Some fixed point results in fuzzy metric space

Abstract

Analysis is the most important branch of mathematics. Among several branches of analysis, functional analysis is the most important part of anal- ysis. Functional analysis is divided into two parts: linear and non-linear. Fixed point theory is an important part of non-linear functional analysis s- ince 1960. Fixed point theory is one of the most dynamic areas of research from last 60 years, with lots of applications in various elds of pure and ap- plied mathematics, as well as, in physical, economic and life sciences. It is a fully developed branch but still continues to be an active and very wide open area of research. It has emerged as one of the major links between abstracts mathematics and its applications. It provides a powerful tool in demonstrating the existence of solutions to a large variety of problems in applied mathematics. It is used mainly in the existence theorem of di erential equations and integral equations. It is al- so used in arti cial intelligence, computer science, decision making, medical diagnosis, neural network, social science and many other related areas. It has very fruitful application in Eigen value problems and boundary value problems. The xed point theory deals with the classical approach to nd the exact solution and to check the stability of the system. In this thesis, we have established some common xed point theorems in metric space and fuzzy metric space which generalizes and improves existing similar results in the literature. It includes basic de nitions and some xed point theorems. We have obtained two common xed point theorems in metric space using reciprocal continuous, compatible mappings of type (E). Also, we have introduced a new compatible mappings of type (K) and obtained a common xed point theorem. some common xed point the- orems in Fuzzy metric spaces using compatible mappings of type (E) and compatible mappings of type (K) which generalizes and improves other similar results in the literature. It includes basic de nitions and those theorems specially having the relevance for the establishment of our theorems. CHAPTER FOUR is intended to obtain some common xed point theo- rems in Intutionistic Fuzzy metric spaces using compatible mappings of type (K) which generalizes and improves other similar results in the literature . It includes conclusion and some future scope. The list of literature consulted has been placed at the end of the thesis as Bibliography. Our original contributions has been contained in chapters 2, 3 and 4. A part of the research work contained in this thesis has been already published in international peer reviewed journal [79], [82],[111], [112], [113],