Solutions Of Fractional Differential Equations With Nonlinear Boundary Value

With the development of science and technology, all kinds of nonlinear prob-lems have attracted more and more attention. Nonlinear analysis has become one of the most important research direction in modern mathematics. The analysis and application of nonlinear problem is an important branch in nonlinear anal-ysis. For the reason that it can explain different kinds of natural phenomenon, more and more papers have studied the problem.Non-linear fractional boundary value problems came from applied Mathe-matics, Physics, Control theory and some other applied disciplines. It is the most active areas in modern applications for nonlinear fractional differential e-quations. The fractional differential equation with nonlinear boundary problems has became the hot topic. Many people had studied this problem, there has been many papers on this problem of studying the existence and the uniqueness of solutions for the equation. By means of some cone theory, fixed point theory and the upper and lower solution method, this paper mainly study kinds of fractional differential equations with nonlinear boundary value problems.The thesis is divided into three sections according to contents.In Chapter 1, By using the cone theory and Banach contraction mapping principle, we mainly discuss the solutions for nth-order nonlinear Caputo frac-tional differential equations with integral boundary.where α, β,γ,δ are nonnegative constants satisfying σ=αγ+αδ+βγ>0, f∈C([0,1]×R+,R+), g, gk, h∈C([0,1],R), R+= [0,+∞). And cDq is the Caputo fractional derivative of order q, q∈R.2≤n= [q]≤q<[q]+1, [q] denotes the integer part of the real number q and R+= [0,+∞).In Chapter 2, We mainly study extremal solution for fractional differential equations with nonlinear boundary conditions Where f∈C([0,1]×R,R), g ∈C(R×R,R), p(t) ∈C[0,1], q(t) ∈C[0,1], n<α< n+1,n≥2,n∈N,η∈(0, 1). CDα stand for the Caputo fractional derivative of order α,α∈R.2≤n= [α]≤α<[α]+1≤n+1, [α] denotes the integer part of the real number a and R+= [0,+∞). By means of studying the linear fractional differential equation we can make the primary equation into specific one which is an upper or lower solutions for the primary equation. By using the upper and lower solutions method and monotone iterative technique we will prove the existence of extremal solutions.In Chapter 3 we mainly discuss solutions for the following fraction differential equationswhere n<α≤n+1, n≥2, n∈N, t∈[0,1],f∈C([0,1]×R×R,R),m≥1 is integer and ζi> 0.