Executive Summary : | Functional Analysis is a recent development of basic mathematical analysis that has significant applications in various fields such as Physics, Chemistry, Biology, Economics, and Engineering. It has two branches: linear functional analysis and nonlinear functional analysis. Fixed point theory, a major part of nonlinear functional analysis, deals with the existence and uniqueness of points x for a self-mapping f on a domain X, where f (x) = x. It can be divided into three branches: metric fixed point theory, topological fixed point theory, and discrete fixed point theory. The Banach contraction principle is a fundamental theorem in metric fixed point theory, with numerous generalizations. Topological fixed point theory is based on schauder's main result. Advancements in calculus have made it easier to understand practical problems and physical systems, allowing for the study of solutions and prediction of output. Fractional calculus, a generalized form of ordinary calculus, includes ordinary differentiation and integration and has numerous applications in various scientific and technological disciplines. Using fixed point techniques, we can study the dynamics of scientific problems more effectively. This proposal aims to establish the existence and uniqueness of solutions to certain types of integral equations and fractional integral equations, as well as the controllability and stability of systems of integral and fractional integral equations. |