Fixed Point Theorem of a Sum Operator in Set and Applications

In the paper, by using the cone theory and monotone iterative method, the sum operator equation Ax + B(x, x) + Cx = x has been considered in P-h,(e), where A is an increasing phi - (h, e)- concave operator, B is a mixed monotone operator and C is a decreasing operator. Then, the existence and uniqueness of solutions for a class of fractional differential equations with nonlinear boundary are discussed as an application. We not only gain a unique solution but also construct two iterative sequences to approximate the solution. The conclusion given by the paper provides new methods to deal with a class of nonlinear differential equations. At last, a concrete example is given to support our results.