In this chapter, we shall establish a survey of the fundamental fixed point theory both for single-valued mapping and multi-valued mappings. Precisely, we study some new variants of fixed point results of multi-valued mapping in many spaces such as Banach space, Banach algebra, Banach algebra satisfying the condition \((\mathscr {P}), \) generalized Banach algebra and RWC-Banach algebra, by using an analog of Dhage’s fixed point theorems and, by using an analog the well-known Perov’s fixed point theorem.