## Graduation Year

2007

## Document Type

Dissertation

## Degree

Ph.D.

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Athanassios Kartsatos, Ph.D.

## Committee Member

Marcus McWaters, Ph.D.

## Committee Member

Arunava Mukherjea, Ph.D.

## Committee Member

Boris Shekhtman, Ph.D.

## Committee Member

Yuncheng You, Ph.D.

## Keywords

Demicontinuous, Eigenvalue, Invariance of domain, Quasimonotone, Strongly quasibounded, Yosida Approximant

## Abstract

In this work, we demonstrate that the Leray-Schauder topological degree theory can be used for the development of a topological degree theory for maximal monotone perturbations of demicontinuous operators of type (S+) in separable reflexive Banach spaces. This is an extension of Berkovits’ degree development for operators as the perturbations above.

Berkovits has developed a topological degree for demicontinuous mappings of type (S_{+}), and has shown that the degree mapping is unique under the assumption that it satisfies certain general properties. He proved that if f is a bounded demicontinous mapping of type (S_{+}), G is an open bounded subset of X, and 0 ∈/ f(∂G), then there exists ε_{0} > 0 such that for every ε ∈ (0, ε_{0}) we have 0 ∈/ (I+ (1/ε)QQ^{∗} (f))(∂G). Here, Q is a compact linear injection from a Hilbert space H into X, such that Q(H) is dense in X, and Q^{∗} its adjoint. The map I+ 1 εQQ^{∗} (f) is a compact displacement of the identity, for which the Leray-Schauder degree is well defined. The Berkovits degree is obtained as the limit of this Leray-Schauder degree as ε tends to zero. We utilize a demicontinuous (S_{+})-approximation of the form T_{t} + f, where T_{t} is the Yosida approximant of T. Namely, we show that if G is an open bounded set in X and 0 ∈/ (T + f)(∂G), then there exist ε_{0} > 0, t_{0} > 0, such that for every ε ∈ (0, ε_{0}), t ∈ (0, t_{0}), we have 0 ∈/ (I + (1/ε)QQ^{∗} (T_{t} + f))(∂G). Our degree is the limit of the Leray-Schauder degree of the compact displacement of the identity I + (1/ε)QQ^{∗} (T_{t} + f) as ε, t → 0. Various extension of the degree has been considered. Finally some properties and applications in invariance of domain, eigenvalue and surjectivity results have also been discussed.

## Scholar Commons Citation

Boubakari, Ibrahimou, "The Leray-Schauder Approach for the Degree of Perturbed Maximal Monotone" (2007). *USF Tampa Graduate Theses and Dissertations.*

https://digitalcommons.usf.edu/etd/641