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In this paper we study a class of sum operator equation \(Ax+Bx+C(x,x)=x\) on ordered Banach spaces, where A is an increasing operator, B is a decreasing operator, and C is a mixed monotone operator. The existence and uniqueness of its positive solution are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems