Study of Existence Uniqueness and Comparison Results Using Some Fixed Point Theorems

Abstract

Fixed point theory is one of the most powerful and important tools of modern mathematics. It is a beautiful mixture of pure and applied mathematics and is considered as a core subject of nonlinear analysis. Fixed point theorems provide efficient tools for proving the existence of solutions to a wide range of mathematical problems. Over the last 70 years, fixed point theory has been a flourishing area of research for many mathematicians. The origin of the fixed point theory can be traced in the nineteenth century. This theory was originated in the use of successive approximations to establish the existence and uniqueness of solutions to differential equations. The fixed point technique is associated with many celebrated mathematicians such as Banach, Brouwer, Caccioppoli, Cauchy, Ciric, Fredholm, Kakutani, Kannan, Liouville, Lipschitz, Picard, and Poincare. The abstract formulation by Banach can be considered as the starting point to metric fixed point theory. newlineInitially, the fixed point theory emerged in 1837, in Liouville s article demonstrating existence of solutions of differential equations. In 1890, Picard improved this technique as a method of successive approximations. Later on, in 1922, Banach extracted and abstracted this technique as a fixed point theorem. Banach fixed point theorem is one of the most significant results in fixed point theory. It ensures the existence and uniqueness of a fixed point and also gives an approximate technique to obtain the fixed point. Numerous extensions and generalizations of the Banach fixed point theorem have been established by several researchers. In past few decades the fixed point theory is a significant area of research in pure and applied mathematics. It has many applications in diverse fields such newlinexv newlineas mathematical physics, economics, computer science, engineering, and control theory. For example, in physics and mechanics, it has been used to study the stability of dynamical systems; in economics, it is essential to study market equilibrium and resource allocatio