Existence Of Solutions For A Class Of Nonlinear Fractional Differential Equations

Along with the development of economy and the progress of sci-ence and technology, people’s widespread interest have been aroused by all kinds of nonlinear problems. Nonlinear analysis and its ap-plication, because of its wide application in the research of various natural phenomenon, has become one of the important branches of modern mathematics. And it diffusely attract the attention of do-mestic and foreign mathematics and natural science. Nonlinear dif-ferential equations and the corresponding boundary value problems root in physics, engineering, chemistry, control theory and other dis-ciplines. It is one of the most animate research fields in nonlinear analysis and Applications. Among which the fractional differential equations has attracted much attention in recent years. Basing on fractional order model is more accuracy than integer order models, the research about boundary value problem of fractional differen-tial equations has become a hot subject recently. It is one of the strategic territory of differential equations research. In this paper, by using the cone theory and the fixed point theory, we discuss the existence and properties of solutions of several fractional differential equations. Meanwhile we apply the main results to the correspond-ing boundary value problem of fractional differential equations.The thesis is divided into four sections according to contents.Chapter1Preference, we introduce the main contents of this paper.Chapter2In this chapter, we use the fixed point theory and Avery-Peterson fixed point theory to investigate the existence of multiple positive solutions to following nonlinear fractional integral boundary value problem: where cD0+αis Caputo fractional derivative of order α,2<α<3,0<λ<2, ξ, ζ∈C(J,J), J=[0,1], ξ(t)> t are delayed and advanced arguments, and f∈C([0,1]×R+×R, R+). We establish the existence result of at least triple positive solutions of boundary value problem (2.1.1), and an example is given to demonstrate the application of our main results.Chapter3In this chapter, we study the delay fractional equa-tion with integral conditions where D0+αis Rieman-Liouville fractional derivative of order a,1<α<2,0<λ<α,0<τ<1, η(ｔ)∈C([-τ,0]), η(t)>0,t∈[-τ,0), and η(0)=0,f∈C([0,1] x [0,∞),[0,∞)). In this chap-ter, we use Guo-Krasnoselskii fixed point theory to investigate the existence of positive solutions of problem (3.1.1), and an example illustrate some results.Chapter4In this chapter, we consider the existence of mul-tiple positive solutions to following system of nonlinear fractional differential equations where n-1<α<n, n>3, n∈N+, and f∈C([0,1]×R+×R, R+). By proving the good properties of Green function, and using Avery-Peterson fixed point theorem we establish the existence result of at least triple positive solutions to the system of fractional differential equation (4.1.1). Example illustrating our main result is also given.