...,n),0 < t1 < ... <tm < 1, 0 < t1 < ... < tn < 1, Ir, Ir (r = j,i)?C(R,R), By using the classical fixed point theorems such as Banach contraction principle and Krasnoselskii's fixed point theorem, we gain the existence and uniqueness of solutions. Compared with the document [9], not only the nonlinearities of the equation (1.1.1) depending on the un-known functions as well as their lower order fractional derivatives, but also adds a integral boundary value conditions. Compared with the document [10], we use the Krasnoselskii fixed point theorem and Banach contraction principlethe to obtain sufficient conditions for existence and uniqueness of solutions to the system, and the boundary conditions of this paper includes impulsive term of unknown functions?, ?.The chapter 2, we study the following the existence and uniqueness of solutions for high order fractional boundary value problemswhere cD0+? cD0+? stand for Caputo fractional derivative, n - 1 < ? ? n, n - 2 < ? ?n - 1, I= [0,1], f is continuous with I × R × R ? R. By using the method of upper and lower solutions and the maximum principle, we gain the existence and uniqueness of solutions. This chapter mainly generalizes the paper [21] in equation form and the boundary conditions. The problem studied by the paper [21] is a special case of this paper, and the nonlinearities of the equation (2.1.1) depending on the unknown functions as well as their lower order fractional derivatives. We use the method of the document[20] to reduce the order of our equations for reference.The chapter 3, we study the following explicit iteration and unbounded solutions for fractional integro-differential equations on an infinite intervalwhere D? is the Riemann-Liouville fractional derivatives,I? is the Riemann-Liouville fractional integral, E is banach space, ? > 0, t ? J = [0, +?), f ? C[J × E × E × E, E],? refers to the zero in the space E,with k(t,s) ? C[D,R], h(t,s) ? C[D0,R], D = {(t,s) ? R2 | 0 ? s ? t}, D0 = {(t,s) ?J × J}. By introducing the monotone iterative technique and Banach fixed principle,we gain the explicit iteration and unbounded solutions of the problem. Compared with the document [32], not only the nonlinearities of the equation (3.1.1) depending on the unknown functions as well as the two nonlinear integral term, but also adds a integral boundary value conditions, and the range of the study is more extensive from a limit-ed interval to an infinite interval. Compared with the document [30], nonlinear term f contains integral operator, and the study is more extensive.