An Introduction to Nonlinear Analysis and Fixed Point Theory

verfasst von: Prof. Dr. h. c. Hemant Kumar Pathak

Verlag: Springer Singapore

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book systematically introduces the theory of nonlinear analysis, providing an overview of topics such as geometry of Banach spaces, differential calculus in Banach spaces, monotone operators, and fixed point theorems. It also discusses degree theory, nonlinear matrix equations, control theory, differential and integral equations, and inclusions. The book presents surjectivity theorems, variational inequalities, stochastic game theory and mathematical biology, along with a large number of applications of these theories in various other disciplines. Nonlinear analysis is characterised by its applications in numerous interdisciplinary fields, ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. Organised into ten chapters, the book shows the elegance of the subject and its deep-rooted concepts and techniques, which provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in diverse applied fields. It is intended for graduate and undergraduate students of mathematics and engineering who are familiar with discrete mathematical structures, differential and integral equations, operator theory, measure theory, Banach and Hilbert spaces, locally convex topological vector spaces, and linear functional analysis.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Fundamentals

The main objective of this chapter is to familiarize the reader to some basic concepts and fundamental results needed for the development of the theory of nonlinear functional analysis.

Hemant Kumar Pathak

Chapter 2. Geometry of Banach Spaces and Duality Mapping

Abstract

In this chapter, we are mainly concern with geometrical structures such as convexity and smoothness of Banach spaces. Indeed, various kind of convexity and smoothness of Banach spaces play an important role in the existence and approximation of fixed points of nonlinear mappings. The necessary concepts of the geometry of normed spaces—strict convexity and uniform convexity—are also discussed. This chapter also deals with useful properties of duality mappings that interplay with these geometrical structures of Banach spaces. In Sect. 2.1, we deal with strict convexity while Sect. 2.2 mainly concern with uniform convexity. In Sect. 2.3, we discuss modulus of convexity. In Sect. 2.4, we mainly concern with smoothness of Banach spaces. Section 2.5 mainly deals with the concept of duality mapping from a Banach space X to its dual \(X^*\). A discussion on these is important for a better understanding of the properties of duality mapping.

Hemant Kumar Pathak

Chapter 3. Differential Calculus in Banach Spaces

Abstract

The differential calculus is one of the fundamental techniques of nonlinear functional analysis. Very often we will use this notion. In this chapter, we develop the calculus in real Banach spaces. Section 3.1 deals with definitions on Gâteaux and Fréchat derivative with illustrative examples. We also give a variant of mean value theorem. Properties of the derivative are discussed in Sect. 3.2, while in Sect. 3.3, we discuss partial derivatives. Section 3.4 deals with higher derivative. Subsequently, we give Taylor’s theorem, inverse function and implicit function theorems.

Hemant Kumar Pathak

Chapter 4. Monotone Operators, Strongly -Accretive Operators and Their Variants

Abstract

In this chapter, we introduce the reader to the theory of monotone operators, \(\phi \)-accretive operators and their generalizations. The concept of monotone operator was first introduced by Minty in his paper of 1962, Minty, Duke Math J 29:341–346, 1962, [400], wherein he gave a surjectivity theorem for such operators. Since then, this theory is widely developed and has found useful applications in the investigation of the solvability of nonlinear operator equations and in particular of partial differential equations and integral equations. Our purpose is to give a systematic treatment (with historical development) of various topics in the theory of monotone operators needed for such an investigation.

Hemant Kumar Pathak

Chapter 5. Fixed Point Theorems

Abstract

In Sect. 5.1, we discuss the Banach’s contraction mapping theorem and some consequences of this theorem. We also deal with contractive mappings considered by Edelstein [212] and certain generalizations of contraction mapping theorem, mainly the ones obtained by Boyd and Wongs [75], Kannan [308, 309], Reich [509] and Husain and Sehgal [283] and others. This section ends with a recent fixed point theorem due to Caristi [128, 129]. Caristi’s theorem finds many applications in the field of nonlinear functional analysis.

