Fixed Point Theorems for Sum Operator with Parameter

This paper develops some new existence and uniqueness theorems of a fixed point for a class of sum operator equations with parameter $\lambda _{1}A(x,x)+\lambda _{2}B(x,x)+\lambda _{3}Cx+\lambda _{4}Dx=x$, where A, B are two mixed monotone operators, C is an increasing operator, D is a decreasing operator. In the case of positive parameters, the results obtained in this paper extend many existing conclusions in the field of study. Furthermore, by using the properties of Green’s function and the above fixed point theorems of sum operator, the unique positive solution a class of fractional differential equations with integro-differential boundary value conditions is given. Application of the results to the study of fractional differential equations is also given in the article.