In the last few decades, the landmarks in the field of functional analysis are mainly the new imaginative ways to use the fundamental tools of functional analysis, either in theory where they had not been applied before or in the construction of more powerful methods to handle functional analysis. This book can be looked at as a study in a part of pure mathematics and application of the same in the Fractal theory. In the first four chapters, the developments of various mappings and iteration processes has been studied and theorems are proved which shows improvements over many well known results. In the fifth chapter a study on Fractal dimension has been done. Where as in the last chapter improvements in Iterated Function System has been explored.