Coincidence and fixed points for multi-valued mappings and its application to nonconvex integral inclusions

In this paper, we consider some problems on coincidence point and fixed point theorems for multi-valued mappings. Applying the characterizations of PP-functions, we establish some new existence theorems for coincidence point and fixed point distinct from Nadler’s fixed point theorem, Berinde–Berinde’s fixed point theorem, Mizoguchi–Takahashi’s fixed point theorem and Du’s fixed point theorem for nonlinear multi-valued contractive mappings in complete metric spaces. Our results compliment and extend the main results given by some authors in the literature. In the sequel, we consider a nonconvex integral inclusion and prove the Filippov type existence theorem by using an appropriate norm on the space of selection of a multi-function and a multi-valued contraction for set-valued mappings.