Nonlinear Fractional Differential Equations Positive Solution

With the development of modern physics and applied mathematics, it de-mands the mathematical ability of analyzing and controling the objective phe-nomena toward to the overall high and precision level, which made the results of the nonlinear analysis was accumulated, and gradually formed an important branch subject of the present analysis mathematics-Nonlinear functional analy-sis. Nonlinear functional analysis is a research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general theories and methods to handle nonlinear problem. Because it can commendably explain all kinds of natural phenomenal, coupled with the widely application in the realistic production and life, it has received highly attention of the domestic and foreign mathematics and natural science field in recent years.The boundary value problem of nonlinear stems from the applied mathe-matics, the physics, the cybernetics and each kind of application discipline. It is an important kind of question in the differential equations, it is one of most active domains of functional analysis studiesin at present.The singular nonlinear differential equation boundary value problem is also the hot spot which has been discussed in recent years. So it attachs more and more attention.Fractional differential equations arise in many engineering and scientific dis-ciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium, poly-mer rheology involves derivatives of fractional order. Fractional differential equa-tions also serve as excellent tool for the description of hereditary properties of various materials and processes. In consequence, fractional differential equations have been of great interest.In this paper, we use the fixed point theory and the nonlinear iterative schemes, to study some kinds of boundary value problems for fractional differen-tial equations.The thesis is divided into three chapters according to contents.In chapter 1, using Schauder fixed point theorem, we discuss the existence of positive solutions to the following nonlinear singular fractional equation with three-point boundary value problem where 1＜α,β＜2,f, g is continue in (0,1]×[0,+∞)×R)â†'[0,+∞), and limt∴0+f(t,·)= limtâ†'0+g(t,·)=+∞, p, q,γ＞0,0＜η＜1,α-q≥1,β-P≥1,γηα-1＜1,γηβ-1＜1,D is the standard Riemann-Liouville fractional derivative.In chapter 2, we study the existence of positive solutions for a multi-point boundary value problem on an infinite interval by using the nonlinear iterative technique and combining with the relevant knowledge of fixed point theroy where J+=(0,∞),1＜α＜2,αi∈J,ξi∈J+,0＜ξ1＜ξ2＜…＜ξm-2＜+∞,0＜∑(?)αi＜1.q(t) is a nonnegative function defined on J, and q(t) doesn't identically vanish on any subintervals of J+, and 0＜∫0 +∞q(s)ds＜+∞.f is continue on J×Râ†'R, and f does not equal to 0 on any subintervals of J, when u is bounded, f(t, (1+t)u) is bounded too.In chapter 3, by employing the Leray-Schauder nonlinear iterative technique and the relevant knowledges, we study the existence of solutions for fractional differential inclusions with boundary conditions where c D is the Caputo derivative, F:J×Râ†'2R\Φ. u0,u1∈R,αi＞0, 0＜ξ1＜ξ2＜…＜ξm-2＜1, d=∑(?)αi＜1(i=1,2,…,m-2).