Solutions Of Fractional Differential Equations Under Nonlinear Boundary Value Conditions

Recently,fractional differential equations have drawn more and more attention by the research communities due to their numerous applications in various fields of science such as engineering,chemistry,physics and mechanics.Boundary value problems of fractional differential equation have been investigated for many years.Now,there are many papers dealt with the problem for different kinds of boundary value conditions,such as two-point,three-point,multi-point boundary condition,integral boundary condition etc.With the continuous research of fractional order derivatives,the fractional differential equations in which the fractional order derivatives are involved in the nonlinearity or in the boundary value conditions have learned by many papers.On the other hand the existence and uniqueness of positive solutions of fractional differential equations for all kinds of boundary value conditions have studied in many papers.Based on this,in this paper,we studied the boundary value problem involving fractional derivatives.By using the fixed point index theory,the cone stretching and compression theory and mixed monotone method,we obtain the existence and uniqueness of the positive solution for the problem.The thesis is divided into two chapters according to contents.In Chapter one,by using the properties of the Green function and the fixed point index theory,we obtain some results on the existence of positive solutions for the singular nonlinear fractional differential equations when the nonlinear term is singular.In Chapter two,by using the fixed point theory in cone we study the existence of positive solutions for the singular fractional differential equation with integral boundary value conditions,and through the mixed monotone method we also obtain the uniqueness of positive solutions when the nonlinear term is singular.