| Differential equations have been studied for a long history, which originates from practical problems, such as gas dynamics, nuclear physics, material mechanics, trajectory calculation, the study of stability of aircraft and missile flight, the research of stability of the chemical reaction process, and so on. In fact, the research history of fractional differ-ential equation is almost as long as the study of integer order differential equations. With the rapid development of high technology, fractional derivatives and fractional differen-tial equations have been widely used in the fields of science, engineering, mathematics, and other fields. Such as they have been successfully applied in the study of viscoelastic material, signal processing, control, fluid dynamics, thermodynamics, biology, etc. It was found that in the process of establishing system model of some practical problems, fractional differential equations may be closer to the actual situation than integer differ-ential equations, the importance of fractional differential equations has been realized by mathematical research, as one of the important branch for fractional differential equa-tion theory, various boundary value problems to fractional differential equations get the attention of researchers in recent years, Many results in this area have been obtained.In this paper, we mainly investigate the existence of solutions to some classes of coupled system of nonlinear fractional differential equations, using nonlinear analysis method, we get the existence of solutions for the coupled system of nonlinear fractional ordinary differential equations. The topics include the existence of solutions when the nonlinear terms have different forms under the three-point boundary value conditions and the anti-periodic boundary conditions.This paper is divided into seven chapters, which are outlined bellow:The first chapter is an introduction. We first present the necessary definitions of fractional integration and fractional differentiation, next we describe the history of frac-tional differential equation research, introduce briefly the related works obtained by the mathematicians both in China and aboard, then we state our problems and draw the main conclusions, at the end we introduce some methods and techniques that we shall use.In chapter2, we study the coupled system of nonlinear fractional differential equa-tions with three-point boundary conditions: where3<α,β<4,m,n,γ>0,0<η<1,α-n≥1,γηα-1＜1,<γηβ-1,γηβ-1<1, Dα is the standard Riemann-Liouville derivative. Here our nonlinearity f1,f2:[0,1]×R×→R are given continuous functions.As strongly coupled system, the nonlinear terms involve both of the unknown func-tions u, v and the derivatives of unknown functions, so we should noted that the skills when constructing the space are very important. We first calculate the Green’s func-tion of corresponding linear equations, then give existence results of (1) by means of the Schauder fixed point theorem, we obtain the uniqueness of solutions for system (1) by the contraction mapping principle, at the end two examples are given to illustrate the applicability of our results.In chapter3, we study the nonlinear fractional differential equation with m point boundary conditions: where m Dα is the standard Riemann-Liouville derivative. Here our nonlin-earity f1,f2:[0,1]×R×→R are given continuous functions.Compared with the problem in chapter2, the problem in this chapter has a more general form. We first calculate the Green’s function of corresponding linear equations, then give existence results of (1) by means of the Schauder fixed point theorem, next we obtain the uniqueness of solutions for system (1) by the contraction mapping principle, at the last two examples are given to illustrate the applicability of our results.In chapter4, we consider the existence and uniqueness of solutions for the coupled system of nonlinear fractional differential equations. As follows: where cDα denotes the Caputo fractional derivative of order a. Here our nonlinearity f:[0, T]×X×X×X→X is given continuous function. And γ,δ:[0, T] x [0, T]→[0,00], φ,ψare integral operators:The features of this anti-periodic boundary value problem is, our nonlinear term contains integral operators about unknown function. By Applying the Schauder fixed point theorem, We get the existence results of solutions, and two examples are given to illustrate the applicability of our results.In chapter5, we consider the existence and uniqueness of solutions for the coupled system of higher-order nonlinear fractional differential equations having generalized form. As follows: where4<α,β≤5, cDα denotes the Caputo fractional derivative of order a. Here our nonlinearity f, g:[O,T] x R→R are given continuous functions.For this anti-periodic boundary value problem, one can easily conclude from [32] that Green’s function (4)(or solution) for an anti-periodic boundary value problem of fractional higher-order contains Green’s function (or solution) for lower-order fractional anti-periodic problems. Hence, our results generalize the existing results on anti-periodic fractional boundary value problems: where i=0,1,2,3. i<α, β≤i+1.We give the existence results of (5) by means of the Schauder fixed point theo-rem, then obtain the uniqueness of solutions for system (5) by the contraction mapping principle, and two examples are given to illustrate the applicability of our results.In chapter6, we consider the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations: where4<α,β≤5, cDα denotes the Caputo fractional derivative of order a. Here our nonlinearity f, g:[O, T]×R×R→R are given continuous functions. We give the existence results of (6) by means of the Leray-Schauder alternative, then we obtain the uniqueness of solutions for system (6) by the contraction mapping principle. We give two examples to illustrate the applicability of our results.In chapter7, we consider the coupled system of nonlinear fractional differential equations with more complex forms: where4<α,β<5,α-q≥1,β-p>1, cDα denotes the Caputo fractional derivative of order α, f, g:[O, T]×R×→R are given continuous functions.As the nonlinear terms containing the derivatives of unknown functions, we need to construct a special space and prove that this space is a Banach space. we prove the existence of solutions to (7) by means of the Schauder fixed point theorem. We obtain the uniqueness of solutions to the system by the contraction mapping principle. At the end, two examples are given to illustrate the applicability of our results.Due to that chapter6and chapter7have the same form of Green’s function as chapter5, one can easily conclude from that the solutions of problems (6),(7) contains the solutions for lower-order fractional anti-periodic problems. Hence, our results in this two chapters generalize the existing results on anti-periodic fractional boundary value problems. |
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