Fixed Point Theory and its Applications to Real World Problems

Fixed-point theory is a mathematical technique that emerged in the 18th century to demonstrate the existence of solutions of differential equations. It was improved as a method of successive approximations by Charles Emile Picard in 1890 and abstracted as a fixed-point theorem in the framework of complete normed space by Stefan Banach in 1922. Fixed-point theorems provide adequate conditions under which there exists a fixed point for a given function and enable us to ensure the existence of a solution of the original problem. The book Fixed Point Theory & its Applications to Real World Problems presents results in fixed point theory that are extensions, improvements, and generalizations of classical and recent results, and touches on distinct research directions within the metric fixed-point theory. It offers new techniques for the existence of superior fixed points, fixed points, near fixed points, fixed circles, near fixed interval circles, fixed discs, near fixed interval discs, coincidence points, common fixed points, coupled common fixed points, amiable fixed sets, strong coupled fixed points, and so on, utilizing minimal conditions. It has novel applications beyond traditional applications and is self-contained and unified, serving as a reference book for researchers seeking novel ideas.

Format: Hardback
Length: 412 pages
Publication date: 01 June 2021
Publisher: Nova Science Publishers Inc

Fixed-point theory, a fundamental concept in mathematics, has a rich history dating back to the second quarter of the 18th century. Initially, it was developed to demonstrate the existence of solutions for differential equations, a breakthrough that was made by Joseph Liouville in 1837. Over time, this theory evolved and was refined through successive approximations, a technique pioneered by Charles Emile Picard in 1890. This method was later abstracted and formulated as a fixed-point theorem within the framework of complete normed spaces, a significant achievement by Stefan Banach in 1922.

One of the key aspects of fixed-point theory is its ability to ensure the presence and uniqueness of a fixed point for a given function. This is achieved by providing adequate conditions under which a solution exists. Moreover, fixed-point theorems offer an approximate technique to locate the fixed point, as well as a priori and a posteriori estimates for the rate of convergence. This theory is widely applicable in various fields, including mathematics, physics, engineering, and computer science.

In the field of mathematics, fixed-point theory plays a crucial role in the study of metric spaces. It provides a framework for analyzing and understanding the behavior of functions and mappings within these spaces. Fixed-point theorems offer sufficient conditions for the existence of a fixed point, which is a point where a function has a unique value. This concept is closely related to the existence of solutions for differential equations and has profound implications in various branches of mathematics, such as topology, analysis, and algebra.

Fixed-point theory has numerous real-world applications. In engineering, it is used to design and optimize systems, such as control systems, communication networks, and transportation networks. By analyzing the behavior of these systems, fixed-point theory helps identify optimal parameters and configurations that ensure stability and efficiency. In physics, fixed-point theory is employed to model physical systems that exhibit non-linear behavior. By finding fixed points, researchers can gain insights into the underlying dynamics and properties of these systems, which can be useful in developing new theories and models.

In computer science, fixed-point theory finds applications in various areas, including graph theory, optimization, and machine learning. Graph theory is a branch of mathematics that studies the structure, properties, and algorithms of graphs, which are mathematical structures consisting of vertices and edges. Fixed-point theory is used to solve problems related to graph connectivity, graph isomorphism, and graph optimization, which are important in various applications, such as network routing, data compression, and social network analysis.

Optimization is a branch of mathematics that focuses on finding the best solution to a given problem. Fixed-point theory is used in optimization to find the optimal parameters or solutions for complex problems. For example, in machine learning, fixed-point theory is used to train neural networks by finding the minimum of a cost function, which is a measure of the error between the predicted and actual outputs.

In conclusion, fixed-point theory is a fundamental concept in mathematics that has a rich history and numerous applications in various fields. It provides a framework for analyzing and understanding the behavior of functions and mappings, and has profound implications in mathematics, engineering, physics, and computer science. Fixed-point theorems offer sufficient conditions for the existence of fixed points, which are essential in many real-world problems. As research in this field continues, fixed-point theory will undoubtedly continue to evolve and contribute to the development of new techniques and models that will have a significant impact on our understanding of the world.

Weight: 679g
ISBN-13: 9781536193367