## Nonlinearity and Functional Analysis Lectures on Nonlinear Problems in Mathematical Analysis by Melvin S Berger pdf

Nonlinearity and Functional Analysis Lectures on Nonlinear Problems in Mathematical Analysis by Melvin S Berger pdf free download. For many decades great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences. However when one drops the assumption of linearity, the associated operator theory and the many concrete problems associated with such a theory represent a frontier of mathematical research.

### Nonlinearity and Functional Analysis Lectures on Nonlinear Problems in Mathematical Analysis by Melvin S Berger pdf

Nonetheless, the fundamental results so far obtained in this direction already form a deep and beautiful extension of this linear theory. Just as in the linear case, these results were inspired by and are highly relevant to concrete problems in mathematical analysis. The object of the lectures represented here is a systematic description of these fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis.

#### Nonlinearity and Functional Analysis Lectures on Nonlinear Problems in Mathematical Analysis by Melvin S Berger pdf

Here I use the term “mathematical analysis” in the broadest possible sense. This usage is in accord with the ideas of Henri PoincarC (one of the great pioneers of our subject). Indeed, by carefully scrutinizing the specific nonlinear problems that arise naturally in the study of the differential geometry of real and complex manifolds, classical and modern mathematical physics, and the calculus of variations, one is able to discern recurring patterns that inevitably lead to deep mathematical results. From an abstract point of view there are basically two approaches to the subject at hand.

The first, as mentioned above, consists of extending specific results of linear functional analysis associated with the names of

Fredholm, Hilbert, Riesz, Banach, and von Neumann to a more general nonlinear context. The second approach consists of viewing the subject matter as an infinite-dimensional version of the differential geometry of manifolds and mappings between them. Obviously, these approaches are closely related, and when used in conjunction with modern topology, they form a mode of mathematical thought of great power. Finally, over and above these two approaches, there are phenomena that are genuinely both nonlinear and infinite dimensional in character.

A framework for understanding such facts is still evolving. The material to be described is divided into three parts, with each part

containing two chapters. Part I is concerned first with the motivation and preliminary mathematical material necessary to understand the context of later developments in the book and second with providing a rudimentary calculus and classification of nonlinear operators. Part I1 deals with local analysis. In Chapter 3, I treat various infinite-dimensional extensions of the classical inverse and implicit function theorems as well as Newton’s, the steepest descent and majorant methods for the study of operator equations. In Chapter 4, I turn attention to those parameterdependent perturbation phenomena related to bifurcation and singular perturbation problems. In this chapter the use of topological (“transcendental”) methods makes its first decisive appearance.

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