The Fixed Point Theorems For Several Sum Operators And Their Applications

By using the properties of cone and fixed point theorems for nonlinear operators, we study several kinds of sum operator equations on ordered Banach spaces, and the existence and uniqueness of their positive solutions are obtained. As applications, we utilize the obtained results to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems. Our results extend and improve the related conclusions in literatures, besides, it provide a new way to study the differential equations. The overall structure of this paper is as follows:In Chapter 1, the background of the discussed problems and the necessity of our study in this paper are introduced briefly. Meanwhile, the main works of this paper are stated in detail.In Chapter 2, by using the properties of cone and fixed point theorems for nonlinear operators with concavity and convexity, we study the existence and uniqueness of the positive solutions for three kinds of sum operator equations Ax+C(x,x)=x Bx+C(x,x)= x Ax+Bx+C(x,x)=x where A is an increasing operator, B is a decreasing operator, C is a mixed monotone operator, and obtain the corresponding fixed point theorems. The research methods are different from that in related literatures.In Chapter 3 and chapter 4, by using the results obtained in this paper, we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation two-point boundary value problems and three-point boundary value problems and Lastly, as applications, we give related examples to illustrate our results respectively.