Bali, Saransh and Manna, Bhakti Bhusan
(2018)
NonLinear Analysis and Application to Partial
Differential Equations.
Masters thesis, Indian Institute of Technology, Hyderabad.
Abstract
In the past couple of decades, when Linear Functional Analysis was quite widely and completely
established, the interest of mathematicians towards Nonlinear Analysis has increased a lot. At one
hand the treatment of various classical problems has been uni�ed, on the other, theories speci�cally
nonlinear one of great signi�cance and applicability have come out.
In Nonlinear Functional Analysis we study the properties of (continuous) mappings between the
normed linear spaces and we describe the methods for solving nonlinear equaions involving such
mappings. For �nding the solution of nonlinear equations there are primarily two major approaches
which are known as topological methods and variational methods. Topological methods are derived
from �xed point theorems and one of the important tools used in this direction are the Topological
Degree and Morse Theory. Variational methods describe the solutions as critical points of a suitable
functional and study ways of locating them. Moreover, there is an important fact to be noted.
The fact is that the problems that are often considered to be di�cult, once they are framed in an
appropriate functional setting, may be faced and solved quite easily.
Here in this project, we provide an introduction to the basic aspect of NonLinear Analysis mainly
those which are based on di�erential calculus in Banach spaces. We have expressed the results here
in geometric style in such a way that they are often a transposition of in�nite dimensions of events,
which are intutive in R2 or R3. A particular nature of NonLinear Analysis is that its theory has
direct applications, especially to those related to di�erntial equations, where the power of nonlinear
methods is expressed in a more striking way like Degree Theory, Bifurcation Theory and Morse
Theory.
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