(untitled) (Journal of Advances and Scholarly Researches in Allied Education, 2020)
In proving solutions of equation, the Banach theorem has several limitations. In particular, mapping should be continuous, and therefore it does not apply to nonlinear issues where the mapping is interruptive. In 1968, Kannan published a certain point theorem that spreads the Banach theorem. This breakthrough took place. He showed it is not necessary to continue the mapping which is contractive. Following the result of Kannan, there have been so many generalizations of fixed point theorems in Banach. The theorems based on Brouwer and Schauder are the most important results in the fixed point theory. These theorems are used to demonstrate the existence of differential, integral and differential solutions. Krasnoselskii, which obtained the set points of sum of two operators, further generalizes those theorems.Common fixed point theorems for general contractions always need a condition of commutativity and the continuity of one of the mappings. One or more of these conditions are weakened by research. In the study of problems of the common fixed points of non-commuting mapping, the notion of compatibility plays an important role.