Hemant Kumar Pathak

Chapter 6. Degree Theory, k-Set Contractions and Condensing Operators

The notion of “degree” of a map was first defined by Brouwer, who showed that the degree is homotopy invariant, and used it to prove the Brouwer fixed point theorem. Note that topological degree theory is a generalization of the winding number of a curve in the complex plane. It is closely connected to fixed point theory and can be used to estimate the number of solutions of an equation. For a given equation, if one solution of an equation is easily found, then degree theory can often be used to prove existence of a second, nontrivial, solution.

Hemant Kumar Pathak

Chapter 7. Variational Methods and Optimization

Abstract

The purpose of this chapter is to give an introduction of variational methods and optimization theory in a rather convincing manner along with results of nonlinear analysis leading to an applied environment. So, we have chosen variational principles as the starting point of our discussion in the framework of Banach space theory that leads to optimization with the observation that by applying the techniques involved in variational methods and optimization one can deal with some real-world problems that arise in nonlinear analysis. We initiate our discussion by presenting some variational principles and their applications. The epicenter of our discussion is the so-called Ekeland variational principle (in short, EVP). Indeed, we show that EVP is equivalent to some other well-known results of nonlinear analysis, notably Takahashi’s minimization theorem.

Hemant Kumar Pathak

Chapter 8. Applications of Monotone Operator Theory

Abstract

We studied so far many nonlinear problems that arose naturally in mathematical physics, differential topology and geometry of manifolds. In mathematical physics, it arises in the problems of elasticity, Newtonian mechanics of particles, theory of gas discharge and quantum fields, etc. In differential topology, it arises in smoothness of surfaces, convex analysis, etc. In differential geometry, nonlinearity arises in study of geodesics on manifolds, surface of constant curvatures, theory of automorphic forms, etc.

Hemant Kumar Pathak

Chapter 9. Applications of Fixed Point Theorems

Fixed point theory is a viable, productive, conclusive and useful to solve problems of existence and uniqueness of solution of a differential equation or an integral equation. Moreover, it encompasses various facets of analysis and a fascinating subject endowed with sophisticated tools with an enormous number of applications in various fields of mathematics. In this chapter, we intend to give some applications of fixed point theorems to obtain existence theorems for nonlinear differential and integral equations. Our treatment includes some standard well-known results as well as some recent ones. We have avoided an extensive discussion on this areas instead we concentrate on a few important problems. As usual, in most cases, the differential equations are transformed into an equivalent operator equations involving integral operators and then appropriate fixed point theorems or degree theoretic methods are invoked to prove the existence of desired solutions by recasting the operator equations into fixed point equations.

Hemant Kumar Pathak

Chapter 10. Applications of Fixed Point Theorems for Multifunction to Integral Inclusions

Abstract

Dynamical systems described by differential equations with continuous right-hand sides were the areas of vigorous steady in the later half of the twentieth century in applied mathematics, in particular, in the study of viscous fluid motion in a porous medium, propagation of light in an optically nonhomogeneous medium, determining the shape of a solid of revolution moving in a flow of gas with least resistance, etc. Euler’s equation plays a key role in dealing with the existence of the solution of such problems. On the other hand, Filippov, Differentsial’nye Uravneniya, 15:1814–1832, 1979, [232] has developed a solution concept for differential equations with a discontinuous right-hand side. In practice, such dynamical systems do arise and require analysis. Examples of such systems are mechanical systems with Coulomb friction modelled as a force proportional to the sign of a velocity, systems whose control laws have discontinuities.

Hemant Kumar Pathak

Backmatter

Titel: An Introduction to Nonlinear Analysis and Fixed Point Theory
verfasst von: Prof. Dr. h. c. Hemant Kumar Pathak
Verlag: Springer Singapore
Electronic ISBN: 978-981-10-8866-7
Print ISBN: 978-981-10-8865-0
DOI: https://doi.org/10.1007/978-981-10-8866-7