Fixed point theorems for multi-valued mappings obtained by altering distances

Recently T. Suzuki showed that the Mizoguchi–Takahashi fixed point theorem is a real generalization of Nadler’s fixed point theorem. Taking inspiration from the result of Mizoguchi and Takahashi and using the ideas of Feng and Liu, Klim and Wardowski obtained some fixed point theorems and showed that their results are different from the Reich point theorem and the Mizoguchi–Takahashi fixed point theorem. Very recently, Pathak and Shahzad introduced a class of functions and generalized some fixed point theorems of Klim and Wardowski by altering distances, i.e., via the mapping TT (from a complete metric space (X,d)(X,d) to the class of nonempty closed subsets of XX). In this paper we introduce a new class of functions which is a subclass of the class introduced by Pathak and Shahzad and improve some results of Pathak and Shahzad by allowing TT to have values in closed subsets of XX.