φ − (h,e)-concave operators and applications

In this article, by introducing a new set and a new concept of φ−(h,e)-concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ−(h,e)-concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems.