Analysis of Common Fixed Point Theorems in Diverse Spaces

Fixed Point (Fixed Point) Theory is a crucial and unexpectedly developing non-linear functional analysis subject matter. Outcomes describing the existence of fixed points are termed as fixed points theorems. Among all the functional parts of Mathematics, the theory of differential equations is of much importance. Its contribution towards Physics and Engineering cannot be neglected. Fixed-Point Theory guarantees the significant solution of these equations. A vast literature on fixed-point theory is available in journals of national and international repute and this is still growing. Fixed Point theorems are used to find solutions to many problems in various disciplines, which provide conditions under which a number of transformations have solutions. Fixed Point theory involves a very transparent and fundamental mathematical setting. This is primarily the logic that made us choose Fixed Point theory as the topic. This Fixed Point Theory is predominantly separated into the three following substantial theories:

Topological Fixed Point Theory
Metric Fixed Point Theory
Discrete Fixed Point Theory

Banach’s and Brouwer's fundamental Fixed Point theory led to the development of two of the scientific main and complementary elements, namely metrical Fixed Point theory and topological Fixed Point theory. The first Fixed Point outcome for a topological space was offered by Dutch mathematician Brouwer in 1912, which state that A continuous self-mapping explained on the closed unit ball in Euclidean space has at least one fixed point. Many authors have proved Fixed Point Theorems either by relaxing the domain or by the function definedon it. Banach in 1922 proved a Fixed Point Theory for contraction mapping, which was defined in complete Metric Space. The Banach theorem states that if it is a complete metric space and is a contraction, then it has a unique fixed point. According to the Brouwer theorem, there must be the closed unit ball in a Euclidean space. But in such a case, the set of all Fixed Points is not compulsorily a one-point set. The point is termed as Fixed Point if, on applying the transformation, the point stands untransformed. There are numerous mathematical issues involving various branches, which can be formulated into equivalent Fixed Point models. The major differentiation between two significant arms that is Topological Fixed Point Theory and Metric Space Set-Point Theory, of Fixed Point Theory is explained by Banach theorem and Brouwer theorem. Due to the wide variety of uses, the Fixed Point Theorems are avidly studied across a multitude of disciplines in the mathematical world. Due to the simplicity and usefulness of Banach's Fixed Point Theorem it came out as a contemporary instrument for the proof of existence and uniqueness of thrs in various boughs of mathematical analysis. In 1968, a contractive condition was introduced by Kannan that hold a unique fixed-point like Banach. However, unlike the Banach situation, a domain wise discontinuity of mappings with fixed points was proved by Kannan. Also, these mappings show a continuous nature at their Fixed Point.

The focus of chapter 2 is to find a fixed point that is common for a rationally contractive pair of mappings in the setting of a complete Complex Valued Metric Space. The result is the generalization of a variety of established theories. C´iri´c -Reich–Rus contraction mapping is described in complex valued Quasi Partial b Metric Space and presented Fixed Point theorem. in the context of complex valued quasi-partial b-metric space in chapter 3. In this study distinct explanation, definitions in addition to notations are used in the discussion of different subsequent. Cyclic C´iri´c-Reich-Rus contraction mapping has been used to show the existence and uniqueness of Common Fixed Point in the setting of complex valued Quasi Partial b Metric Space. The result is supported with suitable examples in this chapter 4.