Research For Solutions Of Boundary Value Problems For Fractional Differential Equations

This thesis mainly studies the existence of positive solutions for a frac-tional boundary value problem on the half-line, the estimation of fractional derivatives at extreme points and the existence of solutions for Riemann-Liouville fractional boundary value problem under Nagumo conditions, the existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions, and the existence of solutions of bound-ary value problem of mixed fractional derivatives. This thesis consists of five chapters and it is arranged as follows.In Chapter one, we introduce the historical background and research sta-tus of fractional differential systems and basic knowledge, which will be used later. Meanwhile, we state our main works of this paper.In Chapter two, we establish the sufficient conditions of the existence of positive solutions for a Riemann-Liouville fractional boundary value problem on the half-line by structuring iterative sequences and using fixed point theory.In Chapter three, we firstly obtain the estimation of Riemann-Liouville fractional derivatives and Caputo fractional derivatives at extreme points re-spectively, then investigate the existence of solutions of Riemann-Liouville frac-tional boundary value problem under Nagumo conditions. Some of our results paly an important role in establishing fractional comparison principles in the next chapter.In Chapter four, we establishing Riemann-Liouville fractional comparison principles of the order q ∈ (1,2) and further study the existence of extremal solutions for Riemann-Liouville fractional boundary value problem with three-points nonlinear boundary conditions by means of the method of upper and lower solutions. Our results are new.In Chapter five, with the aid of calculus of variations and critical point theory, we study the existence of solutions of boundary value problem of mixed fractional derivatives, where the nonlinear term contains a fractional deriva- tive. Our results may be seen as the extension of the classical results of integer order to the case of fractional order.