The Study On Several Classes Of Fractional Differential Equations With Integral Boundary Conditions

In this paper,the existence and uniqueness of solutions for the boundary value problem of three classes fractional differential equations are discussed by the cone and partial order theory as well as fixed point theorem.Meanwhile,the initial value problem for a class of fractional differential equation and its optimal control are also considered.The new results extend and improve some findings in the relevant literatures.The structure of this paper is as follows:In the first chapter,we introduce the background of discussed problem and the necessary of our research in this paper.Moreover,the main results of this paper are stated in detail.In the second chapter,the boundary value problem of fractional differential equation with integral boundary conditions is investigated.we obtain the existence and uniqueness of positive solutions by using the property of green function and the fixed point theorem for increasing operator.In the third chapter,we consider a classes of integro-differential equation with integral boundary conditions.The sufficient conditions that the equation has unique solution are obtained by two different fixed point theorem of sum operator.In the fourth chapter,we study the boundary value problem for impulsive differential equation with integral boundary value conditions.By applying Scheafer fixed point theorem and Banach contraction mapping principle,we get the existence and uniqueness of solution for the equation.In the fifth chapter,we study an initial value problem of fractional differential equa-tion and its optimal control problem.We solve the problem and obtain the existence and uniqueness of positive solutions as well as the existence of an optimal control.