Existence Of Solutions For Boundary Value Problems With Nonlinear Fractional Differential Equations

In this paper,we focus on the existence of solutions for boundary value problems with Riemann-Liouville fractional order derivative.The methods used mainly includes various fixed point theorems.We divide this article into five chapters:In the first chapter,we mainly introduce the research background and current sit-uation of fractional differential equations.The definitions of fractional derivative and integral,some important lemmas are given.Several kinds of boundary value problems for fractional differential equations studied in this paper and the main results obtained are briefly introduced.In the second chapter,we study boundary value problems of fractional differential equations as follows#12 where n-1<α≤n,n≥3,D0+α is Riemann-Liouville fractional derivative,I0+ β is Riemann-Liouville fractional integral.k+1≤δ≤n-2 is a positive integer.k∈{0,1,…,n-3},f∈C([0,1]×R+ k+1,R+).In the third chapter,we study boundary value problems of fractional differential equations as follows#12 where n-1<α<n,n>3,β>0,ρ≥0,0<η<1,ρηα+β-1<Γ(α+β)/Γ(α-n+2),f∈C([0,1]×R+ n-2,R+).In the fourth chapter,we study boundary value problems of fractional differential equations as follows(?)where n-1<α≤n,n≥3,k∈{1,2,…,n-1},0<β2≤β1≤α-k,0<γ<β1,μ≥0,0<ξ<1,μξα-k-β2<Γ(α-k-β2+1)/Γ(α-k-β1+1),(I0+ n-α u)(n-i)represents the(n-i)-order ordinary derivative for(I0+ n-α u).f∈C((0,1)×R2,R).The last chapter,which summarizes the main contents of this paper and prospects the subject.